Annual Progress seminar

Tuesday,14 March 2023, 11.30 am

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Venue: Room 105

Host: Shripad M. Garge

Speaker: Deepkumar Makadiya

Affiliation: IIT Bombay

Title: Twisted Chevalley groups

Abstract: The talk aims at discussing the twisted Chevalley group and its automorphisms. The first part of the talk deals with the existing literature regarding the twisted Chevalley group and we present some results in this direction. The main aim of the second half is to characterize the class of automorphisms of the twisted Chevalley group. En route, we present some useful results to build the setting of the desired characterization.

Number Theory seminar

Tuesday, 14 March 2023, 2.30 pm

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Venue: Google meet: Link: https://meet.google.com/jtv-udzf-bbx

Host: Kummari Mallesham

Speaker: Keshav Aggarwal

Affliation: Alfréd Rényi Institute of Mathematics

Title:  Analytic study of L-functions: Subconvexity bound problem

Abstract: Much of modern number theory revolves around the study of L-functions. The L-functions associated with arithmetic objects called automorphic forms are of great interest and many conjectures have been made about their properties. The best known is the Riemann hypothesis (RH) which has many well-known consequences. One such is the Lindelof hypothesis (LH) which is a conjecture about the growth rate of automorphic L-functions on the critical line. Results achieving partial progress towards LH are known as Subconvexity Bound results and have rich arithmetic applications, like equidistribution of points on certain surfaces, and Quantum Unique Ergodicity. Recent years have seen much activity in this area by combining powerful tools like the Delta method, Motohashi-type formulas, and estimates on moments of L-functions. In this talk, we give an introduction to the subconvexity bound problem and a recent result on bounding a short second moment of GL(3) L-functions. This is joint work with Joseph Leung and Ritabrata Munshi and yields the yet best-known subconvexity bound for GL(3) L-functions in the t-aspect.

Algebraic Groups Seminar

Tuesday, 14 march 2023, 4 pm

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Venue: Room 114

Host: Shripad M. Garge

Speaker: Deepkumar Makadiya

Affiliation: IIT Bombay

Title: Linear algebraic groups, basic themes. IV

Abstract: We complete the discussion around the Jordan decomposition in this lecture.

Algebraic Geometry seminar

Wednesday, 15 March 2023, 11.30 am

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Venue: Ramanujan Hall

Host: Saurav Bhaumik

Speaker: Pinakinath Saha

Affilaition: IIT Bombay

Title: On (weak) Fano $G$-Bott-Samelson-Demazure-Hansen varieties

Abstract: Let $G$ be a semi-simple simply connected algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $W$ be the Weyl group of $G$ with respect to $T$. For an arbitrary sequence $w$ of simple reflections in $W,$ let $Z_{w}$ be the Bott-Samelson-Demazure-Hansen variety (BSDH-variety for short) corresponding to $w.$ Bott-Samelson-Demazure-Hansen-varieties are an important tool in geometric representation theory. They were originally defined as desingularizations of Schubert varieties and were used to describe the geometry of Schubert varieties. There is a natural action of $B$ on $Z_{w}$ given by the left multiplication. Let $\widetilde{Z_{w}}:=G\times^{B}Z_{w}$ be the fibre bundle associated to the principal $B$-bundle $G\to G/B.$ We call it $G$-Bott-Samelson-Demazure-Hansen variety ($G$-BSDH-variety for short). In the first part of the talk, we will describe a basis of the Picard group of $G$-BSDH variety, which we will refer as the $\mathcal{O}(1)$-basis. Then we will characterize the nef, globally generated, ample and very ample line bundles on $G$-BSDH variety in terms of the $\mathcal{O}(1)$-basis. Finally, we will provide a characterization of (weak) Fano $G$-BSDH varieties. We introduce a few more notations for the second part of my talk. Let $G=SO(8n,\mathbb{C})\big/SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the end simple root of its Dynkin diagram. In the second part of the talk, we will discuss the projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle. The first part of the talk will be based on joint work with Saurav Bhaumik. Here is the link of the preprint: \url{https://arxiv.org/abs/2212.10366}. The second part of the talk will be based on joint work with Arpita Nayek. Here are the links of the preprints: \url{https://arxiv.org/abs/2207.01477} and \url{https://arxiv.org/abs/2302.00555}.

