Commutative Algebra seminar
Speaker: Sourjya Banerjee (IMSc)
Host: Manoj Keshari
Title: From Unimodular Rows to Zero Cycles over Real Varieties
Time, day and date: 4:00:00 PM - 5:00:00 PM, Monday, September 1
Venue: Ramanujan Hall
Abstract: A unimodular row of length $n$ over a commutative Noetherian ring $R$ (with $1 \neq 0$) is a row vector $(v_1,\ldots,v_n) \in R^n$ such that the ideal generated by $v_1,\ldots,v_n$ is the whole ring $R$. We discuss unimodular rows and their connections with projective modules, specifically addressing when a unimodular row of length $n$ can be completed to a row of an invertible matrix and how this question arises naturally in the study of projective modules. We present a classical example of a unimodular row of length $(d+1)$ over a $d$-dimensional smooth real variety that cannot be completed to such a row. We then describe a class of $d$-dimensional real varieties where every unimodular row of length $d+1$ is completable to a row of an invertible matrix. We discuss some applications to the study of the $d$-th Euler class group $\mathrm{E}^d(R)$ defined by Bhatwadekar and Raja Sridharan, the Levine--Weibel Chow group of zero cycles $\mathrm{CH}_0(\mathrm{Spec}(R))$, and the natural maps between them. If time permits, we discuss unimodular rows of length $d$ over $d$-dimensional smooth real varieties. The final part is based on ongoing joint work with Jean Fasel.
 

Statistics and Probability seminar
Speaker: Chirag Modi (New York University)
Host: Manas Rachh
Title: ATLAS: Adapting Trajectory Lengths and Step-Size for Hamiltonian Monte Carlo
Time, day and date: 3:00:00 PM - 4:00:00 PM, Tuesday, September 2
Venue: Ramanujan Hall
Abstract: Hamiltonian Monte Carlo (HMC) is the most widely used Markov chain Monte Carlo technique for Bayesian inference in high dimensions. However, the performance of HMC is sensitive to several tuning parameters like step size and trajectory lengths for leapfrog integration that can be difficult to set. As a result, it can still struggle to accurately sample distributions with complex geometries, e.g., varying curvature, due to the constant step size. In this talk, I will present a strategy to locally adapt the step size parameter of HMC at every iteration by evaluating a low-rank approximation of the local Hessian and estimating its largest eigenvalue. I will then combine it with a strategy to similarly adapt the trajectory length by monitoring the no U-turn condition, resulting in an adaptive sampler, ATLAS: adapting trajectory length and step-size. I will further use a delayed rejection framework for making multiple proposals that improve the computational efficiency of ATLAS, and develop an approach for automatically tuning its hyperparameters during warmup. Finally, I will compare ATLAS with NUTS on a suite of synthetic and real-world examples, and show that i) unlike NUTS, ATLAS is able to accurately sample difficult distributions with complex geometries, ii) it is computationally competitive to NUTS for simpler distributions, and iii) it is more robust to the tuning of hyperparameters.
 

Mathematics Colloquium
Speaker: Manoj K Keshari (IIT Bombay)
Title: Positive polynomials and sums of squares
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, September 3
Venue: Ramanujan Hall
Abstract: Given a closed set K in R^n, where R denote the real numbers, it is an important problem to decide whether a given polynomial in n variables is non-negative on K. We will start with two motivations to study non-negative polynomials. Then we will discuss the following descent problem: If f has rational coefficients and there is a certificate over R that f is non-negative over K, can one find such a certificate over rational numbers.
 

Commutative Algebra seminar
Speaker: Tony Puthenpurakal (IIT-B)
Title: F-modules- IV
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, September 4
Venue: Ramanujan Hall
Abstract: We continue our discussion on F-modules
 

Seminar
Speaker: Udit Mavinkurve
Host: Swapneel Mahajan
Title: The Fundamental Groupoid in Discrete Homotopy Theory
Time, day and date: 2:25:00 PM, Monday, September 08
Venue: Room 105
Abstract: In classical homotopy theory, graphs are treated as 1-dimensional CW complexes. But since the classical notions of continuous maps and their homotopies do not respect the discrete nature of graphs, this fails to capture the full combinatorial richness of graph theory. Discrete homotopy theory, introduced around 20 years ago by Barcelo et al., building on the work of Atkin from the mid-seventies, is a homotopy theory specifically designed to study discrete objects like graphs. This theory has found a wide range of applications, including in matroid theory, hyperplane arrangements, and more recently, in topological data analysis.

