The Speakers
Invited Speakers
Prof. Chandan Singh Dalawat (HRI, Allahabad)
Title : Primitive extensions of p-fields
Abstract : It is a fundamental open problem in number theory to classify all finite extensions of a given number field. As a preliminary, one should classify all finite extensions of p-fields (which are completions of number fields at various primes). Every finite extension is a succession of primitive extensions (having no intermediate extensions). We will give a natural parametrisation of the set of primitive extensions of a given p-field and show how various properties of the extension (such as its discriminant) can be read off from its parameter.
Prof. Gadadhar Mishra (ISI, Bangalore)
Title : Simultaneous invariant subspaces of $L^2(\mathbb{D})$
Abstract : Let $\mu=(1-\mid z\mid^2)^{2\lambda-2}dA$ be the weighted area measure on the unit disc $\mathbb{D}$ and let $\mathcal{H}^{(\lambda)}$ be the Hilbert space $L^2(\mathbb{D},d\mu), \lambda>1/2$. Suppose that $\mathcal{M}\subset \mathcal{H}(\lambda)$ is a closed subspace invariant under the unitary representations $(U_{{\phi}^{-1}}f)(z):= \phi^{'\lambda}(z)(f \circ \phi)(z)$, $\phi \in \text{Möb}$, of the Möbius group Möb, and the operator $M$ of multiplication by the coordinate function on $\mathcal{H}(\lambda)$. There is one such, namely, the subspace of holomorphic functions in $\mathcal{H}(\lambda)$. A description of these simultaneous invariant subspaces remain mysterious in general. Never the less, one may go one more step and consider unitary representations of the Möbius group acting on the Hilbert space $ \mathcal{H}(\lambda)\otimes V$, where $V$ is a finite dimensional Hilbert space by replacing the multiplier $\phi^{'\lambda}$ by a suitable multiplier taking values in $\text{GL}(V)$.
These subspaces are in one to one correspondence with subnormal homogeneous operators, namely, those contractive linear operators $T$ such that $\phi(T)$ is unitarily equivalent to $T$ for all $\phi$ in \text{Möb}.
We will discuss some partial answers that we have obtained recently to these questions.
Dr. Surinder Pal Singh Kainth (PU Chandigarh)
Title : HK-integral on Metric Measure Spaces
Abstract : In this talk, we shall discuss the gauge integrals of Henstock-Kurzweil and McShane which generalize the notions of Riemann, Lebesgue, and improper integrals for real-valued functions on a compact real interval. I shall present a quick survey of the basics of this thrust area, along with some of my recent results.
We shall discuss some measure theoretic characterizations of the gauge integrals, corresponding fundamental theorems of calculus and Radon-Nikodym type theorems, Hake’s theorem, relationship among some gauge integrals, and some extensions of absolute continuity.
In [1], we have established some characterizations of compactness and completeness in metric spaces in terms of the Cousin’s lemma, a basic result in gauge integrals. We shall discuss these and some results regarding the differentiability of the primitives of Henstock-Kurzweil integrable functions on metric spaces along with a few consequences. We shall also discuss alternative proofs of generalizations of two standard results on Lebesgue integration; the first one bypasses the Vitali covering while the second one is a version of the fundamental theorem of calculus in the metric space setting.
If time permits, we shall also overview some results for vector valued gauge integrals [3]. These integrals include strong versions of Henstock-Kurzweil and McShane integrals, along with the analogous integrals by restricting to only measurable gauges. These also lead to some fundamental results on Vital sets [2].
References:
1. Surinder Pal Singh Kainth and Narinder Singh, Henstock integral on Metric Spaces Revisited, Real Analysis Exchange, 47(2), 2022, 377-396.
2. Surinder Pal Singh Kainth and Narinder Singh, Some Properties of Vitali Sets, Georgian Mathematical Journal, 30(1), 2023, 117-120.
