Titles and Abstracts

Beyond Abel and Jacobi

Speaker: Prof. Kapil Paranjape, IISER Mohali

Abstract: The classical theorem of Abel and Jacobi characterises the zeroes and poles of a meromorphic function on a compact Riemann surface (or smooth projective variety) using a map which is nowadays called the Abel-Jacobi homomorphism.
Following the work of many algebraic geometers over the past 150 years, this result has been successively refined and studied for higher-dimensional varieties. The most sophisticated version of this refinement is via the theory of "motives" as first proposed by Grothendieck. The most elaborate conjectural refinements of the Abel-Jacobi theorem are those of Bloch and Beilinson.
As with most conjectures, there have been many attempts to construct counter-examples. One method is to construct "minimal" instances of these conjectures for "well-understood" varieties where the conjectures do not (as yet) follow from known results. Such examples were constructed by Griffiths, Mumford-Roitman and many others which led to the more precise formulations of the conjectures that we see today.
In this talk, the speaker will attempt to explain some ideas and examples behind this elaborate framework and present his own work in its context.

Lefschetz theorem in Algebraic Geometry

Speaker: Prof. Jaya Iyer , Institute of Mathematical Sciences

Abstract: In this talk we will discuss the classical Lefschetz hyperplane section theorem, as a comparison theorem of topological structures. We will then pose analogues in Chow theory for algebraic varieties. The talk will be accessible to students as well.

Overdetermined problems: An interaction of PDE and Geometry

Speaker: Prof. Anup Biswas , IISER Pune

Abstract: In a celebrated paper in 1971, Serrin studied the overdetermined problem for the Laplacian operator. The proof relied on a powerful technique, now known as the method of moving planes, a refinement of a reflection principle conceived by Alexandrov. Later this idea of reflection principle was picked-up by Gidas, Ni and Nirenberg in 1979 in their landmark work on the symmetry of positive solutions of elliptic problems. The overdetermined problems were then extended for annulus as well as for unbounded domains. These problems are also related to the parallel surface problems. Recently, similar problems have been studied for nonlocal operators. In this talk, we shall give an overview of these problems and its extension.

Determining coefficients for a fractional p-Laplace equation from exterior measurements

Speaker: Prof. Manas Kar , IISER Bhopal

Abstract: In this talk, we consider an inverse problem of determining the coefficients of a fractional p-Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we will prove an explicit reconstruction formula in the region where the exterior measurements are per- formed. This formula is then used to establish a global uniqueness result for real-analytic coefficients. In addition, we also discuss a stability estimate for the unique determination of the coefficients in the exterior measurement set.

Non-uniqueness results for fluid flow equations

Speaker: Prof. Ujjwal Koley, TIFR Center for Applicable Mathematics (CAM)

Abstract: In this talk, we will discuss about non-uniqueness results for fluid flow equations. In particular, we will concentrate on convex integration technique which plays a pivotal role in this business. This talk is accessible to all students.

Entropy Stable Numerical Schemes for Chew, Goldberger & Low (CGL) rarefied plasma flow equations

Speaker: Prof. Harish Kumar , IIT Delhi

Abstract: We consider CGL equations, which are used to model rarefied plasma flows. The CGL equations are a system of nonconservative hyperbolic system. Hence, the design of stable numerical schemes is highly challenging. In this work, we design high-order entropy-stable finite-difference for the system. The key idea is to rewrite the equations in such a way that the nonconservative part does not contribute to the entropy evolution. We then symmetrize the conservative parts, which is very similar to the MHD equations, using Godunov’s symmetrization process. The final scheme is shown to be entropy stable at the semi-discrete level. Several MHD-inspired test cases are presented to test the stability and performance of the proposed scheme. This is a joint work with Anshu Yadav, Chetan Singh, Deepak Bhoriya, and Dinshaw S. Balsara.

