Lecture 1: November 26, 2020 (Thursday) at 08:00 PM IST

Speaker: Priyadip Mondal (University Of Pittsburgh)

Title: Hidden symmetries and hyperbolic knot complements: Part 1

Lecture 2: December 06, 2020 (Sunday) at 08:00 PM IST

Speaker: Priyadip Mondal (University of Pittsburgh)

Title: Hidden symmetries and hyperbolic knot complements: Part 2 (slides)

Lecture 3: December 07, 2020 (Monday) at 08:00 PM IST

Speaker: Dr. Maggie Miller (Massachusetts Institute of Technology)

Title: 4D light bulb theorem: Part 1 (recording)

Lecture 4: December 11, 2020 (Friday) at 08:00 PM IST

Speaker: Dr. Maggie Miller (Massachusetts Institute of Technology)

Title: 4D light bulb theorem: Part 2 (recording)

Lecture 5: December 16, 2020 (Wednesday) at 08:00 PM IST

Speaker: Anubhav Mukherjee (Georgia Institute of Technology)

Title: Exotic behavior of 4-manifolds: Part 1 (recording)

Lecture 6: December 20, 2020 (Sunday) at 08:00 PM IST

Speaker: Anubhav Mukherjee (Georgia Institute of Technology)

Title: Exotic behavior of 4-manifolds: Part 2

Lecture 7: January 12, 2021 (Tuesday) at 08:00 PM IST

Speaker: Dr. Himalaya Senapati (Chennai Mathematical Institute)

Title: Geometry and the three body problem: Part 1 (recording)

Abstract: The classical three body problem is one of the oldest problems in dynamics and it still continues to throw up surprises. It has received the attention of some of the best physicists and mathematicians, its study contributing to the development of perturbation theory, canonical transformations, singularities and more. It is here where chaos was first discovered. We will briefly review the history of the three body problem leading to the works of Poincaré and the introduction of geometric and topological methods. We will then discuss some recent developments via an approach based on treating Hamiltonian flows as geodesic flows on the configuration space equipped with the Jacobi-Maupertuis metric.

Lecture 8: January 19, 2021 (Tuesday) at 08:00 PM IST

Speaker: Dr. Himalaya Senapati (Chennai Mathematical Institute)

Title: Geometry and the three body problem: Part 2 (recording)

Lecture 9: January 22, 2021 (Friday) at 08:00 PM IST

Speaker: Agniva Roy (Georgia Institute of Technology)

Title: Contact geometry, Legendrians, and open books (recording)

Abstract: We will cover some basic notions of what contact and symplectic manifolds are and how these concepts interplay. Some of the points we will focus on are Legendrian submanifolds and open book decompositions. We will look at several examples and questions in this direction of research.

Lecture 10: January 29, 2021 (Friday) at 08:00 PM IST

Speaker: Agniva Roy (Georgia Institute of Technology)

Title: Legendrian Contact Homology and an application (recording)

Abstract: This talk will discuss the Floer theoretic invariant Legendrian Contact Homology. I will try to skim over the technical jargon and give an idea of how it is computed, and how it has proved useful. Time permitting, I will talk about a project I am working on that involves an application of the same.

Lecture 11: February 12, 2021 (Friday) at 08:00 PM IST

Speaker: Anindya Chanda (Florida State University)

Title: Partially hyperbolic diffeomorphisms and their classification on 3-manifolds: Part 1 (recording)

Abstract: Partial hyperbolicity is a natural generalization of the notion of hyperbolicity of maps on Riemannian manifolds. The study of partially hyperbolic maps is a relatively new area of research in the intersection of manifold theory and dynamical systems. In our discussions we will build the basic foundation in this area and will talk about recent developments towards the classification of partially hyperbolic maps on three manifolds and related open questions.

Lecture 12: February 16, 2021 (Tuesday) at 08:00 PM IST

Speaker: Anindya Chanda (Florida State University)

Title: Partially hyperbolic diffeomorphisms and their classification on 3-manifolds: Part 2 (recording)

Lecture 13: March 09, 2021 (Tuesday) at 09:30 AM IST

Speaker: Mohan Swaminathan (Princeton University)

Title: Gromov compactness for pseudoholomorphic curves -- an effective proof: Part 1 (slides) (recording)

Abstract: Gromov compactness is a fundamental theorem with a lot of applications in symplectic topology. It states (in a special case) that the space of smooth projective curves (more precisely, holomorphic maps from such curves) in a compact Kahler manifold, of a given genus and given homology class, can be compactified by adding certain special types of curves with very mild singularities (simple nodes). I will discuss an effective version of this theorem and its proof, blackboxing the relevant analytical background, and focusing on the more combinatorial aspects of the proof. Time permitting, a potential application (in progress) of these ideas will also be discussed.

