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) 
Abstract: “Those of clear intelligence will understand anything calculated in algebra or arithmetic … But to increase the intelligence of dullwitted 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 socalled 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 nontrivial 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 farreaching 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) 
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 twoparameter 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 BombieriVinogradov 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 onedimensional 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 CohenMacaulay Rmodules. We discuss extremal rays of Betti cone over a standard graded kalgebra 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, ErdösKac, 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 ErdösKac 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 dierencedierential 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 problemdependent 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 nonzero 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 wellknown 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 secondorder, 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 wellposedness. Regularity refers to solution being suiciently smooth. The concept of wellposedness 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 wellknown 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:

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 Atomemission 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 micromelting 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 Atomemission 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 daytoday 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 (postdoctoral 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 wellposed 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 riemannroch theorem prof. sanjay amrutiya date: september 17, 2014 
Abstract: In this talk, we will sketch the proof of the RiemannRoch Theorem for compact Riemann surfaces. The RiemannRoch 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 groupcase 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 (TIFRCAM, 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 
wellposedness, 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 