Speaker: Dr. Subhabrata Majumdar (Head of AI at VIJIL, a US-based startup)

Google Meet joining info
Video call link: https://meet.google.com/pqn-bnit-yqw

The talks will be broadcasted in the Ramanujan hall as well.

Towards Statistical Foundations for Reliable and Defendable Large Language Models

The emergence of Large Language Models (LLMs) has brought in concomitant concerns about the security and reliability of generative AI systems. While LLMs promise powerful capabilities in diverse real-world applications, ensuring that their outputs are resilient to malicious attacks and consistent across similar inputs has significant methodological and computational challenges. This situation calls for the revisiting of modern deep learning architectures through a statistical lens.
I will present on two interconnected themes in this area. First, I will introduce Representation Noising (RepNoise), a defense mechanism that protects the weights of open-source LLMs against malicious uses. RepNoise achieves this through controlled noise injection in the knowledge representations inside a model that makes it harder to recover harmful information later. Second, I will discuss my work on the consistency problem—the equivalent of robustness in LLMs—concerned with measuring and minimizing the sensitivity of LLM outputs to input variations through a combination of controlled synthetic data generation and fine-tuning.
I will conclude by discussing ongoing work at the intersection of AI security and statistics, including the development of statistical bounds for the strength of defense mechanisms like RepNoise, and robustness frameworks for ensuring AI system reliability in high-stakes applications.

Speaker: Dr. Subhabrata Majumdar (Head of AI at VIJIL, a US-based startup)

Google Meet joining info
Video call link: https://meet.google.com/pqn-bnit-yqw

The talks will be broadcasted in the Ramanujan hall as well.

Towards Statistical Foundations for Reliable and Defendable Large Language Models

The emergence of Large Language Models (LLMs) has brought in concomitant concerns about the security and reliability of generative AI systems. While LLMs promise powerful capabilities in diverse real-world applications, ensuring that their outputs are resilient to malicious attacks and consistent across similar inputs has significant methodological and computational challenges. This situation calls for the revisiting of modern deep learning architectures through a statistical lens.
I will present on two interconnected themes in this area. First, I will introduce Representation Noising (RepNoise), a defense mechanism that protects the weights of open-source LLMs against malicious uses. RepNoise achieves this through controlled noise injection in the knowledge representations inside a model that makes it harder to recover harmful information later. Second, I will discuss my work on the consistency problem—the equivalent of robustness in LLMs—concerned with measuring and minimizing the sensitivity of LLM outputs to input variations through a combination of controlled synthetic data generation and fine-tuning.
I will conclude by discussing ongoing work at the intersection of AI security and statistics, including the development of statistical bounds for the strength of defense mechanisms like RepNoise, and robustness frameworks for ensuring AI system reliability in high-stakes applications.

Topology Seminar
Speaker: Sudarshan Gurjar, IIT Bombay
Title: Kodaira Embedding theorem
Time, day and date: 4:00:00 PM – 5:00:00 PM, Friday, October 03
Venue: Ramanujan Hall
Abstract: The Kodaira embedding theorem is a fundamental theorem which can be viewed as a bridge between complex geometry and algebraic geometry.  It answers the question about when a compact complex manifold admits a holomorphic embedding inside complex projective space. The talk will be mostly about explaining this theorem. It will be a mix of complex geometry, algebraic geometry and topology. Basic knowledge of differential topology will be assumed

Topology Seminar
Speaker: Dr. Sumanta Das, IIT Bombay
Host: Rekha Santhanam
Title: Proper Homology in the Spirit of Brown’s Proper Homotopy Theory
Time, day and date: 11:45:00 AM – 12:30:00 PM, Monday, October 06
Venue: Ramanujan Hall
Abstract: We construct a proper homology theory for non-compact spaces, inspired by Brown’s proper homotopy theory. This theory satisfies analogues of the Eilenberg–Steenrod axioms in the proper setting and is functorial with respect to proper maps. It captures the behavior of spaces at infinity and aligns naturally with Brown’s proper homotopy groups. This is ongoing joint work with Rekha Santhanam.

Student Seminar
Speaker: Manish Prasad, IIT Bombay
Host: Santanu Dey
Title: Some Correspondence Results
Time, day and date: 2:00:00 PM – 2:45:00 PM, Monday, October 06
Venue: Room 113
Abstract: We will discuss some Correspondence Results-
             i) Between Algebraic sets and Ideals.
             ii) Between morphism and k -algebra homomorphism.

