Seminars 2013-2019
| A new link between rational dynamics and Kleinian groups Prof. Sabyasachi Mukherjee (TIFR, Mumbai) Date: November 19, 2019 (Block-6/202) |
| Abstract: In the early 80’s, Dennis Sullivan introduced an intriguing dictionary between two branches of conformal dynamics: the theory of Kleinian groups and dynamics of rational maps on the Riemann sphere. Although in some cases, similar techniques can be applied to prove related results in the two fields; in general, there is no direct translation from one world to the other. After reviewing some basic results on rational dynamics and Kleinian groups, we will introduce a novel connection between these two fields. We will also describe a new “mating” framework that combines the features of a rational map and a Kleinian group in a single conformal dynamical system. |
| The Final Problem Prof. Bruce C. Berndt (University of Illinois at Urbana-Champaign, USA) Date: November 6, 2019 |
| Abstract: On page 335 in the volume containing his Lost Notebook, Ramanujan recorded two identities, each involving a double series of Bessel functions. One is connected with the famous circle problem, while the other is connected with the equally famous divisor problem. We briefly review these impenetrable, unsolved problems. Each identity has three different interpretations. Over a period of several years, the speaker, Sun Kim, Junxian Li, and Alexandru Zaharescu examined these identities. Up until slightly over a year ago, the identity connected with the divisor problem in Ramanujan’s original formulation remained to be proved. To the best of our knowledge, this was the last claim in Ramanujan’s lost notebook to be proved. We sketch a proof. A natural question is: Are there similar identities involving other arithmetical functions, e.g., Ramanujan’s τ -function. We discuss the possibility of such identities. |
| LIVING WITH RAMANUJAN FOR 40+ YEARS Prof. Bruce C. Berndt (University of Illinois at Urbana-Champaign, USA) Date: November 5, 2019 |
| Abstract: We begin with a short summary of Ramanujan’s life. This is followed by a description and history of Ramanujan’s (earlier) notebooks. The speaker’s interest in Ramanujan’s notebooks began in 1974. The important people and events in the speaker’s quest to find proofs for Ramanujan’s claims in his notebooks are next related. We then provide a history of Ramanujan’s lost notebook, describe some of its contents, and lastly discuss the final result from the lost notebook to be proved. At the conclusion, we ask some probing questions about the possible loss of some of Ramanujan’s work. |
| What makes a complex exact? Dr. Joydip Saha (IIT Gandhinagar) Date: October 30, 2019 |
| Abstract: In this talk we discuss acyclicity criterion of a complex given by Buchsbaum and Eisenbud. |
| Overdetermined problems for nonlocal operators. Prof. Anup Biswas (IISER Pune) Date: October 23, 2019 |
| Abstract: In this talk, we discuss an extension of the classical overdetermined problems of Serrin to nonlocal operators. We shall see that the analytic problem relies heavily on its probabilistic interpretation. This talk is based on a recent joint work with Sven Jarohs. |
| An Introduction to Weil Conjectures. Prof. Arnab Saha (IIT Gandhinagar) Date: October 17, 2019 |
| Abstract: In 1949, André Weil made conjectures on the behaviour of the number of rational points for varieties (solution spaces of algebraic equations) defined over a finite field. For a given variety X, one can define the Hasse-Weil Zeta function ζ(X, s) which is a generating function that encodes the number of solutions coming from the extensions of the fixed finite field. Then Weil conjectured that ζ(X, s) is rational, satisfies a functional equation and the analogue of the Riemann hypothesis is true. Then the rationality of ζ(X, s) was proved by Dwork (1960), the functional equation by Grothendieck (1965) and finally, the analogue of the Riemann hypothesis by Deligne (1974). This effort that resulted in proving this conjecture has contributed significantly to the foundations of modern algebraic geometry. This will be an expository talk on some aspects of this development. The talk will be self-contained and will not assume any pre-requisite from the audience. |
| On the quasimodularity of a function of Ramanujan. Prof. Ajit Bhand (IISER Bhopal) Date: September 24, 2019 |
| Abstract: Quasimodular forms are holomorphic functions on the upper-half plane that satisfy a certain transformation property with respect to the modular group SL2(Z). In his seminal 1916 paper, Ramanujan described certain identities involving derivatives of such functions. In the same paper, for non-negative integers k and l, he also defined a function Φk,l as a q−series and showed that it is a quasimodular form when (k + l) is odd. In this talk we will discuss properties of Φk,l when (k + l) is even. A converse theorem for quasimodular forms will prove to be an important ingredient in understanding these properties. This is joint work with Karam Deo Shankhadhar. |
