Embedded Files
Upcoming Talks
Upcoming Talks
Past Events
Past Events
2025 : FALL SEMESTER
2025 : FALL SEMESTER
Date: 01 Feb 2025
Speaker: Farhan Shabir
Title: The Moduli space of translation surfaces and its connection with billiards
Abstract: Translation surfaces are geometric objects arising naturally in the study of rational
billiard dynamics. From an algebraic perspective, they correspond to
pairs (X, ω) where X is a Riemann surface and ω is a holomorphic 1-form on
X. We will begin this talk by looking at the various equivalent manifestations
of translation surfaces and the recipe for generating a translation surface associated
to a rational billiard table. We will then introduce the moduli spaces of
translation surfaces and a natural GL2(R) action on this space. We will explain
why it is fruitful to study the moduli spaces of translation surfaces, since the
dynamical behaviour on an individual translation surface has deep connections
with the properties of its GL2(R) orbit. Time permitting, we will also look at
some compactifications of these moduli spaces.
The talk will hopefully be accessible without any advanced prerequisites,
and even if certain results need to be black-boxed, we hope that the general
spirit of the interplay between geometry, algebra, and dynamics is conveyed to
the audience.
Date: 11 Jan 2025 (Saturday )
Speaker: Pankaj Kapdi
Title: Generating the liftable mapping class groups of regular ℤn-covers of the closed and oriented surface Sg of genus g>0.
Abstract:Consider a regular cover p:S→ Sg of the closed and oriented surface Sg of genus g>0 with the Deck group ℤn. The subgroup of the mapping class group Mod(Sg) consisting of mapping classes represented by homeomorphisms that lift under the cover p is called the liftable mapping class group, denoted by LModp(Sg), associated with the cover p. It is known that LModp(Sg) is finitely generated. The question is, can one derive a finite generating set of LModp(Sg). Consider the symplectic representation Ψ:Mod(Sg)→ Sp(2g;ℤ) of Mod(Sg). The kernel of Ψ is known as the Torelli group denoted by ℐ(Sg). For g≠ 2, it is known that ℐ(Sg) is finitely generated. Hence, for g≠ 2, one can obtain a finite generating set for LModp(Sg) by combining a finite generating set of ℐ(Sg) and lifting a finite generating set of Ψ(LModp(Sg)). This approach does not work for g=2 as ℐ(Sg) is not finitely generated. In this talk, I will discuss a method to obtain a finite generating set for LModp(S2).
Date: 18 Jan 2025 (Saturday)
Speaker: Pravin Kumar V
Title: What is “virtual Artin group”?
Abstract: Virtual Artin groups were recently introduced by Bellingeri, Paris, and Thiel as broad generalizations of the (virtual) braid groups. For each (Coxeter) graph, the virtual Artin group is generated by the corresponding Artin group and the Coxeter group, subject to certain mixed relations inspired by the Coxeter group action on its root system. In this talk, I will begin with an overview of (virtual) knots and braids, Coxeter groups and Artin groups before introducing the definition of virtual Artin groups. If time permits, I will discuss some of our recent results on virtual Artin groups, which are collaborative work with Dr. Neeraj Kumar Dhanwani, Dr. Tushar Kanta Naik, and Dr. Mahender Singh.
Date: 25 Jan 2025 (Saturday)
Speaker: Tathagata Nayak
Title: On Character Variety of Anosov Representations
Abstract: Let Γ be the fundamental group of an k-punctured, k ≥ 0, closed connected
orientable surface of genus g ≥ 2. In this talk, it will be shown that the character variety of the (Q+, Q-)-
Anosov irreducible representations, resp. the character variety of the (P +, P −)-Anosov Zariski
dense representations of Γ into SL(n, C), n ≥ 2, is a complex manifold of complex dimension
(2g + k − 2)(n^2 − 1). For Γ = π1(Σg ), these character varieties are holomor-
phic symplectic manifolds. This talk is based on a joint work with Prof. Krishnendu Gongopadhyay.
2024 : spring semester
2024 : spring semester
Date: 25 Nov 2024
Speaker: Ayush Khare
Title: Moduli problems, G.I.T. and stacks
Abstract: Moduli problems in algebraic geometry arise in connection with classification problems. Given a type of geometric object, we aim to classify a collection of objects A up to a suitable notion of equivalence ~, and wanted to give A/ ~ some algebro-geometric structure. Meanwhile G.I.T. is a framework in algebraic geometry that provides a systematic way to construct quotients of algebraic varieties or schemes by group action. Algebraic stacks provide a framework for handling the cases where moduli spaces require a more general structure than scheme.
