Titles and Abstract
Organizing Committee

Titles and Abstracts

Controllability in discontinuous flux

Speaker: Prof. Adimurthi A , IIT Kanpur, formerly TIFR Bangalore

Abstract: In general controllability is not obvious for conservation laws even in one dimension since the method of linearization does not work as in the Navier Stokes Equation. One has to adopt a different method in this case. I will discuss this problem in scalar conservation laws in one dimension withdiscontinuous flux.

Diophantine tuples

Speaker: Prof. Kalyan Chakraborty , Kerala School of Mathematics

Abstract: A Diophantine m-tuple is a collection of m-numbers such that the product of any two distinct entries increased by 1 should be a square in the chosen domain. Diophantine was the first who asked this question and found four rationals \{\frac{1}{16},\frac{33}{16},\frac{17}{4},\frac{105}{16}\} which satisfy this property. Fermat was the first to find a quadruple \{1,3,8,120\} which has this property. Since then the research on this topic has taken its wings along with relating this question to other branches of mathematics. The aim of my talk will be to report a couple of works from my group on this topic and delve on them a bit. I will conclude with listing many interesting doable research problems.

Existence and Uniqueness of Viscosity Solutions of Value function of Local Cahn-Hilliard-Navier-Stokes System

Speaker: Prof. Sheetal Dharmatti , IISER Thiruvananthapuram

Abstract: In this work, we consider the local Cahn-Hilliard-Navier-Stokes equation with regular potential in a two dimensional bounded domain. We formulate a distributed optimal control problem as the minimization of a suitable cost functional subject to the controlled local Cahn-Hilliard-Navier-Stokes system and define the associated value function. We prove the Dynamic Programming Principle satisfied by the value function. Due to the lack of smoothness properties for the value function, we use the method of viscosity solutions to obtain the corresponding solution of the infinite dimensional Hamilton-Jacobi-Bellman equation. We show that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation. The uniqueness of the viscosity solution is established via the comparison principle.

The Wasserstein distance between positive matrices

Speaker: Prof. Tanvi Jain , ISI Delhi

Abstract: We start with a brief discussion on the optimal transport problem and introduce the Wasserstein distance between positive matrices. We then talk about connections of this metric with some other interesting problems and describe its basic properties.
The talk is based on a joint work with Rajendra Bhatia and Yongdo Lim.

A random string in a medium of Poisson traps

Speaker: Prof. Mathew Joseph , ISI Bangalore

Abstract: We consider a random string modeled by the additive stochastic heat equation on a torus. The string evolves in R^n among obstacles of finite width centered around points coming from a Poisson point process. The string dies when any part of it hits an obstacle. We discuss the probability that the string survives up to time T. This talk is based on work with Siva Athreya and Carl Mueller.

Smooth structures on PL-manifolds of dimensions between 8 and 10

Speaker: Prof. Ramesh Kasilingam , IIT Madras

Abstract: The work of Thom, Milnor, Hirsch and Mazur has reduced the problems of smooth-ing theory to homotopy theory and study of the fibration of classifying spaces P L/O → BO → BP L. Let Γ denote the homotopy functor from the category of closed smooth manifolds to the category of abelian groups defined by homotopy classes of maps into P L/O, that is, Γ(M) = [M, P L/O].
In this talk, we give a detailed description of P L/O in low dimensions and some general calculations of Γ(M). In particular, if M is a closed smooth n-manifold of dimension n ≤ 10, we show that Γ(M) is determined by the cohomology of M and the homotopy groups of P L/O. The consequences are twofold : on the one hand, we prove that there exists an exotic n-sphere Σ ∈ Θ_n such that M#Σ is diffeomorphic to M where n = 9, 10 and M is a closed simply connected non-spin n-manifold. On the other hand, for 8 ≤ n ≤ 10, we show that there exists an exotic n-sphere Σ ∈ Θ_n such that the connected sum \mathbb{R}P^n #Σ of the real projective n-space with Σ is not diffeomorphic to the original real projective space \mathbb{R}P^n.
This is joint work with Dr. Samik Basu and Priyanka Magar.

Young's inequality for the twisted convolution

Speaker: Prof. P K Ratnakumar , Harish-Chandra Research Institute, Allahabad

Abstract: The classical generalised Young's inequality says that the convolution inequality \| f \ast g \|_{L^r(\mathbb{R}^n)} \leq \| f \|_{L^p(\mathbb{R}^n)} \| g \|_{L^q(\mathbb{R}^n)} holds for 1\leq p,q,r \leq \infty if and only if \frac{1}{p} + \frac{1}{q}= 1+ \frac{1}{r}. We explore similar inequality for the twisted convolution on \mathbb{C}^n given by f \times_\lambda g (z) = \int_{\mathbb{C}^n} f (z-w) \, g(w) \, e^{i\frac{\lambda}{2} \text{im}(z \cdot \bar{w}) } \, dw, ~ 0 \neq \lambda \in \mathbb{R} which has better mapping properties for \lambda \neq 0. We prove a geometric characterisation of the triples (p,q,r), 1\leq p,q,r \leq \infty for which the bi-linear \lambda-twisted convolution map B_\lambda: (f,g) \to f \times _\lambda \, g is bounded from L^p(\mathbb{C}^n) \times L^q(\mathbb{C}^n) \to L^r(\mathbb{C}^n), leading to an analogue of Young's inequality for twisted convolution.

