IITG Mathematics Seminar Series

Abstract: The role of non-uniform surface mass transfer and non-uniform thermal boundary conditions is investigated in external and internal fluid flow problems, respectively. Non-uniform surface mass transfer distributions are used to avoid discontinuities at the leading and trailing edges of the slot for numerically stable computations. On a similar situation, non-uniform heating is used for smooth thermal boundary conditions in buoyancy driven/ force driven cavity flow problems. The appropriate numerical methods, such as FDM/FEM, are used to obtain numerically stable solutions. It is observed that both non-uniform slot suction and downstream movement of slot are effective in delaying boundary layer separation on external flow problems. In internal flow problems, non-uniform heating enhances thermal mixing within the cavity and maintaining overall lower heat transfer rates at the wall as compared to uniform heating.

Abstract: Kohnen (1991) constructed cusp forms whose Fourier coefficients are given by special values of certain Dirichlet series of Rankin-Selberg type by computing adjoint map of the product map by a fixed cusp form with respect to the Petersson scalar product. The work of Kohnen has been generalized to the case of Jacobi forms by Sakata (1998), to the case Siegel modular forms by Lee (1996). Recently, the work of Kohnen has been generalized by Herrero (2014), where he constructed cusp forms by computing the adjoint map of the map constructed using Rankin-Cohen bracket by a fixed cusp form instead of the product. In this talk, we describe how one can generalize the work Herrero to the case of Hilbert modular forms. This is a joint work with Moni Kumari.

Abstract: Weierstrass during his time surprised many by producing a continuous but no-where differentiable function and further by approximating it by polynomials.However, when it comes to recent applications, such strong topologies do not seem very useful. In fact, motivated by Optimal Design Problems (ODP), we introduce other types of topologies, investigate their properties and mention some of their consequences in Material Science and Fluid Dynamics. This talk will be pitched at a level which is more conceptual than technical.

Abstract: These are well-known polynomials with applications in several directions. Algebraic studies of these polynomial were initiated by Schur. I shall restrict to irreducibility of these polynomials. Apart from givingan account of the known results, I shall give some proofs illustrating the basic ideas and methods in this field.

Abstract: We shall discuss a theorem of DN Bernstein about counting roots of polynomials using mixed volumes of Newton polytopes. This improves the classical theorem of Bezout which counts intersection points of algebraic curves. We shall give an elementary proof for roots of binomials using the Hermite normal form of an integer matrix.

Abstract: The propagation of flexural-gravity waves, generated due to interaction of surface waves and floating thin elastic plate, and propagate along the coupled water surface and plate, is routinely used to model wave interaction with sea-ice as well as very large floating structures (VLFS). Wave blocking which is an analogous effect to surface wave propagation against ocean current and light wave propagation in the curved space-time near a black hole, is shown to exist for flexural gravity waves not only in presence of opposing ocean current but also for high compressive force acting on the plate. Therefore, it provides a novel system to study analogue gravity - a physical system where some theoretical aspects, if not all, of general relativity can be experimentally tested. Hawking radiation, negative energy waves (NEW) are such effects that are mathematically established and shown graphically here for flexural-gravity waves. Another hydrodynamic effect known as deadwater in which existence of high amplitude internal waves halt the motion of ships in apparently still surface water, exists for flexural-gravity waves as well. Few wave simulations will be shown in support of the results.

Abstract: The Sz.-Nagy and Foias characteristic function of a pure contraction is a complete unitary invariant and is an integral part of Sz.-Nagy and Foias analytic model theory. In this talk, we shall find a several variable analogue of characteristic functions. More precisely, an explicit construction of characteristic functions for tuple of pure hypercontractions will be given. We shall see that it is a complete unitary invariant, as in the case of pure contractions. Our explicit construction will help us to obtain a canonical factorization of characteristic functions. This is a joint work with M. Bhattacharjee and J. Sarkar.

Abstract: We study a two-point interpolation problem for two polynomially convex domains in C^n. The first domain is the popular symmetrized polydisc or the symmetrized-n-disk G_n for n = 2. The second domain is the extended symmetrized polydisc \tilde{G}_n. We show that G_n \subset \tilde{G}_n. We obtain a variety of characterizations for the points in \tilde{G}_n and its closure. As a consequence we get a similar set of characterizations for the symmetrized polydisc G_n and its closure. We make some sharp estimates to prove a Schwarz type lemma for \tilde{G}_n from which a Schwarz type lemma for G_n follows. The conditions that are obtained

in the Schwarz lemma for \tilde{G}_n and G_n are necessary conditions for the existence of a two-point interpolating function from D to \tilde{G}_n and G_n. For n = 3, we show that these conditions are sufficient for tilde{G}_n too. We describe all such interpolating functions from D to \tilde{G}_n for n = 3.

Abstract: If a tree is chosen at random among all trees with n vertices, what is the typical distance between a pair of vertices? What is the length of the longest increasing subsequence in a random permutation? What is the chance that the random power series with i.i.d. complex Gaussian coefficients has no zeros in the disk of radius 1/2? These questions seem unrelated, but the common framework of determinantal points processes provides answers to all of them. Determinantal point processes are a class of random discrete sets with a specific form of dependence between points. The definition is motivated by the idea of non-interacting fermions in Quantum physics, but the real motivation is that there are innumerable examples in probability, combinatorics and mathematical physics and that these processes share many common properties. Some examples are the uniform random spanning tree on a graph, the eigenvalues of certain random matrices, zeros of certain random power series, etc. There are also applications to sampling problems in theoretical computer science and possible models (in place of the usual Poisson model) for sensor networks in communication theory. We shall give a survey of this area, focusing on examples. The lecture is aimed to be accessible to anyone with a graduate level understanding of analysis and probability.

