Speakers
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Title: Counting and Randomness.
Abstract: What do counting discrete objects and the behaviour of chaotic gases have to do with each other? I will explain using simple examples.
Abstract: Studying word maps in groups is an active area of research and great progress has been made in the last 3 decades. In this talk, we will study the power maps on finite reductive groups and estimate their image size.
Title: Universal subspaces for Lie groups.
Abstract: Let U be a finite dimensional vector space over ℝ or ℂ, and let ρ:G → GL(U) be a representation of a connected Lie group G. A linear subspace V ⊆ U is called universal if every orbit of G meets V. In this talk, we study universal subspaces for Lie groups, especially compact and complex Lie groups. This is a joint work with Saurav Bhaumik.
Title: Dynamics of actions of automorphisms on a Lie group G and SubG with applications to lattices.
Abstract: On a locally compact group G, the space SubG of closed subgroups of G endowed with the Chabauty topology is compact. There is a natural action of the automorphism group Aut(G) on SubG. Distal maps were introduced by David Hilbert on compact spaces to study non-ergodic maps. After a gentle introduction, we will discuss the distal actions of automorphisms on a connected Lie group G and, characterise those acting distally on SubG for a large class of G. We will present a characterisation of the distal actions of automorphisms of a lattice L in G on SubL for a lattice L. We will briefly discuss expansive actions and show that neither a connected Lie group G nor a lattice L in it admits automorphisms which act expansively on SubG or on SubL unless G is trivial or L is finite respectively. We obtain a useful result on the structure of lattices in a connected Lie group. We will end the talk with a brief introduction to initiatives of Indian Women and Mathematics (IWM).
Title: Arithmeticity and Thinness of Hypergeometric Groups.
Abstract: A hypergeometric group is a subgroup of the general linear group generated by the companion matrices of two monic coprime polynomials. It arises as the monodromy group of a hypergeometric differential equation, and if the defining polynomials are also self-reciprocal and form a primitive pair, then its Zariski closure is either a symplectic or an orthogonal group. In this talk, we will discuss the arithmeticity and thinness of the hypergeometric groups whose defining polynomials also have integer coefficients.
Title: Lp -Asymptotics of solutions of the heat equation
Abstract: It is known that on Euclidean spaces the solution of the heat equation with L1 initial data behaves like a scalar multiple of the fundamental solution, as time goes to infinity. We shall talk about an analog of this result for Riemannian symmetric spaces of noncompact type. This analog involves Herz criteria for convolution operators and the Kunze-Stein phenomena.
Title: Some results on graphical parking functions
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Title: Sequences of integers.
Abstract: There are numerous fascinating sequences of integers, some are relatively easy to study while others are more complicated. By looking at numerous examples of such sequences, the speaker will attempt to justify the (somewhat bold) assertion that *all* of mathematics can be looked at as the study of such sequences. The talk will end with an "Ask me anything" session on sequences of integers.
Reference: The Online Encyclopedia of Integer Sequences.
Title: Self-organization via reinforcement: Urns, Elephant Random Walks and Random Networks.
Abstract: Urn models are classical problems in probability theory. I will discuss some well-known results for urn models and their connection to elephant random walks and preferential attachment random graphs.
Title: Galois theory of differential modules and differential central simple algebras.
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Title: Minimal Embeddings of p-groups in Symmetric Groups.
Abstract: For a finite group G, the minimal faithful permutation representation degree of G, denoted by μ(G), is the smallest positive integer n such that G is embedded in S(n). The quantity c(G) denotes the minimal degree of faithful representation of G by quasi-permutation matrices (square matrices with non-negative integral trace) over the complex field C. In this talk, we shall discuss the quantities μ(G) and c(G) for various classes of non-abelian p-groups including groups with cyclic center, and the groups of order p6, where p is an odd prime.
Title: Decomposing Completely Positive Maps.
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Title: Conjugacy classification in the semigroup of holomorphic isometries of an infinite dimensional complex hyperbolic space.
Abstract: The talk focusses on infinite dimensional complex hyperbolic space. We consider the Hilbert ball model of the hyperbolic space and investigate something called "conjugacy classes" in the semi-group of holomorphic isometries on this model.
Title: Differential Modular Forms.
Abstract: We will introduce differential modular forms, starting with the classical modular forms. Then we will explore the importance of these objects in arithmetic geometry and end with some open questions.
Title: Weyl modules for Toroidal Lie algebras.
Abstract: In this talk, we extend the notion of Weyl modules to toroidal Lie algebras with n variables. Further, using the work of Rao, we identify the level one global Weyl module of toroidal Lie algebra with the suitable submodule of its Fock space representation up to a twist. As an application, we compute the graded character of the level one local Weyl module of toroidal Lie algebra, which generalises the recent work of Kodera.
Title: Knots in RP^3.