Date: 16 Nov 2024
Speaker: Biswadeep Karmakar
Title: An introduction to slice knots and knot concordance
Abstract: In this talk, we will explore the fascinating topics of 'slice knots' and 'knot concordance' within the realm of low-dimensional topology. A 'slice knot' is a knot that bounds a smooth disk in the 4-dimensional ball, offering an important generalization of classical unknots in the 3-dimensional space. We will begin by introducing the basic concepts of slice knots and the related theory of knot concordance, which seeks to classify knots in a different manner than usual. Along the way, we'll discuss some key examples, the connection between slice knots and topological invariants like knot signatures, and how these can shed light on deeper geometric questions such as the existence of exotic structures on smooth manifolds, etc. My hope is that this lecture will provide an accessible and engaging introduction to these topics, while also highlighting the rich interplay between geometry, algebra, and topology that makes the study of knots in higher dimensions so intriguing.
Date: 09 Nov 2024
Speaker: Dr. Arun Kumar
Title: Endomorphisms of Equivariant algebraic K-theory.
Abstract: We will start this talk by giving a brief construction of the equivariant motivic homotopy category. This category is pretty useful in the study of cohomology theories over schemes with group actions. We will then discuss my research (joint work with Girja Tripathi) on the endomorphisms of the equivariant algebraic K-theory space in this category. Our main result is that in the case of actions of finite groups, the endomorphisms are completely determined by the endomorphisms of the 0th K-group presheaf. This is a generalization of a result of Riou which uses the fact that algebraic K-theory is representable by the infinite Grassmannian in the motivic homotopy category.
Date: 26 Oct 2024
Speaker: Dr. Aditya Tiwari
Title: Sturm Liouville Decomposition
Abstract: A Sturm-Liouville equation is a second-order homogeneous linear differential equation. On a given interval, a Sturm-Liouville equation gives rise to a self-adjoint, unbounded operator, referred to as a Sturm-Liouville operator. We will demonstrate the existence of sequence of the eigenvalues and, consequently, a sequence of eigenfunctions of a Sturm-Liouville operator. This sequence of eigenfunctions of a Sturm-Liouvlle operator forms a complete basis for the space of square-integrable functions on the given interval, leading to Sturm-Liouville series, which generalize Fourier series. On a compact Riemannian manifold without boundary, the Laplace-Beltrami operator, like a Sturm-Liouville operator, is a self-adjoint, unbounded operator, and its eigenvalues form an increasing sequence.
Speaker: Dr. Sohan Ghosh
Date: 04 Oct 2024
Title: Fine Selmer groups
Abstract: Let K be any number field. The ideal class group of K, Cl(K) measures how far the ring of integers of K is from being a principal ideal domain. We fix an odd prime p. For each n ≥ 1, let ζp^n be a primitive p n-th root of unity in algebraic closure of Q. Now, consider the (cyclotomic) tower of field extensions Kn/K such that Kn ⊂ S n≥1 K(ζp^n ) and Gal(Kn/K) ∼= Z/pnZ. Let Cl(Kn)(p) denotes the p-primary part of Cl(Kn). Then it is well known that there exists constants λ, μ, ν ∈ N ∪ {0}, depending on p and independent of n, such that #Cl(Kn)(p) = p^ (λn+μpn+ν) (1) for n >> 0. Iwasawa conjectured that μ = 0 in equation (1) i.e; #Cl(K_n)(p) = p^ (λn+ν) for n >> 0. This conjecture was proven for abelian number fields by Ferrero, Washington in 1979 and later by Sinnot in 1984, but it is still an open question for general number fields. In 2005, Coates and Sujatha connected this conjecture with the arithmetic of elliptic curves. More precisely, they found a deep relation between Iwasawa’s μ = 0 conjecture and the structure of p∞-fine Selmer group of an elliptic curve, which is a subgroup of the classical p∞-Selmer group of an elliptic curve. In this talk, we will discuss the properties of the p∞-fine Selmer group of an elliptic curve over certain compact p-adic Lie extensions of global fields.
Date: 21 Sep 2024
Speaker: Debattam Das
Title: Orderibility in $3$-manifolds groups
Abstract: A group is said to be left orderable if there is a left invariant order of the elements, presents in the group. In this talk, We will discuss the orderability on $ 3 $-manifold groups and also the relation between orderbility and generalised $ n $- torsion elements. In the ending note, we will see some recent results on that.
Date: 14 Sep 2024
Speaker: Dr. Pabitra Barman
Title: Moduli spaces of real convex projective structures on surfaces
Abstract: The study of moduli spaces of various geometric structures on surfaces has been a subject of significant interest for many years. In this talk, we will introduce real convex projective structures on surfaces and highlight some key results related to their moduli space. Specifically, we will focus on the Fock-Goncharov coordinate system to parametrize their moduli space for n = 3.
Date: 24 Aug 2024
Speaker: Pravin Kumar V
Title: Ordered Groups and Mapping Class Groups
Abstract: An important class of abstract groups is one that is totally ordered, with the ordering being invariant under left-multiplications. In the first part of the talk, I will introduce the theory of orderable groups and provide a set of examples that arise from topology to illustrate the scope of this subject. In the second part of the talk, I will introduce the mapping class group and discuss the results we obtained from our recent work with Dr. Apeskha Sanghi and Dr. Mahender Singh.