Abstract: Orbifolds are the natural generalization of manifolds, and several topological invariants of orbifolds are computed with rational coefficients. In this talk, I will introduce a few machineries which help to determine the integral (co)homology of orbifolds. This is a joint work with A. Bahri, D. Notbohm and J. Song.

Abstract: One of the fundamental results of Linear Algebra is the Cyclic Decomposition Theorem. Let $A:X\to X$ be a linear operator on a finite dimensional vector space $X$ over a field $F$. The theorem states that $X$ is a direct sum of $A$-invariant subspaces which are generated by a single vector. The special case that $F$ is the field of complex numbers yields the Jordan Canonical Form. We present a short proof of the Cyclic Decomposition Theorem using a result on projections

Abstract: Hamilton-Jacobi equations have wide applications in numerous fields of science such as classical mechanics and geometrical optics in physics. In this talk we emphasis upon both theoretical and numerical perspectives for this first order non-linear partial differential equations especially focussing on the application in the shape from shading i.e., to recover the shape of the 3-dimensional object from 2-dimensional information.

Abstract: The behavior of entropy solutions for large time is one of the subjects which has been studied from the last forty years. Here we present the recent result on how the solutions behave using the shock pockets and reprove a result of Liu and Dafermos-Shrear.

Abstract: In this talk we will discuss two interesting but possibly less studied topics from string combinatorics. We will divide the talk into two parts. In the first part, we will briefly present different combinatorics and algorithmic results on lexicographic string ordering with a goal to introduce and discuss some non-lexicographic ordering and relevant results. In the second part, we will focus on another area where the goal is to infer strings from a given data structure. Stringology literature has been enriched by numerous efficient data structures. We will discuss some interesting results and algorithms to infer strings from some of these.

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 present application is much far and wide. It has applications in composite media, porous domains, laminar structures, domains with rapidly oscillatingboundaries, to name a few. The PDE problems posed on such complicated domains lead to the analysis of homogenization. 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. There are various methods developed in the last 50 years to under-stand the mathematical homogenization theory and some them are; Asymptotic Expansion, Energy Method, Compensated Compactness, Two-scale and multi-scale convergence, Gamma Convergence, Bloch Wave Analysis, Method of Unfolding etc. In the first part, we briefly present various applications of homogenization problems. In the second part, we discuss the optimal control problems posed on a domain with rapidly oscillating coefficients and various homogenization results. We focus on results from my group for the last ten years or so.

Abstract: Phantom morphisms play a central role in Brown Representability Theorem for compactly generated triangulated categories (in particular, the stable module category of K[G], where G is a finite group and K is a field whose characterisic divides the order of the group). In this talk, we will connect this notion with the Theory of Purity of Modules. This will allows us to give a general notion of phantom morphisms connecting two exact structures in addititive categories. We will apply our results to different exact structures. Namely: 1) the pure-exact structure in a Grothendieck category; 2) contractible complexes in Ch(R-Mod); 3) almost split sequences of representations of finite-dimensional algebras; 4) geometrical and categorical purity in the category of quasi-coherent sheaves over a scheme.

Abstract: In many situations, there is a high probability that the norm of a random polynomial or multilinear form is close to the smallest possible value. We show how some of these situations arise through an interplay of functional and harmonic analysis and illustrate some interesting consequences.

Abstract: We shall discuss the difficulty in solving and numerically integrating constrained dynamical systems that are naturally modeled as systems of differential-algebraic equations of the form dx/dt = f(x,u),g(x) = 0 where x is in R^n and u is in R^m and m <= n. In this context we shall introduce a horizontal lift and its exponentiation toward construction of a solution. Especially, the solution and behavior of the algebraic variable is of interest. Cases where u can be rough (belong to fractional Holder space) are of interest. A numerical approximation that can produce useful result in computer simulations will be discussed. As an illustration, we shall show how treating stick-slip problems as constrained dynamics, yields interesting results on the time integrating constrained dynamical systems that are naturally modeled as scales of the stick-slip process. In this context we shall study, in particular, the peeling dynamics of an adhesive tape.

Abstract: Mathematical research is mainly driven by internal questions ( internal to Mathematics). But when it is influenced by questions posed externally, there is no liberty to change the original theme and couple of such questions which are bothering us for the last two decades will be the focus of this talk.In order to understand the context, when mathematical model is formulated to explain approximately certain physical reality, it is important to derive relevant properties mathematically. Say for example in some viscoelastic fluid flows, numerous experiments suggest that after an instantaneous cessation of the external force, the velocity of the fluid dies down with a certain rate. Translating and proving this kind of property pose a challenge in Mathematics research. Starting with a model in a layman's language, in this talk, relevant mathematical issues summarizing our efforts for the last 20-25 years will be briefly discussed. If time permits in the later part of this talk, the focus will be on the following question posed by the user community ` How do we believe the numbers crunched by the machine using an algorithm? Finally, we conclude the talk with some mathematical issues.