Abstract: Classical knot theory is the study of knotted circles in the three-sphere. In order to understand the topology of a three-manifold, one method is to understand the knots inside it. In this talk I will try to give you a glimpse of how the knot theory of RP^3 looks. We will see that there are several new features to projective knots which are not seen in classical knots.
Title: Volume Growth on Manifolds with Multiple Ends.
Abstract: The volume growth function of a Riemannian manifold of bounded geometry is a function having bounded growth of derivative, called a bgd-function. Grimaldi and Pansu showed that any one-ended manifold of finite type admits a metric of bounded geometry such that the volume growth function has the growth type of any given bgd-function. We shall discuss in short the proof of their theorem, and then generalise it to the case of manifolds with multiple ends. The metrics constructed thus are called the Grimaldi-Pansu metrics. In the specific class of manifolds which can be written as connected sums of closed manifolds along a tree, we shall obtain uniform bounds for the volume growth functions of the Grimaldi-Pansu metrics. We shall also look into some other properties of these metrics, like the R.C.A. or R.C.E. conditions and the volume doubling conditions.
Title: Combinatorial Growth of Reciprocal Geodesics in Hecke Groups.
Abstract: A geodesic in hyperbolic orbifold is said to be `reciprocal' if it passes through at least one even ordered cone point. These geodesics correspond to conjugacy classes of reciprocal elements. First, we classify the reciprocal geodesics in the Hecke groups; then, we study the growth of the reciprocal geodesics in terms of word lengths on the hyperbolic orbifolds associated with Hecke groups. In particular, we have completed the results of Marmolejo for the even cases of Hecke groups. Furthermore, we generalise the relation between the low-lying words and geodesic with an excursion of the modular group to Hecke groups.
Title: Residual finiteness of fundamental n-quandles of links.
Abstract: A quandle is an algebraic system with a binary operation satisfying three axioms that are algebraic formulations of the three Reidemeister moves of planar diagrams of links in the 3-space. Link quandles are complete invariants of split links upto orientation of the ambient space. In this talk, we investigate residual finiteness of quandles. The existence of these finiteness properties implies the solvability of the word problem. We will see that the fundamental n-quandle of any link in the 3-sphere is residually finite for each n ≥ 2. This supplements the recent result on residual finiteness of link quandles and the classification of links whose fundamental n-quandles are finite for some n. We also establish several general results on these finiteness properties and give many families of quandles admitting them.
Title: Automorphic Words Maps and Amit--Ashurst Conjecture.
Abstract: In this talk, I will first discuss the construction of a 2-exhaustive set of word images on nilpotent groups of class 2. The second half of the talk will be based on an extention of this idea to Fk(G) automorphic words. In a recent work with Prof. Kulshretha, we proved several results related to Fk(G) automorphic words for nilpotent groups of class 2 and the main motivation behind it was the conjecture by Amit and Ashurst. The idea of Fk(G) automorphic words also enabled us to prove results in this direction including the improvement in the already known bound and proving the conjecture for certain classes of words and groups.
Title: Absence of positive eigenvalues for perturbed sublaplacians.
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Title: Iterations of the functor of naive $\mathbb A^1$-connected components of varieties.
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Title: Simplicial structure on pure twin groups and brunnian twins.
Abstract: Twin groups are planar analogues of Artin braid groups and play a crucial role in the Alexander-Markov correspondence for the isotopy classes of immersed circles on the 2-sphere without triple and higher intersections. These groups admit diagrammatic representations, leading to maps obtained by adding and deleting strands. In this talk, we will explore Brunnian twin groups, which are subgroups of twin groups consisting of twins that become trivial when any of their strands are deleted. We also investigate a simplicial structure on pure twin groups that admits a simplicial homomorphism from Milnor’s construction of the simplicial 2-sphere. This gives a possibility to provide a combinatorial description of homotopy groups of the 2-sphere in terms of pure twins. This is a joint work with Prof. Valeriy G. Bardakov and Prof. Mahender Singh.
Title: A Radon-Nikodym type theorem for Local CP-maps.
Abstract: In this talk, we present the notion of locally C*-algebra and discuss the method to identify it with a projective limit of C*-algebras, and define the notion of local completely positive maps (in short, local CP-maps) on a locally C*-algebra. A Stinespring representation for local CP-maps was proved, in 2008, by A. Dosiev. A suitable notion of minimality condition is given, by Bhat, Ghatak and Pamula in 2021, in order to obtain uniqueness upto unitary equivalence for the associated representation. An ordering on the class of local completely positive maps is defined. Following the minimality condition, a characterisation of local CP maps that are dominated by a fixed map, namely Radon-Nikodym type theorem proved. We discuss the result here. This is an expository talk.
Title: Motivic intersection complex of a threefold.
Abstract: Click Here
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