In this talk, based on joint work with Chris Kapulkin, we introduce the discrete fundamental groupoid, a multi-object generalization of the discrete fundamental group, and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us to extend the existing theory of universal covers to all graphs, and to prove a classification theorem for coverings. We also prove a discrete version of the Seifert¨Cvan Kampen theorem, generalizing a previous result of Barcelo et al. We then use it to solve the realization problem for the discrete fundamental group through a purely combinatorial construction.

Currently, a central open problem in the field is to determine whether the cubical nerve functor, which associates a cubical Kan complex to a graph is a DK-equivalence of relative categories. If true, this would allow the import of results like the Blakers-Massey theorem from classical homotopy theory to the discrete realm. We propose a new line of attack, by breaking it into more tractable problems comparing the homotopy theories of the respective n-types, for each integer n ¡Ý 0. We also solve this problem for the first nontrivial case, n = 1.
 

Statistics/Probability Seminar
Speaker: Promit Ghoshal, University of Chicago
Host: Parthanil Roy
Title: Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond
Time, day and date: 4:00:00 PM – 5:00:00 PM, Tuesday, September 09
Venue: Ramanujan Hall
Abstract: Stein Variational Gradient Descent (SVGD) has emerged as a powerful interacting particle-based algorithm for nonparametric sampling, yet its theoretical properties remain challenging to unravel. This talk delves into two complementary perspectives about SVGD. First, we explore Gaussian-SVGD, a framework that projects SVGD onto the family of Gaussian distributions via a bilinear kernel. We establish rigorous convergence results for both mean-field dynamics and finite-particle systems, demonstrating linear convergence to equilibrium in strongly log-concave settings and unifying recent algorithms for Gaussian variational inference (GVI) under a single framework. Second, we analyze the finite-particle convergence rates of SVGD in Kernelized Stein Discrepancy (KSD) and Wasserstein-2 metrics. Leveraging a novel decomposition of the relative entropy time derivative, we achieve near optimal rates with polynomial dimensional dependence and extend these results to bilinear enhanced kernels.

Seminar
Speaker: Dr. Ramesh Mete (IIT Bombay)
Host: Saikat Mazumdar
Title: Reading seminar on the Yamabe flow
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, September 10
Venue: Room 113
Abstract: In the 1980s, Hamilton proposed a heat flow approach to the Yamabe problem (uniformization theorem in dimension >2). The goal is to start from a given initial metric and deform it to a metric with constant scalar curvature by means of an evolution equation. Hamilton showed that the flow exists for all time t > 0; however, convergence turns out to be highly nontrivial. 
For conformally flat metrics, Ye [1994] showed that the flow converges to a metric of constant scalar curvature. Subsequently, Schwetlick-Struwe [2003] showed that the flow converged in dimensions 2<n<6 under the assumption that the Yamabe energy of the initial metric was "not too large". The energy assumption was then removed by Brendle [2005] to establish unconditional convergence of the flow in low dimensions 2<n<6. Convergence in dimensions n>5 was also shown by Brendle [2007] under a technical hypothesis on the conformal class. 
Our aim in particular is to discuss the work of Schwetlick-Struwe and Brendle in low dimensions.

