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.