| Superimposing theta structure on a generalized modular relation. Prof. Atul Dixit (IIT Gandhinagar) Date: September 18, 2019 |
| Abstract: Modular forms are certain functions defined on the upper half plane that transform nicely under z → −1/z as well as z → z + 1. By a modular relation, we mean that which is governed by z → −1/z only. Equivalently, the relation can be written in the form F (α) = F (β), where αβ = 1. There are many generalized modular relations in the literature such as the general theta transformation F (w, α) = F (iw, β) or other relations of the form F (z, α) = F (z, β) etc. The famous Ramanujan-Guinand formula, equivalent to the functional equation of the non-holomorphic Eisenstein series on SL2(Z), admits a beautiful generalization, which is of the form F (z, w, α) = F (z, iw, β), that is, one can superimpose the theta structure on the Ramanujan-Guinand formula. The current work arose from answering a similar question – can we superimpose the theta structure on a recent modular relation involving infinite series of the Hurwitz zeta function ζ(z, a)? In the course of answering this question in the affirmative, we were led to a surprising new generalization of ζ(z, a). This new zeta function, ζw(z, a), satisfies interesting properties, albeit much more difficult to derive than the corresponding ones for ζ(z, a). In this talk, I will briefly discuss the history of the Riemann zeta function ζ(z), the Hurwitz zeta function ζ(z, a) and then give a sample of the results from the theory of ζw(z, a) that we have developed. This is joint work with Rahul Kumar. |
| Forecasting crude oil futures prices using the Kalman filter and macroeconomic news sentiment. Prof. Paresh Date (IIT Gandhinagar) Date: September 11, 2019 |
| Abstract: This presentation is split into three parts. The first part is a brief tutorial on how a commodity futures market operates, and the role played by stochastics in pricing of financial derivatives, including commodity futures. In the second part, I will introduce a recursive conditional mean estimator for linear Gaussian dynamic systems, popularly called the Kalman filter (KF), and its application in modelling the movement of crude oil futures prices. Finally, I will describe how a KF-based model can be modified using macroeconomic news sentiment to enhance the futures price prediction. I will also present a summary of our findings from numerical experiments on modelling and predicting crude oil futures prices using the Kalman filter and news sentiment scores. |
| Continuous function calculus and its applications. Prof. Bipul Saurabh (IIT Gandhinagar) Date: September 4, 2019 |
| Abstract: Given an operator T and a polynomial p, the operator p(T) can be defined naturally. The question is; can we extend this to more general functions? In particular, given a continuous function f, can we define f(T) in a “natural“ manner? To answer this question, we will introduce continuous function calculus in C∗-algebra framework. If time allows, we will see some of its applications and discuss its connection to other concepts in operator algebra and non-commutative geometry. |
| Fermat’s Two Squares Theorem. Prof. Indranath Sengupta (IIT Gandhinagar) Date: August 28, 2019 |
| Abstract:We will first discuss some elementary number theoretic facts related to the existence of integer solutions of the congruence x^2 ≡ −1(mod p) and the Diophantine equation x^2 + y^2 = p, for prime numbers p. Finally, we will present “A one-sentence proof that every prime p ≡ 1(mod 4) is a sum of two squares”, by D. Zagier, regarded as THE BOOK proof of the Two Square theorem. |
| Zeros on the critical line. Prof. Akshaa Vatwani (IIT Gandhinagar) Date: August 20, 2019 |
| Abstract:The Riemann hypothesis is the conjecture that all non trivial zeros of the Riemann-zeta function ζ(s) lie on the line Re(s) = 1/2, also called the critical line. While this conjecture is still unsolved, it is known that ζ(s) has an infinitude of zeros on this line. We will discuss some proofs of this important result, ranging from Hardy’s method giving an infinitude of zeros, to Selberg’s method which gives a positive proportion of zeros on the critical line. |
| Secant Bundles on Symmetric Power of Curves. Dr. Krishanu Dan (Postdoctoral Fellow, CMI) Date: April 25, 2019 |
| Abstract:Let C be a smooth, projective, irreducible curve over the field of complex numbers, and Cn denotes the n-fold Cartesian product of C. The symmetric group of n elements acts on Cn and let S n(C) be the quotient. This is a smooth, irreducible, projective variety of dimension n, called the n-th symmetric power of C. Given a vector bundle E of rank r on C, one can naturally associate a rank nr vector bundle on S n(C), called the n-th secant bundle of E. In this talk, we will discuss stability conditions of secant bundles on Sn(C). |
| Riemann zeta function (some conjectures and some results) Prof. A. Sankaranarayanan (TIFR Bombay) Date: April 24, 2019 |
| Abstract:We will discuss some conjectures and results about Riemann zeta function. |
| On constancy of second co-ordinate of the gonality sequence Prof. Sarbeswar Pal (IISER-TVM) Date: April 18, 2019 |
