Mathematics Symposium 2023

The Mathematics department, IISER Pune will host its annual in-house symposium on March 23-24, 2023. The members of the department (Faculty, Post-docs, and Students) will present their ongoing research through talks. You are welcome to attend ! 

Venue: Madhava Hall, Maths floor, main building.

Speakers this year:  

Sreedev Manikoth, Prafullkumar Tale, Ayan Mahalanobis, Nupur Patanker, Priyanka Majumder, Debargha Banerjee, Anup Biswas, Jishu Das, Tejas Kalelkar, Puspendu Pradhan, Praphulla Koushik, Pranjal Vishwakarma, Abhrojyoti Sen, Krishna Kaipa, Anupam Singh, Debjit Pal, Anindya Goswami, Rupak Dalai, Chandrasheel Bhagwat, Alok Kumar Sahoo, Tumpa Mahato, and Visakh Narayanan.

Title and Abstract of Talks

1. Anupam Singh. Title: Word maps on groups and algebras.

Abstract: Given a word w in d variables it gives rise to a map on a group G, called word maps defined by evaluation. The main question is to understand what is the fibre and image of such maps. Such questions are motivated by the Waring problem in number theory and Ore's conjecture in group theory. In the last couple of decades, there has been tremendous progress in this area. In a similar spirit to a polynomial in d non-commuting variable, one defines a map on a central simple algebra A and again one wants to understand the nature of the image, like Kaplansky-L'ovo conjecture. In this expository talk, we will see some of the results proved in this direction and the kind of questions being studied. 

2. Sreedev M.    Title: Split-harmonic maps and the interpolation problem for timelike minimal surfaces. Abstract 

3. Anup Biswas.  Title: Nonlocal Ergodic Control Problems.  

Abstract: In this talk, we would present some of the recent results on the nonlocal Hamilton-Jacobi equations related to ergodic control problems.

4.  Ayan Mahalanobis. Title: An attack on the elliptic curve discrete logarithm problem.  Abstract

5. Visakh Narayanan.  Title: Geometry of knots in real projective 3-space.

Abstract: Classically, knot theory was studied in the 3-sphere or the euclidean 3-space. But any three manifold comes with its own knot theory. A manifold very close to the above mentioned two, is the real projective 3-space. In this talk, I would like to discuss how the knot theory of the projective space looks like, through studying some geometric properties of these knots and comparing it with the classical case. In a precise sense, we would see that the knot theory of projective space is an "extension" of classical knot theory.

6. Abhrojyoti Sen.  Title: Fine boundary regularity for fully nonlinear mixed local-nonlocal problems.

Abstract: In this talk,  we will consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation invariant operators. For a bounded $C^2$ domain $\Omega \subset \Rd,$ let $u\in C(\Rd)$ be a viscosity solution of such Dirichlet problem. We obtain global Lipschitz regularity and fine boundary regularity for $u$ by constructing appropriate sub and supersolutions coupled with a \textit{weak version} of Harnack inequality. We apply these results to obtain H\"{o}lder regularity of $Du$ up to the boundary.

7. Vishwakarma Pranjal. Title: Bianchi Modular Forms.

Abstract: A Bianchi modular form, roughly speaking, is a modular form that is defined over SL_2(K) where K is an imaginary qudratic field. Let M_k(Γ) be the space of modular forms of weight k for the subgroup Γ with S_k(Γ) be the subspace of cusp forms. There is a natural complement Ek_(Γ) of S_k(Γ) inside M_k(Γ), Eisenstein subspace of M_k(Γ). In other words, we have a decomposition M_k(Γ) = S_k(Γ) ⊕ E_k(Γ).

In this talk, we will discuss about the  Eisenstein part of cohomology groups, the dimension of the Eisenstein cohomology and the trace on the Eisenstein cohomology groups for complex conjugation using Lefschetz theorem for \Gamma_1(N) congruence subgroups.

