Current Session
Organized by: Rajesh Dey & Pankaj Kapdi
Talk 20. [Premieres 04.08.23]
A brief introduction to hyperbolic and relatively hyperbolic groups
by Dr. Ravi Tomar
Abstract :
In his seminal work, Gromov revolutionized the theory of infinite groups by treating groups as geometric objects. In the same essay, he introduced the notion of hyperbolic and relatively hyperbolic groups. Over the last three decades, these groups have received a tremendous amount of attention, and many of their properties have been understood. In the first talk, I will discuss some basic concepts in geometric group theory and introduce the notion of hyperbolic and relatively hyperbolic groups. In the second talk, I will discuss the notion of the height of a subgroup of a group.
About the speaker :
Ravi Tomar is currently a postdoctoral fellow at IISER Bhopal. His research interest lies in Geometric Group Theory. He has completed his PhD from IISER Mohali under the supervision of Dr. Pranab Sardar.
Talk 19. [Premiered 21.07.23]
Analysis on one-sided full shift spaces
by Dr. Sharvari Tikekar
Abstract :
In this talk we consider one-sided full shift spaces over finite symbols. These are totally disconnected compact metric spaces, homeomorphic to Cantor sets. We will discuss a construction of Laplacian on the space of continuous functions on shift space and look at the analogous Dirichlet boundary value problem for the Laplacian. We will also briefly look at the parallel construction of Laplacian on fractals like Sierpinski gasket due to Kigami, its relation with the shift spaces, and will conclude with some of the open problems in the field. This is a joint work with Shrihari Sridharan.
About the speaker :
Sharvari Tikekar is a postdoctoral fellow at TIFR Mumbai, interested in the areas of dynamical systems and analysis on fractals. She has completed her PhD from IISER Thiruvananthapuram under the supervision of Shrihari Shridharan. She is currently working in the intersection of symbolic dynamics and ergodic theory.
Talk 18. [Premieres 02.06.23]
Nielsen Realization Problem
by Satyajit Maity
Abstract :
For g ≥ 2, let Sg be a closed, connected, and oriented smooth surface of genus g, Mg be the mapping class group, and Tg be the Teichmüller space of the surface Sg. In 1932, Nielsen asked the question of whether a finite subgroup of Mg can act on Sg (by diffeomorphisms). After that, it became a popular problem, and many mathematicians, including Nielsen, Fenchel, and Kravetz, tried and came up with proofs for various particular cases. After 50 years of this, in 1983, Kerckhoff finally gave correct proof that works for the general case. In terms of the action of Mg on Tg, in an equivalent way, Kerckhoff actually showed that any finite subgroup of Mg fixes a point of Tg, and hence the finite subgroup can be realized even as a group acting on the surface by isometries with respect to some hyperbolic metric. In this talk, by starting with a bit of the history of this problem, I will present the main proof that works for the general case.
Reference: Kerckhoff, Steven P., "The Nielsen realization problem", Annals of Mathematics, Second Series, 117 (2), 235–265, March 1983.
About the speaker :
Satyajit is a 3rd year Ph.D. student in our department working on ``Mapping class group action on Teichmüller spaces" in the area of geometry and topology under the supervision of Dr. Atreyee Bhattacharya and Dr. Kashyap Rajeevsarathy.
opf-dedf-ixr (2023-04-28 16:03 GMT+5:30)Talk 17. [Premiered 28.04.23]
Curves on Surfaces: Knots and Virtual Knots
by Pravin Kumar
Abstract :
Virtual knot theory was introduced by Louis H. Kauffman in 1996 as an extension of the classical knot theory. In knot theory, a knot is a smooth embedding of the circle in the 3-space and is represented by a knot diagram obtained by projecting it to a plane. On the other hand, in virtual knot theory, an equivalence relation is defined on virtual knot diagrams by using extended Reidemeister moves. The equivalence class of such virtual knot diagrams is called a virtual knot. From a topological point of view, Virtual knot theory is the same as studying knots in thickened surfaces up to ambient isotopy, surface homeomorphisms, and handle stabilization. In this expository talk, we will start with basic notions of knot theory and motivates the virtual knot theory as an extension of classical knots. Towards the end of the talk, we introduce the virtual braid group and discuss Alexander-Markov's correspondence for virtual knot theory.