3. Stefan Schwabik, Topics in Banach Space Integration, World Scientific, 2005.
In-House Speakers
Dr. Apeksha Sanghi
Title : Metacyclic subgroups of mapping class group
Abstract : Let $\text{Mod}(S_g)$ denote the mapping class group of closed orientable surface $S_g$ of genus $g \geq 2$. In this talk, we will discuss the necessary and sufficient conditions under which two torsion elements in $\text{Mod}(S_g)$ will have conjugates that generate a finite metacyclic subgroup of $\text{Mod}(S_g)$. As an application, we will analyze the liftability of certain periodic mapping class under some finite-sheeted regular cyclic covers of $S_g$. Furthermore, we will see that $4g$ is an upper bound on the order of a non-split metacyclic action on $S_g$. Moreover, we give a complete characterization of the dihedral subgroups of $\text{Mod}(S_g)$ up to a certain equivalence. We conclude the talk by describing nontrivial geometric realizations of some metacyclic actions.
Dr. Chandan Maity
Title : Cohomology (de Rham) of adjoint orbits in the non-compact simple Lie algebra $\mathfrak{g}_2$
Abstract : In this talk, we will describe all the de Rham cohomology groups of the adjoint orbits in the complex simple Lie algebra $\mathfrak{g}_2$.
Mr. Debattam Das
Title : Reciprocity in the Hecke group
Abstract : An element $g$ in a group $G$ is called reciprocal if there exists $h \in G$ such that $g^(−1 )= hgh^(−1)$. The reciprocal elements are also known as ‘real elements’ or ‘reversible elements’ in the literature. We classify the reciprocal elements and parametrize the reciprocal classes in the Hecke groups $\Gamma_p$for p ≥ 3. This generalizes a result by Sarnak for reciprocal
elements in the modular group.
Ms. Divya Setia
Title : Filtration by $V(\xi)$ modules in tensor products
Abstract : Given $\lambda = (\lambda_{1} \geq \lambda_{2} \geq \dots \geq \lambda_{l} > 0)$, Chari and Venkatesh had defined a module $V(\lambda)$ for the current algebra which is a quotient of the local Weyl module $W_{loc}(|\lambda|)$. Another well studied family of modules are Demazure modules. In this talk, I will define the notion of a local Weyl module, $V(\lambda)$-module and Demazure modules for the current algebra. In, simply laced cases the local Weyl module is a level 1 Demazure module. So, we started looking at the structure of the tensor product of a local Weyl module with a $V(\lambda)$-module. I will try to explain the structure of these modules by giving a presentation of $W_{loc}(m \omega) \otimes V(\lambda)$ as an $\mathfrak{sl}_{2}[t]$-module. We have given a filtration by $V(\xi)$-modules in tensor product of a local Weyl module with an irreducible representation of $\mathfrak{sl}_{2}[t]$,where $\xi$ is a hook shape parition. It helps us to decompose $W_{loc}(m \omega) \otimes V(n\omega)$ into direct sum of $V(\xi)$-modules. Using this result, I will also prove the existence of a filtration by $V(\xi)$-modules along with their multiplicity in $W_{loc}(m \omega) \otimes V(n,1^{k})$. At last, I will introduce the graded character of the tensor product of two local Weyl modules of $\mathfrak{sl}_{2}[t]$.
Dr. Gurleen Kaur
Title : On generalized strongly monomial groups
Abstract : A fundamental problem in group algebras is to explicitly determine the algebraic structure of rational group algebras, i.e., the Wedderburn decomposition and complete set of primitive central idempotents. A new direction to this investigation was provided by Olivieri, del R{\'{\i}}o and Sim{\'o}n, in 2004, when they introduced the notion of Shoda pairs and strongly monomial groups. In a joint work with \linebreak Bakshi, we introduced the class of generalized strongly monomial groups which is a generalization of strongly monomial groups and provided an explicit description of the complete algebraic structure of rational group algebra for such groups. In this talk, we will see the vastness of the class of generalized strongly monomial groups. Recently, it has been proved that every finite solvable group can be (isomorphically) embedded in some generalized strongly monomial group with the same derived length (fitting length, supersolvable length).