| Abstract:Given a smooth irreducible projective curve C and an integer r one can associate an integer dr, as the minimal degree of a line bundle with r + 1 sections. Thus, to each curve one can associate a sequence (d1, d2, …) called gonality sequence. In this talk, we will see the behavior of the second co-ordinate of the gonality sequence along a linear system in a K3 surface. |
| Option pricing in a regime-switching stochastic volatility model Dr. Milan Kumar Das (Postdoc, IIT Gandhinagar, Gujarat) Date: April 12, 2019 |
| Abstract:In this talk, we will discuss the locally risk minimizing pricing of a European style basket option pricing under regime switching stochastic volatility model. We will show the well-posedness of the pricing equation which is a non-local degenerate PDE using a probabilistic approach. Finally, we will derive the hedging strategy. |
| Structure of solutions Balance Laws Prof. Adimurthi (TIFR Bangalore) Date: March 29, 2019 |
| Abstract:In this talk I will talk about the hyperbolic conservation laws in space dimension with linear source term depending on the solution. I will discuss the nature of the entropy solutions and show that their behaviour is completely dierent from that of source term being zero. |
| Finite analogue of Andrews’s identity for the smallest part function Dr. Bibekananda Maji (Postdoc, IIT Gandhinagar, Gujarat) Date: March 22, 2019 |
| Abstract:In this talk, we shall discuss a finite analogue of a recent generalization of an identity in Ramanujan’s Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews’s famous identity for spt(n), the number of smallest parts in all partitions of n. The latter motivates us to extend the theory of the restricted partition function p(n, N), namely, the number of partitions of n with largest parts less than or equal to N , by obtaining the finite analogues of rank and crank for vector partitions as well as of the rank and crank moments. This is joint work with Atul Dixit, Pramod Eyyunni and Garima Sood. |
| Pfaffian Orientations and Conformal Minors Dr. Nishad Kothari (Postdoc, Institute of Computing, University of Campinas (UNICAMP), Brazil) Date: February 5, 2019 |
| Abstract:Inspired by a theorem of Lovasz (1983),we took on the task of characterizing graphs that do not contain K4 as a conformal minor — that is, K4-free graphs. In a joint work with U. S. R. Murty (2016), we provided a structural characterization of planar K4-free graphs. The problem of characterizing nonplanar K4-free graphs is much harder, and we have evidence to believe that it is related to the problem of recognizing Pfaffian graphs. In particular, we conjecture that every graph that is K4-free and K3,3-free is also Pfaffian. |
| Semi-linear elliptic problems with singular terms on the Heisenberg group Dharmendra Kumar (PhD Student, IIT Gandhinagar, Gujarat) Date: January 18, 2019 |
| Abstract:We consider the semilinear elliptic problem, We show that there exists a solution u to this problem. The interesting point of the problem under consideration is that it has strong singularity. |
| Some famous conjectures in commutative algebra Dr. Joydip Saha (Postdoc, IIT Gandhinagar) Date: January 11, 2019 |
| Abstract: We will discuss about famous conjectures in commutative algebra. Zariski Lipmann conjecture, set theoretic complete intersection conjecture and New intersection conjecture are main interest of this seminar. |
| Large values of the Riemann zeta function on the 1-line Dr. Kamalakshya Mahatab (Norwegian University of Science and Technology) Date: January 4, 2019 |
| Abstract: In this talk we will discuss an application of the resonance method to compute large values of the Riemann zeta function on the line 1 + it. Also we will see that a similar resonator can be used to compute large values of the Dirichlet L-functions at 1. |
| Vector bundles on hypersurfaces Prof. G. V. Ravindra (Professor, University of Missouri, St.Louis, U.S.A) Date: December 13, 2018 |
| Abstract: I will talk about the problem of extending a bundle on a hypersurface to the ambient projective space. I will explain how this problem is related to a problem of representing powers of a polynomial as determinants of matrices with polynomial entries. |
| Mixed multiplicities of ltrations Prof. Dale Cutkosky (Professor, University of Missouri, Columbia, U.S.A) Date: December 7, 2018 |
| Abstract: We define and explore properties of mixed multiplicities of (not necessarily Noetherian) ltrations of m-primary ideals in a Noetherian local ring R, generalizing the classical theory for m-primary ideals. We construct a real polynomial whose coecients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of R is less than the dimension of R, which holds for instance if R is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of m-primary ideals hold for ltrations, including the famous Minkowski inequalities of Teissier, and Rees and Sharp. This is joint work with Parangama Sarkar and Hema Srinivasan. |
