Seminars


The students and faculty members of the discipline actively cooperate in promoting a strong and vibrant research environment by holding Mathematics Discipline Seminar Series, where they share their insights and latest research developments.  Workshops highlighting cutting edge research in different domains of   mathematics are organized on a regular basis in order to promote collaborative activities and outreach.

 An overview of the history of Indian mathematics

Prof. Michel Danino (Visiting Professor, IIT Gandhinagar)
Date: October 17, 2018

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)
Date: September 26, 2018
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)
Date: September 12, 2018

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:

  • Construction of Cantor ternary set.
  • How certain easily proven properties of the Cantor ternary set, when they are pieced together, help to show the special nature of Cantor sets.
  • Some non-trivial properties ( which, we can’t tell now! )
  • Generalization of the concept of the dimension of a vector space to sets like Cantor set whose dimension is not an integer!
  • Lastly, a brief introduction to fractals and their self-similarity.
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