Geometry and Topology seminar
Speaker: Dr. Gorapada Bera, Simons Center for Geometry and Physics, Stony Brook University
Host: Sandip Singh
Title: Associative submanifolds of G₂-manifolds and adiabatic limits
Time, day and date: 11:30:00 AM - 12:30:00 PM, Monday, March 16
Venue: Online (Google Meet: https://meet.google.com/jzy-kqch-nzy)
Abstract: G₂-manifolds are 7-dimensional Riemannian manifolds with special holonomy, analogous to Calabi–Yau 3-folds, and associative submanifolds are a special class of volume-minimizing 3-dimensional submanifolds, analogous to holomorphic curves or special Lagrangians. Inspired by the counting of holomorphic curves or special Lagrangians in Calabi–Yau 3-folds, Joyce, Doan, and Walpuski have proposals about defining enumerative invariants of G₂-manifolds by counting closed associative submanifolds. In this talk, I will focus on three topics connected to this enumerative theory. First, I will discuss a key ingredient of these proposals: the appearance of conically singular associative submanifolds and their desingularizations. Afterwards, I will describe a method for constructing associative submanifolds in a class of G₂-manifolds known as twisted connected sums. Finally, I will discuss Donaldson’s program on the adiabatic limit of K3-fibered G₂-manifolds, along with conjectures regarding associative submanifolds and potential directions for future research.

Geometry and Topology seminar
Speaker: Dr. Ishan Banerjee, Ohio State University
Host: Sandip Singh
Title: Monodromy of curves in an algebraic surface
Time, day and date: 9:30:00 AM - 10:30:00 AM, Tuesday, March 17
Venue: Online (Google Meet: https://meet.google.com/jzy-kqch-nzy)
Abstract: Given an algebraic surface X embedded into P^N, we can consider the family of smooth curves in X arising from intersecting a degree d hypersurface in P^N with X. This is a fiber bundle with fibers Riemann surfaces of genus g for some g.

Associated to this family of curves we have a monodromy group contained in the mapping class group of a genus g surface. We prove under some ampleness conditions that if X is simply connected this monodromy group is of finite index (we explicitly determine what it is). This is joint work with Nick Salter.

Talk
Speaker: RUDDARRAJU AMRUTHA
Host: Manoj Kumar Keshari
Title: Elementary Symplectic Groups and their Generalizations
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, March 18
Venue: Ramanujan Hall
Abstract: A.A. Suslin proved that the elementary linear group of size atleast 3 is a normal subgroup of the group of invertible matrices. Suslin also proved a relative version of this result w.r.t. an ideal. V.I. Kopeiko proved a symplectic analogue of this result: the elementary symplectic group is a normal subgroup of the symplectic group, where both are defined with respect to the standard skew-symmetric matrix. I will talk about a generalization of Kopeiko's result with respect to any invertible skew-symmetric matrix of Pfaffian 1.

A stronger result holds when the ring is Euclidean or semi-local. Over a Euclidean ring, the elementary symplectic group with respect to an invertible skew-symmetric matrix is equal to the symplectic group with respect to the skew-symmetric matrix. This result also holds for a semi-local ring. I will talk about these equality results.

Finally, I will talk about a generalization of the elementary symplectic group with respect to an invertible alternating matrix, in the case of a projective module, which is the Vaserstein group of a symplectic module.
This Vaserstein group is a normal subgroup of the group of isometries of the symplectic module.

Statistics and Probability seminar
Speaker: Dr. Abhinek Shukla, Centre for Biomedical Data Science, Duke-NUS Medical School
Host: Koushik Saha
Title: Improving Inference in Stochastic Gradient Descent Based on Equal Batch-Size Batch-Means Estimator
Time, day and date: 12:00:00 PM - 1:00:00 PM, Wednesday, March 18
Venue: Online (https://meet.google.com/wjs-deag-ysh)
Abstract: Stochastic gradient descent (SGD) is a widely used technique for solving various optimization problems ranging from regression on a large dataset employed in statistics, to training deep neural networks for high dimensional models in machine learning. Performing tasks such as inference in SGD is a challenging problem due to its time-inhomogeneous Markovian nature. An underlying asymptotic normality of the averaged SGD (ASGD) estimator allows for the construction of a batch-means estimator of the asymptotic covariance matrix. In contrast to the existing increasing batch-size (IBS) strategy proposed for reducing correlation between far-apart batches, we propose a memory efficient equal batch-size (EBS) estimator and show consistency of the proposed batch-means estimator under mild conditions. The proposed EBS technique offers bias-correction of the variance at no additional cost to memory and is also shown to outperform the IBS estimator in extensive simulations. Further, since joint inference for high dimensional problems may be difficult, we present marginal-friendly simultaneous confidence intervals, and demonstrate improved predictions based on the proposed covariance estimators of ASGD.