Mr. Harish kishnani
Title : Word maps on finite nilpotent groups
Abstract : Let $F_d$ denote the free group on $d$ letters and $0 \neq w \in F_d$ be a word. For a group $G$, let $G^d$ denote the group of $d$-tuples in $G$. A \emph{word map} $\tilde{w} : G^d \to G$ is the evaluation map of $w$ on these $d$-tuples. The image of $\tilde{w}$ is denoted by $w(G)$.
In this talk, I will first discuss a collabrative work done with Dr. Dilpreet Kaur and Dr. Amit Kulshrestha. This work is inspired from an article by Alexander Lubotzky, in which he has shown that automorphism invariant subsets of finite simple groups which contain identity are always word images. We observed that this result is not true in general for non-abelian finite simple groups. In fact, we showed that except elementary abelian groups, all finite nilpotent groups have subsets that are automorphism invariant, contain identity of the group, but are not word images. We termed these subsets as \emph{word image impostors}, and classified them for the class of extraspecial $p$-groups. In the course of it, we construct a $2$-exhaustive set of word maps on nilpotent groups of class $2$ and demonstrate its minimality in some cases.
The second half of the talk will be based on an extension of this idea to the idea of \emph{fibre size equivalence}. In a joint work with Dr. Kulshrestha, we proved several results related to fibre size equivalence for nilpotent groups of class $2$ and as an immediate consequence showed that all class $2$ groups are rational. The main motivation behind it was the conjectures by Amit and Ashurst. The idea of fibre size equivalence also enabled us to prove several results in this direction including the improvement in the already known bound and proving the conjecture for certain classes of words.
Dr. Neeraj Kumar Dhanwani
Title : Writing periodic mapping classes into products of Dehn twists
Abstract : Let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this talk, we will discuss various methods for expressing periodic mapping classes as a product of Dehn twists, up to conjugacy. These methods are based on the chain and star relations in the mapping class group, the geometric realisations, and the symplectic representations of periodic elements.
Dr. Neha Malik
Title : Stiefel-Whitney Classes of Representations of Finite Symplectic Groups
Abstract : Orthogonal representations $\pi$ of a finite group $G$ have cohomological invariants $w_i(\pi)$ living in the $i$th degree cohomology group $H^i(G, Z/2Z)$, called Stiefel-Whitney Classes (SWCs). We have computed these classes for several finite groups of Lie type. This talk will give an overview of our results for finite symplectic groups $\Sp(2n,q)$ with $q$ odd. From this calculation, we can answer some interesting questions. For instance, we determine the mod 2 Chern classes of the Weil representations and also obtain a Nakaoka-type result which, in a sense, conveys that the cohomology of $\Sp(2n,q)$ stabilizes.
This is joint work with Prof. Steven Spallone.
Mr. Pravin Kumar
Title : Crystallographic quotients of planar braid groups
Abstract : The symmetric group, $S_n$, has three natural extensions, namely, the braid group $B_n$, the twin group $T_n$ and the triplet group $L_n$. The latter two families of Coxeter groups can be thought of as planar forms of Artin braid groups and have a deep connection with low-dimensional topology, just like braid groups. A discrete subgroup $\Gamma$ of the group of isometries $\operatorname{Isom}(\mathbb R^n)$ of Euclidean $n$-space, with a compact fundamental domain, called an $n$-dimensional crystallographic group. In this talk, we determine some natural crystallographic quotients of twin groups and triplet groups. The content of this talk is part of recent work with Dr. Mahender Singh and Dr. Tushar Kanta Naik.