| Semigroup Rings: Resolutions and Structures Prof. Hema Srinivasan (Professor, University of Missouri, Columbia, U.S.A) Date: December 6, 2018 |
| Abstract: Semigroup rings of embedding dimension 3 have a simple structure and characterization. As a generalization of these, we will elaborate on the concept of gluing and the structure of the resolution of glued semigroups. The talk will be with examples rather than proofs. |
| Counting real zeros of real polynomials Prof. Dilip P. Patil (Professor, Indian Institute Of Science Bangalore, India) Date: November 12, 2018 |
| Brown-Douglas-Fillmore Theorem Prof. Sameer Chavan (Professor, IIT Kanpur) Date: November 12, 2018 |
| Abstract: A bounded linear operator is essentially normal if its self-commutator is compact. By essential equivalence of two operators, we mean unitary equivalence modulo compact operators. The celebrated BDF Theorem classifies all essentially normal operators up to essential equivalence by means of the essential spectrum and index data. The original proof of BDF theorem relies on the idea of associating to every compact Hausdorff space X the group Ext(X) of *-monomorphisms from the continuous functions on X into the Calkin algebra. It turns out that for planar sets, this group is isomorphic to the group of homomorphisms from the rst cohomotopy group into the set of integers. We will discuss two special cases of this theorem. The rest of which, usually known as Weyl-von Neumann theorem, says that two self-adjoint operators are essentially equivalent if and only they have same essential spectrum. In terms of the extension group, this result says that Ext(X) must be trivial for any compact subset X of the real line. The second interesting case identites the extension group of the unit circle with the group of integers. |
| Quiver representations and their applications Prof. Sanjay Amrutiya (Assistant Professor, IIT Gandhinagar) Date: October 31, 2018 |
| Abstract: The theory of quivers and their representations has been an active area of research for many years and found applications in many other branches of Mathematics such as algebra, Lie theory, algebraic geometry, and even in Physics. In this talk, we shall use the geometric methods to study the quiver representations. The aim here is to give an as elementary as possible introduction to this topic. |
| Universal deformations of dihedral representations
Dr. Shaunak Deo (Postdoc, TIFR) |
| Abstract: Given a 2-dimensional dihedral representation of a profinite group over a finite field, we will give necessary and sufficient conditions for its universal deformation to be dihedral. We will then specialize to the case of absolute Galois group of a number field and give sufficient conditions for the universal deformation unramified outside a finite set of primes to be dihedral. We will also see its applications to unramified Fontaine-Mazur conjecture and to an R=T theorem (in the spirit of Calegari-Geraghty) in the setting of Hilbert modular forms of parallel weight one. We will begin with a brief introduction to the deformation theory of Galois representations. This talk is based on joint work with Gabor Wiese. |
| An overview of the history of Indian mathematics
Prof. Michel Danino (Visiting Professor, IIT Gandhinagar) |
| Abstract: “Those of clear intelligence will understand anything calculated in algebra or arithmetic … But to increase the intelligence of dull-witted ones like us, the wise have explained it in many easy rules.” — Bhaskaracharya
This talk will offer a brief overview of some of ancient and medieval India’s landmark achievements in the field of mathematics, beginning with geometry and moving on to algebra and elements of calculus. A few examples will include the so-called Pythagoras theorem, the numeral notation system, concepts of infinity and infinitesimal. The talk will also set the cultural and historical context in which these advances as well as the pragmatic methods Indian mathematicians adopted, in contrast with Greek ones. |
| Boundary value problem with measures for fractional elliptic equations Dr. Mousomi Bhakta (Assistant Professor, IISER Pune) Date: October 3, 2018 |
| Abstract: I’ll discuss universal a priori estimate for positive solutions (and their gradients) of equation (E) (−∆)su = f(u) in any arbitrary domain of RN for a large class of continuous function f and for s ∈ (1/2, 1). Then for C2 bounded domain, I’ll discuss the existence of positive solutions of (E) with prescribed boundary value ν, where ν is a positive Radon measure and discuss regularity property of the solutions. When f(u) = u p, I’ll demonstrate the existence of critical exponent and the multiplicity of positive solutions. It’s a joint work with Phuoc Tai Nguyen. |
| Infinitude of the Zeros of the Riemann Zeta function and its generalization on the critical line
Rahul Kumar (PhD Student, IITGN) |
| Abstract: The Riemann hypothesis is one of the seven Millennium Problems in Mathematics, which is about the zeros of the Riemann zeta function. We will start this talk with an explanation as to why the study of this function and its zeros is important. In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line. In this talk, we first prove Hardy’s result after which we will give its far-reaching generalization. This is joint work with Dixit, Maji and Zaharescu. |
