Department Seminars

Abstract:

When examining dependence in spatial data, it can be helpful to formally assess spatial covariance structures that may not be parametrically specified or fully model-based. That is, one may wish to test for general features regarding spatial covariance without presupposing any particular, or potentially restrictive, assumptions about the joint data distribution. Current methods for testing spatial covariance are often intended for specialized inference scenarios, usually with spatial lattice data. We propose instead a general method for estimation and testing of spatial covariance structure, which is valid for a variety of inference problems (including nonparametric hypotheses) and applies to a large class of spatial sampling designs with irregular data locations. In this setting, spatial statistics have limiting distributions with complex standard errors depending on the intensity of spatial sampling, the distribution of sampling locations, and the process dependence. The proposed method has the advantage of providing valid inference in the frequency domain without estimation of such standard errors, which are often intractable, and without particular distributional assumptions about the data (e.g., Gaussianity). To illustrate, we develop the method for formally testing isotropy and separability in spatial covariance and consider confidence regions for spatial parameters in variogram model fitting. A broad result is also presented to justify the method for application to other potential problems and general scenarios with testing spatial covariance. The approach uses spatial test statistics, based on an extended version of empirical likelihood, having simple chi-square limits for calibrating tests. We demonstrate the proposed method through several numerical studies.

Abstract:

We will begin with a quick review of quantum channels and briefly present the celebrated Choi-Kraus representation theorem. These constitute the key concepts and results that are required to move forward and define a special class of quantum channels called "entanglement breaking channels" --- the quantum channels that admit a Choi-Kraus representation consisting of rank-one Choi-Kraus operators.

We then introduce the entanglement breaking rank of an entanglement breaking channel and define it to be the least number of rank-one Choi-Kraus operators required in its Choi-Kraus representation. We shall show how this rank parameter for a certain map links to an open problem in linear algebra: Zauner's conjecture. In particular, we show that the problem of computing the entanglement breaking rank of the quantum channel, that sends X in M_d to 1/(d+1) [X + Tr(X)I_d] in M_d, is equivalent to the existence problem of SIC POVM in dimension d. This is a joint work with Vern Paulsen, Jitendra Prakash, and Mizanur Rahman.

Abstract:

In this talk, we will discuss the problem whether two regression functions modelling the relation between a response and covariate in two samples differ by a shift in the horizontal and vertical axis. We consider a nonparametric situation assuming only smoothness for the regression functions. A graphical tool based on the derivatives of the regression functions and their inverses is proposed to answer this question and studied in several examples. We also formalize this question in a corresponding hypothesis and develop a statistical test. The asymptotic properties of the corresponding test statistic are investigated under the null hypothesis and under local alternatives and the finite sample properties of the new test are investigated by means of a small simulation study and real data example. This is a joint work with Holger Dette (University of Ruhr, Germany) and Weichi Wu (Tsinghua University, China).

Abstract:

In this talk, we will discuss estimation and prediction inferences for the generalized half normal and Kumaraswamy distributions under different censoring schemes. In particular we will discuss hybrid Type-I censoring, Type II progressive hybrid censoring and adaptive Type-II progressive censoring schemes. Likelihood inference and Bayesian inference will be addressed under different censoring schemes. Further we will discuss about prediction estimates and associated intervals of censored/ future observations using both frequentist and Bayesian approaches. We will numerically compare different estimations and prediction estimates. We will illustrate different methods using various real data sets. Finally we will discuss some ongoing research problem on step stress life testing model.

Abstract:

Lang, Jorgenson, and Kramer have successfully employed techniques coming from theory of heat kernel, to study and estimate various arithmetic invariants. Inspired by their ideas, we describe an approach to study estimates of cusp forms, using theory of heat kernels.

Abstract:

Let G be a connected reductive group over a finite field f of order q. When q is small, we make further assumptions on G. Then we determine precisely when G(f) admits irreducible, cuspidal representations that are self- dual, of Deligne-Lusztig type, or both. Finally, we outline some consequences for the existence of self-dual supercuspidal representations of reductive p-adic groups. This is a joint work with Jeffrey Adler.

Abstract:

Given a smooth function f : [0,1) --> \R, and scalars u_j, v_j in (0,1), I will compute the Taylor (Maclaurin) series of the function F(t) := \det A(t), where A(t) is the 2x2 matrix

f( t u_1 v_1 ) f( t u_1 v_2 )
f( t u_2 v_1 ) f( t u_2 v_2 )

C. Loewner computed the first two of these Maclaurin coefficients, in the thesis of his student R.A. Horn (Trans. AMS 1969). This was in connection with entrywise functions preserving positivity on matrices of a fixed dimension -- the case of all dimensions following from earlier work of Schur (Crelle 1911) and his student Schoenberg (Duke 1942).

It turns out that an "algebraic" family of symmetric functions is hiding inside this "analysis". We will see how this family emerges when one computes the second-order (and each subsequent higher-order) Maclaurin coefficient above. This family of functions was introduced by Cauchy (1800s) and studied by Schur in his thesis (1901). As an application, I will generalize a determinant formula named after Cauchy to arbitrary power series (over any c ommutative ring); the above is the special case f(x) = 1/(1-x) and t=1.

Abstract:

I want to explain the oldest unsolved major problem in Mathematics (called the congruent number problem). It can be traced back at least to the 10th century but it is possibly much older. It turns out to be a beautiful example of the modern theory of the arithmetic of elliptic curves, but it is more accurate to say that this theory grew out of the study of this problem.

In the 17th century, Fermat gave a wonderful proof of the first special case of this problem. It also led Fermat to his so called Last Theorem (now solved by Andrew Wiles). But the original congruent number problem remains unsolved, despite the fact that conjecturally there is a very simple and beautiful answer to it.

Abstract:

Over the last few decades, models based on fluid flow and deformation in porous media are getting a lot of attention because these models have wide range of applications in science and engineering. Specifically, Biot’s consolidation model has many applications which cover the range from geoscience to medicine. It is the aim of this talk to present the a posteriori error analysis for locking free mixed finite element method of Biot's consolidation model. We discuss three novel a posteriori error estimators and show that all three a posteriori error estimators are reliable, efficient and robust. Finally, numerical results are presented to validate the theoretical results.

Abstract:

Estimation of animal abundance and distribution over large regions remains a central challenge in statistical ecology. In our first study, we use a Bayesian smoothing technique based on a conditionally autoregressive (CAR) prior distribution and Bayesian regression to integrate data from reliable but expensive surveys conducted at smaller scales with cost-effective but less reliable data generated from surveys at wider scales to address this problem. We also investigate whether the random effects which represent the spatial association due to the CAR structure have any confounding effect on the fixed effects of the regression coefficients. Next, we develop a novel Bayesian spatially explicit capture-recapture (SECR) model that disentangles the latent ecological process of animal arrival within a detection region from the process of recording this arrival by a given set of detectors. We integrate this into an advanced version of a recent SECR model by Royle (2015) involving partially identified individuals. The above is a joint work with Prof. Mohan Delampady.

Abstract:

This is an informal talk meant for aspiring researchers in Mathematics and related fields such as Theoretical Physics. The purpose of this talk is to convey a sense of what it takes, on a day to day basis, to succeed in research. I would like to lay bare the inner attitude of a researcher. It is based partly on my own experience, but mostly on what I have learnt while collaborating, and generally hobnobbing, with some truly great mathematicians of our times. The talk will be totally non-technical and I hope to make it very interactive.

Abstract:

The representation theory of string algebras started with the work of Gelfand and Ponomarev on Harish-Chandra representations of the Lorentz group. The methods they introduced were adapted and extended to cover the representation theory of a broad class of algebras. The term "string" algebra refers to the combinatorial description of their indecomposable finitely generated representions. I will recall that description, and that of the morphisms between representations, outline the broad picture of the category of finitely generated modules in terms of Auslander-Reiten components and say something about the infinite-dimensional representation theory.

Abstract:

When _lms thin the viscous resistance of the con_ning surfaces controls key aspects of their dynamics. However, those surfaces do more than provide the typical boundary conditions. For example, if one of the boundaries is elastic, there are nonlocal interactions that lead to interesting and challenging mathematical problems and important applications. One unique setting occurs when the free surfaces of most solids approach the bulk melting temperature from below, and they are wet by the melt phase. Mathematically, the uid dynamics of this so-called \premelting" falls under the rubric of a class of higher-order di_usion equations governing the dynamics of viscous current down an incline, viscous gravity currents between a rigid surface and a deformable elastic sheet, wetting/dewetting, and a spate of other thin-_lm settings, to name a few. However, theunderlying forces driving the ow are uniquely associated with intermolecular forces. We discuss aclass of experimentally tested and testable premelting dynamics ows. Finally, through its inuence on the viscosity, the con_nemente_ect implicitly introduces a new universal length scale into the volume ux. Thus, there are a host of thin _lm problems, from droplet breakup to wetting/dewetting dynamics, whose properties (similarity solutions, regularization, and compact support) will changeunder the action of the con_nemente_ect. Therefore, our study suggests revisiting the mathematical structure and experimental implications of a wide range of problems within the framework of the con_nemente_ect.

"The game of Hex gives a constructive proof of the Brouwer's fixed point theorem”
on Feb 21, 2019 at 4:00 PM in FB 567, MATHS Dept. Seminar room by Prof. TES Raghavan from University of Illinois at Chicago

Abstract:

Abstract:

Abstract:

We will discuss results concerning conjugacy-invariant norms on free groups, in particular ones thatare homogeneous and quasi-homogeneous.

On values of the Riemann zeta function at odd positive integers. on January 25, 2019 at 2:30 PM in FB 567, MATHS Dept Seminar room by Prof. Atul Dixit from Gandhinagar

on January 25, 2019 at 11:00 AM in FB 567, MATHS Dept Seminar room by Dr. Soumalya Joardar from INSPIRE faculty of JNCASR}

Abstract:

KMS states are generalization of what is known as Gibbs state for matrices to in_nitedimen-sionalC-algebras. Given an inverse temperature, unlike the Gibbs state for matrix algebra,KMS statesfor general in_nite dimensional Calgebras are far from unique. We discuss how theclassical and quantum symmetry of graphs act as symmetry objects for the corresponding graphCalgebras and the invariance of KMS states on graph Calgebras under such symmetries. Inparticular, we shall give an example of a graph admitting in_nitely many KMS states at criticalinverse temperature all of which are invariant under classical symmetry. But interestingly, thatgraph admits unique quantum symmetry invariant KMS state.

Absract:

A topological dynamical system is a pair (X,T) where T is a homeomorphism of a compact space X. A measure preserving action is a triple (Y, \mu, S) where Y is a standard Borel space, \mu is a probability measure on X and S is a measurable automorphism of Y which preserves the measure \mu. We say that (X,T) is universal if it can embed any measure preserving action (under some suitable restrictions).Krieger’s generator theorem shows that if X is A^Z (bi-infinite sequences in elements of A) and T is the transformation on X which shifts its elements one unit to the left then (X,T) is universal. Along with Tom Meyerovitch, we establish very general conditions under which Z^d (where now we have dcommuting transformations on X)-dynamical systems are universal. These conditions are general enough to prove that 1) A self-homeomorphism with non-uniform specification on a compact metric space (answering a question by Quas and Soo and recovering recent results by David Burguet)2) A generic (in the sense of dense G_\delta) self-homeomorphism of the 2-torus preserving Lebesgue measure (extending result by Lind and Thouvenot to infinite entropy)3) Proper colourings of the Z^d lattice with more than two colours and the domino tilings of the Z^2 lattice (answering a question by Şahin and Robinson) are universal. Our results also extend to the almost Borel category giving partial answers to some questions by Gao and Jackson.The talk will not assume background in ergodic theory and dynamical systems.

Abstract:

The standard isoperimetric inequality states that among all sets with a given fixedvolume(or area in dimension 2) the ball has the smallest perimeter. That is, written here in dimension2, the following infimum is attained by the ball  I will give an elementary overview on this inequality, such as techniques developed for theproof (symmetrization, variational method, Fourier series method in 2 dimensions), higherdimensional version, connections to partial differential equations and some generalizations. Iwill keep the talk basic, requiring no specialized knowledge in differential geometry and partial differential equations.