Monge-Ampere equation
Speaker: Sooraj A P, IIT Bombay
Host: Harsha Hutridurga
Title: On the Monge-Ampere equation
Time, day and date: 2:00:00 PM – 3:00:00 PM, Wednesday, March 18
Venue: Room 215
Abstract: We shall show the existence and Hölder regularity of classical solutions to the Monge-Ampère equation using tools from elliptic regularity theory.

Topology Seminar
Speaker: Sumanta Das, IIT Bombay
Host: Rekha Santhanam
Title: Mapping Class Groups: Classical Foundations
Time, day and date: 3:00:00 PM – 4:00:00 PM, Wednesday, March 18
Venue: Room 215
Abstract: This talk provides an introduction to the mapping class group, an object that algebraically captures the topological symmetries of a surface by considering its orientation-preserving homeomorphisms up to isotopy. The study of these groups naturally bridges fields such as low-dimensional topology, algebraic geometry, algebraic topology, and dynamical systems. In this series of talks, we will discuss some of the connections between mapping class groups and these areas. The first talk will cover the formal definition, various fundamental properties, and the generation of mapping class groups.

Mathematics Colloquium
Speaker: Sudarshan R. Gurjar, IIT Bombay
Title: Belyi-type theorems for vector bundles
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, March 18
Venue: Ramanujan Hall
Abstract: A classical theorem of Belyi states that a non-singular, irreducible, complex projective curve is defined over number field if and only if it admits a non-constant morphism to CP^1 which is branched over at most 3 points. I will discuss analogues theorems for vector bundles and vector bundles with connections and Higgs fields. 

This is based on some joint work with Indranil Biswas.

CACAAG seminar
Speaker: Madhusudan Manjunath, IIT Bombay    
Title: A Gentle Introduction to Non-Archimedean Analytic Geometry à la Berkovich
Time, day and date: 05:30:00 PM – 6:30:00 PM, Wednesday, March 18
Venue: Ramanujan Hall
Abstract: A gentle introduction to Berkovich spaces with an emphasis on intuition, motivations and examples. 

Talk
Speaker: Terrence George, MIT
Host: Niranjan Balachandran
Title: Limit shapes beyond classical tilings
Time, day and date: 9:30:00 AM – 10:30:00 AM, Friday, March 20
Venue: Online (meet.google.com/dyx-hzmn-cbj)
Abstract: The dimer model refers to the study of random dimer covers, or perfect matchings, of a bipartite graph. A remarkable feature of these models is the emergence of limit shapes: in large periodic graphs, a random matching typically concentrates around a deterministic shape. Although limit shapes arise quite generally in dimer models, they are explicitly understood only in two classical cases, namely the hexagonal and square lattices, corresponding to lozenge and domino tilings. In this talk, I will explain how ideas from integrable systems make it possible to compute limit shapes for arbitrary periodic dimer models. This is based on joint work in progress with Tomas Berggren and Alexei Borodin.

Algebraic groups seminar
Speaker: Sabyasachi Dhar, IIT Bombay, Mumbai
Host: Shripad Garge
Title: Parabolic subgroups and Levi subgroups
Time, day and date: 4:00:00 PM - 5:30:00 PM, Friday, March 20
Venue: Room No .215
Abstract: We continue with the remaining part of Chapter 3 of Digne-Michel.

Algebraic groups seminar
Speaker: Patrick Polo, IIT Bombay, Mumbai
Host: Shripad Garge
Title: Preparations for the Harish-Chandra induction
Time, day and date: 5:30:00 PM - 6:30:00 PM, Friday, March 20
Venue: Room No .215
Abstract: We cover preliminaries for the Harish-Chandra induction.