Date: 3 Aug 2024
Speaker: Dr. Rachna Aggarwal
Title: Isometry groups of complex Hyperbolic Spaces
Abstract: In this talk, we will discuss the isometry groups of complex hyperbolic spaces. Specifically,
We will explore the conjugacy classes in the group of isometries of these hyperbolic spaces. We
will begin with this discussion on hyperbolic spaces in two dimensional plane, then in higher
dimensional complex space and further in infinite dimensional complex Hilbert space. We will
use some basics from complex analysis, linear algebra and functional analysis.
This is a joint work with Dr. Mukund Madhav Mishra and Dr. Krishnendu Gongopadhyay.
Date : 11 July 2024
Speaker : Dr Buddhadev Hajra (TIFR Mumbai)
Title : Some Results About Topology of Complex Algebraic Varieties
Abstract : In this talk, I will discuss some well-known topological properties of algebraic varieties. Topological methods are often very effective for proving results in Algebraic Geometry. While I won't provide all the proofs, I will present some interesting examples to illustrate these concepts. I will also introduce an important lemma due to Madhav Nori, commonly known as 'Nori's Lemma' or 'Nori's Exact Sequence', that has been extensively used by many mathematicians in their research. Nori himself employed this result to solve a conjecture posed by Oscar Zariski, which I will address very briefly.
Date : 29 April 2024
Speaker : Dr. Sudip Pandit
Title : The Weil Conjecture after 75 years
Abstract : In 1949, Andre Weil formulated his famous conjecture intending to count the number of points of an algebraic variety over finite fields–it is a theorem now. This conjecture leads to revolutionary developments in algebraic and arithmetic geometry, including Grothendieck's theory of schemes and their cohomology. In this talk, we aim to explore a brief survey of the conjecture and the beautiful ideas used in the proof inspired by the ideas in topology.
Date : 17 April 2024
Speaker : Dr. Shane D'Mello
Title : The topology of real algebraic varieties
Abstract : A study of the topology of real algebraic varieties involves understanding how the real algebraic structure affects the topology of real algebraic varieties in various contexts. We will begin with so-called Hilbert's sixteenth problem and the attempts made to tackle it, and then discuss its generalizations.
Date : 30 March 2024
Speaker : Ronald Huidrom
Title : A brief overview of categorification, Khovanov (co)homology and ramifications
Abstract : A homology theory is a more powerful topological invariant than the Euler characteristic in that it is essentially a functor from the category of spaces to the category of graded modules besides hiding the Euler characteristic as an alternating sum of the ranks of the chain modules. A homology theory is a categorification of the Euler characteristic.In this talk, we briefly look at a categorification of the Jones polynomial, known as the Khovanov homology. Given an oriented link diagram, we shall construct a homology theory whose graded Euler characteristic is the Jones polynomial. This homology theory is invariant under Reidemeister moves and is strictly more powerful than the Jones polynomial.
Date : 23 March 2024
Speaker : Arti Sahu Gangopadhyay
Title : Introduction to Finsler Geometry
Abstract : Finsler space is a pair of smooth manifolds and a smoothly varying family of Minkowski norms. In this talk I will first define the Minkowski norm on a vector space and then give the definition of Finsler manifolds. Then we will discuss some examples
Date : 16 March 2024
Speaker : Vinita Mukund Mulay
Title : Elephant Random Walks
Abstract : The Elephant Random Walk (ERW) is a discrete random walk with unbounded memory. It is one of the few history-dependent processes with known exact moments. Due to its unbounded memory, the ERW shows anomalous diffusion, a behavior seen in various real-life processes. The walk has gained a lot of attention due to this. In the ERW, at every time step, the elephant chooses a step from the past uniformly at random and repeats the chosen step with some fixed probability. Otherwise, it takes a step in the opposite direction of the chosen step. In this talk, we'll formally define the ERW, discuss its various features, and try to understand the techniques used to study its behavior.
Date : 02 March 2024
Speaker : Dr. Lokenath Kundu
Title : Dehn Function and Growth Function of Groups
Abstract : Our main focus will be on introducing the Dehn function and Growth function of any group using simple examples. We'll mention some interesting results, although we might not go into all the detailed proofs. Additionally, we'll highlight a few unresolved problems and conjectures along the way.