About the speaker :
Pravin is a graduate student at IISER Mohali working under Dr. Mahendra Singh. His research interest lies in knot theory, Coxeter groups, Artin groups, and mapping class groups of surfaces.
opf-dedf-ixr (2023-04-21 09:59 GMT+5:30)Talk 16. [Premiered 21.04.23]
Hyperbolicity of the Curve Graph
by George Shaji
Abstract :
We will briefly explore the "Guessing Geodesics Lemma", and see its immense usefulness by applying it to the curve graph. Specifically, we shall show that the Curve Graph of a finite type surface (of sufficient complexity) is hyperbolic using "Unicorn Paths" or "Bicorn Paths". This method provides an extremely beautiful, diversely applicable and significantly simpler alternative to the Masur-Minsky machinery used to prove the same fact much earlier.
About the speaker :
George is a graduate student at the University of Utah working under Prof. Mladen Bestvina. His research interest lies in CAT(0) Cube Complexes, Coarse Median Spaces, and Mapping Class Groups/Curve Graphs of Infinite type Surfaces. He completed his bachelor's from St. Stephens College, University of Delhi, and his master's from IISER Mohali.
opf-dedf-ixr (2023-04-14 16:02 GMT+5:30)Talk 15. [Premiered 14.04.23]
Distances in Curve Complex
by Kuwari Mahanta
About the speaker : Kuwari Mahanta is currently doing her Ph.D. at IIT Guwahati, under the supervision of Dr. Sree Krishna Palaparthi.
Kuldeep TalkTalk 14. [Premiered 24.03.23]
Open books and embedding of smooth manifolds
by Kuldeep Saha
Abstract :
Embedding of manifolds is a classical and fundamental problem in Topology. We shall discuss some explicit constructions of smooth embeddings of manifolds using open-book decomposition.
About the speaker : Kuldeep Saha is currently a Post-Doctoral fellow at TCG CREST, Kolkata. He completed his Ph.D. from CMI under the supervision of Dr. Dishant M. Pancholi.
opf-dedf-ixr (2023-03-17 16:01 GMT+5:30)Talk 13. [Premiered 17.03.23]
On stability of (weak) Haagerup property under graph product of groups
by Shubhabrata Das
Abstract :
Haagerup property (or Gromov's a-T-menability) and weak amenability are approximation properties in a group, which are both generalisations of amenability. Weak Haagerup property is another metric approximation property, further generalising both weak amenability and the Haagerup property. The notion of a graph product in a sense interpolates between the direct product and the free product of groups, and induces an action on a CAT(0) cube complex coming from a canonical wall space structure on the group. In this talk we will discuss that graph products of groups with weak Haagerup property also satisfy weak Haagerup property. We will also sketch a different proof that the Haagerup property is stable under graph products.
Homepage : https://presiuniv.ac.in/web/staff.php?staffid=201
Talk 12. [Premiered 17.02.23]
FLOYD METRICS ON FINITELY GENERATED GROUPS
by Suman Paul
About the speaker : Suman Paul is currently a Post-Doctoral fellow at IISERB. He completed his Ph.D. from IIT Kanpur under the supervision of Dr. Abhijit Pal.
opf-dedf-ixr (2023-02-10 16:04 GMT+5:30)Talk 11. [Premiered 10.02.23]
Surfaces of infinite-type are non-Hopfian
by Sumanta Das
About the speaker : Sumanta Das is a fourth-year Ph.D. student at IISc, Bangalore, interested in geometric and algebraic topology, specifically, the study of maps between non-compact surfaces. His research supervisor is Prof. Siddhartha Gadgil.
Abstract
In this seminar, Sumanta will talk about a joint work with Prof. Gadgil where they have shown that finite-type surfaces are characterized by a topological analogue of the Hopf property. Namely, an orientable surface Σ is of finite-type if and only if every proper map f : Σ → Σ of degreeone is homotopic to a homeomorphism.
Talk 10. [Premiered 20.01.23]
On isometries of Riemannian Manifolds
by Sannidhi A.
About the speaker : Mr. Sannidhi is a 5th-year Ph.D. student of our institute. He works in the field of Symplectic Geometry under the guidance of Dr. Dheeraj Kulkarni.
Abstract
A Riemannian isometry between Riemannian manifolds is a diffeomorphism of the underlying manifolds which preserves the Riemannian metric. In 1939, Myers and Steenrod proved two results about the isometries of Riemannian manifolds. In this talk, we will look at one of them, which states that any distance-preserving surjective map between Riemannian manifolds is smooth, and hence an isometry.