Abstract:

The dynamical setting of a non-autonomous system is related to the Bedford conjecture. Further,Short C^k's arises as basins of attraction of a fixed point in the non-autonomous setting. In this talk, I will discuss the relation between non-autonomous holomorphic dynamics, Short C^k's and their importance in the context of the Levi-problem. Also, I will give some construction of Short C^k's with pathological properties and finally discuss some related results and questions.

Abstract:

In multivariate linear regression, one of the major challenges is to model dependent responses from correlated predictors in a compact and interpretable manner. One promising approach is to model the coefficient matrix as a jointly sparse and low­-rank matrix. Learning such a decomposition is, however, very challenging due to the simultaneous presence of orthogonality and sparsity constraints.Here, we introduce a divide and conquer strategy to infer such a coefficient matrix from data. In the divide step, we decompose the coefficient matrix into a sum of unit-­rank matrices whose left and right singular vectors are sparse. In the conquer step, each unit­-rank matrix is estimated either by a sequential(greedy) approach or an exclusive extraction approach. We show that the proposed algorithm is guaranteed to converge to an accumulation point and provide an efficient implementation in the R package secure. We demonstrate the efficacy of the procedure in simulation studies and two applications in genetics.

On January 18, 2019 at 02:00 PM in FB567,MATHS Dept Seminar room by Dr. Arbaz Khan (postdoctoral fellow) from University of Manchester-UK}

Abstract:

Over the last few decades, models based on fluid flow and deformation in porous media are getting a lot of attention because these models have wide range of applications in science and engineering. Specifically, Biot’s consolidation model has many applications which cover the range from geoscience to medicine. It is the aim of this talk to present the a posteriori error analysis for locking free mixed finite element method of Biot's consolidation model. We discuss three novel a posteriori error estimators and show that all three a posteriori error estimators are reliable, efficient and robust. Finally, numerical results are presented to validate the theoretical results.

Abstract:

The problem of estimating quantiles has received considerable attention by several researchers in the recent past from classical as well as decision theoretic point of view. It is noted that, exponential quantiles are widely used in the study of reliability, life testing and survival analysis, and related fields. Suppose there are k(≥2) normal populations with a common mean and possibly different variances. The problem of estimation of quantiles of the first population is considered with respect to a quadratic loss function. The concept of invariance is introduced to the problem. As a result, sufficient conditions are derived to improve affine and location equivariant estimators. Consequently, some complete class results have been obtained. Next, the problem of estimating ordered quantiles of two exponential distributions has been considered assuming equality on location parameters. Using order restriction on the quantiles, several new estimators have been proposed including the restricted maximum likelihood estimator. Using quadratic loss function, a sufficient condition has been obtained to improveequivariant estimators under the order restriction. Consequently, some improved estimators have been obtained. All the proposed estimators have been compared numerically. It has been seen that the percentage of risk reduction after using order restriction is quite significant.

Lack-of-fit of a parametric AR(1) model in the presence of measurement error.On January 14, 2019 at 11:00 AM in FB567, MATHS Dept Seminar room by Prof. Hira Koul from Department of Statistics & Probability, Michigan State University

Abstract:

Recently, power law (also known as Zipf's law) has been turned out to be very natural in modelling and inferring about networks, finance, environment studies, rare events. In this seminar, Hill's estimator (best known to estimate index of regular variation) will be discussed with examples. We shall talk about the advantages and disadvantages of using Hill's estimator once population deviates from pure power law. We shall also raise some issues which are yet to be resolved.

Abstract:

We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the n-th generation converges weakly to a Cox cluster process. This is a joint work with Rajat Subhra Hazra and Parthanil Roy.

Abstract:

We will start by defining a G_2 and a nearly G_2 structure on a seven dimensional manifold M. After surveying known results about G_2 manifolds and explaining the importance of such manifolds, we will talk about hypersurfaces of a G_2 manifold, which inherits an SU(3) structure from the ambient G_2 structure. We will give a necessary and sufficient condition for a hypersurface of a nearly G_2 manifold to be nearly Kahler.After that, we will focus our attention on minimal hypersurfaces (zero mean curvature) and will discuss some new results in that direction.

Abstract:

Sequential multiple assignment randomized trials (SMART) are used to construct data-driven optimal treatment strategies for patients based on their treatment and covariate histories in di_erent branches of medical and behavioral sciences where a sequence of treatments are given to the patients; such sequential treatment strategies are often called dynamic treatment regimes (DTR). In the existing literature, the ma- jority of the analysis methodologies for SMART studies assume a continuous primary outcome. However, ordinal outcomes are also quite common in clinical practice; for example, the quality of life is often measured in an ordinal scale (e.g., poor, moderate, good). In this work, _rst, we develop the notion of dynamic generalized odds-ratio (dGOR) to compare two dynamic treatment regimes embedded in a 2-stage SMART with an ordinal outcome. We propose a likelihood-based approach to estimate dGOR from SMART data. Next, we discuss some combinatorial properties of dGOR and derive the asymptotic properties of its estimate. We discuss some alternative ways to estimate dGOR using concordant-discordant pairs and multi-sample U-statistic. Then, we extend the proposed methodology to a K-stage SMART. Furthermore, we propose a basic policy search algorithm that uses dGOR to _nd an optimal DTR within a _nite class. A simulation study shows the performance of the estimated dGOR in terms of the estimated power corresponding to the derived sample size. We analyze data from Sequenced Treatment Alternatives to Relieve Depression (STAR*D), a multi-stage randomized clinical trial for treating major depression, to illustrate the proposed methodology. A freely available online tool using R statistical software is provided to make the proposed methodology accessible to other researchers and practitioners.

Abstract:Quantile regression is a more robust, comprehensive, and flexible method of statistical analysis than the commonly used mean regression methods. Quantile function completely characterizes a distribution. More and more data is getting generated in the form of multivariate, functional, and multivariate functional data and the quantile analysis gets more challenging as the data complexity increases. We propose a set of quantile analysis methods for multivariate data and multivariate functional data. In plant science, the study of salinity tolerance is crucial to improving plant growth and productivity and we apply our methods to barley field data for a salinity tolerance analysis. We apply our methods to radiosonde wind data to illustrate our proposed quantile analysis methods for visualization, outlier detection, and prediction. We model multivariate functional data for flexible quantile contour estimation and prediction.The estimated contours are flexible in the sense that they can characterize non-Gaussian and non-convex marginal distributions. We aim to perform spatial prediction for non-Gaussian processes.

Abstract:

Abstract:

Arens irregularity of a Banach algebra is due to elements in its Banach dual which are not weakly almost periodic. The usual algebras in harmonic analysis (such as the group algebra or the Fourier algebra, when known) turned out to be all Arens irregular (even in an extreme way, as we shall see in the talk). To start, more than sixty years ago, Richard Arens himself proved that I_1 is irregular, and was followed by Mahlon Day proving the same result for many discrete groups including the abelian ones. A long exciting story followed after. The talk is an attempt to tell you the story and to explain the combinatorial reason (on the group or its dual) creating such irregularity.

Abstract:These will be an informal talk on the use of Carleman estimates for formally determined inverse problems for hyperbolic PDEs.

Abstract: In this talk, we will discuss the existence theory of distributional solutions solving.

On a bounded domain Ω. The nonlinear structure A(x, t, Δu) is modelled after the standard parabolic p-Laplace operator. In order to do this, we develop suitable techniques to obtain a prior estimates between the solution and the boundary data. As a consequence of these estimates, a suitable compactness argument can be developed to obtain the existence result. An interesting ingredient in the proof is the careful use of the boundedness of a new type of Maximal function defined on negative Sobolev spaces. The a priori estimates proved discussed in this talk are new even for the heat equation on bounded domains.

Abstract:

Bounded sequences in the Sobolev space of non-compact manifold $M$ converge weakly in $L^p$ only after subtraction of countably many terms supported in the neighborhood of infinity. These terms are defined by functions on the manifolds-at-infinity of $M$, defined by a gluing procedure, and the sum of their respective Sobolev energies is dominated by the energy of the original sequence. Existence of minimizers in isoperimetric problems involving Sobolev norms is therefore dependent on comparison between Sobolev constants for $M$ and its manifolds-at-infinity. This is a joint work with Leszek Skrzypczak.

Abstract:

In this introductory talk, we describe an extension of the probabilistic model of diffusion that was introduced by K.Itô using his method of stochastic differential equations'. In our model, which is based on ideas from stochastic partial differential equations, the evolving system of (solute) particles is modeled by a tempered distribution. Our method can also be viewed as an extension of the ‘method of characteristics’ in partial differential equations.

Abstract:

Since Lovasz introduced the idea of associating complexes with graphs to solve problems in Graph Theory, the properties that these complexes enjoy have been a topic of study. In this talk, I will be discussing the complexes associated with some classes of graphs, including complete graphs, Kneser and Stable Kneser graphs. The connectedness of some of the complexes associated with the graphs provide good lower bounds for the chromatic number of these graphs.

Abstract:Geometric invariant theory (GIT) provides a construction of quotients for projective algebraic varieties equipped with an action of a reductive algebraic group. Since its foundation by Mumford, GIT plays an important role in the construction of moduli (or parameter) spaces. More recently, its methods have been successfully applied to problems of representation theory, operator theory, computer vision and complexity theory. In the first part of the talk, I will give a very brief introduction to GIT and in the second part I will talk about some geometric properties of the torus and finite group quotients of Flag varieties.

Abstract:A branching random walk is a system of growing particles that starts from one particle at the origin with each particle branching and moving independently of the others after unit time. In this talk, we shall discuss how the tails of progeny and displacement distributions determine the extremal properties of branching random walks. In particular, we have been able to verify two related conjectures of Eric Brunet and Bernard Derrida in many cases that were open before.This talk is based on a joint works with Ayan Bhattacharya (PhD thesis work at Indian Statistical Institute (ISI), Kolkata, presently at Centrum Wiskunde & Informatica, Amsterdam), Rajat Subhra Hazra (ISI Kolkata), Souvik Ray (M. Stat dissertation work at ISI Kolkata, presently at Stanford University) and Philippe Soulier (University of Paris Nanterre).

Abstract:

During the last four decades, it has been realized that some problems in number theory and, in particular, in Diophantine approximation, can be solved using techniques from the theory of homogeneous dynamics. We undertake this as the theme of the talk. We shall first give a broad overview of the subject and demonstrate how some Diophantine problems can be reformulated in terms of orbit properties under certain flows in the space of lattices, following the ideas of G. A. Margulis, S. G. Dani, D. Y. Kleinbock and G. A. Margulis, E.Lindenstrauss. Subsequently, we will discuss the joint work of the speaker with Victor Beresnevich, Anish Ghosh and Sanju Velani on Inhomogeneous dual approximation on affine subspaces.

Abstract:

Randomized experiments have long been considered to be a gold standard for causal inference. The classical analysis of randomized experiments was developed under simplifying assumptions such as homogeneous treatment effects and no treatment interference leading to theoretical guarantees about the estimators of causal effects. In modern settings where experiments are commonly run on online networks (such as Facebook) or when studying naturally networked phenomena (such as vaccine efficacy) standard randomization schemes do not exhibit the same theoretical properties. To address these issues we develop a randomization scheme that is able to take into account violations of the no interference and no homophily assumptions. Under this scheme, we demonstrate the existence of unbiased estimators with bounded variance. We also provide a simplified and much more computationally tractable randomized design which leads to asymptotically consistent estimators of direct treatment effects under both dense and sparse network regimes.

Abstract:

Step-stress life testing is a popular experimental strategy, which ensures an efficient estimation of parameters from lifetime distributions in a relatively shorter period of time. In our analysis, we have used two different stress models, viz., the Cumulative Exposure and Khamis-Higgins models, with respect to the one and two-parameter exponential distributions, and the two-parameter Weibull distributions, respectively, under various censoring schemes. Both Bayesian and frequentist (wherever possible) approaches have been applied for the estimation of parameters and construction of their confidence/credible intervals. Under Bayesian analysis, estimation with order restriction on the mean lifetimes of units has been considered as well. In yet another attempt to analyze lifetime observations, we construct optimal variable acceptance sampling plans, an instrument to test the quality of manufactured items with a crucial role in the acceptance or rejection of the lot. Assuming the one-parameter exponential lifetime distribution, in presence of Type-I and Type-I hybrid censoring, we propose decision-theoretic approach based plans with a new estimator of the scale parameter. The optimal plans are obtained by minimizing the Bayes’ risk under a well-defined loss function.

