- Time:
- 11:30am
- Location:
- Room No. 105, Department of Mathematics
- Description:
**Annual Progress seminar**Tuesday,14 March 2023, 11.30 am

=========================

**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.

- Time:
- 2:30pm
- Location:
- Venue: Google meet: Link: https://meet.google.com/jtv-udzf-bbx
- Description:
**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.

- Time:
- 4:00pm
- Location:
- Room No. 114, Department of Mathematics
- Description:
**Algebraic Groups Seminar**Tuesday, 14 march 2023, 4 pm

========================

**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.

- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Algebraic Geometry seminar**Wednesday, 15 March 2023, 11.30 am

=============================

**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}.

- Time:
- 2:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Geometry and Topology Seminar**Wednesday, 15 March 2023, 2.30 pm

=============================

**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.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Mathematics Colloquium**15 March 2023, 4 pm

====================

**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.

- Time:
- 2:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
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.

- Time:
- 3:00pm
- Location:
- Venue: Online, link will be sent later
- Description:
**Statistics and Probability seminar****Thursday, 16 March 2023, 3 pm****============================****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.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Commutative algebra seminar**Thursday, 16 March 2023, 4 pm

====================

**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.

- Time:
- 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Geometry seminar**Thursday, 16th March 2023, 5 pm

===============================

**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.

- Time:
- 10:00am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**CACAAG seminar**Friday, 17 March, 2023, 10 am

======================

**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.

- Time:
- 11:00am-11:50am
- Location:
- Room 113, Mathematics Department
- Description:
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

- Time:
- 12:30pm-1:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
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

- Time:
- 6:00pm
- Location:
- Venue: meet.google.com/oes-jruv-qup
- Description:
**Virtual Commutative algebra seminar****Friday,**17 March 2023, 6.30 pm=============================

**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.