Commutative Algebra seminar
Host: Tony J Puthenpurakal
Title: Intersection theorem in char p > 0.  III
Time, day and date: 11:30:00 AM – 12:30:00 PM, Monday, August 17
Venue: Room 215
Abstract: -

APS
Speaker: Sagnik Roy, IIT Bombay
Host: Saurav Bhaumik
Title: Orientation of Manifolds and Poincaré Duality
Time, day and date: 3:30:00 PM – 4:30:00 PM, Monday, August 17
Venue: Room 215
Abstract: This presentation explores the geometric and topological foundations of manifold orientation and its theoretical culmination in Poincaré Duality. It establishes orientability criteria using differential and topological frameworks, contrasting orientable manifolds like spheres with non-orientable spaces such as the Möbius strip and Klein bottle. By formalizing local orientations, the talk introduces the fundamental class and the cap product. These mathematical tools establish the canonical Poincaré Duality isomorphism for closed manifolds. Finally, the presentation extends this duality to non-compact manifolds by introducing singular cohomology with compact support alongside inductive proof techniques.

Seminar on the Plancherel measure
Speaker: Ravi Raghunathan, IIT Bombay
Title: On the support of the Plancherel measure (d'après Bernstein)
Time, day and date: 5:00:00 PM – 6:15:00 PM, Monday, August 17
Venue: Room 215
Abstract: -

APS
Speaker: Lavanya, IIT Bombay
Host: Shripad M. Garge
Title: Affine algebraic groups are linear
Time, day and date: 11:00:00 AM – 12:00:00 PM, Tuesday, August 18
Venue: Room 215
Abstract: -

Statistics and Probability seminar
Speaker: Soham Bonnerjee, University of Pennsylvania
Host: Debraj Das
Title: Fast Watermark Segmentation in LLM generated texts (and other application of epidemic change-points)
Time, day and date: 5:00:00 PM - 6:00:00 PM, Tuesday, August 18
Venue: Ramanujan Hall
Abstract: Epidemic change-points has seldom received attention in statistics community; however, it has  important, and somewhat surprising applications in critically important and modern problems. In this talk, we primarily focus on the "watermark segmentation" problem. With the growing use of large language models, concerns over content authenticity have spurred a variety of watermarking schemes. These schemes use secret keys to detect machine-generated text while remaining imperceptible to readers. Detection typically reduces statistical hypothesis testing for the presence of watermarks, a topic that is now well studied. In contrast, the finer grained task of localizing which segments of a text are watermarked is much less explored; existing approaches often lack scalability or guarantees robust to paraphrasing and post-editing. We bring a new perspective to this segmentation problem through the lens of epidemic change-points and, by exploiting this connection, propose WISER, a novel and computationally efficient watermark segmentation algorithm. We establish finite-sample error bounds and consistency for detecting multiple watermarked segments in a single text. Complementing these theoretical results, our extensive numerical experiments show that WISER outperforms state-of-the-art baseline methods, both in terms of computational speed as well as accuracy, on various benchmark datasets embedded with diverse watermarking schemes. Together, these theoretical and empirical results position WISER as an effective tool for watermark localization. Towards the end of the talk, we will briefly discuss spatial anomaly detection as another important application of the epidemic change-points in "space", and show how similar ideas lead to computationally fast algorithms with optimal detection-accuracy results for single and multiple patches. Based on joint works with Subhrajyoty Roy (Washington University at St. Louis), Sayar Karmakar (University of Florida), and George Michailidis (UCLA).

Partial Differential Equations seminar
Speaker: Aritro Pathak, visiting fellow, TIFR Mumbai
Host: Harsha Hutridurga
Title: Uniform domains with Ahlfors-David regular boundaries, and uniform rectifiability.
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, August 19
Venue: Ramanujan Hall
Abstract: Uniform rectifiability is a quantitative notion of rectifiability that is closely related to the (L^2) boundedness of Riesz transforms. Uniform domains satisfy quantitative openness and quantitative path connectivity conditions. When the Ahlfors-David regular boundary of a uniform domain is uniformly rectifiable, one obtains solvability of the Dirichlet problem for the Laplacian with rough boundary data, as well as quantitative absolute continuity of harmonic measure. We will discuss some recent results in this active area at the interface of geometric measure theory and harmonic analysis. In particular, we will present a new result concerning the Weak Geometric Lemma and uniform rectifiability for Ahlfors-David regular sets whose complements are uniform domains.

Mathematics Colloquium
Speaker: Prof(Retd.). Sadanand D. Agashe, Electrical Engineering, IIT Bombay
Host: Sivaji Ganesh Sista
Title: New Formulas and Results for 3-Dimensional Vector Fields
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, August 19
Venue: Ramanujan Hall
Abstract: New formulas are derived for once-differentiable 3-dimensional fields, using the operator (x d/dx + y d/dy + z d/dz). This new operator has a property similar to that of the Laplacian operator; however, unlike the Laplacian operator, the new operator requires only once-differentiability. A simpler formula is derived for the classical Helmholtz decomposition. Orthogonality of the solenoidal and irrotational parts of a vector field, the uniqueness of the familiar inverse-square laws, and the existence of solution of a system of first-order PDEs in 3 dimensions are proved. New proofs are given for the Helmholtz Decomposition Theorem and the Divergence theorem. The proofs use the relations between the rectangular-Cartesian and spherical-polar coordinate systems. Finally, an application is made to the study of Maxwell’s equations.
Keywords: Spherical-Polar Coordinates, Helmholtz Decomposition, Divergence Theorem, Orthogonality, Maxwell’s Equations

Talk
Speaker: Arkadev Ghosh, Chennai Mathematical Institute (CMI)
Host: Saurav Bhaumik
Title: GIT quotients of Schubert varieties and Hessenberg varietiesmodulo a one-dimensional torus
Time, day and date: 5:00:00 PM – 6:00:00 PM, Wednesday, August 19
Venue: Online (https://meet.google.com/imq-fzbd-vez)
Abstract: In this talk, we present some results on Geometric Invariant Theory (GIT) quotients of Schubert varieties and Hessenberg varieties in flag varieties, for the action of a one-dimensional torus.