Abstract:

The Cauchy dual subnormality problem asks when the Cauchy dual operator of an m-isometry is subnormal. This problem can be considered as the non-commutative analog of the fact that the reciprocal of a Bernstein function is completely monotone. We discuss this problem, its connection with a Hausdorff moment problem, its role in a problem posed by N. Salinas dating back to 1988, and some instances in which this problem can be solved. This is a joint work with A. Anand, Z. Jablonski, and J. Stochel.

Abstract:

One of the central questions in representation theory of finite groups is to describe the irreducible characters of finite groups of Lie type, namely matrix groups over finite fields. The theory of character sheaves, initiated by Lusztig, is a geometric approach to this problem. I will give a brief overview of this theory.

Abstract:

Consider a non-parametric regression model y = μ(x) + ε, where y is the response variable, x is the scalar covariate, ε is the error, and μ is the unknown non-parametric regression function. For this model, we propose a new graphical device to check whether v-th (v ≥ 1) derivative of the regression function μ is positive or not, which includes checking for monotonicity and convexity as special cases. An example is also presented that demonstrates the practical utility of the graphical device. Moreover, this graphical device is employed to formulate a class of test statistics to test the aforementioned assertion. The asymptotic distribution of the test statistics are derived, and the tests are implemented on various simulated and real data.

Abstract:

Mathematical modelling of complex ecological interactions is a central goal of research in mathematical ecology. A wide variety of mathematical models are proposed and relevant dynamical analysis is performed to understand the complex interaction among various trophic levels. Existing models are modified as well to remove their ecological discrepancies. Main objective of this talk is to provide a overview of current research trend in mathematical ecology and how dynamical complexities among interacting populations can be captured and analyzed with the help of mathematical models involving ODEs, PDEs, DDEs, SDEs and their combinations.

Abstract:

In this talk, we study the configuration of systoles (minimum length geodesics) on closed hyperbolic surfaces. The set of all systoles forms a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface, we call these admissible.There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first result characterises admissibility.It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, to the classical result that there are only two minimal non-planar graphs).

Abstract:

Lag windows are commonly used in the time series, steady state simulation, and Markov chain Monte Carlo (MCMC) literature to estimate the long range variances of ergodic averages. We propose a new lugsail lag window specifically designed for improved finite sample performance. We use this lag window for batch means and spectral variance estimators in MCMC simulations to obtain strongly consistent estimators that are biased from above in finite samples and asymptotically unbiased. This quality is particularly useful when calculating effective sample size and using sequential stopping rules where they help avoid premature termination.Further, we calculate the bias and variance of lugsail estimators and demonstrate that there is little loss compared to other estimators. We also show mean square consistency of these estimators under weak conditions. Our results hold for processes that satisfy a strong invariance principle, providing a wide range of practical applications of the lag windows outside of MCMC. Finally, we study the finite sample properties of lugsail estimators in various examples.

Abstract:

Intractable integrals appear in many areas in statistics; for example, generalized linear mixed

models and Bayesian statistics. Inference in these models relies heavily on the estimation of said integrals. In this talk, I present Monte Carlo methods for estimating intractable integrals. I introduce the accept-reject sampler and demonstrate its use on an example. Although useful, the accept-reject sampler is not effective for estimating high-dimensional integrals. To this end, I present Markov Chain Monte Carlo (MCMC) methods, like the Metropolis-Hastings sampler, which allow estimation of high-dimensional integrals. I discuss important theoretical properties of MCMC methods and some statistical challenges in its practical implementation.

Abstract:

Intractable integrals appear in many areas in statistics; for example, generalized linear mixed

models and Bayesian statistics. Inference in these models relies heavily on the estimation of said integrals. In this talk, I present Monte Carlo methods for estimating intractable integrals. I introduce the accept-reject sampler and demonstrate its use on an example. Although useful, the accept-reject sampler is not effective for estimating high-dimensional integrals. To this end, I present Markov Chain Monte Carlo (MCMC) methods, like the Metropolis-Hastings sampler, which allow estimation of high-dimensional integrals. I discuss important theoretical properties of MCMC methods and some statistical challenges in its practical implementation.

Abstract:

We talk about the mod 2 cohomology ring of the Grassmannian $\widetilde{G}_{n,3}$ of oriented 3-planes in $\mathbb{R}^n$. We first state the previously known results. Then we discuss the degrees of the indecomposable elements in the cohomology ring. We have an almost complete description of the cohomology ring. This description provides lower and upper bounds on the cup length of $\widetilde{G}_{n,3}$. This talk is based on my work with Somnath Basu.

Abstract:

How can we determine whether a mean-square continuous stochastic process is finite-dimensional, and if so, what its precise dimension is? And how can we do so at a given level of confidence? This question is central to a great deal of methods for functional data, which require low-dimensional representations whether by functional PCA or other methods. The difficulty is that the determination is to be made on the basis of iid replications of the process observed discretely and with measurement error contamination. This adds a ridge to the empirical covariance, obfuscating the underlying dimension. We build a matrix-completion inspired test statistic that circumvents this issue by measuring the best possible least square fit of the empirical covariance’s off-diagonal elements, optimised over covariances of given finite rank. For a fixed grid of sufficient size, we determine the statistic’s asymptotic null distribution as the number of replications grows. We then use it to construct a bootstrap implementation of a stepwise testing procedure controlling the family-wise error rate corresponding to the collection of hypothesis formalising the question at hand. Under minimal regularity assumptions we prove that the procedure is consistent and that its bootstrap implementation is valid. The procedure involves no tuning parameters or pre-smoothing, is indifferent to the omoskedasticity or lack of it in the measurement errors, and does not assume a low-noise regime. An extensive study of the procedure’s finite-sample accuracy demonstrates remarkable performance in both real and simulated data.

This talk is based on an ongoing work with Victor Panaretos (EPFL, Switzerland).

Abstract:

Virtual element methods(VEM) is a recently developed technology considered as generalization of FEM hav- ing firm mathematical foundations, simplicity in implementation, efficiency and accuracy in computations. Unlike FEM which allows element like triangle and quadrilateral only, VEM permits very general shaped polygons including smoothed voronoi, random voronoi, distorted polygons, nonconvex elements. Basis func- tions in VEM are constructed virtually and can be computed from the informations provided by degrees of freedom(DoF) associated with the VEM space.

Moreover, basis functions are solution of some PDEs which determine the dimension of VEM space. Furthermore, we have two projection operators on VEM space; orthogonal L2projection operator Π0k,K and elliptic projection operator Π∇k,K.Both operators are definedlocally element-wish on K ∈ Th , where Th and K denote mesh partition, polygon respectively and project basis function to computable polynomial subspace sitting inside VEM space. Basically, in abstract VEMformulation, we split the bilinear form into two parts polynomial part and non-polynomial part or stabi- lization part. Polynomial part can be computed directly from degrees of freedom and non-polynomial part can be approximated from DoF ensuring same scaling as polynomial part. However, the above mentioned framework will not work in case of non-linear problems. The primary reason is that term involving nonlinearfunction e.g. (f (u)∇u • ∇v)K can not be split into polynomial and non-polynomial parts. Hence discreteform will not be computable from DoF. In view of this difficulty, we introduce a graceful idea of employingorthogonal projection operator Π0k,Kin order to discretise nonlinear term. Exploiting this technique, weencounter semi-linear parabolic and hyperbolic problems ensuring optimal order of convergence in L2 and H 1 norms. However, we assert that this technique can be employed to discretize general nonlinear type of problem.

Abstract:

Main objects of study in model theory are definable subsets of structures. For example, the definable sets in the field of complex numbers are precisely the (boolean combinations of) varieties. The (model-theoretic) Grothendieck ring of a structure aims to classify definable sets up to definable bijections. The Grothendieck ring of varieties is central to the study of motivic integration, but its computation is a wide open problem in the area. The problem of classification is simplified to a large extent if the theory of the structure admits some form of elimination of quantifiers, i.e., the complexity of the formulas describing the definable sets is in control.

In this double talk'', we will begin by describing the construction of the Grothendieck ring and by giving a survey of the known Grothendieck rings. Then we will present the results and techniques used to compute the Grothendieck ring in the case of dense linear orders (joint with A. Jain) and atomless boolean algebras. On our way we will also state the implicit function theorem’’ for boolean varieties.

Abstract:

We begin by presenting a spectral characterization theorem that settles Chevreau's problem of characterizing the class of absolutely norming operators --- operators that attain their norm on every closed subspace. We next extend the concept of absolutely norming operators to several particular (symmetric) norms and characterize these sets. In particular, we single out three (families of) norms on B(H,K): the Ky Fan k-norm(s)", the weighted Ky Fan (\pi, k)-norm(s)", and the (p, k)-singular norm(s)", and thereafter define and characterize the set of absolutely norming operators with respect to each of these three norms.

We then restrict our attention to the algebra B(H) of operators on a separable infinite-dimensional Hilbert space H and use the theory of symmetrically normed ideals to extend the concept of norming and absolutely norming from the usual operator norm to arbitrary symmetric norms on B(H). In addition, we exhibit the analysis of these concepts and present a constructive method to produce symmetric norm(s) on B(H) with respect to each of which the identity operator does not attain its norm.

Finally, we introduce the notion of "universally symmetric norming operators" and "universally absolutely symmetric norming operators" and characterize these classes. These refer to the operators that are, respectively, norming and absolutely norming, with respect to every symmetric norm on B(H).

In effect, we show that an operator in B(H) is universally symmetric norming if and only if it is universally absolutely symmetric norming, which in turn is possible if and only if it is compact. In particular, this result provides an alternative characterization theorem for compact operators on a separable Hilbert space.

Abstract:

In this talk, we address the question of identifying commutant and reflexivity of the multiplication d-tuple M_z on a reproducing kernel Hilbert space H of E-valued holomorphic functions on Ω, where E is a separable Hilbert space and Ω is a bounded domain in C^d admitting bounded approximation by polynomials. In case E is a finite dimensional cyclic subspace for M_z, under some natural conditions on the B(E)-valued kernel associated with H , the commutant of M_z is shown to be the algebra H^{∞}_ B(E)(Ω) of bounded holomorphic B(E)-valued functions on Ω, provided M_z satisfies the matrix-valued von Neumann’s inequality. Also, we show that a multiplication d-tuple M_z on H satisfying the von Neumann’s inequality is reflexive. The talk is based on joint work with Sameer Chavan and Shailesh Trivedi.

Abstract:

Although we can trace back the study of epidemics to the work of Daniel Bernoulli nearly two and a half centuries ago, the fact remains that key modeling advances followed the work of three individuals (two physicians) involved in the amelioration of the impact of disease at the population level a century or so ago: Sir Ronald Ross (1911) and Kermack and McKendrick (1927). Ross' interests were in the transmission dynamics and control of malaria while Kermack and McKendrick's work was directly tied in to the study of the dynamics of communicable diseases. In this presentation, I will deal primarily with the study of the dynamics of influenza type A, a communicable disease that does not present a “fixed” target. The study of the short-term dynamics of influenza, single epidemic outbreaks, makes use of extensions/modifications of the models first introduced by Kermack and McKendrick while the study of its long-term dynamics requires the introduction of modeling modifications that account for the continuous emergence of novel influenza variants: strains or subtypes. Here, I will briefly review recent work on the dynamics of influenza A/H1N1, making use of single outbreak models that account for the movement of people in the transmission process over various regions within Mexico. This research has been carried in collaboration with a large number of researchers over a couple of decades.

From a theoretical perspective, I will observe that over the past 100 years modeling epidemic processes have been based primarily on the use of the mass action law. What have we learned from this approach and what are the limitations? In this lecture, I will revisit old and “new” modeling approaches in the context of the dynamics of vector borne, sexually-transmitted and communicable diseases.