Initial boundary value problem for 1D scalar balance laws with strictly convex flux

Speaker: Prof. Manas Sahoo, NISER

Abstract: By introducing a suitable boundary functional, we obtain a type formula for the initial-boundary value problem for a balance law. We show that the explicit solution given by formula is entropy admissible and satisfies the initial condition in a strong sense and boundary condition in the sense of Bardos, le Roux, and

Glimpses of mathematics at the frontiers of Science & Technology

Speaker: Prof. B. V. Ratish Kumar , IIT Kanpur

Abstract: In this talk, we will take an illustrative tour on how mathematics is helping us in dealing with the problems from science and technology and thereby enriches the quality of life. Also, we will look at a real life problem and the role of mathematics in solving it.

Lattice point counting and the spectrum of the Laplacian

Speaker: Prof. Satadal Ganguly , Indian Statistical Institute, Kolkata

Abstract: Abstract: I will start by discussing some classical lattice point counting problems and then move on to some generalities. Finally, I will discuss a new result on counting lattice points on determinant surfaces.

Graded shifts in a pure minimal free resolution

Speaker: Prof. K. N. Raghavan , Institute of Mathematical Sciences

Abstract: Can any increasing sequence of integers occur as the graded shifts in a pure minimal free resolution (of some graded module)? As Eisenbud-Floystad-Weyman showed, the question becomes easier to answer (we know where to look for a solution) if more is demanded, namely, equivariance (for the general linear group). Despite the technical nature of this abstract, the talk will attempt to be a widely accessible introduction to the ideas involved.

Geometry emerging from spectra

Speaker: Prof. Walter van Suijlekom , Radboud University

Abstract: We give a gentle introduction to the spectral approach to geometry, where we replace spaces by commutative algebras, and capture the metric by combining it with the vibration spectrum of a suitable operator on the geometric space. We will give many examples and also show how to reconstruct geometry from this spectral data. This will allow us to generalize to noncommutative spaces, which we illustrate by some of the key examples. We will conclude by establishing some convergence results on the emerging geometric spaces when an increasing part of the spectrum is available.

Genuinely Ramified maps and Stability

Speaker: Prof. A. J. Parameswaran, ISI Tezpur

Abstract: A morphism of smooth projective curves f:X\to Y is called Genuinely ramified if it does not factor through an etale cover of Y . We will discuss the equivalence of this condition in terms surjectivity of the fundamental group, pull back of stable bundles remain stable, beta subbundle of the direct image of the structure sheaf and most interestingly on the connectivity of the fibre product. We will also discuss that this equivalent to the connectivity of the closure of the complement of the diagonal in the fibre product.

Wavelet Analysis on Local Fields of Positive Characteristic

Speaker: Prof. Biswaranjan Behera, Indian Statistical Institute

Abstract: This talk is an overview of wavelet analysis on local fields of positive characteristic. After a brief introduction to Fourier analysis on local fields, we will introduce the concepts of multiresolution analysis (MRA) and wavelets on a local field K of positive characteristic. Next, we will discuss about the characterization of various functions associated with wavelet analysis, namely, scaling functions, wavelets, MRA-wavelets and low-pass filters. We will also find sufficient conditions on a set of basic wavelets so that the corresponding wavelet system forms an unconditional basis for L^p(K), 1 \leq p \leq \infty , and the Hardy space H^1(K) .

q -partition algebras

Speaker: Prof. Geetha Thangavelu, IISER Thiruvananthapuram

Abstract: Let V be a $n$-dimensional vector space over a field. There are two actions of symmetric group on V\otimes r . The symmetric group S_n acts by letter action and the S_r acts by place permutations. The action by place permutations can be $q$-deformed to the action of Iwahori-Hecke algebras and is in Schur-Weyl duality with the natural quantum GL_n action. But there is no obvious $q$-deformed letter action of S_n. In this talk, we will see a construction of q-deformed letter action and q-partitions algebras. We also see how this partition algebras are isomorphic to the q-partition algebras defined by Halverson and Thiem. This is a joint work with Richard Dipper.