Lecture 14: March 12, 2021 (Friday) at 09:30 AM IST

Speaker: Mohan Swaminathan (Princeton University)

Title: Gromov compactness for pseudoholomorphic curves -- an effective proof: Part 2

Lecture 15: March 27, 2021 (Saturday) at 08:00 PM IST

Speaker: Dr. Suman Paul (IIT Kanpur)

Title: Combination problem for conformally hyperbolic groups: part 1 (recording) (slides)

Abstract: available here

Lecture 16: April 03, 2021 (Saturday) at 08:00 PM IST

Speaker: Dr. Suman Paul (IIT Kanpur)

Title: Combination problem for conformally hyperbolic groups: part 2 (recording) (slides)

Lecture 17: April 09, 2021 (Friday) at 08:00 PM IST

Speaker: Rahul Pandey (IIT Kanpur)

Title: Contracting boundary of a cusped space: part 1 (recording)

Abstract: available here

Lecture 18: April 12, 2021 (Monday) at 08:00 PM IST

Speaker: Rahul Pandey (IIT Kanpur)

Title: Contracting boundary of a cusped space: part 2 (recording)

Lecture 19: April 23, 2021 (Friday) at 10:00 AM IST

Speaker: Achinta Nandi (Oklahoma State University)

Title: On homotopy equivalence for proper holomorphic maps

Abstract: available here

Lecture 20: May 11, 2021 (Tuesday) at 08:00 PM IST

Speaker: Dr. Haritha Cheriyath (IISER Bhopal)

Title: Subshifts of finite type with a hole: properties and applications : part 1 (slides) (recording)

Abstract: A dynamical system consists of a state space and a map defined on it. It is broadly classified into a closed and an open system. In a (traditional) closed system, the orbit of a point lies in the state space for all time, whereas in an open system, the orbit of a point may eventually escape from the state space through a hole. In this talk, we consider an important class of dynamical systems known as the subshifts of finite type. We study the average rate at which the orbits escape into the hole. This problem turns out to be an interesting application of a combinatorial problem which is counting the number of words of fixed length not containing any of the words from a fixed collection as subwords. In the second part of the talk, we present several applications of our results including computing the Perron eigenvalues and eigenvectors of any binary (0-1) matrix and obtaining a combinatorial expression for a well-known invariant measure on the subshift of finite type.

Lecture 21: May 14, 2021 (Friday) at 08:00 PM IST

Speaker: Dr. Haritha Cheriyath (IISER Bhopal)

Title: Subshifts of finite type with a hole: properties and applications : part 2 (slides) (recording)

Lecture 22: May 17, 2021 (Monday) at 07:00 PM IST

Speaker: Deniz Kutluay (Indiana University)

Title: Knotoids and Khovanov homology of knotoids (recording)

Abstract: The theory of knotoids was introduced by Turaev in 2010. Knotoids can be considered as open-ended knot-type diagrams. In the first part of the talk, we will define knotoids, show how they generalize knots and explain how Turaev polynomial of knotoids is defined. In the second part, we will give an outline of the construction of Khovanov homology of knots, and how it is generalized to knotoids.

Lecture 23: May 20, 2021 (Thursday) at 06:00 PM IST

Speaker: Deniz Kutluay (Indiana University)

Title: Winding homology (recording)

Abstract: Turaev defined a two-variable polynomial invariant of knotoids which encompasses a generalization of the Jones knot polynomial to knotoids. In this talk, we will define a triply-graded homological invariant of knotoids categorifying the Turaev polynomial, called winding homology. Forgetting one of the three gradings gives a generalization of the Khovanov knot homology to knotoids. We will review basic properties of the winding homology, and give examples of pairs of knotoids that are distinguished by the winding homology but not by the Khovanov knotoid homology or the Turaev polynomial.

Lecture 24: May 21, 2021 (Friday) at 08:00 PM IST

Speaker: Braeden Reinoso (Boston College)

Title: Capping off open books and fractional Dehn twist coefficients : part 1 (recording)

Abstract: The fractional Dehn twist coefficient of an open book decomposition is a number which roughly measures the twisting of the fiber surface along a boundary component. Given an operation that takes one open book to another (e.g. stabilization, adding Dehn twists, murasugi sums, capping off a boundary component with a disk, etc.) it is a natural question from the contact-geometric perspective to ask how the fractional Dehn twist coefficients change. Under some of these operations we understand completely the behavior of fractional Dehn twist coefficients, but for capping off the question is difficult. I will describe some recent results using Heegaard Floer homology to bound the change in fractional Dehn twist coefficients under capping off, and mention some applications to studying the Heegaard Floer homology of branched covers. In the first talk, I will introduce open book decompositions, fractional Dehn twist coefficients, and some of the basics of Heegaard Floer homology. In the second talk, I will introduce Heegaard Floer homology with local coefficients and discuss the proofs of my results.