Abstract:

Let F(x; y) that belongs to Z[x; y] be a homogeneous and irreducible with degree 3. Consider F(x; y) = h for some fixed nonzero integer h. In 1909, Thue proved that this has only finitely many integral solutions. These eponymous equations have several applications. Much effort has been made to obtain upper bounds for the number of solutions of Thue equations which are independent of the size of the coefficients of F. Siegel conjectured that the number of solutions could be bounded only in terms of h and the number of non-zero coefficients of F. This was settled in the affirmative by Mueller and Schmidt. However, their bound doesn’t have the desired shape. In this talk, we present some instances when their result can be improved. This is joint work with N. Saradha.

Abstract:

Suppose p(t)=X_0+X_1t+...+X_n t^n is a polynomial where each X_i is randomly chosen to be +1 or -1. How many real roots does the polynomial have, on average? Turns out that the answer is of order \log(n). More generally, given a subset of the complex plane, how many roots are in the given subset (on average, say)? Turns out that the roots are almost all close to the unit circle, and distributed roughly uniformly in angle. We survey basic results answering these questions. The talk is aimed to be accessible to advanced undergraduate and graduate students.

Abstract:

Let F(x; y) that belongs to Z[x; y] be a homogeneous and irreducible with degree 3. Consider F(x; y) = h for some fixed nonzero integer h. In 1909, Thue proved that this has only finitely many integral solutions. These eponymous equations have several applications. Much effort has been made to obtain upper bounds for the number of solutions of Thue equations which are independent of the size of the coefficients of F. Siegel conjectured that the number of solutions could be bounded only in terms of h and the number of non-zero coefficients of F. This was settled in the affirmative by Mueller and Schmidt. However, their bound doesn’t have the desired shape. In this talk, we present some instances when their result can be improved. This is joint work with N. Saradha.

Abstract:

Penalized regression techniques are widely used in practice to perform variable selection (also known as model selection). Variable selection is important to drop the covariates from the regression model, which are irrelevant in explaining the response variable. When the number of covariates is large compared to the sample size, variable selection is indeed the most important requirement of the penalized method. Fan and Li(2001) introduced the Oracle Property as a measure of how good a penalized method is. A penalized method is said to have the oracle property provided it works as well as if the correct sub-model were known (like the Oracle who knows everything beforehand). We categorize different penalized regression methods with respect to oracle property and show that bootstrap works for each category. Moreover, we show that in most of the situations, the inference based on bootstrap is much more accurate than the oracle based inference.

Abstract:

The Adaptive Lasso (Alasso) was proposed by Zou (2006) as a modification of the Lasso for the purpose of simultaneous variable selection and estimation of the parameters in a linear regression model. Zou (2006) established that the Alasso estimator is variable-selection consistent as well as asymptotically Normal in the indices corresponding to the nonzero regression coefficients in certain fixed-dimensional settings. Minnier et al. (2011) proposed a perturbation bootstrap method and established its distributional consistency for the Alasso estimator in the fixed-dimensional setting. In this paper, however, we show that this (naive) perturbation bootstrap fails to achieve the desired second order correctness [i.e. with uniform error rate o(n^{-1/2})] in approximating the distribution of the Alasso estimator. We propose a modification to the perturbation bootstrap objective function and show that a suitably studentized version of our modified perturbation bootstrap Alasso estimator achieves second-order correctness even when the dimension of the model is allowed to grow to infinity with the sample size. As a consequence, inferences based on the modified perturbation bootstrap is more accurate than the inferences based on the oracle Normal approximation. Simulation results also justifies our method in finite samples.

Abstract:

Let g be a simple finite dimensional Lie algebra. Let A be the Laurent polynomial algebra in n + 1 commuting variables. Then g ⊗ A is naturally a Lie algebra. We now consider the universal central extension g ⊗ A ⊕ ΩA/dA. Then we add derivations of A, Der(A) and consider τ = g ⊗ A ⊕ ΩA/dA ⊕ Der(A). τ is called full toroidal Lie algebra. In this talk, we will explain the classification of irreducible integrable modules for the full toroidal Lie algebra. In the first half of the lecture, we will recall some general facts of toroidal Lie algebras and then we will go on into the technical part.

Abstract:

We briefly discuss the concept of quantum symmetry and mention how it fits into the realm of noncommutative geometry. We take a particular noncommutative topological space coming from connected, directed graph which is called graph C*-algebra and introduce a notion of quantum symmetry of such noncommutative space. A few concrete examples of such quantum symmetry will also be discussed.

Abstract:

Let Gq be the q-deformation of a simply connected simple compact Lie group G of type A, C or D and Oq(G) be the algebra of regular functions on Gq. In this talk, we show that the Gelfand-Kirillov dimension of Oq(G) is equal to the dimension of the underlying real manifold G. If time allows then we will discuss some applications of this result.

Abstract:

Cryptographic protocols base their security on the hardness of mathematical problems. Discrete Logarithm Problem (DLP) is one of them. It is known to be computationally hard in the groups of cryptographic interest. Most important among them are the multiplicative subgroup of finite fields, group of points on an elliptic curve and the group of divisor classes of degree 0 divisors (Jacobian) on a hyperelliptic curve.

Abstract:

Let g be a simple finite dimensional Lie algebra. Let A be the Laurent polynomial algebra in n + 1 commuting variables. Then g ⊗ A is naturally a Lie algebra. We now consider the universal central extension g ⊗ A ⊕ ΩA/dA. Then we add derivations of A, Der(A) and consider τ = g ⊗ A ⊕ ΩA/dA ⊕ Der(A). τ is called full toroidal Lie algebra. In this talk, we will explain the classification of irreducible integrable modules for the full toroidal Lie algebra. In the first half of the lecture, we will recall some general facts of toroidal Lie algebras and then we will go on into the technical part.
In this talk, I will discuss some of the index calculus algorithms for solving the discrete logarithm problem on these groups. More specifically, I will discuss the tower number field sieve algorithm (TNFS) for solving the discrete logarithm problem in the medium to large characteristic finite fields.

Abstract:

Multivariate two-sample testing is a very classical problem in statistics, and several methods are available for it. But, in the current era of big data and high-dimensional data, most of the existing methods fail to perform well, and they cannot even be used when the dimension of the data exceeds the sample size. In this talk, I will propose and investigate some methods based on inter-point distances, which can be conveniently used for data of arbitrary dimensions. I will discuss the merits and demerits of these methods using theoretical as well as numerical results.

Abstract:

Let F be a non-Archimedean local field F with ring of integers O and a fi nite residue fi eld k of odd characteristic. In contrast to the well-understood representation theory of the fi nite groups of Lie type GL_n(k) or of the locally compact groups GL_n(F), representations of groups GL_n(O) are considerably less understood. For example, the uniqueness of Whittaker model is well known for the complex representations of both GL_n(k) and GL_n(F) but is not known for GL_n(O).

Abstract:

In this talk we will see two possible generalizations, due to A. Connes and Frohlich et al., of the de-Rham calculus on manifolds to the noncommutative geometric context. Computations of both these will be highlighted for a class of examples provided by the quantum double suspension, which helps to compare these two generalizations in a very precise sense.

Abstract:

In this presentation, we analyze a semi-discrete finite difference scheme for a stochastic balance laws driven by multiplicative L´evy noise. By using BV estimate of approximate solutions, generated by finite difference Scheme, and Young measure technique in stochastic setting, we show that the approximate solutions converges to the unique BV entropy solution of the underlying problem. Moreover, we show that the expected value of the L^1-difference between approximate solutions and the unique entropy solution converges at a rate O(√∆x), where ∆x being a spatial mesh size.

Abstract:

We will give an overview of some techniques involved in computing the mod p reduction of p-adic Galois representations associated to certain cusp forms of GL(2).

Abstract:

We study the congruences between Galois representations and their base-change along a p-adic Lie extension. We formulate a Main conjecture arising out of these Galois representations which explains how the congruences are related to values of L-functions. This formulation requires the Galois representations to satisfy some conditions and we provide some examples where the conditions can be verified.

Abstract:

The Bellman and Isaac equations appear as the dynamic programming equations for stochastic control and differential games. This talk is concerned with the error estimates for monotone numerical approximations with the viscosity solutions of such equations. I will be focusing on both local and non-local scenario. I will discuss the most general results for equations of order less than one (non-local) and first order (local) equations. For second order equations, convex and non-convex cases (partial results!) will be treated separately. I will explain why the methods are quite different for convex and non-convex cases. To the end, I will discuss the recent developments on error estimates for non-local Isaac equations of order greater than one.

Abstract:

Transport phenomenon is of fundamental as well as practical importance in a wide spectrum of problems of different length and time scales, viz., enhanced oil recovery (EOR), carbon-capture and storage (CCS), contaminant transport in subsurface aquifers, and chromatographic separation. These transport processes in porous media feature different hydrodynamic instabilities [1, 2]. Viscous

Abstract:

This talk will present a class of tests for fitting a parametric model to the regression function in the presence of Berkson measurement error in the covariates without specifying the measurement error density but when validation data is available. The availability of validation data makes it possible to estimate the calibrated regression function non-parametrically. The proposed tests are based on a class of minimized integrated square distances between a nonparametric estimate of the calibrated regression function and the parametric null model being fitted. The asymptotic distributions of these tests under the null hypothesis and against certain alternatives are established. Surprisingly, these asymptotic distributions are the same as in the case of known measurement error density. In comparison, the asymptotic distributions of the corresponding minimum distance estimators of the null model parameters are affected by the estimation of the calibrated regression function. A simulation study shows desirable performance of a member of the proposed class of estimators and tests. This is co-authored with Pei Geng.

Abstract:

Residual empirical processes are known to play a central role in the development of statistical inference in numerous additive models. This talk will discuss some history and some recent advances in the Asymptotic uniform linearity of parametric and nonparametric residual empirical processes. We shall also discuss their usefulness in developing asymptotically distribution free goodness-of-fit tests for fitting an error distribution functions in nonparametric ARCH(1) models. Part of this talk is based on some joint work with Xiaoqing Zhu.

Abstract:

Passive transport models are equations of advection-diffusion type. In most of the applications involving passive transport, the advective fields are of greater magnitude compared to molecular diffusion.
This talk attempts to present a novel theory developed by myself, Thomas Holding (Imperial) and Jeffrey Rauch (Michigan) to address these strong advection problems. Loosely speaking, our strategy is to recast the advection-diffusion equation in moving coordinates dictated by the flow associated with the advective field. Crucial to our analysis is the introduction of a fast time variable and the introduction of some new notions of weak convergence along flows in Lp spaces. We also use ideas from the theory of “homogenization structures” developed by Gabriel Nguetseng.
Our asymptotic results show the following dichotomy:

• If the Jacobian matrix associated with the flow satisfies certain structural conditions (loosely speaking, boundedness in the fast time variable) then the strong advection limit is a non-degenerate diffusion when seen along flows.

• On the other hand, when the Jacobian matrix associated with the flow fails to satisfy the aforementioned structural conditions, then the strong advection limit is a parabolic problem with a constraint. Here we show the appearance of an initial layer where there is an enhanced dissipation along flows.

Our results have close links to

• the Freidlin-Wentzell theory on perturbations of dynamical systems.

• the theory of Relaxation enhancing Lipschitz flows.

This talk will illustrate the theoretical results via various interesting examples. We address some well-known advective fields such as the Euclidean motions, the Taylor-Green cellular flows, the cat’s eye flows and some special class of the Arnold-Beltrami-Childress (ABC) flows. We will also comment on certain examples of hyperbolic or Anosov flows. Some of the results to be presented in this talk can be found in the following

Publication:

T. Holding, H. Hutridurga, J. Rauch. Convergence along mean flows, SIAM J Math.
Anal., Volume 49, Issue 1, pp. 222–271 (2017).

Abstract:

In the problem of selecting a linear model to approximate the true unknown regression model, some necessary and/or sufficient conditions will be discussed for the asymptotic validity of various model selection procedures including Akaike’s AIC, Mallows’ Cp, Schwarz’ BIC, generalized AIC, etc. We shall see that these selection procedures can be classified into three distinct classes according to their asymptotic behaviour. Under some fairly weak conditions, the selection procedures in one class are asymptotically valid if there exists fixed dimensional correct models; while the selection procedures in another class are asymptotically valid if no fixed dimensional correct model exists. On the other hand, the procedures in the third class are compromises of the procedures in the first two classes.