Date : 17 February 2024
Speaker : Dr. Ayush Udeep
Title : Minimal Embedding of p-groups in Symmetric Groups
Abstract : Minimal Embedding of p-groups in Symmetric Groups
Date : 10 February 2024
Speaker : Shreepad Agarwal
Title : Lattices in Euclidean Spaces
Abstract : Lattices in Euclidean Spaces
Date : 03 February 2024
Speaker : Dr. Visakh Narayanan
Title : The Shape of our Universe
Abstract : For so many centuries, we thought our earth was flat. This was basically because that is how it "feels to our senses"! Now with the development of technology, we understand that we were wrong and our earth looks like a round sphere. This is an important lesson, "sometimes, what is obvious to our senses is not the truth"! Now a question that arises in the same line of thought is, "what is the shape of our universe"? It feels like we are in "flat infinite three-dimensional space". But as in the above example, there is no reason why it should be. In the past century, a subject has been developing which could potentially be used to make some progress towards this question. It is called "Topology". Right now, we have an entire "jungle of possible models" for the shape of the universe. The wild animals in this jungle are called "three-manifolds". Most likely we would find the real shape of our universe in this jungle! It is to be noted that in recent years, we are experiencing a merger between Physics and Mathematics through the study of this jungle. The main players in this merger are called "Quantum invariants". The talk is an introduction to this jungle and its close connections with Physics. It is not going to be about the abstract nonsense which goes into this study, but we will concentrate on its philosophical aspects. It is aimed at motivating the daring ones to start their own journey through the jungle. The only prerequisite would be curiosity!
Date : 24 January 2024
Speaker : Divya Setia
Title : Representation theory of Lie algebras and its connection with algebraic combinatorics
Abstract : In this talk, I will introduce the notion of the representation theory of Lie algebras and provide the classification of finite-dimensional irreducible representations of finite-dimensional Lie algebras over C. The primary focus will be on irreducible representations of the Lie algebra sl_n (n × n trace 0 matrices), demonstrating that their character corresponds to the Schur polynomial. This result aids in establishing connections between the representation theory of Lie algebras and algebraic combinatorics.Towards the end of the talk, I will introduce the notion of representations of the current Lie algebra, particularly local Weyl modules and Chari Venkatesh modules. I will also present a few results from my recent paper titled, ”Demazure filtration of tensor product modules and character formula” (joint work with Dr. Tanusree Khandai).
Date : 20 January 2024
Speaker : Pravin Kumar
Title : Infinite-type spaces and their mapping class group
Abstract : A surface is said to be of finite-type if its fundamental group is finitely generated; otherwise, it is said to be of infinite-type. The mapping class group, MCG(S), of a surface S, is the group of isotopy classes of homeomorphisms of S onto itself. A mapping class group is a topological group with the topology inherited from the quotient map of Homeo(S) equipped with the compact-open topology. The MCG(S) is countable and discrete for finite-type surfaces. Interestingly, the topology of MCG(S) exhibits more intriguing properties for an infinite-type surface; it becomes uncountable, totally disconnected, and has the structure of a Polish group. In this talk, we will explore surface classification and the topology of mapping class groups of surfaces. Lastly, some recent results in this area—an outcome of collaborative work with Dr. Mahender Singh and Dr. Apeskha Sanghi. The talk will be accessible to a broader audience and contain more pictures.
2023 : Fall SEmester
2023 : Fall SEmester
Date : 02 December 2023
Speaker : Manujith K Michel
Title : Differential central simple algebras and projective representations
Abstract : Differential central simple algebras and projective representations
Date : 18 November 2023
Speaker : Debattam Das
Title : Reciprocal geodesics in Hecke Groups
Abstract : An element in a group is called reciprocal if there exists h such that g^{-1} = hgh^{-1}. The reciprocal elements are also known as real elements or reversible elements in the literature. The geodesics coming out of the reciprocal elements are called reciprocal geodesics. In this talk, we consider Hecke groups, which are Fuchsian groups of first kind, and will show the classification of reciprocal elements of the Hecke group. Then, we will discuss the reciprocal geodesics of the Hecke group. This is a joint work with my supervisor, Prof. Krishnendu Gongopadhyay.
Date : 4 November 2023
Speaker : Dr. Nishant Rathee
Title : Algebraic Structures Related to Set-Theoretic Solutions of the Yang-Baxter Equation
Abstract : We explore set-theoretic solutions of the Yang-Baxter equation by providing essential background and motivation. We then discuss the algebraic structures associated with these solutions, focusing primarily on skew braces and braces. We discuss their applications and the reasons for our interest, highlighting recent developments in this area. Additionally, we will see some open problems. I will further discuss Rota-Baxter operators and relative Rota-Baxter operators, along with their connection with skew braces. This is a joint work with Professor Mahender Singh.
Date : 20 October 2023
Speaker : Harish Kishnani
Title : A Survey on Word Maps and Future Directions
Abstract : A Survey on Word Maps and Future Directions
Date : 7 October 2023
Speaker : Dr. Santosh Kumar Pamula
Title : Moment Dilations
Abstract : In this talk we present the notion of dilation of bounded linear operators on complex Hilbert space (by viewing the given operator as "a part of" another, well understood operator) and extending it to completely positive maps defined on C*-algebras. In addition, we recall a few well-known dilation results from the literature. In the second part of the talk, we consider the operator analogue of the classical moment problem. This is to identify the sequence of bounded operators on a Hilbert space that admit a dilation (moment dilation). Our discussion will focus on obtaining a necessary and sufficient condition for the existence of moment dilations.