Student Seminar
Speaker: Rohit Jana, IIT Bombay
Host: Santanu Dey
Title: TBA
Time, day and date: 2:45:00 PM, Monday, October 06
Venue: Room 113
Abstract: TBA

Seminar
Speaker: Dr. Divya Kappara, Mathematics Department, IIT Bombay
Host: Siuli Mukhopadhyay
Title: Spatial prediction and risk mapping: a generalized linear model approach with applications to disease modeling
Time, day and date: 4:00:00 PM - 5:00:00 PM, Monday, October 06
Venue: Ramanujan Hall
Abstract: Epidemiological data typically appear as overdispersed count observations sampled from limited spatial locations. This sparsity motivates the use of spatial statistical methods to predict disease burden in unsampled regions while handling non-normality. A central task in disease modeling is to identify high-risk areas where disease counts exceed critical thresholds. Such events can signal conditions under which outbreaks are more likely or sustained transmission may occur. We propose using a generalized linear model approach for predicting spatially referenced count data. The response variable is assumed to be conditioned on a weakly stationary latent spatial process accounting for both overdispersion and spatial correlation structure. The Model estimates are used to generate predictions at new locations, quantify prediction uncertainty, and estimate the odds of threshold exceedance. The uncertainty around the predictions and odds of an outbreak are quantified using resampling methods adapted for spatial data under a GLM setup. We illustrate the proposed method through real data analysis, also studying the effect of varying sampling procedures.

Speaker: Sumit Chandra Mishra, Department of Mathematics, IIT Indore

Host: Sandip Singh

Title: Ruled residue theorem for function fields

Time, day and date: 4:00 PM - 5:00:00 PM, Tuesday, October 07

Venue: Room No. 215, Mathematics Department

Abstract: Let E be a field with a valuation v. In 1983, Ohm proved that for any extension of v to the rational function field E(X)  in one variable, the corresponding residue field extension is either algebraic or ruled, i.e., it is the rational function field in one variable over a finite extension of the residue field of E. This is called the Ruled Residue Theorem. More generally, one can consider the function field F of a curve over E and ask if for all extensions of v to F, the corresponding residue field extension is either algebraic or ruled? If not, is there any bound on the number of extensions of v to F where this fails? I will mention known results for the function fields of conics. Later on, I will discuss the case of function fields of elliptic curves and hyperelliptic curves( joint work with Prof. Karim J. Becher and Dr. Parul Gupta, and joint work with Dr Parul Gupta, respectively).

All are welcome to attend the seminar.

Mathematics Colloquium
Speaker: Sundaram Thangavelu, Indian Institute of Science, Bangalore
Host: Sanjoy Pusti
Title: On points of contention between two families of operators
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, October 08
Venue: Ramanujan Hall
Abstract: Let $ \mathcal{H} $ be a Hilbert space and $ \mathcal{F}_j, j =1,2 $ two families of bounded linear operators acting on $ \mathcal{H}.$ We say that a vector $ u \in \mathcal{H} $ is a point of contention between $ \mathcal{F}_1$ and $ \mathcal{F}_2 $ if for any $ T \in \mathcal{F}_1 $ and $ S \in \mathcal{F}_2, $ the condition $ Tu = Su $ forces both $ T $ and $ S $ to be contstant multiple of the identity operator. We explain this phenomenon when $ \mathcal{H} $ is either the Labesgue space $ L^2(\R^n) $ or the Fock space $ \mathcal{F}(\C^n).$

Student Seminar
Speaker: Garima Khetawat, IIT Bombay
Host: Santanu Dey
Title: Independence attractors of graphs
Time, day and date: 2:00:00 PM – 2:45:00 PM, Thursday, October 09
Venue: Room 113
Abstract: By an independent set in a simple graph G, we mean a set of pairwise non-adjacent vertices in G. The independence polynomial of G is defined as IG(z) = a0+a1z+a2z 2+ · · · + aβz β , where ai is the number of independent sets in G with cardinality i and β denotes the cardinality of a largest independent set in G, known as the independence number of G. Let Gm denote the m-times lexicographic product of G with itself. The independence attractor of G, denoted by A (G), is defined as A (G) = limm→∞{z : IGm(z) = 0}, where the limit is taken with respect to the Hausdorff metric on the space of all compact subsets of the plane. In this talk, we discuss some results (without proofs) regarding the possibility of independence attractors being circles or line segments.

