Title: Dirichlet Domains in Complex Hyperbolic Bidisks

Abstract: The geometry of bidisks -- the product of two hyperbolic planes --  has been widely studied. In this talk, we explore a complex analogue: the product of two complex hyperbolic planes. Focusing on cyclic group actions, we investigate the structure of Dirichlet domains in this space. We describe the behaviour of equidistant surfaces, introduce the notions of visibility and invisibility in this context, and demonstrate that for loxodromic actions, the Dirichlet domain remarkably has exactly two faces. Along the way, we highlight the interplay between the product metric, holomorphic sectional curvature, and the boundary behaviour of the level sets. This exploration opens new avenues for understanding discrete group actions in higher-rank complex hyperbolic geometry. This is a joint work with Prof. Krishnendu Gongopadhyay and Dr. Lokenath Kundu.

Title: Euler Characteristics of weight 1 modular forms

Abstract: Let E be an elliptic curve over Q. The Selmer group is a fundamental cohomological tool capturing deep arithmetic information about elliptic curves. Fix an odd prime p, and let Zp denote the ring of p-adic integers. For each n ≥ 1, let ζ_p^n be a primitive p^n-th root of unity in \bar{Q ̄}. The cyclotomic Z_p-extension, denoted by Q_cyc, is the unique extension of Q in Q(\cup_{n≥1} ζ_p^n ) whose Galois group Γ := Gal(Qcyc/Q) ∼= Z_p. 

When E has good ordinary reduction at p, the Euler characteristic of the p∞-Selmer group over Q_cyc is conjecturally related to the special value L_E(1)/Ω_E, under the Birch and Swinnerton-Dyer conjecture and some suitable assumptions. Here, L_E(s) denotes the Hasse–Weil (complex) L-function of E over Q, and Ω_E is the smallest positive real period of E.

In this talk, we raise questions about how similar results might extend to modular forms of weight 1.

Title: Dominating Surface-Group Representations in PSL_n(C) and PU(2, 1).

Abstract: Let S be a connected, oriented, punctured surface of negative Euler characteristic. In this talk, we present two comparison results for representations of π_1(S) into higher-rank Lie groups. First, we show that a generic representation ρ : π_1(S) → PSLn(C) is dominated by a ‘positive’ representation ρ_0 : π_1(S) → PSLn(R) in both the Hilbert length spectrum and the translation length spectrum in the symmetric space X_n = PSLn(C)/PSU(n), while preserving the lengths of the peripheral curves. This is a joint work with Subhojoy Gupta.

Second, we extend this perspective to complex hyperbolic geometry: we show that a T-bent representation ρ : π_1(S) → PU(2, 1) is dominated by a discrete and faithful representation ρ_0 : π_1(S) → PO(2, 1) in the Bergman translation length spectrum, again preserving the lengths of peripheral curves. This is a joint work with Krishnendu Gongopadhyay.

These results offer new insights into the role of positivity in higher-rank Teichm ̈uller theory and complex hyperbolic geometry.

Title: Generating the Liftable Mapping Class Groups

Abstract: For a closed orientable surface S_g, the mapping class group Mod(S_g) is defined as isotopy classes of orientation-preserving homeomorphisms of S_g. A mapping class is called liftable if it has a representative homeomorphism that lifts under a regular cover. In this talk, we discuss a generating set for a liftable mapping class group under k-sheeted regular free covers p:S_{k(g-1)+1}--> S_g.

Title: The Dehn Function of Palindromic Automorphism Group of Free Group

Abstract: The palindromic automorphism group ΠA(F_n) of a free group of rank n is a subgroup of Aut(F_n). The Dehn function of the group Aut(F_n) and Out(F_n) is exponential for n>=3 and is known from the work of Bridson and Vogtmann. In this talk, we present a joint work with Krishnendu Gongopadhyay in which we determine the Dehn function of the palindromic automorphism group of Aut(F_n) for n>=3.

Title: Studying descent of Quadratic forms in characteristic 2

Abstract: Most courses on quadratic forms over fields start with the hypothesis of characteristic being different than 2. In this talk, we will work with quadratic forms over characteristic 2 fields. After reviewing some basic properties, we'll address the question of Descent. The descent problem seeks conditions under which a K-form (quadratic or bilinear) is defined over F for a field extension K/F. We'll answer it for some small dimensional cases when K is the function field of a quadric and address the difficulties we encounter while answering this question in general. 