Date : 20 August 2023
Speaker : Dr. Neeraj K. Dhanwani
Title : Quandles : A Review
Abstract : We begin the talk with a definition and examples of quandle. We shall see its connections with other branches of mathematics. Then, we focus on quandles arising from closed surfaces, known as Dehn quandles. By analysing the characterization of Dehn quandles, we generalise their construction. We discuss generating sets of Dehn quandles, their automorphism groups, and their canonical quotients. We conclude the talk by discussing approaches to writing presentations of these quandles.
2023 : Spring Semester
2023 : Spring Semester
Date: 1st April 2023
Speaker : Hari Prasad Poilath
Title : Character Methods In Word Maps
Abstract : Character Methods in Word Maps
Date: 25th March 2023
Speaker : Dr. Chandan maity
Title: Cohomology (de Rham) of adjoint orbits in the non-compact simple Lie algebra of type g_2.
Abstract: The talk aims at emphasizing basic notions of the cohomology of homogeneous space. Towards the end, we will describe all the de Rham cohomology groups of the adjoint orbits in the complex simple Lie algebra g_2(C).
Date: 23rd February 2023
Speaker: Ravi Tomar
Title: An Introduction to Relatively Hyperbolic Groups
Abstract: In his seminal work, Gromov revolutionized the theory of infinite groups by treating groups as geometric objects. In the same article, 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 this talk, I will introduce the notion of relatively hyperbolic groups and discuss some properties of them.
Date: 29th January 2023
Speaker: Pronay Kumar Karmakar
Title: Euler systems and Iwasawa Main Conjecture.
Abstract: Historically the idea of Euler systems for elliptic curves come from the work of Victor Kolyvagin. In this talk we are going to discuss about Euler systems (mainly Bipartite Euler systems) to show that how it can be used to prove some the Iwasawa main conjectures for elliptic curves.
Date: 21st January 2023
Speaker: Apeksha Sanghi
Title: Cyclic actions on surfaces.
Abstract: In this talk, we will see the geometric realization of cyclic actions on surfaces. We will begin with mapping class groups and its relation with group action on surfaces. We will cover the basics of orbifold theory required for this talk, followed by some examples. We will describe combinatorial view point of cyclic actions on surfaces which will help us to understand the geometric realization of such actions on surfaces.
Date: 14th January 2023
Speaker: Rakesh Halder
Title: An introduction to the (Gromov) hyperbolic spaces (groups) and A Combination Theorem for Spaces of Tree-Graded Spaces.
Abstract: The first half of this talk will briefly introduce the (Gromov) hyperbolic spaces, hyperbolic groups, and some stuff related to the second half. The combination problem is a well-known problem in Geometric Group Theory to find new hyperbolic spaces (Groups) out of given ones. In 1992, Bestvina-Feighn proved such a combination theorem for trees of metric spaces, and later in 2012, Mj-Sardar proved one such theorem for metric bundles. In this second half, I will discuss my work, A Combination Theorem for Spaces of Tree-Graded Spaces, which generalizes the above theorems. On the other hand, this theorem is also one step towards proving a combination theorem asked by M. Kapovich.
2022 : Fall Semester
2022 : Fall Semester
Date: 26th November 2022
Speaker: Bhavneet Singh
Title: Elementary proof of Conway's theorem
Abstract: In this talk, we introduce the concept of tangles and hence, Rational tangles. We build operations on Rational tangles and associate a Rational number to the Rational tangle. Conway's theorem states that two rational tangles are ambient isotopic if and only if they have the same fraction. If time permits, we'll also talk about an application of this formulation in studying recombination processes in DNA.
Date: 12th November, 2022
Speaker:Dr. Rijubrata Kundu.
Title: Covering the alternating groups by products of cycle classes.
Abstract: Link to abstract
Date: 5th November, 2022
Speaker: Dipankar Maity
Title: Topology in higher category theory
Abstract: Link to abstract
Date: 24th September, 2022
Speaker: Tejbir Lohan
Title: Reversibility of Isometries.
Abstract: Link to abstract
Slides: Link to slides
Date: 17th September, 2022
Speaker: Anusha Bhattacharya
Title: Suslin's problem
Abstract: Whenever we encounter a statement in mainstream mathematics, a common practice is to either prove it or look for counter-examples in ZFC (assuming its consistency). However, there is no guarantee that either of the two practices would always work. There are statements that can neither be proved nor can be disproved in ZFC. These statements are said to be 'independent of ZFC'.
Our aim is to investigate a statement called the Suslin's problem. Cantor showed that any totally ordered unbounded dense set which is complete and separable is order isomorphic to R. Suslin's problem asks whether separability can be replaced by a property called topologically c.c.c. A totally ordered set is said to be topologically c.c.c. if there is no uncountable collection of disjoint open intervals. It has been shown that the Suslin's problem is independent of ZFC using some objects called Suslin lines. In this talk, we will use the Diamond principle (independent of ZFC) and show that "Diamond principle holds implies Suslin lines exist". We will also see an outline of the non-existence of Suslin lines.