Mathematics Colloquium
Speaker: Manas Rachh, IIT Bombay
Title: Complex scattering makes for simple numerics
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, September 10
Venue: Ramanujan Hall
Abstract: Scattering problems involving unbounded interfaces occur frequently in physics and engineering settings. Due to this prevalence, there exist many numerical methods for solving such problems. Unfortunately, the complicated behavior of solutions in the vicinity of infinite interfaces can make it challenging to derive explicit error bounds for these methods. Many of these methods also require a large computational domain, and so require a large number of discretization points to accurately solve the problem.
In this talk, we will present a class of decomposable scattering problems. For this class of problems, the PDE domain can be decomposed into a collection of simple subdomains. The fundamental solutions for these simple regions can then be used to reduce the scattering problem into an integral equation on the interfaces between these subdomains. These integral equations can then be analytically continued into the complex plane, where they can be safely truncated with controllable accuracy. We demonstrate this procedure for the example of two dielectric waveguides meeting at an interface. For this problem, we show that the fundamental solutions and densities decay exponentially in the complex plane, and so the analytically continued integral equation can be truncated with exponential accuracy.

Commutative Algebra seminar
Speaker: Tony Puthenpurakal
Title: F-modules IV
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, September 11
Venue: Ramanujan Hall
Abstract: We continue our study of F-modules

Geometry and Topology Seminar
Speaker: Sumanta Das, IIT Bombay
Host: Rekha Santhanam
Title: Geometric Kernels of Proper Maps Between Non-Compact Surfaces
Time, day and date: 11:30:00 AM, Friday, September 12
Venue: Ramanujan Hall
Abstract: A map between 2-manifolds is said to have a geometric kernel if it sends a non-contractible simple loop to a null-homotopic loop. For compact 2-manifolds without boundary, every map that is not injective on the fundamental group admits such a kernel. In contrast, maps between compact 2-manifolds with boundary, or between non-compact 2 manifolds, need not admit a geometric kernel. In this talk, I will present a sufficient condition, formulated in terms of Brown’s proper fundamental group, which guarantees that a degree-one map between non-compact 2-manifolds without boundary admits a geometric kernel.

Statistics and Probability seminar
Speaker: Dr. Kaartick Adhikari, IISER Bhopal
Host: Koushik Saha
Title: The spectrum and local weak convergence of sparse random uniform hypergraphs.
Time, day and date: 4:00:00 PM - 5:00:00 PM, Friday, September 12
Venue: Ramanujan Hall
Abstract: The notion of the local weak convergence, also known as Schramm convergence, for a sequence of graphs was introduced by Benjamini and Schramm. It is well known that the local weak limit of the sparse Erdos-Renyi graphs is the Galton-Watson measure with Poisson offspring distribution almost surely. Recently, Adhikari, Kumar, and Saha showed that the local weak limit of the line graph of the sparse Linial-Meshulam complexes is the d-block Galton-Watson measure almost surely. In this talk, we discuss the local weak convergence of a unified model, namely, the weighted line graphs of sparse k-uniform random hypergraphs on n vertices.

APS
Speaker: Imran Hussain (IIT Bombay)
Host: Saikat Mazumdar
Title: Rigidity results for the Q curvature equation on Einstein manifolds
Time, day and date: 11:30:00 AM – 12:30:00 PM, Monday, September 15
Venue: Ramanujan Hall
Abstract: In this seminar, we classify the constant Q-curvature conformal metrics(cQcc metrics) on Einstein manifolds. We show that on Einstein manifolds cQcc metrics are rigid in the sense that if (M,g) is Einstein then cQcc metrics in the conformal class of g are all of the form "a times g" where "a" is constant

Seminar
Speaker: Prof. Sanghoon Kwon (Catholic Kwandong University, Republic of Korea)
Host: Dipendra Prasad
Title: Non-uniform quotient of Bruhat-Tits buildings: spectral geometry and representation theory
Time, day and date: 2:30:00 PM, Monday, September 15
Venue: Room 105
Abstract: In this talk, we explore the rich interplay between the geometry of Bruhat–Tits buildings and the spectral theory of their non-uniform arithmetic quotients over global function fields. Beginning with an introduction to the Bruhat–Tits tree for PGL2 and higher dimensional buildings for groups, we investigate how these combinatorial and geometric objects give rise to Ramanujan complexes through the theory of automorphic representations.
We emphasize non-uniform lattices arising from function fields, highlighting how Eisenstein series and the residual spectrum contribute to the spectral decomposition. Special attention will be paid to the structure of the Laplacian spectrum on these quotients and the role of the Hecke operators. No special background is assumed. Part of this presentation is based on the joint work with Soonki Hong.