Talk 09. [Premiered 13.01.23]
Enumeration of words and its applications
by Dr. Haritha C.
About the speaker : Dr. Haritha is a former BS-MS and Ph.D. student of our institute. Currently, she is doing her post-doctorate at TIFR-Mumbai.
Abstract
We start with an enumeration problem studied by Guibas and Odlyzko in 1979 and its applications in seemingly unrelated scenarios including game theory, pattern matching algorithm, and graph theory. One of the main objects of our study is a subshift of finite type, which is a tool to model a large class of dynamical systems. We discuss its correspondence with an edge labeled multigraph and hence with its associated adjacency matrix. We see how some properties of a subshift of finite type are studied using this correspondence and a generalized version of the enumeration problem.
Talk 08. [Premiered 09.12.22]
On Homogenous Spaces
by Ms. Sayoojya Prakash
About the speaker : Sayoojya Prakash is a second-year Ph.D. student. She is working in the field of Riemannian manifolds under the supervision of Dr. Atreyee Bhattacharya.
Abstract
Homogeneous spaces are manifolds on which Lie group acts transitively. A special class of homogeneous spaces is symmetric spaces. They were introduced by Cartan in 1925 in his attempt to classify Riemannian manifolds whose curvature satisfies the property ∇R = 0. In this talk, we will see the definition of homogeneous spaces, symmetric spaces, and some of their properties.
opf-dedf-ixr (2022-11-25 16_02 GMT+5_30).mp4Talk 07. [Premiered 22.11.22]
On Three Theorems of Surface Homeomorphisms
by Mr. Ajay Nair
About the speaker : Ajay Kumar Nair is currently a 4th Year Ph.D. student at the Indian Institute of Science, Bengaluru and his advisor is Prof. Subhojoy Gupta. His research interests include Geometric structures, Surface group representations, and Low-dimensional geometry-topology. He did his MS thesis under the guidance of Dr. Kashyap Rajeevsarathy on the topic "KNOTS, 3-MANIFOLDS, AND THE LICKORISH WALLACE THEOREM".
Abstract
Given an orientable, compact, and connected surface S_g with genus g > 0, we define the mapping class group as isotopy classes of self-homeomorphisms of S_g. Dehn-Lickorish Theorem, Nielsen-Thurston Classification theorem, and Dehn-Nielsen-Baer theorem are three of the most important results in the surface theory. In this talk, we would try to motivate these three results by looking at the case when g = 1, i.e., the torus. In the case of torus, these results are easier to see. If time permits, I would try to give a basic sketch of the proofs of at least one of the theorems.
Talk 06. [Premiered 18.11.22]
Expansive actions of automorphisms on locally compact groups and related compact spaces
by Prof. Riddhi Shah ( JNU Delhi )
About the speaker :
Riddhi Shah is a Professor of Mathematics at JNU Delhi, formerly at TIFR Mumbai. She has been the Chairperson, now a member, of the Indian Women and Mathematics (funded by NBHM). She is recently inducted as a fellow of the National Academy of Sciences, India (NASI). Homepage : https://jnu.ac.in/index.php/content/rshahTalk 05. [Premiered 11.11.22]
On the Yamabe problem
by Dr. Atreyee Bhattacharya
About the speaker :
Dr. Atreyee Bhattacharya is an Assistant Professor in the Department of Mathematics, IISER Bhopal.Homepage : https://sites.google.com/iiserb.ac.in/homepage-atreyee-bhattacharya/home
Abstract
The Yamabe Problem is often viewed as an n-dimensional generalization of the uniformization theorem for closed Riemannian surfaces and one of the classic problems in Geometric analysis. The geometric version of the uniformization theorem says that every closed surface admits a Riemannian metric with constant Gaussian curvature. As a natural follow-up, one would like to ask if there is an n-dimensional analog of the uniformization theorem. On a Riemannian manifold of dimension at least three, there are several different notions of curvature (all of which coincide in dimension two), most of which, when constant, impose some strong topological restrictions on the manifold and thus fail to provide an n-dimensional uniformization theorem. Surprisingly, the weakest notion of curvature in dimension three or more, known as the scalar curvature, seems to work in this case. In fact, the Yamabe problem resolved in the 1980s, in particular, says that every closed Riemannian manifold admits a Riemannian metric with constant scalar curvature. This was proved by the collective work of a number of renowned names and finally settled by Richard Schoen in 1984 as an application of his proof of the celebrated positive energy theorem in general relativity. In this talk, we will try to provide an overview of the Yamabe problem touching upon its connection with the geometric analysis and general relativity.