| Raabe’s cosine transform and a generalization of Ramanujan’s formula for Zeta (2m + 1) Rajat Gupta (PhD Student, IITGN) Date: September 19, 2018 |
| Abstract: A comprehensive study of a generalized Lambert series is undertaken, transformations of this series are derived by investigating Raabe’s cosine transform. Using this, we will obtain a two-parameter generalization of Ramanujan’s formula for ζ(2m + 1), where ζ(s) is the Riemann zeta function. This is joint work with Prof. Atul Dixit, Rahul Kumar and Dr. Bibekananda Maji. |
| Twin Primes and the parity problem
Prof. Akshaa Vatwani (IITGN) |
| Abstract: The twin prime conjecture is one of the oldest open problems in number theory. In this talk, we formulate an analogue of the Bombieri-Vinogradov theorem for the Möbius function evaluated at shifts of primes in an arithmetic progression and indicate how this leads to the infinitude of twin primes. This is joint work with Prof. Ram Murty. |
| Lüroth’s Theorem & Rational Parametrization Prof. Indranath Sengupta (IITGN) date: August 29, 2018 |
| Abstract: We will start with the basic notion of field extensions and transcendental extensions of fields. We will then discuss the Lüroth’s Theorem, which is a classical result in Algebra and says that every subfield (containing the field k) of a simple transcendental extension k(X) is also a simple transcendental extension of k. The theorem, which is purely algebraic in nature has a very deep geometric consequence, viz., every one-dimensional variety which is unirational is also birational. |
| Extremal Rays of Betti Cones Rajiv Garg (Indian Institute of Technology Dharwad, Karnataka) date: April 3, 2018 |
| Abstract: In 2009, Eisenbud and Schreyer proved that extremal rays of Betti cone over a polynomial ring are spanned by Betti diagrams of pure Cohen-Macaulay R-modules. We discuss extremal rays of Betti cone over a standard graded k-algebra and their purity. In particular, we show that Koszulness is a necessary condition for the purity of extremal rays. |
| Stability Conditions in Algebra Dr. Umesh Dubey (HRI, Allahabad) date: March 8, 2018 |
| Abstract: In this talk, we will describe the notion of stability condition for representations of a quiver. We will also give some variants in other contexts. As an application of this notion, we will mention some classification problems in Algebra. This notion in the context of representations was first introduced by A. King motivated from a similar notion due to D. Mumford. If time permits, we will discuss some applications towards constructing certain moduli spaces in Algebraic Geometry. |
| Introduction to Stochastic Models Prof. Raj Srinivasan (University of Saskatchewan, Canada) date: February 7, 2018 |
| Abstract: Stochastic models have been found to be useful in predicting performance measures of computer, communication and manufacturing networks. In this talk, I will provide an overall big picture of some aspects of stochastic modelling involving queues and queueing networks. This is an introductory talk aimed at M.Sc. and Ph.D. students. |
| On the distribution of the number of prime factors – variation of the classical theme Prof. Krishnaswami Alladi (University of Florida) date: January 4, 2018 |
| Abstract: Although the study of prime numbers goes back to Greek antiquity, it was only in the early twentieth century that the first systematic study of ν(n), the number of prime factors of n was made by Hardy and Ramanujan. Subsequently, several fundamental results on ν(n) were proved by Turan, Erdös-Kac, Landau, Sathe and Selberg, including the close study of the number of integers with a fixed number of prime factors. In the course of proving the celebrated Erdös-Kac theorem, the truncated function νy(n), the number of prime factors of n which are < y, plays a crucial role. However, not much is known about the number of integers up to x for which νy(n) takes a fixed value, with y varying as a function of x. In studying this problem recently, I noticed a very interesting variation of the classical theme which I shall describe. In doing so, we will encounter a variety of analytic techniques involving the Riemann zeta function, sieve methods, and dierence-dierential equations. Details of the analysis has been carried out in a 2016 PhD thesis of my student Todd Molnar. |
| Concentration Bounds for Stochastic Approximation with Applications to Reinforcement Learning Gugan Thoppe (Postdoctoral Fellow, Technion,Israel Institute of Technology) date: November 8, 2017 |