Abstract:

We consider the problem of computationally-efficient prediction from high-dimensional and highly correlated predictors in challenging settings where accurate variable selection is effectively impossible. Direct application of penalization or Bayesian methods implemented with Markov chain Monte Carlo can be computationally daunting and unstable. Hence, some type of dimensionality reduction prior to statistical analysis is in order. Common solutions include application of screening algorithms to reduce the regressors, or dimension reduction using projections of the design matrix. The former approach can be highly sensitive to threshold choice in finite samples, while the later can have poor performance in very high-dimensional settings. We propose a Targeted Random Projection (TARP) approach that combines positive aspects of both the strategies to boost performance. In particular, we propose to use information from independent screening to order the inclusion probabilities of the features in the projection matrix used for dimension reduction, leading to data-informed sparsity. Theoretical results on the predictive accuracy of TARP is discussed in detail along with the rate of computational complexity. Simulated data examples, and real data applications are given to illustrate gains relative to a variety of competitors.

Abstract:

Let G be a simple algebraic group over the field of complex numbers and B be a Borel subgroup of G. Let X_w be a Schubert variety in the flag variety G/B corresponding to an element w of the Weyl group of G, and let Z_w be the Bott–Samelson variety, a natural desingularization of X_w. In this talk we discuss the classification of the "reduced expressions of w" such that Z_w is Fano or weak Fano.

Abstract:

GATE is a big exam used for PG admissions by academic institutes as well as hiring by PSUs. In 2015, more than 8 lakh people appeared for GATE, all subjects combined.

GATE uses formula scoring with negative marking for multiple choice questions or MCQs, e.g., 1, 0, and -1/3 marks for correct, omitted, and wrong answer, respectively. Some questions have 2 marks, with -2/3 for wrong answers. Some have numerical answers, with no negative marking. The number of distinct scores possible is small (below 400), and the number of candidates is large (lakhs).

A modern statistical approach to evaluating MCQ exams uses item response theory (IRT; also called latent trait models). In this approach, each question has some parameters, called “difficulty” or “discrimination” etc., written abstractly as vector a, and each student has an ability or talent attribute, written abstractly as a scalar theta. The probability that a given question (with vector a) will be answered correctly is taken to be some specified function f(theta,a).

The definition of a, and choice of f, are modeling decisions. Later calculations, though complex, are routine. The aim is to estimate a for every question and theta for every candidate.

Two common IRT models are the Rasch model and the 2-parameter logistic (2PL) model, which I will describe. Since our question outcomes are not dichotomous (right/wrong) but polytomous (right/ omitted/wrong), we use the generalized partial credit model (GPCM), which I will describe. GPCM results are poor. The estimated abilities have low correlations with formula scores; these correlations vary across disciplines; and there are also clear conceptual problems in applying GPCM to GATE.

I will then present our new two-step IRT model, where the candidate first decides whether or not to attempt the question, and then (if attempting) gets it right or wrong. The corresponding mathematical model is simple, and aligned with how we believe GATE works. Results are better. The correlation with formula scores is higher, and near-constant across disciplines.

The policy implications of our model are positive. We now have a two-dimensional score of each candidate’s performance. The formula score represents an overall knowledge score, which may appeal to industry. The IRT ability estimate represents an academic potential estimate, which may appeal to academic institutes. If admission and hiring processes are no longer based on the same measure, both may benefit. A minor extra advantage is the possibility of awarding rationally derived ranks to individual candidates, with very few clashes.

As part of the talk, I will also discuss the estimation methods and numerical implementation. However, the emphasis will be on model statement, results, and policy implications. I hope that most of the talk will be accessible to all stakeholders in GATE.

Abstract:

Henry-Parusinski proved that the Lipschitz right classification of function germs admits continuous moduli. This allows us to introduce the notion of Lipschitz simple germs and list all such germs. We will present the method of the classification in this talk and mention some open problems.

Abstract:

In this talk, the likelihood construction is explained under different censoring schemes. Further, the techniques for estimation of the unknown parameters of the survival model are discussed under these censoring schemes.

Abstract:

In this talk, the change point problem in hazard rate is considered. The Lindley hazard change point model is discussed with its application to model bone marrow transplant data. Further, a general hazard regression change point model is discussed with exponential and Weibull distribution as special cases.

Abstract:

A {\em simplicial cell complex} K of dimension d is a poset isomorphic to the face poset of a d-dimensional simplicial CW-complex X. If a topological space M is homeomorphic to X, then K is said to be a {\em pseudotriangulation} of M. In 1974, Pezzana proved that every connected closed PL d-manifold admits a (d+1)-vertex pseudotriangulation. For such a pseudotriangulation of a PL d-manifold one can associate a (d+1)-regular colored graph, called a crystallization of the manifold.
Actually, crystallization is a graph-theoretical tool to study topological and combinatorial properties of PL manifolds. In this talk, I shall define crystallization and show some applications on PL d-Manifolds for d=2, 3 and 4.

In dimension 2, I shall show a proof of the classification of closed surfaces using crystallization. This concept has some important higher dimensional analogs, especially in dimensions 3 and 4. In dimensions 3 and 4, I shall give lower bounds for facets in a pseudotriangulation of a PL manifolds. Also, I shall talk on the regular genus (a higher dimensional analog of genus) of PL d-manifolds. Then I shall show the importance of the regular genus in dimension 4. Additivity of regular genus has been proved for a huge class of PL 4-manifolds. We have some observations on the regular genus, which is related to the 4-dimensional Smooth Poincare Conjecture.

Abstract:

Rayleigh-Bénard convection is a classical extended dissipative system which shows a plethora of bifurcations and patterns. In this talk, I'll present the results of our investigation on bifurcations and patterns near the onset of Rayleigh-Bénard convection of low-Prandtl number fluids. Investigation is done by performing direct numerical simulations (DNS) of the governing equations. Low dimensional modeling of the system using the DNS data is also done to understand the origin of different flow patterns. Our investigation reveals a rich variety of bifurcation structures involving pitchfork, Hopf, homoclinic and Neimar-Sacker bifurcations.

Abstract:

In this talk, we will discuss some recent results on the existence and uniqueness of strong solutions of certain classes of stochastic PDEs in the space of Tempered distributions. We show that these solutions can be constructed from the solutions of "related" finite dimensional stochastic differential equations driven by the same Brownian motion. We will also discuss a criterion, called the Monotonicity inequality, which implies the uniqueness of strong solutions.

Abstract:

The investigation of solute dispersion is most interesting topic of research owing to its outspread applications in various fields such as biomedical engineering, physiological fluid dynamics, etc. The aim of the present study is to know the different physiological processes involved in the solute dispersion in blood flow by assuming the relevant non-Newtonian fluid models. The axial solute dispersion process in steady/unsteady non-Newtonian fluid flow in a straight tube is analyzed in the presence and absence of absorption at the tube wall. The pulsatile nature of the blood is considered for unsteady flow. Owing to non-Newtonian nature of blood at low shear rate in small vessels, non-Newtonian Casson, Herschel-Bulkley, Carreau and Carreau-Yasuda fluid models which are most relevant for blood flow analysis are considered. The three transport coefficients i.e., exchange, convection and dispersion coefficients which describe the whole dispersion process in the system are determined. The mean concentration of solute is analyzed at all time. A comparative study of the solute dispersion is made among the Newtonian and non-Newtonian fluid models. Also, the comparison of solute dispersion between single- and two-phase models is made at all time for different radius of micro blood vessels.

Abstract:

Quasilinear symmetric and symmetrizable hyperbolic system has a wide range of applications in engineering and physics including unsteady Euler and potential equations of gas dynamics, inviscid magnetohydrodynamic (MHD) equations, shallow water equations, non-Newtonian fluid dynamics, and Einstein field equations of general relativity. In the past, the Cauchy problem of smooth solutions for these systems has been studied by several mathematicians using semigroup approach and fixed point arguments. In a recent work of M. T. Mohan and S. S. Sritharan, the local solvability of symmetric hyperbolic system is established using two different methods, viz. local monotonicity method and a frequency truncation method. The local existence and uniqueness of solutions of symmetrizable hyperbolic system is also proved by them using a frequency truncation method. Later they established the local solvability of the stochastic quasilinear symmetric hyperbolic system perturbed by Levy noise using a stochastic generalization of the localized Minty-Browder technique. Under a smallness assumption on the initial data, a global solvability for the multiplicative noise case is also proved. The essence of this talk is to give an overview of these new local solvability methods and their applications.

Abstract:

In this talk, we present the moduli problem of rank 2 torsion free Hitchin pairs of fixed Euler characteristic χ on a reducible nodal curve. We describe the moduli space of the Hitchin pairs. We define the analogue of the classical Hitchin map and describe the geometry of general Hitchin fibre. Time permits,talk on collaborated work with Balaji and Nagaraj on degeneration of moduli space of Hitchin pairs.

Abstract:

The formalism of an abelian category'' is meant to axiomatize the operations of linear algebra. From there, the notion of derived category'' as the category of complexes upto quasi-isomorphisms'' is natural, motivated in part by topology. The formalism of t-structures allows one to construct new abelian categories which are quite useful in practice (giving rise to new cohomology theories like intersection cohomology, for example).
In this talk we want to discuss a notion of punctual (=point-wise'') gluing of t-structures which we formulated in the context of algebraic geometry. The essence of the construction is classical and well known, but the new language leads to several applications in the motivic world.

Abstract:

Using equivariant obstruction theory we construct equivariant maps from certain classifying spaces to representation spheres for cyclic groups, product of elementary Abelian groups and dihedral groups.
Restricting them to finite skeleta constructs equivariant maps between spaces which are related to the topological Tverberg conjecture. This answers negatively a question of \"Ozaydin posed in relation to weaker versions of the same conjecture. Further, it also has consequences for Borsuk-Ulam properties of representations of cyclic and dihedral groups.
This is joint work with Samik Basu.

Abstract:

Abstract:

Quantile regression provides a more comprehensive relationship between a response and covariates of interest compared to mean regression. When the response is subject to censoring, estimating conditional mean requires strong distributional assumptions on the error whereas (most) conditional quantiles can be estimated distribution-free. Although conceptually appealing, quantile regression for censored data is challenging due to computational and theoretical difficulties arising from non-convexity of the objective function involved. We consider a working likelihood based on Powell's objective function and place appropriate priors on the regression parameters in a Bayesian framework. In spite of the non-convexity and misspecification issues, we show that the posterior distribution is strong selection consistent. We provide a “Skinny” Gibbs algorithm that can be used to sample the posterior distribution with complexity linear in the number of variables and provide empirical evidence demonstrating the fine performance of our approach.

Abstract:

Abstract:

Given a closed smooth Riemannian manifold M, the Laplace operator is known to possess a discrete spectrum of eigenvalues going to infinity. We are interested in the properties of the nodal sets and nodal domains of corresponding eigen functions in the high energy (semiclassical) limit. We focus on some recent results on the size of nodal domains and tubular neighbourhoods of nodal sets of such high energy eigenfunctions (joint work with Bogdan Georgiev).

Abstract:

I will define affine Kac-Moody algebras, toroidal Lie algebras and full toroidal Lie algebras twisted by several finite order automorphisms and classify integrable representations of twisted full toroidal Lie algebras.

Abstract:

In recent years, one major focus of modeling spatial data has been to connect two contrasting approaches, namely, the Markov random field approach and the geostatistical approach. While the geostatistical approach allows flexible modeling of the spatial processes and can accommodate continuum spatial variation, it faces formidable computational burden for large spatial data. On the other hand, spatial Markov random fields facilitate fast statistical computations but they lack in flexibly accommodating continuum spatial variations. In this talk, I will discuss novel statistical models and methods which allow us to accommodate continuum spatial variation as well as fast matrix-free statistical computations for large spatial data. I will discuss an h-likelihood method for REML estimation and I will show that the standard errors of these estimates attain their Rao-Cramer lower bound and thus are statistically efficient. I will discuss applications on ground-water Arsenic contamination and chlorophyll concentration in ocean. This is a joint work with Debashis Mondal at Oregon State University

Abstract:

It is a well-known result from Hermann Weyl that if alpha is an irrational number in [0,1) then the number of visits of successive multiples of alpha modulo one in an interval contained in [0,1) is proportional to the size of the interval. In this talk we will revisit this problem, now looking at finer joint asymptotics of visits to several intervals with rational end points. We observe that the visit distribution can be modelled using random affine transformations; in the case when the irrational is quadratic we obtain a central limit theorem as well. Not much background in probability will be assumed. This is in joint work with Jon Aaronson and Michael Bromberg.