APS
Speaker: Bittu Singh (IIT Bombay)
Host: Rekha Santhanam
Title: Equivariant formal spaces and maps
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, September 17
Venue: Ramanujan Hall
Abstract: D. Sullivan developed a framework for studying rational homotopy theory, in which the rational homotopy type of a space is characterized by commutative differential graded algebras (CDGAs). In this work, we explore the equivariant version of rational homotopy theory, where the equivariant rational homotopy type of a $G$-space is encoded in terms of functors to CDGAs. An especially interesting situation occurs when the rational homotopy type of a space can be completely determined by its cohomology algebra—a property known as formality. This perspective extends further: maps induced on cohomology may determine maps between rational spaces, and such maps are called formal maps.

APS
Speaker: Hamidul Ahmed, IIT Bombay
Host: Bata Krishna Das
Title: Multiplier varieties and multiplier algebras of CNP Dirichlet
Time, day and date: 5:00:00 PM – 6:00:00 PM, Thursday, September 18
Venue: Ramanujan Hall
Abstract: In this talk, I will explore complete Nevanlinna–Pick (CNP) Dirichlet series kernels and the isomorphism problem for their multiplier algebras. To begin with, I will briefly recall key results from my earlier APS presentation, where we identified the multiplier variety as the common zero set of certain polynomials, before presenting new developments. We shall examine the isomorphism problem for a significant class of CNP Dirichlet series kernels, and show that any algebraic isomorphism in this class is automatically an isometric isomorphism.  In particular, we shall address an open question posed by McCarthy and Shalit, resolving it in the negative.

Algebra Seminar
Speaker: Prof. Gurmeet Bakshi (Panjab University, Chandigarh)
Host: Sudhir R. Ghorpade
Title: Classical Problems to Modern Perspectives: Units in Integral Group Rings
Time, day and date: 4:00:00 PM - 5:00:00 PM, Monday, September 22
Venue: Ramanujan Hall
Abstract: The question of determining the unit group of an integral group ring has a long and fascinating history of more than eighty years. Despite important contributions by Higman, Bass, Milnor, Ritter, Sehgal, and many others, a full description remains a challenge even for cyclic groups. In this talk, I will trace the development of the subject from its classical origins to recent progress. Central to modern approaches are Shoda pairs and their refinements, which provide effective tools to analyze the rational group algebra and construct explicit units. I will present results on generalized strongly monomial groups, highlighting their role in understanding primitive idempotents, central units, and finite index subgroups of unit groups. Alongside these theoretical advances, I will also touch upon computational aspects. We will also see how old problems continue to inspire new algebra.

Special Colloquium
Speaker: N.S. Narasimha Sastry, Kerala School of Mathematics
Host: Dipendra Prasad
Title: `Monster is fabulous'
Time, day and date: 5:15:00 PM - 6:15:00 PM, Monday, September 22
Venue: Ramanujan Hall
Abstract: So said Simon Norton! True Indeed!! This enigmatic, Monster finite simple group, emerged during the last phase of the classification of finite simple groups, one of the major achievements of twentieth century mathematics. The existence of Monster ties together many unique situations in Mathematics including highly transitive permutation groups (Mathieu groups) , perfect sphere packings in finite vector spaces and Golay codes, the exceptional 24-dimensional Leech lattice, some exceptional non split group extensions, special strongly regular graphs, a unique decomposition of Lie algebra of type E_8 and so on. Further, twenty of the sporadic simple groups and non-split group extensions (many, then unknown),, apart from few classical groups appear as quotients of subgroups of the Monster. Also, it seems to highlight the role of extraspecial groups (finite p-groups which are finite analogue of the Heisenberg group) in the emergence of sporadic simple groups..  
Apart from Its enormous size, its permutation or linear representations only possible on objects of very large size; and its very intricate structure, it has deep connections with many other mathematical structures including modular functions, affine E_8-root system, Vertex operator algebras; and string theory.