Student Seminar
Speaker: Aditya Dwivedi, IIT Bombay
Host: Santanu Dey
Title: TBA
Time, day and date: 2:45:00 PM – 3:30:00 PM, Thursday, October 09
Venue: Room 113
Abstract: TBA

Seminar
Speaker: Professor Ori Davidov (Department of Statistics, The University of Haifa)
Host: Siuli Mukhopadhyay
Title: Safe hypotheses testing with application to order restricted inference
Time, day and date: 3:00:00 PM – 4:00:00 PM, Thursday, October 09
Venue: Ramanujan Hall
Abstract: Hypothesis testing problems are fundamental to the theory and practice of statistics. It is well known that when the union of the null and the alternative does not encompass the full parameter space the possibility of a Type III error arises, i.e., the null hypothesis may be rejected when neither the null nor the alternative are true. In such situations, common in the context of order restricted inference, the validity of our inferences may be severely compromised. The study of the geometry of the distance test, a test widely used in constrained inference, illuminates circumstances in which Type III errors arise and motivates the introduction of \emph{safe tests}. Heuristically, a safe test is a test that is free of Type III errors at least asymptotically. A novel safe test is proposed and studied. The new testing procedure is associated with a \emph{certificate of validity}, a pre--test indicating whether the original hypotheses are consistent with the data. Consequently, Type III errors can be addressed in a principled way, and constrained tests can be carried out without fear of systematically incorrect inferences. Although we focus on testing problems arising in order restricted inference, the underlying ideas are more broadly applicable. The benefits associated with the proposed methodology are demonstrated through simulations and analysis of illustrative examples

Commutative Algebra seminar
Speaker: Prof. R. V. Gurjar, TIFR Bombay (Retd.)
Host: Tony Puthenpurakal
Title: Resolution of singularities of algebraic or analytic varieties.
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, October 09
Venue: Ramanujan Hall
Abstract: First I will show how to resolve curve singularities (this must be well-known to many). Then discuss Jung-Hirzebruch argument to get the equation of a surface in three variables in a very simple form, and then resolve its singularities. This will complete the proof of resolution of singularitiies of surfaces. Next I will discuss some easy but useful results about resolution type results (resolution of indeterminacies, principalization, general notion of resolution of singularities, and their relations to each other).

Geometry and Topology seminar
Speaker: Sudarshan Gurjar, IIT Bombay
Host: Rekha Santhanam
Title: Kodaira Embedding Theorem
Time, day and date: 11:30:00 AM - 12:30:00 PM, Monday, October 13
Venue: Ramanujan Hall
Abstract: This is a continuation of my earlier talk. After recalling the theorem, I will outline a proof of it.

Ph.D thesis defense
Speaker: Samarendra Sahoo
Date: 13 October 2025
Day: Monday
Venue: Ramanujam Hall
Time: 4-5 pm

Title: Minimal free resolutions, $I$-stable filtrations and some lower
bounds of Hilbert coefficients.

Abstract: Let $(A, \mathfrak{m})$ be a Cohen-Macaulay local ring and $M$ a
Cohen-Macaulay $A$-module. We study certain lower bounds for the Hilbert
coefficients of $M$ and investigate the conditions under which the
associated graded module $G_{\mathfrak{m}}(M)$ is Cohen--Macaulay. Let
$M_i$ denote the $i$-th syzygy of $M$, and suppose that $(A,
\mathfrak{m})$ is a complete intersection ring. We examine the asymptotic
behavior of $e_1(M_i)$, the first Hilbert coefficient of $M_i$, and
$\operatorname{reg}(G_{\mathfrak{m}}(M_i))$, the Castelnuovo--Mumford
regularity of the associated graded module, for sufficiently large $i$.
Furthermore, for an $\mathfrak{m}$-primary ideal $I$ and an $I$-stable
filtration $F = \{I_n\}_{n \ge 0}$, we analyze the situation when $\dim
A(F)/A(I) = \dim A$, where $A(F)$ denotes the Rees algebra with respect to
the filtration $F$.