Geometry and Topology Seminar

Wednesday, 15 March 2023, 2.30 pm

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Venue: Ramanujan Hall

Host: Sandip Singh

Speaker: Arghya Mondal

Affiliation: CMI, Chennai

Title: A higher dimensional generalization of Margulis' construction of expander graphs

Abstract: Expanders are a family of finite graphs whose vertex sizes go to infinity but edge sizes grow at most linearly in vertex sizes, while still remaining highly connected. The first explicit construction of such graphs was by Margulis using discrete groups having Property (T), a rigidity property defined in terms of unitary representations. In recent years various higher dimensional generalizations of expanders, replacing graphs by simplicial complexes of a fixed dimension, have been considered. We will discuss a group theoretic construction of one such generalization, which is an extension of Margulis' construction to higher dimensions.

Mathematics Colloquium

15 March 2023, 4 pm

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Venue: Ramanujan Hall

Host: Mayukh Mukherjee

Speaker: Arunima Ray

Affiliation: MPI, Bonn

Title: Embedding surfaces in 4-manifolds

Abstract: Manifolds are fundamental objects in topology since they locally model Euclidean space. Within a given ambient manifold, we are often interested in finding embedded submanifolds, which would then enable cutting and pasting operations, such as surgery. The study of surfaces in 4-dimensional manifolds has led to breakthroughs such as Freedman's proof of the 4-dimensional Poincare conjecture. Important open questions on 4-manifolds can also be reduced to the question of finding certain embedded surfaces.

I will consider the following question: When is a given map of a surface to a 4-manifold homotopic to an embedding? I will give a survey of related results, including the celebrated work of Freedman and Quinn, and culminating in a general surface embedding theorem, arising in joint work with Daniel Kasprowski, Mark Powell, and Peter Teichner.

Speaker: Bikram Bir, IIT Bombay

Date and Time (of the first lecture): Thursday 16 March 2023 at 02.30PM

Venue: Ramanujan Hall, Department of Mathematics.

Title: Oldroyd Model of Order One: Theory and Numerics

Abstract:  In these lectures, we discuss about a few finite element methods
for the equations of motion arising in the two-dimensional Oldroyd model
of order one; a model that represents linear viscoelastic fluid flows and
that can be viewed as an integral perturbation of the Navier-Stokes
equations.
First, we study existence and uniqueness of the continuous and weak
solution. Then, we discretize the space variable based on standard
Galerkin finite element method and the temporal variable based on
different time-discrete schemes. We next employ different finite element
methods like the two-grid method, the penalty method, the grad-div
stabilization method, the projection method and the nonconforming finite
element method. In all these cases, our main aim will be to obtain an
optimal error estimate for the velocity and the pressure. Finally, we
present some
numerical experiments to validate our theoretical findings.
 

Statistics and Probability  seminar

Thursday, 16 March 2023, 3 pm

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Venue: Online, link will be sent later

Host: Debraj das

Speaker: Sandipan Roy

Affiliation: University of Bath

Title: Statistical Inference in Complex Data with Network Structure

Abstract: New technological advancements have allowed the collection of datasets of large volume and different levels of complexity. Many of these datasets have an underlying network structure. Networks are capable of capturing dependence relationships among a group of entities and hence analyzing these datasets unearth the underlying structural dependence among the individuals. Examples include gene regulatory networks, understanding stock markets, protein-protein interaction within the cell, online social networks etc. We present two important aspects of large high-dimensional data with network structure. The first one focuses on a data with a network structure that evolves over time. Examples of such data sets include time course gene expression data, voting records of legislative bodies etc. Traditionally, the estimation of Gaussian graphical models (GGM) is performed in an i.i.d setting. More recently, such models have been extended to allow for changes in the distribution, but primarily where changepoints are known a priori. In this work, we study the Group-Fused Graphical Lasso (GFGL) which penalizes partial correlations with an L1 penalty while simultaneously inducing block-wise smoothness over time to detect multiple changepoints. The other aspect that we examine is heterogeneity in a network structure and how we can use such heterogenous features in a predictive model. We use a linear latent variable model viz. PCA and its extensions to learn an underlying network structure from data varying over time. We then employ the learned network as a feature in a predictive model to perform the downstream task in the test data. Neuroimaging-driven prediction of brain age, defined as the predicted biological age of a subject using only brain imaging data, is an exciting avenue of research. In this work, we seek to build models of brain age based on functional connectivity while prioritizing model interpretability and understanding. This way, the models serve to both provide accurate estimates of brain age as well as allow us to investigate changes in functional connectivity that occur during the aging process.