We begin with GIT quotients of Schubert varieties in partial flag varieties, for ample line bundles linearized with respect to a one-parameter subgroup dual to a simple root. We establish a criterion for the existence of semistable points. In the case of the Grassmannian Gr(r,n), we identify the unique minimal-dimensional Schubert variety admitting semistable points. Further, we describe the corresponding GIT quotient as a projective space of matrices, with the natural action of a product of two special linear groups arising as the semisimple part of a Levi subgroup.

We then extend these techniques to study the action of a higher-dimensional subtorus on certain Schubert varieties in Gr(r,n), determined by the peak set of the associated permutation. Under a natural divisibility condition on r and n, we describe the semistable locus explicitly and show that the GIT quotient is an iterated projective space bundle over a projective space.

In another direction, we study the action of a one-parameter subgroup on the irreducible components of semisimple Hessenberg varieties associated with a diagonal matrix having two distinct eigenvalues. We show that, for suitable line bundles, the resulting GIT quotients are isomorphic to homogeneous fiber bundles over a flag variety whose fibers are products of projective lines. Furthermore, we describe the space of global sections of the descended line bundle as a module over a suitable Levi subgroup.

Topology seminar
Speaker: Udit Mavinkure, IIT Bombay
Host: Rekha Santhanam
Title: Quasicategories vs. Segal Spaces, Part 3
Time, day and date: 2:30:00 PM – 3:30:00 PM, Thursday, August 20
Venue: Room 215
Abstract: This talk is the third installment in a four-part expository series on higher category theory. The goal of the series is to establish the following result of Joyal and Tierney: the Joyal model structure for quasicategories and the Rezk model structure for complete Segal spaces are Quillen equivalent. (NB: Newcomers are welcome; background from the April and May talks will not be assumed!)

We will begin with a brief recap of quasicategories: their definition, and how they generalize ordinary categories. We will then discuss the Joyal model structure, and its specific properties that enable a straightforward description of its simplicial completion, à la Cisinski.

PDE and Numerical Analysis seminar
Speaker: Ketan Savla, University of Southern California
Host: Harsha Hutridurga
Title: LQR for finite extent linear evolution PDEs.
Time, day and date: 4:00:00 PM – 5:00:00 PM, Thursday, August 20
Venue: Ramanujan Hall
Abstract: We consider the linear quadratic regulator (LQR) problem for linear evolution PDEs on a bounded spatial interval, with control entering as an additive forcing term and arbitrary boundary data. Classical frequency-domain approaches to LQR for PDEs rely on the real-frequency Fourier transform, and work cleanly only on unbounded domains or under homogeneous boundary conditions; applied naively on a finite interval, they produce a control law that depends on future, unspecified boundary values. We resolve this using the unified transform of Fokas, which extends the Fourier transform on a bounded domain to complex frequencies. Decoupling the PDE in this complex frequency variable and solving the resulting family of finite-dimensional LQR problems yields a closed form control law; deforming the associated frequency integrals onto suitable contours in the complex plane then eliminates all dependence on the unknown boundary data. For the reaction-diffusion equation, we show that the resulting integral representation is equivalent to a state-feedback law whose kernel splits into a Toeplitz part, as on the whole line, and a new Hankel part generated purely by the boundary.

Algebraic groups seminar
Speaker: Sabyasachi Dhar, IIT Bombay, Mumbai
Host: Shripad Garge
Title: Iwahori-Hecke Algebras. III
Time, day and date: 5:00:00 PM - 6:30:00 PM, Thursday, August 20
Venue: Room No .215
Abstract: We continue with the study of the Iwahori-Hecke algebras. We will be doing some examples in this lecture.

Student Seminar
Speaker: Aditya Dwivedi, IIT Bombay
Host: Suman Kumar Sahoo
Title: The Algebra of Magic Squares
Time, day and date: 3:30:00 PM – 4:30:00 PM, Friday, August 21
Venue: Ramanujan Hall
Abstract: Magic squares have been studied by thinkers for centuries, often appearing in classical texts across cultures, such as the Lo Shu square, Buduh square, or the bhadra chaturashra. The problem of counting magic squares was also pursued by many thinkers over centuries, often leading to calculations in specific cases (see Al-Ashabb et al., Mandadi et al., Mattingly et al., Dvivedi et al.).
But what happens when we move beyond these specific historical constructions and ask a generalised question: exactly how many n x n magic squares exist for a given line sum r?
This talk explores the surprising bridge between this ancient combinatorial problem and modern commutative algebra. After reviewing the elementary 1 x 1 and 2 x 2 matrices, we will introduce the celebrated Anand-Dumir-Gupta conjectures. Using the Birkhoff-von Neumann theorem, we will construct a specific subring generated by permutation matrices. By analysing the Hilbert series of this ring, we arrive at the elegant final result: the number of n x n magic squares is simply a polynomial in the line sum r.