| Abstract: Stochastic Approximation (SA) is useful in finding optimal points, or zeros of a function, given only noisy estimates. In this talk, we will review our recent advances in techniques for SA analysis. In the first part, we will see a motivating application of SA to network tomography and also discuss the convergence of a novel stochastic Kaczmarz method. Next, we shall discuss a novel tool based on Alekseev’s formula to obtain rate of convergence of a nonlinear SA to a specific solution, when there are multiple locally stable solutions. In the third part, we shall extend the previous tool to the two timescale but linear SA setting, also discussing how this tool applies to gradient Temporal Dierence (TD) methods such as GTD(0), GTD2, and TDC used in reinforcement learning. For much of the foregoing analysis, the initial step size must be chosen suiciently small, depending on unknown problem-dependent parameters. Since this is oen impractical, we finally discuss a trick to obviate this in context of the one timescale, linear TD(0) method, and also provide a novel expectation bound. |
| on the arithmetic nature of the values of the riemann zeta function dr. bibekananda maji (post doctral fellow, IIT gandhinagar) date: september 13, 2017 |
| Abstract: In this talk, we are going to discuss the work of Kanemitsu, Tanigawa, and Yoshimoto on some generalized Lambert series. We have extended results of Kanemitsu et al. and found a new generalization of Ramanujan’s famous formula for odd zeta values. For any odd positive integer N and any non-zero integer m, this generalization gives a relation between ζ(2m + 1) and ζ(2Nm + 1) by way of N + 1generalized Lambert series. For example, it gives a relation between ζ(3) and ζ(7). Several important corollaries of these generalizations, which include as special cases some well-known results in the literature, are obtained along with results on transcendence of certain values. This is a joint work with Atul Dixit. |
| the classical maximum principle and solvability of the dirichlet problem in generalized sense dharmendra kumar (phd student, IIT gandhinagar) date: august 17, 2017 |
| Abstract: We prove weak and strong maximum principles, including a Hopf lemma for solutions to equations defined by second-order, linear elliptic partial dierential operator. |
| variants of equidistribution in arithmetic progressions problem akshaa vatwani (postdoctoral fellow at university of waterloo, canada) date: august 10, 2017 |
| Abstract: It is well known that the prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of multiplicative functions. We derive some variants of such results and give a few applications. We also discuss an interesting application that relates to the Chowla conjecture on correlations of the Möbius function, and show its relevance to the twin prime conjecture. |
| When is an apparently small eect not negligible Saleh Tanveer (The Ohio State University) date: May 15, 2017 |
| Abstract: Description of a physical phenomenon through a mathematical model invariably involves dropping some terms that are considered negligible. Nonetheless, a simplified model must have some mathematical properties in order to faithfully model physics. These include regularity and well-posedness. Regularity refers to solution being suiciently smooth. The concept of well-posedness includes existence, uniqueness and continuous dependence on initial data and boundary data in an appropriate norm. Singularities in a solution suggests that terms ignored in a model can be important–this has immediate consequence on the smallest predicted scales seen in experiment. If a mathematical problem is not well posed, it cannot be physically relevant since in the real world, we only know measure initial and boundary conditions to finite precision. If a small regularising term is included in an otherwise unstable model, the near structural instability can manifest itself in the highly unusual phenomena where disparate length scales interact. We will illustrate these notions through a series of example problems from fluid mechanics and other physical problems. |
| discontinuous galerkin finite element method for the elliptic obstacle problem prof. kamana porwal date: april 19, 2017 |
| Abstract: In this talk, I will present a new error analysis of discontinuous Galerkin (DG) finite element methods for an elliptic obstacle problem. In a recent work of Gaddam and Gudi, 2016, a bubble enriched conforming finite element method is introduced and analyzed for the obstacle problem in dimension 3. Using the localized behavior of DG methods, we have obtained an optimal order a priori error estimates for linear and quadratic DG methods in dimension 2 and 3 without the inclusion of the bubble function. We have also derived a reliable and eicient a posteriori error estimator and the analysis is carried out in a unified setting which holds for a class of DG methods. |
| a new proof to infinitude of primes dr.bibekananda maji date: march 31, 2017 |
| Abstract: In this talk, we discuss some interesting proofs of the infinitude of prime numbers and provide a new way to construct infinite sequence of pairwise relatively prime natural numbers. |
| solvable primitive extensions prof. chandan dalawat (HRI, allahabad) date: march 03, 2017 |
| 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 characterize the extensions D of F which are the galoisian closures of some solvable primitive extension E of F. The talk will be aimed at a general audience and the background in group theory and field theory will be recalled. |
