IITG Mathematics Seminar Series

 Lecture Number: 337 Title: Determinantal Point Processes: A Survey Speaker: Dr. Manjunath Krishnapur Affiliation: Indian Institute of Science, Bangalore Date: 25th April, 2019 (Thursday) Time: 05:00 PM

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.

 Lecture Number: 336 Title: On integral homology of orbifolds Speaker: Dr. Soumen Sarkar Affiliation: IIT Chennai Date: 22nd March, 2019 (Friday) Time: 04:00 PM

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.

 Lecture Number: 335 Title: Projection Lemma and the Cyclic Decomposition Theorem Speaker: Prof. Michael Karow Affiliation: TU Berlin, Germany Date: 20th March, 2019 (Wednesday) Time: 03:00 PM

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

 Lecture Number: 334 Title: Applications of Hamilton-Jacobi equations in Shape from Shading Speaker: Prof. G. D. Veerappa Gowda Affiliation: TIFR CAM Bangalore Date: 6th March, 2019 (Wednesday) Time: 11:00 AM

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.

 Lecture Number: 333 Title: Structure of solutions for scalar conservation laws in space dimension Speaker: Prof. A. Adimurthi Affiliation: TIFR CAM, Bangalore Date: 5th March, 2019 (Tuesday) Time: 11:00 AM

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.

 Lecture Number: 332 Title: String Inference and (non)Lexicographic Ordering Speaker: Prof. M. Sohel Rahman Affiliation: BUET, Bangladesh Date: 4th March, 2019 (Monday) Time: 4:30 PM

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.

 Lecture Number: 331 Title: Optimal Control Problems in Domains with Oscillating Boundary and Homogenization Speaker: Prof. A. K. Nandakumaran Affiliation: Indian Institute of Science, Bangalore Date: 25th February, 2019 (Monday) Time: 10:00 AM

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.

 Lecture Number: 330 Title: Phantom morphisms and purity Speaker: Prof. Pedro A. Guil Asensio Affiliation: University of Mursia, Spain Date: 19th February, 2019 (Tuesday) Time: 04:00 PM

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.

 Lecture Number: 329 Title: When averages are extreme: Probabilistic techniques in functional and harmonic analysis Speaker: Prof. Andrew Tonge Affiliation: Kent State University Ohio, USA Date: 11th February, 2019 (Monday) Time: 10:00 AM

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.

 Lecture Number: 328 Title: Constrained Dynamical System: Rough Paths and Simulation Speaker: Prof. Soumyendu Raha Affiliation: Indian Institute of Science, Bangalore Date: 6th February, 2019 (Wednesday) Time: 4:00 PM

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.

 Lecture Number: 327 Title: Through eyes of viscoelastic fluids: How can an external stimulus influence research in theoretical and computational PDEs Speaker: Prof. Amiya Kumar Pani Affiliation: Indian Institute of Technology Bombay Date: 16th January, 2019 (Wednesday) Time: 4:00 PM

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.

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