8.  Tejas Kalelkar. Title: An algorithm to recognise hyperbolic manifolds.

Abstract:  A Pachner move is a local combinatorial change to the triangulation of a manifold. Any two geometric (ideal) triangulations of a (cusped) complete hyperbolic 3-manifold M are related by a sequence of Pachner moves through topological triangulations. We give a bound on the length of this sequence in terms of the total number of tetrahedra and a lower bound on the dihedral angles in the cusped case and an upper bound on the lengths of edges in the compact case. This leads to an effective algorithm to check the equivalence of hyperbolic manifolds (given geometric ideal triangulations of their complements). This is joint work with Advait Phanse and Sriram Raghunath.

9. Prafullkumar. Title: Ways to Solve Hard Computational Problems.

Abstract: To solve any real-life problem using a computer, one needs to encode it as a computational problem. Consider the case when we want to distribute medical kits such that a person or one of his/her friends has one. Graphs, because of their tremendous expressive power, are often used in such encodings. The above-mentioned problem is the same as finding a so-called `dominating set' in the graph.

The design of algorithms to solve such problems is important in computer science. Unfortunately, the general belief is that there are no efficient algorithms to solve these problems. In this talk, we will see various frameworks developed to cope with this hardness, focusing on parameterized complexity. We will see the applicability of these frameworks to various types of graph problems.

10. Debargha Banerjee. Title: Modular forms and p-adic local Langlands. 

Abstract:- I will give a brief introduction to p-adic local Langlands programs and talk briefly about my recent work on the occurrence of p-adic local langlands in the ordinary or supersingular part of the etale cohomology of modular curves. 

11.  Praphulla Koushik.  Title: On certain vector bundles over Dirac manifolds.

Abstract: Motivated by the study of Poisson vector bundles over Poisson manifolds by Viktor Ginzburg in their work “Grothendieck Groups of Poisson Vector Bundles”(https://dx.doi.org/10.4310/JSG.2001.v1.n1.a4), we introduce the notion of a certain vector bundle over Dirac manifolds and study certain properties. This will be based on our work in progress.

12. Nupur Patanker.  Title: On reversible $\mathbb{Z}_2$-double cyclic codes.     Abstract

13. Chandrasheel Bhagwat. Title: Spectra of geometric spaces.

Abstract: I would briefly describe various notions of 'spectra' (algebraic, geometric, analytic, combinatorial) associated to special type of geometric spaces called the locally symmetric spaces. I would give some idea how these notions are related to each other and describe some of my own work in this context (joint work with C. S. Rajan, Supriya Pisolkar, Ayesha Fatima, Gunja Sachdeva). It is a short talk, and will be accessible to a wider mathematical audience.

14. Puspendu Pradhan. Title: Blocking sets of certain line sets with respect to a quadric in PG(3,q).     Abstract

15. Jishu Das. Title:  A discrepancy result for Hilbert modular forms.

Abstract: Let F be a totally real number field, $r= [F: Q]$, and N be an integral ideal. Let $A_k(N, ω)$ be the space of holomorphic Hilbert cusp forms with respect to $K_1(N)$, weight $k = (k_1, ..., k_r)$ with $k_j> 2$, $k_j$ even for all j and central Hecke character ω. For a fixed level N, we study the behavior of the Petersson trace formula for the Hecke operators acting on $A_k(N, ω)$ as $k_0 → ∞$ where $k_0 = min(k_1, ..., k_r)$ subjected to a given condition. We give an asymptotic formula for the Petersson formula under certain conditions. As an application, we generalize a discrepancy result for classical cusp forms by Jung-Sardari to Hilbert cusp forms for F with the ring of integers O having odd narrow class number 1, and the ideals being generated by numbers belonging to $\mathbb{Z}$. This is joint work in progress with Baskar Balasubramanyam and Kaneenika Sinha.  