Commutative algebra seminar

Thursday, 16 March 2023, 4 pm

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Venue: Ramanujan Hall

Host: Tony Puthenpurakal

Speaker: Prof. R. V. Gurjar

Affiliation: Former Professor, IIT Bombay

Title: Positively Graded domains

Abstract:  I will continue my lectures on this topic. Following results will be discussed.  1. Demazure's construction of normal affine positively graded domains. Some applications of this will be discussed.  2. Flenner and Keiichi Watanabe's rationality of singularities criterion for positively graded affine domains.  3. A very general result I conjectured around 1990 and proved by O.Mathieu In 2002 will be discussed. It has some new consequences for rings of invariants of reductive algebraic group action on an affine space.  4. Divisor Class Groups of positively graded domains. Works of Brieskon Flenner, Samuel, Scheja-Storch, Anurag Singh, etc, will be mentioned. The connection with the Topology of these results will be discussed.

Geometry seminar

Thursday, 16th March 2023, 5 pm

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Venue: Ramanujan Hall

Host: Mayukh Mukherjee

Speaker: Mitul Islam

Affiliation: Heidelberg University

Title: Relatively hyperbolic groups and convex projective structures

Abstract: Studying discrete subgroups of linear groups using a preserved geometric structure has a long tradition. For instance, using real hyperbolic geometry to study discrete subgroups of SO(n,1). Convex projective structures, a generalization of real hyperbolic structures, has recently received much attention in the context of studying discrete subgroups of PGL(n). In this talk, I will discuss convex projective structures and discuss results (joint with A. Zimmer) on relatively hyperbolic groups that preserve convex projective structures. In particular, I will discuss a complete characterization of relative hyperbolicity in terms of the geometry of the projective structure. 

CACAAG seminar

Friday, 17 March, 2023, 10 am

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Venue: Ramanujan Hall

Host: Madhusudan Manjunath

Speaker: Madhusudan Manjunath

Affiliation: IIT Bombay

Title: Unimodality and Log concavity in Algebra, Geometry and Combinatorics: Take II.

Abstract:  We will start with a recap and take a more conceptual approach to this topic (with the goal of touching upon recent developments).  We will not assume any particular background and encourage students and those who missed out last week to attend.

Date and time:  Friday, March 17th, 2023

Venue: 11:00 a.m. - 11:50 a.m.

Host: Krishnan Sivasubramanian

Speaker:  Rakesh Jana

Affiliation: IIT Bombay

Title: Distance Matrices of Trees Inspired by Buneman’s Four-Point Condition

Date and time:  March 17, 2023, 12:30 to 1.30 pm

Venue: Ramanujan hall

Host: Prof. P. Vellaisamy

Speaker:  Mostafizar

Affiliation: IIT Bombay

Title: Generalized counting process: its non-homogeneous and          time-changed versions

Virtual Commutative algebra seminar

Friday, 17 March 2023, 6.30 pm

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Host: J. K. Verma

Venue: meet.google.com/oes-jruv-qup

Speaker: Christine Berkesh

Affiliation: University of Minnesota, USA

Title: Differential operators, retracts, and toric face rings 

Abstract: Toric face rings, introduced by Stanley, are simultaneous generalizations of Stanley–Reisner rings and affine semigroup rings, among others. We use the combinatorics of the fan underlying these rings to inductively compute their rings of differential operators. Along the way, we discover a new differential characterization of the Gorenstein property for affine semigroup rings. Our approach applies to a more general class of rings, which we call algebras realized by retracts. This is joint work with C-Y. Chan, P. Klein, L. Matusevich, J. Page, and J. Vassilev.