Abstract:

The action of Gl_n(F_q) on the polynomial ring over n variables has been studied extensively by Dickson and the invariant ring can be explicitly described. However, the action of the same group on the ring when we go modulo Frobenius powers is not completely solved. I'll talk about some interesting aspects of this modified version of the problem. More specifically, I'll discuss a conjecture by Lewis, Reiner and Stanton regarding the Hilbert series corresponding to this action and try to prove some special cases of this conjecture.

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One way to understand representations of a group is to ‘restrict’ the representation to its various subgroups, especially to those subgroups which give multiplicity one or finite multiplicity. We shall discuss a few examples of restriction for the representations of p-adic groups. Our main examples will be the pairs $(GL_2(F), E^*) and (GL_2(E), GL_2(F))$, where $E/F$ is a quadratic extension of $p$-adic fields. These examples can be considered as low-rank cases of the well known Gross-Prasad conjectures, where one considers various ‘restrictions’ simultaneously. Further, we consider a similar ‘restriction problem’ when the groups under consideration are certain central extensions of $F$-point of a linear algebraic groups by a finite cyclic group. These are topological central extensions and called ‘covering groups’ or ‘metaplectic groups’. These covering groups are not $F$-point of any linear algebraic group. We restrict ourselves to only a two fold covers of these groups and their ‘genuine’ representations. Covering groups naturally arise in the study of modular form of half-integral weight. Some results that we will discuss are outcome of a joint work with D. Prasad.

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It is a well-accepted practice in experimental situations to use auxiliary information to enhance the accuracy of the experiment i.e., to reduce the experimental error. In its simplest form of use of auxiliary information, data generated through an experiment are statistically modeled in terms of some assignable source(s) of variation, besides a chance cause of variation. The assignable causes comprise ‘treatment’ parameters and the ‘covariate’ parameter(s). This generates a family of ‘covariate models’ - serving as a ‘combination’ of ‘varietal design models’ and ‘regression models’. These are the well-known Analysis of Covariance
(ANCOVA) Models. Generally, for such models, emphasis is given on analysis of the data [in terms of inference on treatment effects contrasts] and not so much on the choice of the covariate(s) values. In this presentation, we consider the situation where there is some flexibility in the choice of the experimental units with specified values of the covariates. The notion of 'optimal' choice of values of the covariates for a given design set-up so as to minimize variance for parameter estimates has attracted attention of researchers in recent times. Hadamard matrices and Mixed orthogonal array have been conveniently used to construct optimum covariate designs with as many covariates as possible.

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Recent increase in the use of 3-D magnetic resonance images (MRI) and analysis of functional magnetic resonance images (fMRI) in medical diagnostics makes imaging, especially 3-D imaging very important. Observed images often contain noise which should be removed in such a way that important image features, e.g., edges, edge structures, and other image details should be preserved, so that subsequent image analyses are reliable. Direct generalizations of existing 2-D image denoising techniques to 3-D images cannot preserve complicated edge structures well, because, the edge structures in a 3-D edge surface can be much more complicated than the edge structures in a 2-D edge curve. Moreover, the amount of smoothing should be determined locally, depending on local image features and local signal to noise ratio, which is much more challenging in 3-D images due to large number of voxels. In this talk, I will talk about a multi-resolution and locally adaptive 3-D image denoising procedure based on local clustering of the voxels. I will provide a few numerical studies which show that the denoising method can work well in many real world applications. Finally, I will talk about a few future research directions along with some introductory research problems for interested students. Most parts of my talk should be accessible to the audience of diverse academic background.

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We shall discuss a new method of computing (integral) homotopy groups of certain manifolds in terms of the homotopy groups of spheres. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we verify a conjecture of J. C.
Moore. This is joint work with Somnath Basu.

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The theory of pseudo-differential operators provides a flexible tool for treating certain problems in linear partial differential equations. The Gohberg lemma on unit circle estimates the distance (in norm) from a given zero-order operator to the set of the compact operators from below in terms of the symbol. In this talk, I will introduce a version of the Gohberg lemma on compact Lie groups using the global calculus of pseudo-differential operators. Applying this, I will obtain the bounds for the essential spectrum and a criterion for an operator to be compact. The conditions used will be given in terms of the matrix-valued symbols of operators

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Let $F$ be a $p$-adic field. The restriction of an irreducible admissible representation of $GL_{2}(F)$ to its maximal tori was studied by Tunnell and Saito; and they provide a very precise answer. In particular, one gets multiplicity one. This can be considered as the first case of the Gross-Prasad conjectures.

We will discuss a metaplectic variation of this question. More precisely, we will talk about the restriction of an irreducible admissible genuine representation of the two fold metaplectic cover $\widetilde{GL}_2(F)$ of $GL_2(F)$ to the inverse image in $\widetilde{GL}_2(F)$ of a maximal torus in $GL_2(F)$. We utilize a correspondence between irreducible admissible genuine supercuspidal representations of the metaplectic group widetilde{SL}_2(F)$to irreducible admissible supercuspidal representations of linear group$SL_2(F)$. This is a joint work with D. Prasad. Abstract: We will describe the problem of mod p reduction of p-adic Galois representations. For two dimensional crystalline representations of the local Galois group Gal(Q¯ p |Qp ), the reduction can be computed using the compatibility of p-adic and mod p Local Langlands Correspondences; this method was first introduced by Christophe Breuil in 2003. After giving a brief sketch of the history of the problem, we will discuss how the reductions behave for representations with slopes in the half- open interval [1, 2). In the relevant cases of reducible reduction, one may also ask if the reduction is peu or tres ramifi´ee. We will try to sketch an answer to this question, if time permits. (Joint works with Eknath Ghate, and also with Sandra Rozensztajn for slope 1.) Abstract: Abstract: Let M be a compact manifold without boundary. Define a smooth real valued function of the space of Riemannian metrics of M by taking Lp-norm of Riemannian curvature for p >= 2. Compact irreducible locally symmetric spaces are critical metrics for this functional. I will prove that rank 1 symmetric spaces are local minima for this functional by studying stability of the functional at those metrics. I will also show examples of irreducible symmetric metrics which are not local minima for it. Abstract: Unfolding operators have been introduced and used to study homogenization problems. Initially, it was introduced for problems with rapidly oscillating coefficients and porous domains. Later, this has been developed for domains with oscillating boundaries, typically with rectangular or pillar type boundaries which are classified as non-smooth. In this talk, we will demonstrate the development of generalized unfolding operators, where the oscillations of the domain can be smooth and hence it has wider applications. We will also see the further adaptation of this new unfolding operators for circular domains with rapid oscillations with high amplitude of O(1). This has been applied to homogenization problems in circular domains as well. Abstract: We describe three approaches to the classical p-completion(or localization) of a topological space: as spaces, through cosimplicial space resolutions, and through mapping algebras – and show how they are related through appropriate "universal" systems of higher cohomology operations. All terms involved will be explained in the talk. Abstract: We deal with the following eigenvalue optimization problem: Given a bounded open disk$B$in a plane, how to place an obstacle$P$of fixed shape and size within$B$so as to maximize or minimize the fundamental eigenvalue$lambda_1$of the Dirichlet Laplacian on$B setmunus P$. This means that we want to extremize the function$rho \rightarrow lambda_1(B setminus rho(P))$, where$rho$runs over the set of rigid motions such that$rho(P) subset B$. We answer this problem in the case where$P$is invariant under the action of a dihedral group$D_{2n}$, and where the distance from the center of the obstacle$P$to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of$P$coincide with a diameter of$B$. The maximizing and the minimizing configurations are identified. Abstract: We deal with the following eigenvalue optimization problem: Given a bounded open disk B in a plane, how to place an obstacle P of fixed shape and size within B so as to maximize or minimize the fundamental eigenvalue λ1 of the Dirichlet Laplacian on B \ P . This means that we want to extremize the function ρ → λ1 (B \ ρ(P)), where ρ runs over the set of rigid motions such that ρ(P) ⊂ B. We answer this problem in the case where P is invariant under the action of a dihedral group D 2n, and where the distance from the center of the obstacle P to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of P coincide with a diameter of B. The maximizing and the minimizing configurations are identified. Abstract: Matched case-control studies are popular designs used in epidemiology for assessing the effects of exposures on binary traits. Modern investigations increasingly enjoy the ability to examine a large number of exposures in a comprehensive manner. However, risk factors often tend to be related in a non-trivial way, undermining efforts to identify true important ones using standard analytic methods. Epidemiologists often use data reduction techniques by grouping the prognostic factors using a thematic approach, with themes deriving from biological considerations. However, it is important to account for potential misspecification of the themes to avoid false positive findings. To this end, we propose shrinkage type estimators based on Bayesian penalization methods. Extensive simulation is reported that compares the Bayesian and Frequentist estimates under various scenarios. The methodology is illustrated using data from a matched case-control study investigating the role of polychlorinated biphenyls in understanding the etiology of non-Hodgkin's lymphoma. Abstract: Cowen and Douglas have shown that the curvature is a complete invariant for a certain class of operators. Several ramifications of this result will be discussed. Abstract: It is well known that the center of U(glN) is _nitely generated as an algebra. Gelfand de_ned central elements (Gelfand invariants) Tk for every positive integer k. It is known that the _rst N generates the center. The decomposition of tensor product modules for a reductive Lie algebra is a classical problem. It is known that each Gelfand invariant acts as a scaler on an irreducible submodule of a tensor product module. In this talk, for each k we de_ne several operators which commute with glN action but does not act as a scalar. This means these operators take one highest weight vector to another highest weight vector. Thus it is a practical algorithm to produce more highest weight vectors once we known one of them. Further the sum of these operators is Tk. If time permits we will de_ne some of these operators in the generality of Kac-Moody Lie algebras. Abstract: Recent increase in the use of 3-D magnetic resonance images (MRI) and analysis of functional magnetic resonance images (fMRI) in medical diagnostics makes imaging, especially 3-D imaging very important. Observed images often contain noise which should be removed in such a way that important image features, e.g., edges, edge structures, and other image details should be preserved, so that subsequent image analyses are reliable. Direct generalizations of existing 2-D image denoising techniques to 3-D images cannot preserve complicated edge structures well, because, the edge structures in a 3-D edge surface can be much more complicated than the edge structures in a 2-D edge curve. Moreover, the amount of smoothing should be determined locally, depending on local image features and local signal to noise ratio, which is much more challenging in 3-D images due to large number of voxels. In this talk, I will talk about a multi-resolution and locally adaptive 3-D image denoising procedure based on local clustering of the voxels. I will provide a few numerical studies which show that the denoising method can work well in many real world applications. Finally, I will talk about a few future research directions along with some introductory research problems for interested students. Most parts of my talk should be accessible to the audience of diverse academic background. Abstract: The theory of pseudo-di_erential operators has provided a very powerful and exible tool for treating certain problems in linear partial differential equations. The importance of the Heisenberg group in general harmonic analysis and problems involving partial di_erential operators on manifolds is well established. In this talk, I will introduce the pseudo-di_erential operators with operator-valued symbols on the Heisenberg group. I will give the necessary and su_cient conditions on the symbols for which these operators are in the Hilbert-Schmidt class. I will identify these HilbertSchmidt operators with the Weyl transforms with symbols in L2(R2n+1 _ R2n+1). I will also provide a characterization of trace class pseudo-di_erential operators on the Heisenberg group. A trace formula for these trace class operators would be presented. Abstract: A finite separable extension E of a field F is called primitive if there are no intermediate extensions. It is called a solvable extension if the group of automorphisms of its galoisian closure over F is a solvable group. We show that a solvable primitive extension E of F is uniquely determined (up to F-isomorphism) by its galoisian closure and characterise the extensions D of F which are the galoisian closures of some solvable primitive extension E of F. Abstract: The genesis of “zero” as a number, that even a child so casually uses today, is a long and involved one. A great many persons concerned with the history of its evolution, today accepts that the number “zero”, in its true potential, as we use it in our present day mathematics, has its root, conceptually as well as etymologically, in the word ‘´S¯unya’ of the Indian antiquity. It was introduced in India by the Hindu mathematicians, which eventually became a numeral for mathematical expression for “nothing”, and via the Arabs, went to Europe, where it survived a prolonged battle with the Church (which once banned the use of ‘zero’ !) through centuries. However, the time frame of its origin in Indian antiquity is still hotly debated. Furthermore, some recent works even try to suggest that a trace of the concept, if not in total operational perspective, might have a Greek origin that traveled to India during the Greek invasion of the northern part of the country. However, from the works on Vedic prosody by Pi ˙ngala (Pi ˙ngalacchandah. s¯utra) [3rd Century BC] to the concept of “lopa” in the grammarian Panini, (As. t.¯adhy¯ay¯ı) (400-700 BC, by some modern estimates) it appears very likely that the thread of rich philosophical and socio academic ambiances of Indian antiquity was quite pregnant with the immensity of the concept of ‘´S¯unya’ - a dichotomy as well as a simultaneity between nothing and everything, the ‘zero’ of void and that of an all pervading ‘fathomless’ infinite. A wide variety of number systems were used across various ancient civilizations, like the Inca, Egyptian, Mayan, Babylonian, Greek, Roman, Chinese, Arab, Indian etc. Some of them even had ‘some sort of zero’ in their system! Why then, only the Indian zero is generally accepted as the ancestor of our modern mathematical zero? Why is it only as late as in 1491, that one may find the first ever mention of ‘zero’ in a book from Europe? In this popular level lecture, meant for a general audience, based on the mindboggling natural history of ‘zero’, we would like to discuss, through numerous slides and pictures, the available references to the evolution and struggle of the concept of place-value based enumeration system along with a “zero” in it, in its broader social and philosophical contexts. Abstract: Logistic regression is an important and widely used regression model for binary responses and is extensively used in many applied fields. In the presence of misclassified binary responses using internal validation data, a pseudo-likelihood method is proposed for estimation of logistic regression parameters. Under minimal assumptions we establish rigorous asymptotic results for the pseudo-likelihood based estimators. A bootstrapped version of the pseudo likelihood based estimators is also proposed and its distributional consistency is proved enabling us to effectively use bootstrap method for statistical inference. The results of the simulation studies clearly indicate the superiority of pseudo-likelihood based estimators to the full likelihood based estimators, and the likelihood estimators based on misclassified binary responses only. Also, inferences on the regression parameters using asymptotic distribution of pseudo-likelihood estimators and its bootstrap version are found to be similar. Joint work with T. Bandyopadhyay (IIM-A) and Sumanta Adhya (WBSU). Abstract: High dimension, low sample size data pose great challenges for the existing statistical methods. For instance, several popular methods for cluster analysis based on the Euclidean distance often fail to yield satisfactory results for high dimensional data. In this talk, we will discuss a new measure of dissimilarity, called MADD and see how it can be used to achieve perfect clustering in high dimensions. Another important problem in cluster analysis is to find the number of clusters in a data set. We will see that many existing methods for this problem can be modified using MADD to achieve superior performance. A new method for estimating the number of clusters will also be discussed. We will present some theoretical and numerical results in this connection. This presentation is based on joint works with Anil K. Ghosh. Abstract: Most of todays experimentally verifiable scientific research, not only requires us to resolve the physical features over several spatial and temporal scales but also demand suitable techniques to bridge the information over these scales. In this talk I will provide two examples in mathematical biology to describe these systems at two levels: the micro level and the macro (continuum) level. I will then detail suitable tools in homogenization theory to link these different scales. The first problem arises in mathematical physiology: swelling-de-swelling mechanism of mucus, an ionic gel. Mucus is packaged inside cells at high concentration (volume fraction) and when released into the extracellular environment, it expands in volume by two orders of magnitude in a matter of seconds. This rapid expansion is due to the rapid exchange of calcium and sodium that changes the cross-linked structure of the mucus polymers, thereby causing it to swell. Modelling this problem involves a two-phase, polymer/solvent mixture theory (in the continuum level description), together with the chemistry of the polymer, its nearest neighbor interaction and its binding with the dissolved ionic species (in the micro-scale description). The problem is posed as a free-boundary problem, with the boundary conditions derived from a combination of variational principle and perturbation analysis. The equilibrium-states of the ionic gels are analyzed. In the second example, we numerically study the adhesion-fragmentation dynamics of rigid, round particles clusters subject to a homogeneous shear flow. In the macro level we describe the dynamics of the number density of these cluster. The description in the micro-scale includes (a) binding/unbinding of the bonds attached on the particle surface, (b) bond torsion, (c) surface potential due to ionic medium, and (d) flow hydrodynamics due to shear flow. Abstract: In this talk, I shall consider the high-dimensional moving average (MA) and autoregressive (AR) processes. My goal will be to explore the asymptotics for eigenvalues of the sample autocovariance matrices. This asymptotics will help in the estimation of unknown order of the high-dimensional MA and AR processes. Our results will also provide tests of different hypotheses on coefficient matrices. This talk will be based on joint works with Prof. Arup Abstract: McKay correspondence relates orbifold cohomology with the cohomology of a crepant resolution. This is a phenomenon in algebraic geometry. It was proved for toric orbifolds by Batyrev and Dais in the nineties. In this talk we present a similar correspondence for omnioriented quasitoric orbifolds. The interesting feature is how we deal with the absence of an algebraic or analytic structure. In a suitable sense, our correspondence is a generalization of the algebraic one. Orbifolds will be developed from scratch. Abstract: We show that if a modular cuspidal eigenform f of weight 2k is 2-adically close to an elliptic curve E over the field of rational numbers Q, which has a cyclic rational 4-isogeny, then the n-th Fourier coefficient of f is non-zero in the short interval (X, X + cX^{1/4}) for all X >> 0 and for some c > 0. We use this fact to produce non-CM cuspidal eigenforms f of level N>1 and weight k > 2 such that i_f(n) << n^{1/4} for all n >> 0$.