Title: Schur Multiplier and Schur Covers of Relative Rota-Baxter Groups

Abstract: Relative Rota-Baxter groups are generalizations of Rota-Baxter groups and share a close connection with skew left braces. These structures are well-known for offering bijective non-degenerate set-theoretical solutions to the Yang-Baxter equation. In this talk, we will discuss about the recently introduced extension theory and low-dimensinal cohomology of relative Rota-Baxter groups \cite{BRS2025}. We see an analogue of the Hochschild-Serre exact sequence for central extensions of relative Rota-Baxter groups. We introduce the Schur multiplier M_{RRB}(\mathcal{A}) of a relative Rota-Baxter group \mathcal{A}=(A,B,\beta,T), and prove that the exponent of M_{RRB}(\mathcal{A}) divides |A||B| when both A and B are finite. We define weak isoclinism of relative Rota-Baxter groups, introduce their Schur covers, and prove that any two Schur covers of a finite bijective relative Rota-Baxter group are weakly isoclinic. The results align with recent results of Letourmy and Vendramin \cite{LV2024} for skew left braces. This talk is based on joint works with Dr. Nishant Rathee and Prof. Mahender Singh \cite{BRS2025},\cite{BRS2024}.

Title: Products of Cycle-Classes of Symmetric Groups

Abstract: Abstract. Let G be a finite group and C_1, C_2, · · · , C_k be non-trivial conjugacy classes of G. The product of these classes defined by C_1C_2 · · · C_k := {x_1x_2 · · · x_k|x_i ∈ C_i, 1 ≤ i ≤ k} is invariant under conjugation and thus a union of conjugacy classes of G. In particular, for an integer k ≥ 2, the k-th power of a conjugacy class C of G, denoted by C_k, is simply the set CC · · · C (k-times). The main question is, for a given collection of non-trivial conjugacy classes of a finite group G, how large is the set C_1 · · · C_k?

In particular, when does this set cover the entire group G? It is a well-established theme in finite group theory which is very active at the current moment. The works of Bertram, Herzog, Lev, Kaplan, Guralnick, Malle, etc. are good sources of motivation for this problem. In this talk, I will first discuss some already known results in this direction. Then, a joint work with Dr. Rijubrata Kundu and Dr. Sumit C. Mishra will be discussed. In the symmetric group, we take the power of a conjugacy class of cycles of a fixed length and determine conditions under which it will cover the alternating group. We will provide a complete answer to both the conjectures of Herzog, Lev and Kaplan [Herzog, Marcel; Kaplan, Gil; Lev, Arieh, Covering the alternating groups by products of cycle classes. J. Combin. Theory Ser. A 115 (2008), no. 7, 1235–1245].

Title: Preferential Attachment Based on Reinforcement

Abstract: We consider a preferential attachment random graph model with self-reinforcement. Each time a new vertex is added, it attaches to an existing vertex with a probability proportional to the sum of the degrees of that vertex across all previous time steps. The resulting growing graph forms a random tree whose vertex degrees increase at a polynomial rate over time. We compute the growth exponent of this model and demonstrate that it is strictly larger than the growth exponent in models without self reinforcement. Our analysis provides insights into how self-reinforcement influences network evolution and accelerates degree growth. This is joint work with Prof. Frank den Hollander.

Title: Graph Discretization of Riemannian Manifolds for Spectral Approximation and Learning Applications

Abstract: This talk discusses the approximation of eigenvalues and eigenfunctions of the Laplace–Beltrami operator on compact Riemannian manifolds without boundary via weighted graph discretization. Building on the work of Burago et al, I present a joint work with Dr. Soma Maity which generalizes their results to a broader class of manifolds satisfying lower bounds on the Ricci curvature and injectivity radius and upper bounds on diameter and sectional curvature. The construction yields uniform spectral convergence in this class. I will conclude by illustrating connections to spectral graph neural networks, where such discretizations underpin the use of graph Laplacian to approximate manifold-based learning architectures.

Title: On Character Variety of Anosov Representations

Abstract: Let Γ be the fundamental group of a 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 = π_1(Σ_g), these character varieties are holomorphic symplectic manifolds. This talk is based on a joint work with Prof. Krishnendu Gongopadhyay.

Title: Character space and Gelfand type representation of locally C^∗-algebra

Abstract: In this talk, we present a suitable notion of character space of commutative unital locally C^∗-algebra using the notion of inductive limit of topological spaces and study its properties. We prove that every commutative unital locally C^∗-algebra is isomorphic (as a topological ∗-algebra) to the space of continuous functions on its character space. As a consequence, we define unital locally C^∗-algebra generated by a locally bounded normal operator T and its character space is homeomorphic to the local spectrum of T. Also, the functional calculus and spectral mapping theorem are proved in this framework. The talk is based on a joint work with Dr. Santhosh Kumar Pamula.

Title: Orderability of big mapping class groups.