Ms. Priya Rani
Title : Real algebraic knots in $\mathbb{RP}^{4}$
Abstract : Classification of knots is a classical problem in Knot theory. In $\mathbb{R}^{3}$, we study knots upto smooth isotopy whereas real rational knots are classified up to rigid isotopy in projective spaces $\mathbb{RP}^{n}$. In $\mathbb{RP}^{3}$, real rational knots are classified upto degree 6. In $\mathbb{RP}^{4}$, there are two quadrics upto projective transformations with signature $(-1,1,1,1,1)$ and $(-1,1,1,1,-1)$. In this talk, we are going to discuss real algebraic knots upto degree 4 which lies on the quadric of signature $(-1,1,1,1,-1)$. This talk is based on an ongoing work with Dr. Shane D'Mello.
Mr. Pronay Kumar Karmakar
Title : Bipartite Euler systems for certain Galois representations
Abstract : Let $E/\Q$ be an elliptic curve with ordinary reduction at a prime $p$, and let $K$ be an imaginary quadratic field. The anticyclotomic Iwasawa main conjecture, depending upon the sign of the functional equation of $L(E/K,s)$, predicts the behavior of Selmer group of $E/\Q$ along the anticyclotomic tower of $K$. Some of the crucial ideas of Bertolini and Darmon on this conjecture have been abstracted by Howard into an axiomatic set-up through a notion of Bipartite Euler systems, assuming that $E[p]$ is an irreducible representation of $G_{K}$. We generalize this work by assuming only $(E[p])^{G_K}=0$. We use the results of Howard, Nekov\'a\v{r} and Castella \emph{et al}., along with those of Mazur and Rubin on Kolyvagin systems to show one divisibility of the anticyclotomic main conjecture, for both the signs. The other divisibility can be reduced to proving the nonvanishing of sufficiently many $p$-adic $L$-functions attached to a family of congruent modular forms.
Mr. Rakesh Halder
Title : Cannon-Thurston (CT) maps
Abstract : We will introduce Cannon-Thurston maps in this talk and sketch a brief history of it. Then we will discuss on some recent developments.
Dr. Riju Basak
Title : Oscillatory Multipliers on Lebesgue and Hardy Spaces
Abstract : Oscillatory multiplier operators play a significant role in studying Cauchy problems for Schr\"odinger and wave equations. In this talk, I shall discuss the boundedness of oscillatory multiplier operators on $L^p$ and Hardy spaces.
Dr. Rijubrata Kundu
Title : Generating functions for the powers in $\text{GL}(n,q)$
Abstract : Let $G$ be a group and $M \geq 2$ be an integer. Consider the power map $\omega_M:G \to G$ defined by $g \mapsto g^M$. Let $G^M:=\omega_M(G)$ denote the image of $\omega_M$. In other words, $G^M$ is the set of all elements in $G$ which admits an $M$-th root in $G$. In this talk, we will consider $G:=\text{GL}(n,q)$, the group of all $n\times n$ invertible matrices over the finite field $\mathbb{F}_q$ and derive a generating function for $\frac{|\text{GL}(n,q)^M|}{|\text{GL}(n,q)|}$ in the argument $n$, fixing $q$. For simplicity we will assume $M$ is a prime and $(M,q)=1$. We will give a combinatorial formula for $\frac{|\text{GL}(n,q)^M|}{|\text{GL}(n,q)|}$ under certain conditions, which can be derived using the aforementioned generating function. This is joint work with Prof. Anupam Singh.
Dr. Sandipan Dutta
Title : Limit sets of cyclic quaternionic Kleinian groups
Abstract : In this talk, we consider the natural action of $\text{SL}(3,\mathbb{H})$ on the quaternionic projective space $\mathbb{P}_\matbb{H}^2$. Under this action, we investigate limit sets for cyclic subgroups of of $\text{SL}(3,\mathbb{H})$. We compute two types of limit sets, which were introduced by Kulkarni and Conze-Guivarc'h, respectively.