Abstract:

Given an irreducible polynomial f(x) with integer coefficients and a prime number p, one wishes to determine whether f(x) is a product of distinct linear factors modulo p. When f(x) is a solvable polynomial, this question is satisfactorily answered by the Class Field Theory. Attempts to find a non-abelian Class Field Theory lead to the development of an area of mathematics called the Langlands program.
The Langlands program, roughly speaking, predicts a natural correspondence between the finite dimensional complex representations of the Galois group of a local or a number field and the infinite dimensional representations of real, p-adic and adelic reductive groups. I will give an outline of the statement of the local Langlands correspondence. I will then briefly talk about two of the main approaches towards the Langlands program - the type theoretic approach relying on the theory of types developed by Bushnell-Kutzko and others; and the endoscopic approach relying on the trace formula and endoscopy. I will then state a couple of my results involving these two approaches.

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Affine Kac-Moody algebras are infinite dimensional analogs of semi-simple Lie algebras and have a central role both in Mathematics and Mathematical Physics. Representation theory of these algebras has grown tremendously since their independent introduction by R.V. Moody and V.G.
Kac in 1968. Extended affine Lie algebras are natural generalization of affine Kac-Moody algebras. Centerless Lie tori play an important role in explicitly constructing the extended affine Lie algebras; they play similar role as derived algebras modulo center in the realization of affine Kac-Moody algebras.
In this talk we consider the universal central extension of a centerless Lie torus and classify its irreducible integrable modules when the center acts non-trivially. They turn out to be highest weight modules for the direct sum of finitely many affine Lie algebras upto an automorphism. This is a joint work with E. Rao.

Abstract:

Given an irreducible polynomial f(x) with integer coefficients and a prime number p, one wishes to determine whether f(x) is a product of distinct linear factors modulo p. When f(x) is a solvable polynomial, this question is satisfactorily answered by the Class Field Theory. Attempts to find a non-abelian Class Field Theory lead to the development of an area of mathematics called the Langlands program.
The Langlands program, roughly speaking, predicts a natural correspondence between the finite dimensional complex representations of the Galois group of a local or a number field and the infinite dimensional representations of real, p-adic and adelic reductive groups. I will give an outline of the statement of the local Langlands correspondence. I will then briefly talk about two of the main approaches towards the Langlands program - the type theoretic approach relying on the theory of types developed by Bushnell-Kutzko and others; and the endoscopic approach relying on the trace formula and endoscopy. I will then state a couple of my results involving these two approaches.

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The aim of this lecture is to consider a singularly perturbed semi-linear elliptic problem with power non-linearity in Annular Domains of R^{2n} and show the existence of two orthogonal S^{n−1} concentrating solutions. We will discuss some issues involved in the proof in the context of S^1 concentrating solutions of similar nature.

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Let E1 and E2 be elliptic curves defined over the field of rational numbers with good and ordinary reduction at an odd prime p, and have irreducible, equivalent mod p Galois representations. In this talk, we shall discuss the variation in the parity of ranks of E1 and E2 over certain number fields.

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The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this talk we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the Cowen-Douglas class.

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In this talk we will introduce the theory of p-adic families of modular forms and more generally p-adic family of automorphic forms. Notion of p-adic family of modular forms was introduced by Serre and later it was generalized in various directions by the work of Hida, Coleman-Mazur, Buzzard and various other mathematicians. Study of p-adic families play a crucial role in modern number theory and in recent years many classical long standing problems in number theory has been solved using p-adic families. I'll state some of the problems in p-adic families of automorphic forms that I worked on in the past and plan to work on in the future.

Abstract:

Fractal Interpolation Function (FIF) - a notion introduced by Michael Barnsley - forms a basis of a constructive approximation theory for non-differentiable functions. In view of their diverse applications, there has been steadily increasing interest in the particular flavors of FIF such as Ḧolder continuity, convergence, stability, and differentiability. Apart from these properties, a good interpolant/approximant should reflect geometrical shape properties that are described mathematically in terms of positivity, monotonicity, and convexity. These properties act as constraints on the approximation problem.
In this talk, we discuss certain shape preserving aspects of polynomial and rational FIFs. It has been observed that the notion of FIF provides a bounded linear operator on the space of all continuous functions and this operator acts as a medium through which the theory of fractal functions interacts with various other branches of mathematics. The talk is intended also to explore further on this fractal operator and to introduce a new class of approximants. If time permits, we shall skim through the notion of fractal Fourier series which has recently been introduced to the literature.

Abstract:

A Subordinated stochastic process X(T(t)) is obtained by time-changing a process X(t) with a positive non-decreasing stochastic process T(t). The process X(T(t)) is said to be subordinated to the driving process X(t) and the process T(t) is called the directing process.
Subordinated processes demonstrate interesting probabilistic properties and have applications in finance, economics, statistical physics and fractional calculus
The aim of this talk is to discuss the concept of subordinated processes and to explore the applications of these processes in different fields with reference to my research work.

Abstract:

Many physical phenomena can be modeled using partial differential equations. In this talk, applications of PDEs, in particular hyperbolic conservation laws will be shown to granular matter theory and crowd dynamics.
An explicit finite volume Godunov–type, well-balanced numerical scheme using the idea of discontinuous flux for hyperbolic conservation laws for a 2 × 2 system of hyperbolic balance laws, modeling the growth of a sandpile under the action of a vertical source on a flat bounded table was proposed in [1, 2]. In such a system, an eikonal equation for the standing layer of the pile is coupled to an advection equation for the rolling layer. The key steps are including the source term as a part of convection term and decoupling the system into an uncoupled system of conservation laws with discontinuous coefficients. The performance of the proposed numerical scheme is dis- played through the numerical experiments presented for different choices of boundary conditions considered in the papers of Falcone, Vita et al [3, 4].
The recent literature has introduced models based on nonlocal conservation laws in several space dimensions, e.g., see [5] for crowd dynamics applications. Construction of a Lax–Friedrichs type numerical algorithm for such systems is shown for such systems and proved to be converging to the exact solution. The key step in the convergence proof is providing strong BV estimates on the approximate solutions. A new nonlocal model of crowd dynamics aiming to capture the phenomenon of so–called lanes formation, when two groups of people move in opposite directions, is also presented. Numerical integrations show the convergence rate, lanes formations as well as various qualitative properties of the class of equations considered, see [6, 7].