Date: 10th September, 2022
Speaker: Dr. Neeraj K. Dhanwani
Title: Liftable Mapping Class Group
Abstract: Given a covering space of surface S, it is interesting to understand the group of self-homeomorphisms of S that lift under the cover. This talk discusses this problem in the case of regular free cyclic covers of a closed and oriented hyperbolic surface S. The mapping class group Mod(S) of a closed-oriented surface S is the group of isotopy classes of orientation-preserving self homeomorphisms on S. Using the symplectic representation of Mod(S), we give sufficient and necessary conditions for elements in Mod(S) to lift under the regular cyclic cover. We provide a generating set for the group of liftable mapping classes using this criterion.
Date: 3rd September, 2022
Speaker: Debattam Das
Title: Generators of mapping class group
Abstract: In this talk, we will see a generating set for mapping class groups of a closed surface using Dehn twists. We begin the lecture by introducing the definition of mapping class group, Dehn twist, and some properties of Dehn Twists. In the talk, I will show various generating sets for MCG of a closed surface. This includes Dehn-Lickorish generators, Humphries generators, and torsion generators. At the end of the talk, I will mention some open problems in this topic.
2022 : Spring Semester
2022 : Spring Semester
April 9, 2022
Speaker: Harish Kishnani
Title: Word maps on finite nilpotent groups
Abstract: Abstract
April 2, 2022
Speaker: Dr. Gurleen Kaur
Title: Monomial groups and their group algebras
Abstract: Abstract
March 26, 2022
Speaker: Sundara Narasimhan
Title: Primes in arithmetic progressions
Abstract: Abstract
March 12, 2022
Speaker: Nilendu Das
Title: Teichmuller space of closed surfaces
Abstract: Abstract
March 5, 2022
Speaker: Kirti Taneja
Title: Irreducible representations of metacyclic groups
Abstract: Abstract
February 26, 2022
Speaker: Soumya Dey
Title: Playing with presentations
Abstract: We shall discuss some classical tools from combinatorial group theory, and learn to play with some hands-on examples.
February 19, 2022
Speaker: Sukrit Dubey
Title: Projection valued measure
Abstract: Abstract
February 13, 2022
Speaker: Ramanujan Srihari
Title: Norm forms of cyclic extensions
Abstract: Abstract
February 5, 2022 : 11am
Speaker: George Shaji
Title: Word and Conjugacy problems in hyperbolic groups
Abstract: I'll be introducing hyperbolic spaces. Specifically, hyperbolic groups, and I'll show that the word problem and conjugacy problem turn out to be solvable within this class of groups.
January 29, 2022
Speaker: Arushi Agarwal
Title: Geometric Quantization
Abstract: Abstract
January 15, 2022
Speaker: Abhigyan Writwik Medhi
Title: Introduction to Pfister Forms and Function fields of quadratic forms.
Abstract: Abstract
2021 : Fall Semester
2021 : Fall Semester
November 20, 2021
Speaker: Nidhi Gupta
Title: Fundamental group of rational and unirational varieties
Abstract: It is generally hard to distinguish unirational varieties from rational varieties. In this talk, we will see that even fundamental group doesn't differentiate them. First, we will discuss how rational varieties over complex numbers have trivial fundamental group, and then we will establish the same for unirational varieties using theorem of Serre.
November 13, 2021
Speaker: Dr. Chandan Maity
Title: On the cohomology (de Rham) of homogeneous spaces
Abstract: Abstract
November 6, 2021
Speaker: Dr. Sushil Buniya
Title: Conjugacy and reversibility problems in groups.
Abstract: Abstract
October 23, 2021
Speaker: Arpan Dutta
Title: Extensions of valuations
Abstract: Extensions of valuations is a remarkably deep and open problem which is significant from both algebraic and geometric points of view. The theory of ramification is developed to study extensions to algebraic extensions. However, a comprehensive theory to understand extensions to transcendental extensions is yet to be completely developed. In this talk we will give a gentle introduction to some aspects of the classical works, and mention some recent developments.
The talk is aimed at a general audience and does not need any particular prerequisite.
October 16, 2021
Speaker: Divya Setia
Title: Representation Theory of finite dimensional semisimple Lie algebras.
Abstract: Abstract
October 9, 2021
Speaker: Rakesh Halder
Title: Construction of new Hyperbolic Groups out of known ones.
Abstract: We will discuss theoretical approaches to know new examples of Hyperbolic groups from given ones. In order to get new examples, graph of groups (more generally, complexes of groups) and metric bundle have been developed over time. In this talk, we will see the metric bundle approach, and graph of groups if time permits.