Mathematics Colloquium
Speaker: Saikat Mazumdar, IIT Bombay
Title: Conformal Powers of the Laplacian, Q-Curvature, and the Compactness Problem
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, October 15
Venue: Ramanujan Hall
Abstract: The conformal powers of the Laplacian, known as the GJMS operators, are a family of conformally invariant differential operators with leading term a power of the Laplacian, and the Q-curvature is the associated scalar invariant. They arise naturally and provide a unifying framework connecting important developments in geometry, analysis, and mathematical physics.
In this talk, I will consider the question of compactness of constantQ-curvature metrics on a closed Riemannian manifold. This is the higher-order analogue of the compactness problem for conformal metrics with constant scalar curvature. I will outline the background, motivations, and key challenges, and present some of my results in this direction. This is a recent work with Bruno Premoselli (ULB Brussels).

Geometry and Topology seminar
Speaker: Pritthijit Biswas, IIT Bombay
Host: Rekha Santhanam
Title: Brill-Noether loci inside the moduli space of stable vector bundles over curves
Time, day and date: 11:30:00 AM - 12:30:00 PM, Thursday, October 16
Venue: Ramanujan Hall
Abstract: Let X be a smooth projective curve of genus g over the field C. Let M_{X}(2, L)denote the moduli space of stable rank 2 vector bundles on X with fixed determinant L of degree 2g-1. Consider the Brill-Noether subvariety W ^{1}_{X} (2, L) of M_{X} (2, L) which parametrises stable vector bundles having at least two linearly independent global sections. In this talk, for generic X and L, I would outline a proof of the fact that W ^{1}_{X} (2, L) is stably-rational when g = 3 and unirational when g = 4. This is a part of joint work with Jaya NN Iyer.

Student Seminar
Speaker: Saibur Alom, IIT Bombay
Host: Santanu Dey
Title: Fourier Transform of Hilbert Transform on R^n
Time, day and date: 2:00:00 PM – 2:45:00 PM, Thursday, October 16
Venue: Room 113

Student Seminar
Speaker: Subir Dakshi, IIT Bombay
Host: Santanu Dey
Title: The Hardy-Littlewood maximal function
Time, day and date: 2:45:00 PM – 3:30:00 PM, Thursday, October 16
Venue: Room 113

Commutative algebra seminar
Speaker: Prof. R. V. Gurjar
Date: 16 Oct 2025
Day Thursday
Venue: Ramanujam Hall
Time 4-5 pm
Title: Resolution of singularities of algebraic or analytic varieties.

Abstract.
First I will show how to resolve curve singularities (this must be
well-known to many). Then discuss Jung-Hirzebruch argument to get the
equation of a surface in three variables in a very simple form, and then
resolve its singularities. This will complete the proof of resolution of
singularitiies of surfaces. Next I will discuss some easy but useful
results about resolution type results (resolution of indeterminacies,
principalization, general notion of resolution of singularities, and their
relations to each other).

Statistics/Probability Seminar
Speaker: Mayukh Choudhury, IIT Bombay
Host: Debraj Das
Title: Asymptotic Theory of $K$-fold Cross-validation in Lasso and the Validity of Bootstrap
Time, day and date: 5:00:00 PM – 6:00:00 PM, Thursday, October 16
Venue: Ramanujan Hall (meet.google.com/bgr-tffn-cem)
Abstract: 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). However, inferential properties, such as the variable selection consistency and $n^{1/2}-$consistency, of the $K-$fold CV based Lasso estimator and validity of the Bootstrap approximation are still unknown. In this talk, we will discuss about $n^{1/2}-$consistency of the $K$-fold CV based penalty and utilizing that we explore the aforementioned inferential properties of the underlying Lasso estimator. Additionally, we establish the validity of Bootstrap in approximating the distribution of the $K-$fold CV based Lasso estimator. We validate our Bootstrap method in finite samples based on simulations.

Analysis seminar
Speaker: Prof. S. Thangavelu
Host: Sanjoy Pusti
Title: On Hardy's inequality for fractional powers of the sublaplacian
Time, day and date: 4:00:00 PM – 5:15:00 PM, Friday, October 17
Venue: Ramanujan Hall
Abstract: In these lectures we plan to give an introduction to Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group. We will recall the case of the standard Laplacian on $ \R^n $ and give a proof of Hardy's inequality for $ (-Delta)^s $ using extension problem and trace Hardy inequality. We will then develop necessary background for studying the modified extension problem for the sublaplacian, establish an analogue of trace Hardy inequality and deduce Hardy's inequality.