Abstract:

Moduli of vector bundles on a curve was constructed and studied by Mumford, Seshadri and many others. Simpson simplified and gave general construction of moduli of pure sheaves on higher dimensional projective varieties in characteristic zero. Langer extended it to the positive characteristics. Alvarez-Consul and King gave another construction by using moduli of representations of Kronecker quiver.
In this talk we'll briefly describe the functorial approach of Alvarez-Consul and King. We'll present a generalisation of this approach to moduli of equivariant sheaves by introducing the notion of Kronecker-McKay quiver. This is obtained in a joint work with Sanjay Amrutiya.
If time permits we will give some application of our construction to equivariant theta functions.

Abstract:

Homogenization is a branch of science where we try to understand microscopic structures via a macroscopic medium. Hence, it has applications in various branches of science and engineering. This study is basically developed from material science in the creation of composite materials though the contemporary applications are much far and wide. It is a process of understanding the microscopic behavior of an in-homogeneous medium via a homogenized medium. Mathematically, it is a kind of asymptotic analysis.
We plan to start with an illustrative example of limiting analysis in 1-D for a second order elliptic partial differential equation. We will also see some classical results in the case of periodic composite materials and oscillating boundary domain. The emphasis will be on the computational importance of homogenization in numerics by the introduction of correctors.
In the second part of the talk, we will see a study on optimal control problems posed in a domain with highly oscillating boundary. We will consider periodic controls in the oscillating part of the domain with a model problem of Laplacian and try to understand their optimality and asymptotic behavior.

Abstract:

In this talk we will discuss certain aspects of vector bundles over complex projective spaces and projective hypersurfaces. Our focus will be to find conditions under which a vector bundle can be written as a direct sum of smaller rank bundles or when it can be extended to a larger space.
We will mention some open conjectures in this area. We will also discuss some recent work. The talk should be accessible to a graduate student.

Abstract:

Enumerative geometry is a branch of mathematics that deals with the following question: "How many geometric objects are there that satisfy certain constraints?" The simplest example of such a question is "How many lines pass through two points?". A more interesting question is "How many lines are there in three dimensional space that intersect four generic lines?". An extremely important class of enumerative question is to ask "How many rational (genus 0) degree d curves are there in CP^2 that pass through 3d-1 generic points?" Although this question was investigated in the nineteenth century, a complete solution to this problem was unknown until the early 90's, when Kontsevich-Manin and Ruan-Tian announced a formula. In this talk we will discuss some natural generalizations of the above question; in particular we will be looking at rational curves on del-Pezzo surfaces that have a cuspidal singularity. We will describe a topological method to approach such questions. If time permits, we will also explain the idea of how to enumerate genus g curves with a fixed complex structure by comparing it with the Symplectic Invariant of a manifold (which are essentially the number of curves that are solutions to the perturbed d-bar equation) .

Abstract:

For Gaussian process models, likelihood based methods are often difficult to use with large irregularly spaced spatial datasets due to the prohibitive computational burden and substantial storage requirements. Although various approximation methods have been developed to address the computational difficulties, retaining the statistical efficiency remains an issue. This talk focuses on statistical methods for approximating likelihoods and score equations. The proposed new unbiased estimating equations are both computationally and statistically efficient, where the covariance matrix inverse is approximated by a sparse inverse Cholesky approach. A unified framework based on composite likelihood methods is also introduced, which allows for constructing different types of hierarchical low rank approximations. The performance of the proposed methods is investigated by numerical and simulation studies, and parallel computing techniques are explored for very large datasets. Our methods are applied to nearly 90,000 satellite-based measurements of water vapor levels over a region in the Southeast Pacific Ocean, and nearly 1 million numerical model generated soil moisture data in the area of Mississippi River basin. The fitted models facilitate a better understanding of the spatial variability of the climate variables.

Ying Sun is an Assistant Professor of Statistics in the Division of Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) at KAUST. She joined KAUST after one-year service as an Assistant Professor in the Department of Statistics at the Ohio State University (OSU). Before joining OSU, she was a postdoctorate researcher at the University of Chicago in the research network for Statistical Methods for Atmospheric and Oceanic Sciences (STATMOS), and at the Statistical and Applied Mathematical Sciences Institute (SAMSI) in the Uncertainty Quantification program.

Abstract:

In this talk, the LAD estimation procedure and related issues will be discussed in the non-parametric convex regression problem. In addition, based on the concordance and the discordance of the observations, a test will be proposed to check whether the unknown non-parametric regression function is convex or not. Some preliminary ideas to formulate the test statistics of the test along with their properties will also be investigated.

Abstract:

The general philosophy of Langlands' functoriality predicts that given two groups H and G, if there exists a 'nice' map between the respective L-groups of H and G then using the map we can transfer automorphic representations of H to that of G. Few examples of such transfers are Jacquet-Langlands' transfer, endoscopic transfer and base change. On the other hand, by the work of Serre, Hida, Coleman, Mazur and many other mathematicians, we can now construct p-adic families of automorphic forms for various groups. In this talk, we will discuss some examples of Langlands' transfers which can be p-adically interpolated to give rise to maps between appropriate p-adic families of automorphic forms.

Abstract:

Over the last few years, O-minimal structures have emerged as a nice framework for studying geometry and topology of singular spaces. They originated in model theory and provide an axiomatic approach of characterizing spaces with tame topology. In this talk, we will first briefly introduce the notion of O-minimal structures and present some of their main properties. Then, we will consider flat currents on pseudomanifolds that are definable in polynomially bounded o-minimal structures. Flat currents induce cohomology restrictions on the pseudomanifolds and we will show that this cohomology is related to their intersection cohomology.

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The Grothendieck ring, K0(M), of a model-theoretic structure M was defined by Krajicˇek and Scanlon as a generalization of the Grothendieck ring of varieties used in motivic integration. I will introduce this concept with some examples and then proceed to define the K-theory of M via a symmetric monoidal category. Prest conjectured that the Grothendieck ring of a non-zero right module, MR, is nontrivial when thought of as a structure in the language of right R-modules. The proof that such a Grothendieck ring is in fact a non-zero quotient of a monoid ring relies on techniques from simplicial homology, combinatorics, lattice theory as well as algebra. I will also discuss this result that settled Prest’s conjecture in the affirmative..

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We know how to multiply two real numbers or two complex numbers. In both cases it is bilinear and norm preserving. It is natural to ask which of the other R^n admits a such multiplication. We will discuss how this question is related to vector fields on sphere and the answer given by famous theorem of J. F. Adams.

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Euler system is a powerful machinery in Number theory to bound the size of Selmer groups. We start from introducing a brief history and we will explain the necessity of generalizing this machinery for the framework of deformations as well as the technical difficulty of commutative algebra which happens for such generalizations. If time permits, we talk about ongoing joint work on generalized Euler system with Shimomoto.

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Monodromy group of a hypergeometric differential equation is defined as image of the fundamental group G of Riemann sphere minus three points, namely 0, 1, and the point at infinity, under some certain representation of G inside the general linear group GL_n. By a theorem of Levelt, the monodromy groups are the subgroups of GL_n generated by the companion matrices of two monic polynomials f and g of degree n.

If we start with f, g, two integer coefficient monic polynomials of degree n, which satisfy some "conditions" with f(0)=g(0)=1 (resp. f(0)=1, g(0)=-1), then the associated monodromy group preserves a non-degenerate integral symplectic form (resp. quadratic form), that is, the monodromy group is a subgroup of the integral symplectic group (resp. orthogonal group) of the associated symplectic form (resp. quadratic form).

In this talk, we will describe a sufficient condition on a pair of the polynomials that the associated monodromy group is an arithmetic subgroup (a subgroup of finite index) of the integral symplectic group, and show some examples of arithmetic orthogonal monodromy groups.

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Systems of Conservation laws which are not strictly hyperbolic appear in many physical applications. Generally for these systems the solution space is larger than the usual BVloc space and classical Glimm-Lax Theory does not apply. We start with the non-strictly hyperbolic system

For n = 1, the above system is the celebrated Bugers equation which is well studied by E. Hopf. For n = 2, the above system describes one dimensional model for large scale structure formation of universe. We study (n = 4) case of the above system, using vanishing viscosity approach for Riemann type initial and boundary data and possible integral formulation, when the solution has nice structure. For certain class of general initial data we construct weak asymptotic solution developed by Panov and Shelkovich.

As an application we study zero pressure gas dynamics system, namely,

where _ and u are density and velocity components respectively.

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I will recall basic definitions and facts of algebraic geometry and geometry of quadrics and then i will explain the relation of vector bundles and Hitchin map with these geometric facts.

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I will discuss about all the cases in which product of two eigenforms is again an eigenform. This talk is based on one of my works together with my recent work with Soumya Das.

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In this talk, we will discuss injectivity sets for the twisted spherical means on $\mathbb C^n.$ Specially, I will explain the following recent result. A complex cone is a set of injectivity for the twisted spherical means for the class of all continuous functions on $\mathbb C^n$ as long as it does not completely lay on the level surface of any bi-graded homogeneous harmonic polynomial on $\mathbb C^n.$

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In this talk we will survey recent developments in the analysis of partial differential equations arising out of image processing area with particular emphasis on a forward-backward regularization. We prove a series of existence, uniqueness and regularity results for viscosity, weak and dissipative solutions for general forward-backward diffusion flows.

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In this talk we will survey recent developments in the analysis of partial differential equations arising out of image processing area with particular emphasis on a forward-backward regularization. We prove a series of existence, uniqueness and regularity results for viscosity, weak and dissipative solutions for general forward-backward diffusion flows.

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Fuzzy logic is one of the many generalizations of Classical logic, where the truth values are allowed to lie in the entire unit interval [0, 1], as against just the set {0, 1}. Fuzzy implications are a generalization of classical mplication from two-valued logic to the multivalued setting. In this presentation, we will talk about a novel generative method called the composition, of fuzzy implications that we have proposed. Denoting the set of all fuzzy implications defined on [0, 1], by I, the composition on I can be looked in two different ways, viz.,

(i) a generating method of fuzzy implications, and

(ii) a binary operation on the set I.

The rest of the talk will be a discussion of the composition on I along these two aspects. Firstly, we will discuss the closures of fuzzy implications with respect to some desirable properties. Then the effect of the composition on fuzzy implications that can be obtained from other generating methods of fuzzy implications, namely, (S,N)-, R-, f-, g- implications will also be discussed.

Secondly, looking at the composition as a binary operation on the set I, we will show that it forms I a lattice ordered monoid. Since it cannot be made a group, we determine the largest subgroup, denoted by S, obtained in I and we propose some group actions on I employing S. Finally, we demonstrate that, using one such group action, we have obtained, for the first time, representations of the Yager’s families of fuzzy implications.

Abstract:

Fuzzy logic is one of the many generalizations of Classical logic, where the truth values are allowed to lie in the entire unit interval [0, 1], as against just the set {0, 1}. Fuzzy implications are a generalization of classical implication from two-valued logic to the multivalued setting. In this presentation, we will talk about a novel generative method called the composition, of fuzzy implications that we have proposed. Denoting the set of all fuzzy implications defined on [0, 1], by , the composition on can be looked in two different ways, viz.,

(i) a generating method of fuzzy implications, and

(ii) a binary operation on the set .

The rest of the talk will be a discussion of the composition on along these two aspects. Firstly, we will discuss the closures of fuzzy implications with respect to some desirable properties. Then the effect of the composition on fuzzy implications that can be obtained from other generating methods of fuzzy implications, namely, (S, N) -, R-, f -, g- implications will also be discussed.

Secondly, looking at the composition as a binary operation on the set , we will show that it forms a lattice ordered monoid. Since it cannot be made a group, we determine the largest subgroup, denoted by , contained in and we propose some group actions on employing . Finally, we demonstrate that, using one such group action, we have obtained, for the first time, representations of the Yager’s families of fuzzy implications.

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In the first part of the talk, we study infection spread in random geometric graphs where n nodes are distributed uniformly in the unit square W centred at the origin and two nodes are joined by an edge if the Euclidean distance between them is less than . Assuming edge passage times are exponentially distributed with unit mean, we obtain upper and lower bounds for speed of infection spread in the sub-connectivity regime,

In the second part of the talk, we discuss convergence rate of sums of locally determinable functionals of Poisson processes. Denoting the Poisson process as N, the functional as f and Lebesgue measure as , we establish corresponding bounds for

in terms of the decay rate of the radius of determinability.