October 2, 2021
Speaker: Suneel Mourya
Title: Brief introduction to p-adic Galois representation.
Abstract: Abstract
September 25, 2021
Speaker: Pravin Kumar V
Title: What is an orderable group?
Abstract: An important class of abstract groups is one that consists of totally ordered groups whose order are invariant under left (and right) group multiplications. In this talk, I will introduce the theory of orderable groups and give a set of examples to illustrate the scope of this subject. Then I will mention some algebraic properties of orderable groups and Archimedean property.
September 18, 2021
Speaker: Pronay Kumar Karmakar
Title: Introduction to the Iwasawa theory for elliptic curves.
Abstract: In this talk, we shall define the Selmer groups and p-adic measures. Then we shall give a brief sketch about the p-adic L-functions. At the end we shall discuss about the Main Conjectures of Iwasawa theory.
September 11, 2021
Speaker: Biswadeep Karmakar
Title: An introduction to Higher Homotopy Theory
Abstract: The study of Higher Homotopy Groups is a natural generalization of the notion of Fundamental Groups. Computing them provides intricate pieces of information on the topology of the concerned space. This talk will basically aim to introduce the subject and look at some interesting theorems and problems in this area. The basic knowledge equivalent to a first course in Algebraic Topology shall be assumed for the talk.
2021 : Spring Semester
2021 : Spring Semester
April 7, 2021
Speaker: Dr. Chetan Balwe
Title: Universal R-triviality of standard norm varieties
Abstract: I will briefly review rationality and near-rationality properties of varieties and sketch a proof of the fact that standard norm varieties over an algebraically closed field of characteristic zero are universally R-trivial. This talk is based on recent joint work with Amit Hogadi and Anand Sawant.
March 31, 2021
Speaker: Sushil Bhunia
Title: A glimpse on big mapping class groups.
Abstract: I will give an introduction to mapping class groups of infinite type surfaces. More precisely, a surface S is said to be of finite-type if its fundamental group is finitely generated; otherwise it is of infinite-type. The mapping class group of S is the group of isotopy classes of homeomorphisms of S and is denoted by MCG(S). When the surface is of infinite type, we call the mapping class group a big mapping class group. There has been a recent surge of activity aimed at understanding infinite-type surfaces and their mapping class groups. I will give an algebraic property of big MCG(S) (namely the -property) and explain some ideas from recent joint work with Swathi Krishna.
March 24, 2021
Speaker : Dr. Anirban Bose
Title: Twisted conjugacy in linear algebraic groups.
Abstract: Let G be a group and φ an automorphism of G. The φ- twisted conjugacy action of G on itself is given by (g,x) → gxφ(g)^-1, for all g,x ∈ G. In this talk we will consider this notion in the realm of linear algebraic groups and discuss some recent results. This talk will be based on a joint work with Sushil Bhunia.
Video, Board-Work
March 17, 2021
Speaker : Alok Kumar
Title: The Notorious Collatz conjecture
Abstract: In this talk, we will discuss an exposition of the most elementary and dangerous Collatz conjecture, and partial results towards proving the conjecture. I will use one of the most prolific slides prepared by Prof. Terrence Tao on this conjecture.
March 10, 2021
Speaker: Simran Tinani
Title: Introduction to k-normal elements over the finite fields.
Abstract: Recently, the concept of k-normal elements over a finite field was introduced as an extension of the well-known concept of normal elements. The questions of the existence and cardinalities of k-normal elements comprise an active research avenue, and in full generality remain open problems. In this talk, I will first describe normal elements, some key results on them, and their significance. I will then define k-normal elements and present some results on their existence and numbers, along with brief outlines of the methods employed in the proofs.
Video, Slides
February 24, 2021
Speaker: Yogesh
Title: An introduction to urn models
Abstract: I will give a brief introduction to Urn models, discuss their applications to real life problems, and talk about some mathematical tools necessary for studying them.
Video, Slides
February 17, 2021
Speaker : Dr. Sugandha Maheshwari
Title: An introduction to algebraic coding theory
Abstract: In this seminar, I would like to introduce the subject of algebraic coding theory, assuming no prerequisites. I would also like to talk about the applications of coding theory. Further, I would also like to share research perspective(s) in this direction.
Video, Slides
February 10, 2021
Speaker : Rakesh Pawar
Title: Rational functions on P^1 and Bilinear forms
Abstract: I will discuss the classification of rational functions on the Projective line P^1 over a field k up to 'algebraic deformations' in terms of symmetric matrices/bilinear forms over k. The talk will be based on the article of C. Cazanave.
Video, Slides
27 Jan, 2021
Topic: An introduction to Weil Conjectures.
Speaker: Dr. Vaibhav Vaish
Abstract: Weil conjectures (now theorems), proposed by Andre Weil in 1949, motivated significant advances in algebraic geometry since then. Formulated as certain claims on local zeta functions - generating functions involving number of roots of polynomials (over finite fields) - they are tied closely to algebraic geometry, number theory, and indeed, algebraic topology. This talk is intended as an elementary introduction to the topic.