In this talk, I will try to present some striking features of this group,, mention some (group theoretic) foundational steps in the classification, state its connection to modular functions (the so-called `Monster Moonshine'), the affine E_8 connection, and some brief remarks on the initial steps leading to its construction. (Paucity Of knowledge prevents me from making any references to its connections between the Monster group and String theory which seems to be very substantial.) 

I will try to make the talk as simple (if not, simpler!) as possible.. .

APS
Speaker: Surajit Pal, IIT Bombay
Host: Debanjana Mitra
Title: Controllability of the Heat Equation and the Reachable Space
Time, day and date: 11:00:00 AM – 12:00:00 PM, Tuesday, September 23
Venue: Ramanujan Hall
Abstract: In this talk, I will prove the null controllability of the heat equation using Carleman estimate for parabolic operator. While the heat equation is null controllable, it is not exactly controllable due to smoothing effect of heat operator. Consequently, we will focus on characterizing the set where the reachable states lie

APS
Speaker: Animesh Sinha, IIT Bombay
Host: Koushik Saha
Title: Concentration Bounds for Random Hermitian Matrices with Independent Half-Rows
Time, day and date: 3:00:00 PM – 4:00:00 PM, Tuesday, September 23
Venue: Ramanujan Hall
Abstract: We shall use a version of McDiarmid’s inequality to derive concentration bounds for both the empirical spectral distribution (ESD) & the resolvent of random Hermitian matrices with independent half-rows. We shall also show how these bounds can possibly help us in showing weak almost sure convergence of a sequence of ESDs to a limiting spectral distribution (LSD).

Topology Student Seminar
Speaker: Advaith Nair, IIT Bombay
Host: Rekha Santhanam
Title: --
Time, day and date: 11:00:00 AM – 12:00:00 PM, Wednesday, September 24
Venue: Room 113
Abstract: Poincare duality is a result on an n-dimensional orientable manifold. It is an important tool useful in various calculations such as cohomology ring of several spaces and also in proving several other results in the subject. I will start by defining the fundamental class of a manifold and then eventually formulate and prove the main theorem

APS
Speaker: Lavanya V, IIT Bombay
Host: Rekha Santhanam
Title: Irreducible Representation of the Unitary Group
Time, day and date: 4:00:00 PM – 5:00:00 PM, Wednesday, September 24
Venue: Room 114
Abstract: In this seminar, we will characterize the irreducible representations of the unitary group U(n). First, we show that every finite dimensional representation of U(n) has a special vector called a highest weight vector. And we will derive Weyl Character Formula. By combining this formula with the Weyl Integration Formula, we show a one-to-one correspondence between irreducible representations and a set of dominant weights

APS
Speaker: Soumyajit Acharya, IIT Bombay
Host: Chandan Biswas
Title: Uniform estimates for Fourier restriction to polynomial curves
Time, day and date: 10:00:00 AM – 11:00:00 AM, Thursday, September 25
Venue: Ramanujan Hall
Abstract: We study uniform L p (R d ) → L q (λγ dt) bounds for Fourier restriction to degree-N polynomial curves γ equipped with affine arclength, in the range q = (d(d + 1)/2) p ′ and q > (d^2 + d + 2)/2 , with constants depending only on d, N, p. The method uses the extension formulation, a local decomposition with control of Lγ, a Littlewood–Paley square function, and almost-orthogonality across dyadic shells. We also note the link to adjoint restriction for the paraboloid.

Special Colloquium
Speaker: Venky Krishnan, TIFR-CAM Bangalore
Host: Suman Sahoo
Title: Boundary rigidity problem, tensor tomography and microlocal analysis
 Time, day and date: 12:00:00 PM - 1:00:00 PM, Thursday, September 25
Venue: Ramanujan Hall
Abstract: Boundary rigidity problem, loosely speaking, asks whether it's possible to determine a Riemannian metric in a bounded domain from the knowledge of the associated boundary distance function. This problem is closely related to the integral geometry problem of determining a symmetric 2-tensor field from the knowledge of its integrals along geodesics connecting boundary points. Microlocal analysis naturally enters into the study of integral geometry problems. The talk will survey some known results and some interesting open problems.