Low order finite element approximations of a sixth order problem

Speaker: Prof. Bishnu Lamichhane , University of Newcastle, Australia

Abstract: We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation. We follow the Ciarlet-Raviart formulation for the biharmonic problem to formulate our saddle point problem and the finite element method. The new formulation allows us to use the H1-conforming Lagrange finite element spaces to get the approximate the solution. We prove a priori error estimates for our approach. Numerical results will be presented for linear and quadratic finite element methods.

Lowest-order equivalent nonstandard finite element methods for fourth-order plates

Speaker: Prof. Neela Nataraj , IIT Bombay

Abstract: The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the C^0 interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side with F \in H^{-2} (\Omega) replaced by F(JIM) and then are quasi-optimal in their respective discrete norms. The smoother JIM is defined for a piecewise smooth input function by a (generalized) Morley interpolation IM followed by a companion operator J. An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined for linear and semi-linear problems with quadratic nonlinearity. Applications include the biharmonic plate bending problem, stream function vorticity formulation of incompressible 2D Navier-Stokes problem, and the von Kármán plate bending problem.
This is a joint work with C. Carstensen, G.C. Remesan, and D. Shylaja.

On endomorphism of Fano manifolds of Picard number one whose co-tangent bundle is algebraically completely integrable system

Speaker: Sarbeshwar Pal , IISER Thiruvananthapuram

Abstract: Let X be a projective Fano manifold of Picard number one. There is a folklore that any non constant endomorphism of X is an isomorphism. The statement was proved in some cases, for example homogeneous spaces, hypersurfaces of the projective space and for Fano manifolds containing a rational curve with trivial normal bundle. In this talk we will study the problem when the co-tangent bundle of X is algebraically completely integrable system and the tangent bundle of X is not nef. Unfortunately, there are no examples of X known (other than the moduli space of vector bundles on curves) such that T^*X is algebraically completely integrable. In this talk we will give a new example of such an X

On Burger's Equation

Speaker: Prof. B Rajeev , IISER Thiruvananthapuram

Abstract: In this talk we outline a proof of the existence, uniqueness and regularity of solutions to the deterministic Burger's equation in dimension one or higher, when the initial conditions are smooth. We use a stochastic method involving a Brownian motion and a change of measure.

Shimura map on a class of functions

Speaker: Prof. Balakrishnan Ramakrishnan, ISI Tezpur

Abstract: The correspondence between the spaces of modular forms of half-integral weight and integral weight, developed by Shimura, plays a major role in the theory of automorphic forms. A. Selberg observed a special property of this correspondence on a class of functions (H. Cohen also made similar observations for the Shimura-Kohnen map). Later the observation of Selberg was extended to a wider class of functions. In this talk, we give a brief account of these results, including our joint work with Manish Kumar Pandey.

Commutant lifting theorem on the polydisc

Speaker: Prof. Jaydeb Sarkar , ISI Bangalore

Abstract: Sarason's one variable commutant lifting theorem is a key result in the theory of linear operators, complex analysis, and Hilbert function space theory, which has a stellar reputation in its application to classical results like Nevanlinna-Pick interpolation, Caratheodory-Fejer interpolation problem, Nehari interpolation problem, von Neumann inequality, isometric dilations, just to name a few. The expanded list easily includes control theory and electrical engineering. However, Sarason's lifting theorem does not hold in the setting of polydisc in general. Comprehending the subtleties of the lifting theorem on the polydisc is considered to be one of the challenging problems.
In the first half of this talk, we will provide a quick historical overview (within the span of little more than a century), present an introduction to the commutant lifting theorem, and explore how it interacts with the Nevanlinna-Pick interpolation. The second half of the talk will go over some recent advances in the commutant lifting theorem on the polydisc and its applications to interpolation and perturbation problems.

Conics, cubics and algebraic cycles

Speaker: Prof. Ramesh Sreekantan , ISI Bangalore

Abstract: In this talk we will relate some classical problems in enumerative geometry to the construction of algebraic cycles on K3 surfaces and Abelian surfaces.

What are the Bloch-Beilinson Conjectures?

Speaker: Prof. Vasudevan Srinivas , TIFR Bombay

Abstract: The Bloch-Beilinson Conjectures are among the deepest open questions today, relating aspects of algebraic geometry, algebraic K-theory and number theory.
The conjectures have roots, on the one hand, in classical results (Euler, Riemann, Dedekind, Hilbert, Artin, etc.) on special values and zeroes of zeta functions, in the period upto the early 20th century. Another source, somewhat more recent (going upto the mid 1970's) is work of Tate, Iwasawa, Lichtenbaum, Quillen and Borel, which brought in the role of algebraic K-theory.
A more recent inspiration, beginning with several key calculations of Bloch, relate these to algebraic geometry. Bloch's vision was articulated in a general, more precise form by Beilinson, around 1982, resulting in what are now called the Bloch-Beilinson Conjectures. There are also refinements (e.g. the Bloch-Kato conjectures). In fact there is tantalising, but rather meagre, evidence to support these conjectures, inspite of about 40 years of effort by interested mathematicians.
My lecture will attempt to give an accessible introduction to this important circle of ideas.

