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.

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.

Abstract:

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.

Abstract:

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.

Abstract:

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.

Abstract:

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.

Abstract:

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.

About the Speaker:

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.

Abstract:

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..