Lecture 25: May 28, 2021 (Friday) at 08:00 PM IST

Speaker: Braeden Reinoso (Boston College)

Title: Capping off open books and fractional Dehn twist coefficients : part 2 (recording)

Lecture 26: June 18, 2021 (Friday) at 08:00 PM IST

Speaker: Dr. Gianluca Faraco (Max Planck Institute for Mathematics, Bonn)

Title: Flat geometries on surfaces: part 1 (recording)

Abstract: In these two lectures I will talk about geometric structures on finite type surfaces with particular regard to flat structures. It is classical to see that any kind of geometric structure determines a holonomy representation, however the reverse problem of realising a given representation as the holonomy of some geometric structure is more arduous, uphill and not always possible. After a general introduction, I discuss about the relationship between a geometric structure and its holonomy representation by including motivations, known results and open problems.

Lecture 27: June 25, 2021 (Friday) at 08:00 PM IST

Speaker: Dr. Gianluca Faraco (Max Planck Institute for Mathematics, Bonn)

Title: Flat geometries on surfaces: part 2 (recording)

Lecture 28: July 09, 2021 (Friday) at 08:00 PM IST

Speaker: Dr. Yvon Verberne (Georgia Institute of Technology)

Title: Finite quotients of the braid group and its generalizations : part 1 (recording)

Abstract: In this talk, we will introduce the notion of totally symmetric sets and give examples of these sets in different groups. In detail, we will discuss how these sets are used to give bounds for finite quotients of the braid group.

Lecture 29: July 16, 2021 (Friday) at 08:00 PM IST

Speaker: Dr. Yvon Verberne (Georgia Institute of Technology)

Title: Finite quotients of the braid group and its generalizations : part 2 (recording)

Abstract: In this talk, we will continue our focus on using totally symmetric sets to bound finite quotients, this time for the welded and virtual braid groups.

Lecture 30: July 23, 2021 (Friday) at 08:00 PM IST

Speaker: Dr. Hitesh Gakhar (University of Oklahoma)

Title: An Introduction to Persistent Homology (recording)

Abstract: Persistent homology is an algebraic and computational tool from topological data analysis. This talk is meant to be a self-sufficient introduction to persistent homology. We will discuss the general algebraic framework, specific notions of persistence, and a few real-world applications.

Lecture 31: July 30, 2021 (Friday) at 08:00 PM IST

Speaker: Dr. Hitesh Gakhar (University of Oklahoma)

Title: Persistent Künneth formulae and Applications (recording)

Abstract: The classical Künneth formula in algebraic topology provides a relationship between the homology of a product space and that of its factors. In this talk, we will give similar results for persistent homology. That is, we will provide relationships between persistent homology of two different notions of product filtrations and that of their factor filtered spaces. We will also present some applications of these formulae.

Lecture 32: August 06, 2021 (Friday) at 08:00 PM IST

Speaker: Ravi Tomar (IISER Mohali)

Title: Acylindrical complexes of hyperbolic groups and Cannon-Thurston map: part 1 (recording)

Abstract: Complexes of groups are the generalization of graphs of groups. In the first talk, I will discuss basics of complexes of groups. We will try to set up a background for the second talk. If time permits, I will state our main results.

Lecture 33: August 13, 2021 (Friday) at 08:00 PM IST

Speaker: Ravi Tomar (IISER Mohali)

Title: Acylindrical complexes of hyperbolic groups and Cannon-Thurston map: part 2 (recording)

Abstract: In the second talk, We will discuss technical details of the proof of our main results. Let G(Y) be an acylindrical developable complex of hyperbolic groups with qi embedding conditions over a finite simplicial complex Y and let G(Y_1) be a subcomplex of groups obtaining by restricting G(Y) to G(Y_1). Under some extra hypotheses, we prove that there exists Cannon-Thurston map from the fundamental group of G(Y_1) to fundamental group of G(Y). In the second part of the talk, We will discuss a combination theorem for developable complex of hyperbolic groups with finite edge groups over a finite simplicial complex.

Lecture 34: August 20, 2021 (Friday) at 08:00 PM IST

Speaker: Avijit Nath (IISER Tirupati)

Title: On generalized Dold manifolds

Abstract: Dold introduced the notion of Dold manifolds to describe the odd dimensional generators of the unoriented cobordism ring. Classical Dold manifolds were defined as the orbit space of $\mathbb{Z}_2$-action on the product of a sphere and a complex projective space where $\mathbb{Z}_2$ acts on the sphere by antipodal involution and the complex projective space by complex conjugation. We obtain a generalization of the Dold manifolds where we replace the complex projective space by an almost complex manifold $X$ admitting a complex conjugation. We call them generalized Dold manifolds. In this talk, I will give the description of its tangent bundle under the assumption that $H^1(X,\mathbb{Z}_2)=0$ and a criteria for orientability and spin structures as an applications of simple computation of first and second Stiefel-Whitney classes. Using the description of its tangent bundle, I would like to discuss our estimates for span and stable span of generalized Dold manifold and a very general criterion for (non)vanishing of cobordism classes of it. I will also illustrate our results by taking almost complex manifolds as complex Grassmann manifolds, more generally as complex flag manifolds. This talk is based on joint work with Parameswaran Sankaran.