A sequence of operator algebras converging to odd spheres in the quantum Gromov-Hausdorff distance

Speaker: Prof. Tirthankar Bhattacharya , Indian Institute of Science

Abstract: Marc Rieffel had introduced the notion of the quantum Gromov-Hausdorff distance on compact quantum metric spaces and found a sequence of matrix algebras that converges to the space of continuous functions on 2-sphere in this distance. One finds applications of similar approximations in many places in the theoretical physics literature. We shall recall Rieffel’s work and define a compact quantum metric space structure on the sequence of Toeplitz algebras on generalized Bergman spaces and prove that the sequence converges to the space of continuous function on odd spheres in the quantum Gromov-Hausdorff distance. We shall start from scratch. So, a large portion of the talk will be accessible to final year M.Sc. and first year Ph.D. students.

GROTHENDIECK INEQUALITY

Speaker: Prof. Samya Kumar Ray , IISER Thiruvananthapuram

Abstract: In this talk we shall discuss a fundamental inequality in functional analysis proved by Grothendieck in 1953. In the first half of our discussion, we shall talk about a close connection between the Grothendieck inequality and von Neumann inequality. Let us set C_2(n) =\sup\{\|P(T )\|:\|P\|_{\mathbb{D}^n,\infty}\leq 1,\|T\|_\infty\leq 1\}, where the supremum is taken over all polynomials P\in\mathbb{C}[z_1, . . . , z_n] of degree at most 2 and commuting n-tuples T := (T_1, . . . , T_n) of contractions on some Hilbert space. In 1976, Varopoulos proved the remarkable inequality K_G^{\mathbb{C}}\leq \lim_{n\to\infty}C_2(n)\leq 2K_G^{\mathbb{C}}, where K_G^{\mathbb{C}} is the complex Grothendieck constant. We shall discuss solution of an old question of Varopoulos who asked whether \lim_{n\to\infty}C_2(n)=K_G^{\mathbb{C}}? In the later part of the talk we revisit the original formulation of the Grothendieck inequality and propose
a generalization of the Grothendieck inequality by generalizing the notion of the Grothendieck constant for any pair of finite dimensional Banach spaces. We also show how our study is related to a very old conjecture of Pisier in the geometry of Banach spaces.

Student Presenters

An Energy Stable and AP Scheme for the Barotropic Euler System

Speaker: Mr. Mainak Kar , IISER Thiruvananthapuram

Abstract: In this talk we present an energy stable and asymptotic preserving (AP) scheme for barotropic Euler equations scaled by the Mach number. The Mach number characterises the different flow regimes in the asymptotic limit, which are automatically recognised and captured by the AP scheme. The energy stability of the scheme is achieved by introducing a velocity shift proportional to the pressure gradient in the momentum flux. We also present the results of several numerical case studies to substantiate the robustness and efficiency of the proposed scheme.

Permutation modules of the walled Brauer algebras

Speaker: Ms. Sulakhana Chowdhury , IISER Thiruvananthapuram

Abstract: In this talk, we will discuss the permutation modules and Young modules of the group algebras of the direct product of symmetric groups KSa,b, and the walled Brauer algebras Br,t(δ). In the dual Specht-filtered modules category, if the characteristic of the field is neither 2 nor 3, then the permutation modules are dual Specht filtered, and the Young modules are relative projective cover of the dual Specht modules. We show that the restriction of the cell modules of Br,t(δ) to the group algebras of the direct product of the symmetric groups is dual Specht filtered, and the Young modules act as the relative projective cover of the cell modules of Br,t(δ). Finally, we shall prove our main goal that if char K ̸= 2,3, then the permutation module of Br,t(δ) can be written as a direct sum of indecomposable Young modules.

Upper bounds on the size of the Schur multiplier of some classes of groups p-groups

Speaker: Mr. Tony Mavely , IISER Thiruvananthapuram

Abstract: The Schur multiplier of a given group is the second homology group with coefficients in Z, and there has been substantial interest in obtaining the size of the Schur multiplier. Using the maximum number of triangles in a graph and some linear algebra, we 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 = d, but also the bounds for 2 intermediate values of k. We also provide a bound for p-groups of nilpotency class greater than or sequal to 3.