Abstract: An orderable group is a group that is totally ordered, with the ordering being invariant under left-multiplications. In this talk, we will see that a generalized ideal arc system for surfaces determines a left ordering on the mapping class group of a connected, oriented infinite-type surface with a non-empty boundary. We will also explore the conditions under which two generalized ideal arc systems induce the same left ordering. Finally, we will conclude the talk by comparing the usual topology of mapping class groups with the order topology. This is a joint work with Prof. Mahender Singh and Dr. Apeksha Sanghi.

Title: Kulkani Limit Sets for Cyclic Quaternionic Projective Groups

Abstract: In this talk, I will present results on the dynamics of cyclic subgroups of PSL(n + 1, H) acting on the quaternionic projective space P^n_H, with particular focus on their Kulkarni limit sets. The elements of such groups have been classified as elliptic, loxodromic, or parabolic based on their spectral properties and Jordan forms. Building on this classification, we describe the structure of the corresponding Kulkarni limit sets and the associated domains of discontinuity. This is a joint work with Krishnendu Gongopadhyay and Sandipan Dutta.

Title: A^1-Connected Components of Affine Quadrics

Abstract: Let X be a variety over a field k. Denote by π^A1_0(X) the sheaf of A^1-connected components of X, and by S(X) the sheaf of naive A^1-connected components of X. A result of Balwe, Rani, and Sawant shows that for any field extension F/k, the canonical map

π^A1_0(X)(F) → lim_n S^n(X)(F)

is an isomorphism. In this talk, we show that for any field F/k and any smooth affine quadratic hypersurface X, this isomorphism stabilizes at n=2. That is,

π^A1_0(X)(F) = S^2(X)(F). 

As an application, we combine this result with Morel’s characterization of A^1-connected spaces in terms of the triviality of field-valued sections of π^A1_0 to give a complete characterization of A^1-connected affine quadrics. This talk is based on joint work with Dr. Chetan Balwe.

Title: Picard-Vessiot Extensions with Unipotent Differential Galois Groups

Abstract: A differential field is a field F equipped with a derivation D : F → F, an additive map satisfying the Leibniz rule: D(ab) = D(a)b + aD(b) for all a, b ∈ F. Analogous to the classical Galois theory for polynomial equations, differential Galois theory addresses a similar theory for linear differential equations. Informally, a Picard-Vessiot extension E of a differential field F is a differential field extension of F generated by the solutions of a linear homogeneous differential equation with coefficients in F. The group of all differential automorphisms of E that fix F and commute with the derivation D is called the differential Galois group of the extension. For a differential field F having an algebraically closed field of constants, we analyze the structure of Picard-Vessiot extensions of F whose differential Galois groups are unipotent linear algebraic groups, and as an application of these results we will give a generalization of Liouville’s theorem [Ros68]. This talk is based on the joint work with Dr. Matthias Seiss and Dr. Varadharaj R. Srinivasan [SSS25].

Title: New and General Type Meromorphic 1−forms on Curves.

Abstract: In this talk, we discuss the existence of new and of general type pairs (X, ω) where X is a curve and ω is a meromorphic 1−form on X. These pairs arise in the study of first order autonomous differential equations and exhibit unique algebraic dependency relations among their solutions.

Title: Splitting iterative derivations on central simple algebras

Abstract: The splitting of a derivation d on a k−central simple algebra A by a Picard-Vessiot extension of (k, d|k) is well studied in the case that the constants of (k, d|k) is an algebraically closed field of characteristic zero. In fact in this case the Picard-Vessiot extension K for the differential module A is the smallest no new constant extension splitting d. One can also characterize the structure of the differential algebra A using the structure of the differential Galois group of K. Similar results are obtained in positive characteristic if we use iterative derivations and Picard-Vessiot theory of ID−modules. More precisely, we show that if d is an iterative derivation on a k−central simple algebra A such that the constants of (k, d|k) is algebraically closed then the Picard-Vessiot extension K for the ID−module A is the smallest no new constant extension splitting d. We also characterise the structure of the ID−algbera A when the differential Galois group of K is linearly reductive or solvable.

Title: Motivic Intersection Complex of a Threefold

Abstract: The triangulated category of motivic sheaves DM(X) over a scheme X, with rational coefficients, admits a realization functor to the bounded derived category of mixed constructible sheaves D^b_m(X, Q_l). The latter is equipped with a perverse and a weight structure, as well as the S. Morel’s t-structure. In this talk, we explore an analogue of Morel’s t-structure within a suitable subcategory of DM(X). A key object in the theory of perverse sheaves is the intersection complex IC_X, which appears as a simple object in the heart of the perverse t-structure and can also be characterized using Morel’s weight truncation functors. Motivated by this perspective, we define its motivic analogue EM_X for a three-dimensional variety X, using the motivic version of Morel’s truncation. We show that EM_X defines a pure motive, and more precisely a Chow motive offering a parallel to the classical intersection complex in the motivic setting.