Complex Geometry Talk
Speaker: Dr. Ramesh Mete, IIT Bombay
Host: Saikat Mazumdar
Title: Twisted and coupled cscK metrics on ruled surfaces
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, October 22
Venue: Ramanujan Hall
Abstract: First we recall the standard canonical metrics in K\"{a}hler geometry. A K\"{a}hler metric is called a constant scalar curvature K\"{a}hler (cscK in short) metric if its scalar curvature is constant. The set of all cscK metrics include the set of all K\"{a}hler-Einstein metrics. Stoppa (2009) introduced the twisted cscK equation by twisting the cscK equation with another equation, called the J-equation. Datar and Pingali (2020) introduced coupled cscK metrics generalizing coupled K\"{a}hler-Einstein metrics defined by Hultgren and Witt Nystr\"{o}m (2019). We study the existence problem of twisted (and coupled) cscK metrics on ruled surfaces using Calabi ansatz. As an application, we give a bound for the Chen-Cheng (2021) invariant on ruled surfaces

Student Seminar
Speaker: Annu Sonkar, IIT Bombay
Host: Santanu Dey
Title: TBA
 Time, day and date: 2:00:00 PM – 2:45:00 PM, Thursday, October 23
Venue: Room 113

Student Seminar
Speaker: Advaith R Nair, IIT Bombay
Host: Santanu Dey
Title: Sard’s Theorem
Time, day and date: 2:45:00 PM – 3:30:00 PM, Thursday, October 23
Venue: Room 113

Commutative Algebra seminar
Speaker: Prof. R. V. Gurjar, TIFR (retd).
Host: Tony Puthenpurakal
Title: Resolution of singularities of algebraic or analytic varieties.
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, October 23
Venue: Ramanujan Hall
Abstract: First I will show how to resolve curve singularities (this must be well-known to many). Then discuss Jung-Hirzebruch argument to get the equation of a surface in three variables in a very simple form, and then resolve its singularities. This will complete the proof of resolution of singularitiies of surfaces. Next, I will discuss some easy but useful results about resolution type results (resolution of indeterminacies, principalization, general notion of resolution of singularities, and their relations to each other).

Analysis seminar
Speaker: Prof. S. Thangavelu
Host: Sanjoy Pusti
Title: On Hardy's inequality for fractional powers of the sublaplacian
Time, day and date: 4:00:00 PM – 5:15:00 PM, Friday, October 24
Venue: Ramanujan Hall
Abstract: In these lectures we plan to give an introduction to Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group. We will recall the case of the standard Laplacian on $ \R^n $ and give a proof of Hardy's inequality for $ (-Delta)^s $ using extension problem and trace Hardy inequality. We will then develop necessary background for studying the modified extension problem for the sublaplacian, establish an analogue of trace Hardy inequality and deduce Hardy's inequality.

Geometry and Topology seminar
Speaker: Sudarshan Gurjar, IIT Bombay
Title: Kodaira Embedding Theorem 
Time, day and date: 11:30:00 AM - 12:30:00 PM, Monday, October 27
Venue: Ramanujan Hall
Abstract: This is the fourth talk in the series. After briefly recalling the previous lectures, I will complete the proof of the theorem.

Student Seminar
Speaker: Neeraj Rawat, IIT Bombay
Host: Santanu Dey
Title: One phase Stefan problem in one dimension
Time, day and date: 2:00:00 PM – 2:45:00 PM, Monday, October 27
Venue: Room 113

Student Seminar
Speaker: Sayantan Sinha, IIT Bombay
Host: Santanu Dey
Title: TBA
Time, day and date: 2:45:00 PM – 3:30:00 PM, Monday, October 27
Venue: Room 113

Analysis seminar
Speaker: Prof. S. Thangavelu
Host: Sanjoy Pusti
Title: On Hardy's inequality for fractional powers of the sublaplacian
Time, day and date: 11:00:00 AM – 12:15:00 PM, Tuesday, October 28
Venue: Ramanujan Hall
Abstract: In these lectures we plan to give an introduction to Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group. We will recall the case of the standard Laplacian on $ \R^n $ and give a proof of Hardy's inequality for $ (-Delta)^s $ using extension problem and trace Hardy inequality. We will then develop necessary background for studying the modified extension problem for the sublaplacian, establish an analogue of trace Hardy inequality and deduce Hardy's inequality.