Talk 4Talk 04. [Premiered 21.10.22]
The Fenchel-Nielsen and the Shear Coordinates on the Teichmuller space
by Pabitra Barman
About the speaker :
Mr. Pabitra Barman is a final year Integrated Ph.D. student of IISc, working on the Teichmuller space, under the supervision of Dr. Subhojoy Gupta.
Abstract
Every closed orientable surface of genus greater than 1 admits a pants decomposition. Such decomposition allows us to construct a very natural coordinate system on its Teichmuller space known as the Fenchel-Nielsen Coordinates. In this talk, we will introduce the Fenchel-Nielsen Coordinates on the Teichmuller space of a closed orientable surface of genus greater than 1, starting with a brief overview of the space itself. Following it, we will introduce the Shear coordinates on the Teichmuller space of a finite volume complete punctured hyperbolic surface using a fixed ideal triangulation.
Talk 03. [Premiered 14.10.22]
Infinite metacyclic subgroups of the mapping class group (Pt. 2)
by Pankaj Kapdi
About the speaker :
Mr. Pankaj Kapdi is a 5th-year Integrated Ph.D. student of IISER Bhopal, working on Mapping class group of surfaces, under the supervision of Dr. Kashyap Rajeevsarathy.
Abstract
For g ≥ 2, let Mod(Sg) be the mapping class group of the closed orientable surface Sg of genus g. In this talk, we will discuss a complete characterization of the infinite metacyclic subgroups of Mod(Sg) up to conjugacy. In particular, we discuss equivalent conditions under which a pseudo-Anosov mapping class generates a metacyclic subgroup of Mod(Sg) with another mapping class. As an application to our main results, we show the existence of certain infinite metacyclic subgroups of Mod(Sg) explicitly. If time permits, we derive bounds on the order of a periodic generator of an infinite metacyclic subgroup of Mod(Sg).
2nd Talk.mp4Talk 02. [Premiered 07.10.22]
Infinite metacyclic subgroups of the mapping class group (Pt. 1)
by Pankaj Kapdi
About the speaker :
Mr. Pankaj Kapdi is a 5th-year Integrated Ph.D. student of IISER Bhopal, working on Mapping class group of surfaces, under the supervision of Dr. Kashyap Rajeevsarathy.
Abstract
For g ≥ 2, let Mod(Sg) be the mapping class group of the closed orientable surface Sg of genus g. In this talk, we will discuss a complete characterization of the infinite metacyclic subgroups of Mod(Sg) up to conjugacy. In particular, we discuss equivalent conditions under which a pseudo-Anosov mapping class generates a metacyclic subgroup of Mod(Sg) with another mapping class. As an application to our main results, we show the existence of certain infinite metacyclic subgroups of Mod(Sg) explicitly. If time permits, we derive bounds on the order of a periodic generator of an infinite metacyclic subgroup of Mod(Sg).
cnz-jvmt-xbs (2022-09-30 16:11 GMT+5:30)Talk 01. [Premiered 30.09.22]
Kirby Moves
by Prerak Deep
About the speaker :
Prerak Deep is a fourth-year Ph.D. student of IISER Bhopal. He is working in the field of contact topology, particularly on a problem that involves 'Kirby's theorem' under the supervision of Dr. Dheeraj Kulkarni.
Abstract
In 1960s, Lickorish and Wallace (independently) proved that any closed, orientable smooth 3-manifolds can be obtained by performing an integral surgery on a link in S3. Since the conserve is trivial, we get a correspondence between framed links in S3 and closed, oriented smooth 3-manifold. But this correspondence is not one-to-one, we may have two framed links corresponding to the same 3-manifold.
In 1970, Kirby proved that given such a pair of framed links there is a finite sequence of moves to obtain one link from another. These moves are called Kirby Moves, and there are two types of such moves. In this, I will talk about the Kirby moves. We will see some examples of these moves in action. If time permits, we may talk about the contact geometric analogs of the Lickorish-Wallace theorem, and Kirby moves.