16. Debjit Pal. Title: GC structures on certain principal torus bundles.

Abstract: A certain type of  principal torus bundle over a complex  manifold admits a family of generalized complex (GC) structures. We show that such a generalized complex structure is equivalent to the product of the complex structure on the base and the symplectic structure on the fiber in a tubular neighborhood of the fiber. This has consequences for the generalized Dolbeault cohomology of the bundle. This is a joint work with my supervisor Prof. Mainak Poddar.

17. Priyanka Majumder. Title: Asymptotics for Arakelov self intersection number.

Abstract: I will talk about an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf associated with certain modular curves. This Arakelov invariant is basically the sum of a geometric part that encodes the finite intersection of divisors coming from the cusps, and an analytic part that is given in terms of the canonical Green’s function evaluated at these cusps. In my PhD thesis we have computed the analytic contribution, and the computations for the geometric contribution was done jointly with Debargha Banerjee and Chitrabhanu Chaudhuri.

18. Krishna Kaipa. Title: Waring rank of binary forms over finite fields.

Abstract: Let F be a degree d binary form i.e.  a homogeneous polynomial of degree d in two variables X and Y over a field K. The Waring problem is to express  F as a sum of d-th powers of linear forms. The minimum number of linear forms required to do this is called the Waring rank of F.  If the field K is algebraically closed and if d is odd, then the problem is well understood and the solution goes back to Sylvester (early 1900s). If d is even and further if K is not algebraically closed, then the problem is difficult. We will focus on the case d=4 and K a finite field motivated by some problems in algebraic coding theory.

19. Rupak Kumar Dalai. Title: Hardy–Littlewood maximal function on Modulation spaces.

Abstract: The Hardy–Littlewood maximal function is a significant non-linear operator, a classical tool in harmonic analysis. However, it has recently been widely used in studying partial differential equations. The celebrated theorem of Hardy, Littlewood, and Wiener asserts that the maximal function is bounded in L^p(R^n) for 1 < p ≤ ∞, one of the cornerstones of harmonic analysis.

Feichtinger introduced the modulation spaces in the early 1980s, proposing a theory that parallels the better-known Besov spaces. Modulation spaces are a family of Banach spaces defined by the behavior of the short-time Fourier transform.

In this talk, we study some elementary properties of the maximal function and the modulation spaces. And we will talk about the boundedness of the Hardy–Littlewood maximal function on modulation spaces. 

20. Anindya Goswami. Title: Inference of Binary Regime Models with Jump Discontinuities.

Abstract: Identifying the instances of jumps in a discrete-time-series sample of a jump-diffusion model is a challenging task. Using a threshold method, we have developed a novel statistical technique for jump detection and volatility estimation in return time series data. The consistency of the volatility estimator has been obtained. Since we have derived the threshold and the volatility estimator simultaneously by solving an implicit equation, we have obtained unprecedented accuracy across a wide range of parameter values. Using this method, the increments attributed to jumps have been removed from a large collection of historical data on Indian sectoral indices. Subsequently, we have tested the presence of regime-switching dynamics in the volatility coefficient using a new discriminating statistic. The statistic has been shown to be sensitive to the transition kernel of the regime-switching model. We perform the testing using the Bootstrap method and find a clear indication of presence of multiple regimes of volatility in the data. A link to all Python codes is given in the conclusion. The methodology is suitable for analyzing high-frequency data and may be applied to algorithmic trading.

21. Alok Kumar Sahoo. Title: "Existence of Sign-changing solutions to a Hamiltonian elliptic System."

Abstract: For a strongly coupled Hamiltonian elliptic System, the first step is understanding the existence and qualitative properties of non-trivial solutions. The situation becomes more interesting when we consider a sign-changing solution.  This talk is based on the existing part. As we are looking for a sign-changing solution, group action plays a crucial role. We prove the results in two different cases depending on the size orbit of the group action.

22. Tumpa Mahato. Title: Polynomially Knotted 2-Spheres in R^4.

Abstract: We  present polynomial parametrization of a family of knotted spheres ,known as Spun Knots in 4 dimensional space. In general polynomial maps give non-compact knots but here we see that these are compact images.