APS
Speaker: Adarsh Gupta, IIT Bombay
Host: Sudarshan Gurjar
Title: Homotopy groups of smooth projective surfaces
 Time, day and date: 12:45:00 PM – 1:45:00 PM, Thursday, September 25
Venue: Room 215
Abstract: In this seminar, I will talk about the second homotopy group of smooth projective surfaces. When the universal cover of these surfaces are holomorphically convex. Then these groups are free abelian groups.

APS
Speaker: Lal Bahadur Sahu, IIT Bombay
Host: Sandip Singh
Title: Bruhat Decomposition
Time, day and date: 3:00:00 PM – 4:00:00 PM, Thursday, September 25
Venue: Room 215
Abstract: In this seminar, we will discuss the Bruhat decomposition, a fundamental structural result in the theory of reductive algebraic groups. It asserts that such a group can be expressed as a disjoint union of double cosets of a chosen Borel subgroup, with each coset uniquely determined by an element of the Weyl group. As a corollary, we will also derive a normal form for elements of reductive algebraic groups, illustrating how this decomposition provides both conceptual clarity and computational tools in the study of algebraic groups.

APS
Speaker: Suraj Panigrahi, IIT Bombay
Host: Kummari Mallesham
Title: Rational Numbers represented by cubic forms in 11 variables
Time, day and date: 4:00:00 PM – 5:00:00 PM, Thursday, September 25
Venue: Conference Room
Abstract: Given a rational cubic form C(X) and a non-zero rational number r, We show that the equation C(X)=r has a rational solution provided C(X) is a non-degenerate cubic form in at least 11 Variables, This is an improvement upon earlier result by Tim Browning

Commutative Algebra seminar
Speaker: Tony Puthenpurakal, IIT Bombay
Title: F-modules 5
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, September 25
Venue: Ramanujan Hall
Abstract: We continue our discussion on F-modules

APS
Speaker: Mayukh Choudhury, IIT Bombay
Host: Debraj Das
Title: Asymptotics of Cross-validation in Lasso and High Dimensional GLMs
Time, day and date: 5:00:00 PM – 6:00:00 PM, Thursday, September 25
Venue: Ramanujan Hall
Abstract: In the first half of this talk, we will discuss the asymptotic theory of $K$-fold Cross Validation in Lasso. Lasso is one of the widely used regularization methods in regression. Statisticians usually implement Lasso in practice by choosing the penalty parameter in a data-dependent way, the most popular being the $K-$fold cross-validation (or $K-$fold CV). In this case, we establish that only under some moment type conditions, Lasso with $K$-fold CV based penalty is $n^{1/2}-$consistent, but not variable selection consistent. Additionally, we establish the validity of Bootstrap in approximating the distribution of the $K-$fold CV based Lasso estimator. Therefore, our results theoretically justify the use of $K-$fold CV based Lasso estimator to perform statistical inference. We validate our Bootstrap method in finite samples based on simulations.

In the later half, we shift our focus to Generalized linear models or GLMs. With continuation to earlier reports, so far, we have presented asymptotic results for properly centered and scaled Lasso estimator and its Bootstrapped version, when the underlying dimension $d$ was fixed. Now we will observe the asymptotic traits when $d$ can grow with $n$. We will first discuss the asymptotic distribution of properly centered and scaled GLM estimator when parameter dimension $d$ increases at a slower rate than $n$. In a later regime, when $d>>n$, we employ sparsity in the model, typically Lasso in GLM. When the penalty sequence $n^{-1/2}\lambda_n\to\infty$, considering Lasso to be VSC, we establish that Gaussian approximation fails to this end for properly centered and scaled Lasso estimator. As an alternative, a PB-Lasso estimator is defined and asymptotic results are established to mimic the original distribution of Lasso.