Student Presenters

Bounds on the Eigenvalues of Matrix Rational Functions

Speaker: Mr. Shrinath Hadimani , IISER Thiruvananthapuram

Abstract: Upper and lower bounds on absolute values of the eigenvalues of a matrix polynomial are well studied in the literature. As a continuation of this we derive, in this manuscript, bounds on absolute values of the eigenvalues of matrix rational functions using the following techniques/methods: the Bauer-Fike theorem, a Rouch ́e theorem for matrix- valued functions and by associating a real rational function to the matrix rational function. Bounds are also obtained by converting the matrix rational function to a matrix polynomial. Comparison of these bounds when the coefficients are unitary matrices are brought out. Numerical calculations on a known problem are also verified.
This is a joint work with Ms. Pallavi. B and Dr. Sachindranath Jayaraman.

On the existence of a non-principal Euclidean ideal class in biquadratic fields with class number two

Speaker: Mr. Sunil Kumar Pasupulati , IISER Thiruvananthapuram

Abstract: In 1979, Lenstra introduced the definition of the Euclidean ideal which is a generalization of Euclidean domain.
Definition 1. Let R be a Dedekind domain and \mathbb{E} be the set of non zero integral ideals of R. If C is an ideal of R, then it is called Euclidean if there exists a function \psi : \mathbb{E} → \mathbb{N} , such that for every I \in \mathbb{E} and x \in I^{-1}C \ C there exist a y \in C such that

\psi ((x - y)IC^{-1})

H.Graves constructed an explicit biquadratic field \mathbb{Q}(\sqrt{2}, \sqrt{35}) which has a non-principal Euclidean ideal class. C. Hsu generalized the result by Graves and proved suppose K = \mathbb{Q}(\sqrt{q}, \sqrt{kr}) ,then K has a non-principal Euclidean ideal class whenever h_K = 2. Here the integers q, k, r are all primes ≥ 29 and are all congruent to 1 modulo 4. The family of biquadratic field given by C. Hsu is extended by Chattopadhyay and Muthukrishnan and proved that, If K = \mathbb{Q}(\sqrt{q}, \sqrt{kr}), where q ≡ 3 and k, r ≡ 1 (mod 4) are prime numbers. Suppose that h_K = 2. Then K has a Euclidean ideal class. In this talk, I will prove that a new family of biquadratic fields having non principal Euclidean ideal class whenever the class number of biquadratic field equal to two. This is joint work with Srilakshmi Krishnamoorthy.

Using graph theory to obtain the size of the Schur multiplier of special p-groups

Speaker: Mr. Tony Mavely , IISER Thiruvananthapuram

Abstract: The Schur multiplier of a given group is the second homology group with coefficients in \mathbb{Z}, and there has been considerable interest in the study of the size of the Schur multiplier. We provide a novel approach of using the maximum number of triangles in a graph to obtain an upper bound for the size of the Schur multiplier of special p-groups of rank k. This not only gives us the existing bounds for the two extreme cases of k = 2 and k = \begin{pmatrix} d \\ 2 \end{pmatrix} but also the bounds for intermediate values of k. We also provide a bound for groups of nilpotency class greater than or equal to 3.

Bounding the exponent of a finite group and its commutator subgroup and Schur's exponent conjecture

Speaker: Ms. Komma Patali , IISER Thiruvananthapuram

Abstract: Let G be a finite p-group and S be a Sylow p-subgroup of Aut(G) with exp(S) = q. We prove that if the nilpotency class of G is c, then exp(Aut(G)) | p^{[log_p c]} q^3 , and if G is a metabelian p-group of class at most 2p - 1, then exp(G) | pq^3. We also prove that exp(\gamma_2(G)) | p^{[log_p c] - 1} exp(G/Z(G)) if the nilpotency class of G is at most c, and exp(\gamma_2(G)) | exp(G/Z(G)) if G is a metabelian p-group of class at most 2p - 1. We discuss the progress made towards the Schur's exponent conjecture in recent years, and we will describe our contribution towards this conjecture.

Residual-Based a Posteriori Error Estimator for a Multi-scale Cancer Invasion Model

Speaker: Mr. Nishant Ranwan , IISER Thiruvananthapuram

Abstract: The residual-based a posteriori error estimation of the multiscale cancer invasion model, a system of three non-stationary reaction-diffusion equations consisting of nonlinear reaction terms and sensitivity functions, is studied. First we derive the upper bound for the error in terms of the given initial data and residuals. Then we derive the upper bound for the residuals by splitting them into spatial and temporal residuals. Thus we obtained a residual-based a posteriori error estimator for the coupled system that is reliable and efficient concerning the small perturbations of parameters in the model. The numerical results were demonstrated for two sets of parameters where the first set creates a stiffer system than the second one. Using the first set of parameters, we obtained the experimental order of convergence as 0.78, and using the second set of parameters, it is 0.88.