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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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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..
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