Student Seminar
Speaker: Samiun Ali Molla, IIT Bombay
Host: Keshav Aggarwal
Title: Bounds for Riemann zeta in the critical strip
Time, day and date: 11:30:00 AM – 12:10:00 PM, Tuesday, October 28
Venue: Room 114

Student Seminar
Speaker: Divye Goyal, IIT Bombay
Host: Keshav Aggarwal
Title: Distribution of zeros of Riemann zeta
Time, day and date: 12:15:00 PM – 12:55:00 PM, Tuesday, October 28
Venue: Room 114

Special Colloquium
Speaker: Uwe Franz, Universite Marie et Louis Pasteur, France
Host: Sutanu Roy
Title: What can Levy processes tell us about compact quantum groups?
Time, day and date: 4:00:00 PM - 5:00:00 PM, Tuesday, October 28
Venue: Ramanujan Hall
Abstract: Schurmann has defined Levy processes on involutive bialgebras, a generalization of the notion of Levy processes with values in groups or semigroups, to the setting of noncommutative stochastic processes. This talk will provide an introduction to these processes. We will then apply this theory to compact quantum groups in the sense of Woronowicz, and discuss how Levy processes can provide geometric information, like a spectral dimension or a notion of connected component of the unit, on these quantum spaces.

Mathematics Colloquium
Speaker: Mayukh Mukherjee, IIT Bombay
Title: A \Delta-criterion, Green geometry and the strong Liouville property on groups
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, October 29
Venue: Ramanujan Hall
Abstract: We develop a unified framework linking harmonic function growth, random-walk geometry, and heat kernel asymptotics on finitely generated groups. The central analytic device is a potential theory-based boundary functional whose vanishing characterises strong Liouville property, and we give general heat-kernel envelope conditions that force vanishing without requiring sharp two-sided estimates. This yields new, checkable criteria for strong Liouville on broad classes of groups (including polynomial growth and many subGaussian settings). In the opposite direction, we prove that on exponentially growing groups $\Delta$ does \emph{not} decay on balls (under mild on-diagonal bounds), forcing the existence of non-constant positive harmonic functions. On the geometric side, we show that a trivial Martin boundary collapses the Green geometry and the Green speed vanishes along any path with finite word-speed. Two speed results refine the picture on nilpotent groups. First, there are natural heavy-tail regimes with speed vanishing in probability. Second, under a uniform cone lower bound on jump directions, any positive word-speed implies finite first moment - and by symmetry the drift is then zero almost surely.

This is based on joint work with Soumyadeb Samanta and Soumyadip Thandar

Student Seminar
Speaker: Sourav Khatua, IIT Bombay
Host: Santanu Dey
Title: Morita Equivalence and C*-Correspondence and its application to Graph C*-algebras
Time, day and date: 2:00:00 PM – 2:45:00 PM, Thursday, October 30
Venue: Room 113

Commutative Algebra seminar
Speaker: Sayed Sadiqul Islam, IIT Bombay
Host: Tony Puthenpurakal
Title: ON THE MATLIS DUALS OF LOCAL COHOMOLOGY MODULES
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, October 30
Venue: Ramanujan Hall
Abstract: Let R be a regular local ring of positive characteristic. Let M be a local cohomology module H^i_I(R)​ for any non-zero ideal I of R. We show that if M is non-zero then the support of D(M), where D is the Matlis-dual functor, is the whole Spec(R). The proof is due to Lyubeznik and Yildirim and uses techniques from F-module theory.

Topology and Related Topics Seminar
Speaker: Omkar Ramdas, IIT Bombay
Host: Rekha Santhanam
Title: An introduction to Lawvere theory and their applications
Time, day and date: 11:00:00 AM – 12:00:00 PM, Friday, October 31
Venue: Room 105
Abstract: Lawvere theory is a unique categorical framework which unifies the ideas of algebraic structures like groups, rings, etc into one category using their "logical signatures". I am planning to give two talks on this.
In this first talk, I'll cover the logical prerequisites needed and then we'll go on to define what Lawvere theories are. This talk will be very basic and accessible to all.

Analysis seminar
Speaker: Prof. S. Thangavelu
Host: Sanjoy Pusti
Title: On Hardy's inequality for fractional powers of the sublaplacian
Time, day and date: 4:00:00 PM – 5:15:00 PM, Friday, October 31
Venue: Ramanujan Hall
Abstract: In these lectures we plan to give an introduction to Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group. We will recall the case of the standard Laplacian on $ \R^n $ and give a proof of Hardy's inequality for $ (-Delta)^s $ using extension problem and trace Hardy inequality. We will then develop necessary background for studying the modified extension problem for the sublaplacian, establish an analogue of trace Hardy inequality and deduce Hardy's inequality.