Ms. Shushma Rani
Title : Tensor product multiplicities and Chari-Venkatesh modules for current Lie algebra of $\mathfrak{sl}_3$
Abstract : The concept of Chari-Venkatesh modules, defined by generators and relations, was first introduced in 2015. In this presentation, we will introduce an exact sequence of a specific category of Chari-Venkatesh modules for Lie algebra of type $A_2$. This sequence not only enables us to compute the graded character for these modules but also demonstrates their isomorphism to the corresponding fusion product modules. As an application, we will obtain Littlewood-Richardson coefficients, which yield the tensor product multiplicities.
Dr. Sudipta Mukherjee
Title : Classification of irreducible integrable Harish-Chandra modules for loop affine-Virasoro algebras
Abstract : Affine Kac-Moody Lie algebras and Virasoro algebras are very fundamental in both Mathematics and Physics. The Virasoro algebra acts on the derived algebra of an affine Kac-Moody algebra by derivations. The emerging semidirect product called the Affine-Virasoro algebra turns out to be an important Lie algebra of study as it has applications to conformal field theory, number theory and soliton theory. On the other hand, loop algebras are also studied extensively for simple Lie algebras, affine Kac-Moody Lie algebras and Virasoro algebras. In this talk, we will classify the irreducible integrable modules with finite dimensional weight spaces for loop affine-Virasoro algebras.
Dr. Sumit Chandra Mishra
Title : Ruled Residue Theorems for function fields of curves
Abstract : Let $E$ be a field with a valuation $v$. In 1983, Ohm proved that for any extension of $v$ to the rational function field $E(X)$ in one variable, the corresponding residue field extension is either algebraic or ruled, i.e., it is the rational function field in one variable over a finite extension of the residue field of $E$. This is called the Ruled Residue Theorem. More generally, one can consider the function field $F$ of a curve over $E$ and ask if for all extensions of $v$ to $F$, the corresponding residue field extension is either algebraic or ruled? If not, is there any bound on the number of extensions of $v$ to $F$ where this fails? I will mention known results for the function fields of conics. Later on, I will discuss the case of function fields of elliptic curves, which is a joint work with Prof. Karim J. Becher and Dr. Parul Gupta.
Mr. Suneel Kumar
Title : On the local constancy of certain mod p Galois representations
Abstract : Let $f$ be a normalized cuspidal eigenform of weight $k\geq 1$, character $\psi$ and level $\Gamma_1(N)$ such that $p\nmid N$. The work of Deligne, Deligne-Serre, and Eichler-Shimura associates to $f$ a $p$-adic Galois representation $\rho_f: \text{Gal}\left(\bar{\mathbb{Q}}/\mathbb{Q}\right)\rightarrow \text{GL}_2(\bar{\mathbb{Q}}_p)$. The classical problem is to determine the local structure of $\rho_f$ at any prime $l$. However, our focus is on studying the local structure of $\rho_f$ at the prime $l=p$. In this talk, firstly we will discuss some existing results on a local constancy of mod p reduction $\bar{\rho}_f$ of $\rho_f$ restricted to the local group of $\text{Gal}\left(\bar{\mathbb{Q}}/\mathbb{Q}\right)$ at prime $p$. Then we will present our results, which prove local constancy in the weight space by giving an explicit lower bound on the radius of local constancy centered around weights up to $(p-1)^2+3$ and the slope in $(0,p-1)$ satisfying certain constraints. This is joint work with my PhD advisor Dr. Abhik Ganguli.
Mr. Tejbir Lohan
Title : Reversibility of affine transformations
Abstract : Reversible elements in a group are those elements that are conjugate to their own inverses. They are closely related to strongly reversible elements, which can be expressed as a product of two involutions. In this talk, we will classify reversible and strongly reversible elements in the affine group $\mathrm{GL}(n,\mathbb{D}) \ltimes \mathbb{D}^n$, where $\mathbb{D} = \mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$. This is a joint work with Prof. Krishnendu Gongopadhyay and Dr. Chandan Maity.