Video, Board-work
20 Jan, 2021
Topic : Algebraic and Transcendental Solutions of first order non-linear ODE
Speaker: Urshashi Roy
Abstract : Consider the ODE y= f(x,y), where f(x,y) ∈ C(x)[y]. We want to answer whether the above ODE has transcendental solutions in a Liouvillian extension of C(x). By algebraic solution we mean a solution that lies in an algebraic extension of C(x). For example √ x is an algebraic solution of y'= y/2x .In fact c√ x are also solutions for any constant c. Finding all the algebraic solutions of an ODE might not be possible always. Our aim is to find relations between algebraic and transcendental solutions (if they exist). We will discuss few examples and the challenges to solve the above problem.
2020 : Fall Semester
2020 : Fall Semester
21 Nov, 2020
Topic : Representations of groups of order p^3
Speaker: Dr. Kapil Hari Paranjape
Abstract : We will introduce the notion of (finite) dimensional representations of a finite group. An important question is whether a group is determined by its representations. We will use the existence of two different groups of order p^3 to provide an example where the groups have the "same" representations but are not isomorphic.
Video Passcode: jC=b?16. Slides, Notes
18 Nov, 2020
Topic : Introduction to Computational Topology
Speaker: Abhijit Bhalachandra
Abstract : With the explosion in data and the availability of unorganized, high-dimensional, and large datasets, topology-based tools are gaining a more and more relevant role in extracting insights from these datasets. The insights specifically have to do with the qualitative and shape descriptive aspects of the data. In this regard, persistent homology has proven to be a game-changer. It enables tracking the changes in the homological features of a dataset, and is an effective pre-processing tool in exploratory data analysis. This talk aims at introducing persistent homology, generalizing the method (multi-dimensional persistence), and its computational challenges.
Video Passcode: 0%Lb6&tI Slides
11 Nov, 2020
Topic : Introduction to Hyperbolic Spaces/Groups
Speaker : Rakesh Halder
Abstract : We will define hyperbolic spaces/groups and then introduce some other characteristics of this space (with examples). Then we will see notions that are used in this area; define boundary of this space and its topology . In the meantime we will see how the study of this space helps to understand groups (though it's vice versa).
Video Passcode: p#&qh=7H Slides
4 Nov, 2020
Topic : Introduction to Graphons
Speaker : Puneeth Deraje
Abstract : Graphons are essentially an extension of the concept of graphs from a finite vertex set to an uncountable set. They can be thought of as functions from the unit square to the unit interval. In this talk I will introduce and formalize the notion of these entities. Further, we will see how this notion allows us to turn the space of graphs into a metric space with some nice properties. I will outline the proof of one such property, namely completeness.
Video Passcode: 4$a&nL0# Slides
28 Oct, 2020
Topic : Geometric Structures on manifolds
Speaker : Dr. Krishnendu Gangopadhyay
Abstract : Many interesting structures on a manifold can be viewed as a locally homogeneous geometric structures modelled on a pair (X,G) where X is a geometric space and G is a lie group acts by automorphisms of X . In this talk I shall give a brief introduction to these ideas.
Video Passcode: l8G$#@4$
21 Oct, 2020
Topic : Computing the optimal policy of a Markov Decision Process (MDP)
Speaker : Kausthub Keshava
Abstract : Markov decision processes are an extension of Markov process and are widely used in the field of robotics and machine learning. In the talk, I intend to prove the existence of a deterministic optimal policy for any finite state MDP. The next obvious step is to calculate the optimal policy, and Value iteration is a method to do the same. We shall look at how this method utilizes the Banach fixed point theorem to converge to an optimal policy.
Video Passcode: ^EfW3Jii Slides
14 Oct, 2020
Topic : Resolution Of Singularities
Speaker : Shikha Bhutani
Abstract : Resolution of singularities exist for varieties in all dimension over an algebraically closed field of characteristic zero. However, for positive characteristics, it is still an open problem for dimension greater than or equal to 4. In this talk, I intend to discuss the resolution of curves and surfaces in characteristic zero using blow-ups.
Video Passcode: i1q?.ns? Slides
30 Sept, 2020
Topic: Introduction to topological K-theory
Speaker: Ahina Nandy
Abstract: I will introduce the construction of K groups for (compact, hausdroff) topological spaces and explain how the construction can be extended to a generalized co-homology theory. I will introduce Bott periodicity theorem for complex K-theory and some of its application.
Video Passcode: g?&DeXD4 Slides
23 Sept, 2020
Topic : Kubota-Leopoldt p-adic L-function
Speaker : Pronay Karmakar
Abstract : L-functions play an important role in number theory. In Iwasawa theory, p-adic analog of these L-functions (called p-adic L-functions) is very crucial. In this talk we are going to construct the p-adic analog of the Riemann zeta function.
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