| advances in frame theory: optimal frames for erasers prof. ram n. mohapatra date: february 03, 2017 |
| Abstract: Frames were introduced by Duin and Schaeer in 1952 and as the wavelet theory evolved, it became clear that Frames can be used for signal transmission. Frames were studied in Hilbert spaces and have found their way to Banach spaces. We have studied Frames in Hilbert modules. In this talk we shall introduce the concept erasers and will discuss optimal frames for erasers. We shall mention some interesting open problems. |
| tranformation involving $r_k(n)$ and bessel functions prof. atul dixit date: january 24, 2017 |
| Abstract: Let rk(n) denote the number of representations of the positive integer n as the sum of k squares, where k ≥ 2. In 1934, the Russian mathematician A. I. Popov obtained a beautiful transformation between two series involving rk(n) and Bessel functions. Unfortunately, Popov’s proof appears to be defective. In this work, we give a rigorous proof of Popov’s result by observing that N. S. Koshliakov’s ingenious proof of the Voronoi summation formula circumvents these diiculties. In the second part of our talk, we will obtain a proof of a more general summation formula for rk(n) due to A. P. Guinand and apply it to obtain a new transformation of a series involving rk(n) and a product of two Bessel functions. This transformation can be considered as a massive generalization of many well-known results in the literature. This is joint work with Bruce C. Berndt, Sun Kim and Alexandru Zaharescu. |
| division algorithm in polynomial rings ranjana mehta (ph. d. student at IIT gandhinagar) date: january 16, 2017 |
| Abstract: We learn the division of polynomials in one variable in high school algebra. The division algorithm of polynomials in more than one variable is not a straight forward generalization of one variable case. In this talk, we will present the algorithm for the multivariable case and show that it is related to a fundamental problem called the ideal membership problem. In the next talk, we will discuss the division algorithm in the power series rings. |
| a result of steinhaus on difference set prof. n. r. ladhawala date: november 10, 2016 |
| Abstract: In this talk, we will discuss a result of Steinhaus, “for a set of real numbers with positive Lebesgue measure, the difference set contains an open interval”. Some good consequences of this result of Steinhaus will also be discussed |
| regularity questions to elliptic partial differential equations ram baran verma (ph. d. student at IIT gandhinagar) date: october 10, 2016 |
| Abstract: In this presentation, the definition of the Newtonian potential and it’s properties will be o presented. Furthermore, interior Schauder estimate and as a consequence of it, compactness 0 result for the bounded solution of the Poission’s equation, will also be discussed. |
| on subadditivity of maximal shifts in the resolutions of graded algebras prof. hema srinivasan (university of missouri, columbia, USA) date: june 15, 2016 |
| branch of logarithm rahul kumar (ph. d. student, IIT gandhinagar) date: march 16, 2016 |
| Abstract: In this talk, the definition and some properties such as continuity, analyticity etc. of logarithm function of complex variable will be discussed. We shall also compare the real and complex logarithm function. In the end, concept of branch of multivalued function will be explained with the help of an example. |
| presenting cantor set served with its beautiful and paradoxical attributes amogh and deepak (b. tech, IIT gandhinagar) date: march 30, 2016 |
| Abstract: In this talk, we’ll see:
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| measure zero and sard’s theorem dharmendra kumar (ph. d. student, IIT gandhinagar) date: february 09, 2016 |
| ramanujan expansions and twin primes prof. ram murty (queen’s university, kingston, canada) date: january 01, 2016 |
| on locally lipschitz functions prof. gerald beer (emeritus professor at california state university, los angeles (csula), USA date: april 09, 2015 |
| fourier transforms and application to signal processing prof. v. d. pathak (the m. s. university of baroda, vadodara) date: february 12, 2015 |
| Abstract: An Atom-emission spectrometer is used to identify the various elements present in a given metallic sample. Such a metallic sample is exposed to a heat source due to which micro-melting occurs in the sample and an optical signal is generated. A holographic diffraction grating is then used to split the composite optical signal into spectral components. Then a transducer is used to convert the optical information to an electrical signal, which is captured in the analogue form. By analysing this signal the composition of the metallic sample is determined. In this lecture, we will see how the techniques of Fourier analysis can be used to identify and resolve some of the problems arising in the functioning of the Atom-emission spectrometer. |