APS
Speaker: Amanpreet Singh, IIT Bombay
Host: Saurav Bhaumik
Title: Dessins d’Enfants and the Absolute Galois Group
Time, day and date: 11:30:00 AM – 12:30:00 PM, Friday, September 26
Venue: Room 113
Abstract: In this seminar, I will talk about dessins d’enfants, a special kind of bipartite graph embedded on compact connected Riemann surfaces. We will see how such a dessin d’enfant gives a Belyi pair. Using Belyi’s theorem, we will explore how the absolute Galois group of \mathbb{Q} acts on dessins d’enfants, and this action is faithful.To summarize, we will see how dessins d’enfants provide a bridge between combinatorics, topology, algebraic geometry, and number theory

APS
Speaker: Amal Das, IIT Bombay
Host: Saikat Mazumdar
Title: Hardy-Sobolev inequality with full boundary singularity and obtainability of its sharp constant
Time, day and date: 2:15:00 PM – 3:15:00 PM, Friday, September 26
Venue: Ramanujan Hall
Abstract: We will study the Hardy-Sobolev inequality with full boundary singularity and also show that the best constant is achieved by some nonzero function under some (local) geometric conditions on the boundary of the domain. This is equivalent to a variational problem, and a solution can be obtained as an extremal for this variational problem. The core issue is to show the convergence of minimizing sequence, which we will see holds, provided one is strictly below a threshold energy level given by the best Hardy-Sobolev constant in $\mathbb{R}^n_ +$.

APS
Speaker: Kshitij Sinha, IIT Bombay
Host: Harsha Hutridurga
Title: Mathematical study of few questions in quantitative homogenization
Time, day and date: 3:30:00 PM – 4:30:00 PM, Friday, September 26
Venue: Ramanujan Hall
Abstract: In this talk, we study rates in the context of periodic homogenization of parabolic problems with large lower order terms (both drift and potential). We demonstrate that the solution is a product of three terms: (i) a function of time, (ii) the ground-state of an exponential cell eigenvalue problem and (iii) the solution to a parabolic equation with zero effective drift. For the latter, we derive L^2 rates in the homogenization limit. We will also be discussing a question regarding the interior controllability of wave equations with highly oscillatory coefficients.

Finite Geometry and Coding Theory Seminar
Speaker: Puspendu Pradhan, IIT Bombay
Host: Sudhir R. Ghorpade
Title: PGL_2(q)-orbits of lines of PG(3,q) and Higher weight spectra of some Reed-Muller codes
Time, day and date: 11:30:00 AM - 12:15:00 PM, Monday, September 29
Venue: Ramanujan Hall
Abstract: (https://drive.google.com/open?id=1mASoUTg688KdSHa7n-RVYKl51v32qIb8)
We shall divide the talk into two parts. In the first part, we focus on the problem of classifying the lines of the projective 3 space PG(3, q) over a finite field Fq into orbits under the action of the group PGL2(q), which represents the linear symmetries of the twisted cubic C. While this problem has been addressed in the literature for characteristic 2, in this talk we shall consider the case of characteristic 6= 2. We reduce this question to the classification of binary quartic forms over Fq into PGL2(q)-orbits. This part of the talk is based on joint works with Krishna Kaipa and Nupur Patanker. For non-negative integers ν and m, the Reed–Muller code RMq(ν, m) is defined as the subspace obtained by evaluating all polynomials of degree at most ν in m variables, with coefficients in Fq, at the points of the affine space AG(m, q). In the second part of the talk, we will present progress on determining the higher weight spectra of the Reed–Muller code RMq(2, 3) over the ternary field F3. This part of the talk is based on an ongoing joint work with Sudhir Ghorpade, Trygve Johnsen, Rati Ludhani, and Rakhi Pratihar.

APS
Speaker: Chayan Karmakar, IIT Bombay
Host: Ravi Raghunathan
Title: Geometry of the Tensor Product for SL(3,C)
Time, day and date: 4:00:00 PM – 5:00:00 PM, Monday, September 29
Venue: Ramanujan Hall
Abstract: We aim to study the geometric structure of tensor products of irreducible representations of $\mathrm{SL}(3, \mathbb{C})$. By plotting the highest weights of the irreducible components in the weight lattice, we observe that they form some convex regions. We also study additional structural properties of the tensor products, including a stability result in the resulting tensor diagrams