| some insights into mathematical transforms: a layman’s perspective prof. k.v.v. murthy date: november 19, 2014 |
| Abstract: “During the study of various subjects in science and engineering, one comes across several Transforms, like Fourier-, Laplace-, Wavelet-, etc., I will make an attempt to present the manifestation of quite a few of these, in day-to-day encounters from which one can understand a large class of them, in a unified way”. My presentation may not be rigorous from the view of mathematicians, but could be a motivation for a beginner to see the significance of Transforms. |
| eigenvalues and eigenvectors – in action prof. mohan joshi date: november 12, 2014 |
| Abstract: This is part of an exercise of a group of faculty members from Mathematics and Engineering disciplines to bring in to sharp focus the applicability potential of Mathematics. In this talk we shall display a real life application of one of the celebrated theorem in matrix theory- spectral theorem for symmetric matrices. |
| modeling the dynamics of hepatitis c virus with combined antiviral drug therapy: interferon and ribavirin prof. sandeep banerjee (iit roorkee) date: november 10, 2014 |
| an introduction to finite element method dr. akanksha srivastava (post-doctoral fellow, IIT gandhinagar) date: october 15, 2014 |
| Abstract: The objective of this talk is to provide an introduction to the mathematical theory underlying the finite element method (FEM) with special emphasis on theoretical questions such as its accuracy and reliability. Practical issues concerning the development of efficient finite element algorithms will also be discussed. We will begin by considering linear second order elliptic PDEs in both one and two dimensions and show how our problem is well-posed in the sense defined by Hadamard. If time permits, we will also discuss the application of FEM for solving nonlinear problems. |
| symmetric polynomials and hilbert’s 14th problem prof. indranath sengupta date: october 08, 2014 |
| Abstract: Hilbert proposed a list of 23 mathematical problems in the second ICM at Paris in 1900. The fourteenth in the list, famous as Hilbert’s 14th was disproved by Nagata in the year 1958. Subsequently many counter examples were produced by Paul Roberts (1900), Kuroda, Mukai etc. This problem initiated the study of a very rich branch called Invariant Theory. In our lecture, we will start with Symmetric Polynomials and prove the fundamental theorem of symmetric polynomials due to Gauss (1816). We will see how Hilbert’s 14th problem appears naturally from Gauss’ result. We will present the affirmative answer to the problem for finite group actions due to E.Noether (1926). We will also see Nagata’s counter example to the problem. At the end, if time permits we will mention a result of Kurano & Matsuoka (2009) to show how the problem is still being studied from the perspective of commutative algebra and algebraic geometry. |
| the riemann-roch theorem prof. sanjay amrutiya date: september 17, 2014 |
| Abstract: In this talk, we will sketch the proof of the Riemann-Roch Theorem for compact Riemann surfaces. The Riemann-Roch Theorem is central in the theory of compact Riemann surfaces. Roughly speaking it tells us how many linearly independent meromorphic functions are there having certain restrictions on their poles. |
| a probability sampler prof. chetan pahlajani date: september 10, 2014 |
| Abstract: The goal of this talk is to provide a taste of some of the ideas of modern Probability Theory. We will discuss several fundamental concepts, relate them to other areas of Mathematics, and look at some classical asymptotic results. We will also introduce stochastic processes, together with some of the mathematical tools used to study them. |
| differential operators on hopf algebras prof. uma iyer (bronx community college) date: august 14, 2014 |
| introduction to lie algebra and quantum group-case sl_2 prof. uma iyer (bronx community college) date: august 12, 13, 2014 |
| mountain pass theorem and its application to elliptic partial differential equations mr. gaurav dwivedi (research scholar, IIT gandhinagar) date: april 11, 2014 |
| control of systems of differential equations prof. mythily ramaswamy (TIFR-CAM, bangalore ) date: march 28, 2014 |
| conditioning of bases of finite dimensional normed spaces prof. b.v.limaye (emeritus fellow at iit bombay) date: march 11, 2014 |
| maximal ideals of the polynomial algebra over an algebraically closed field prof. surjeet kour date: march 21, 2014 |
| on optimality conditions for constrained problem prof. anulekha dhara date: march 14, 2014 |
| sequential estimation of gini index prof. bhargab chattopadhyay (the university of texas at dallas) date: november 22, 2013 |
| degree theory and its applications prof. jagmohan tyagi date: october 24, 2013 |
| well-posedness, regularization, and viscosity solutions of minimization problems prof. d. v. pai date: october 10, 2013 |
| gibbs phenomenon prof. n.r.ladhawala date: september 19, 2013 |
| satellite control to heat control prof. m.c. joshi date: september 12, 2013 |