August 2022 semester
12/10/2022: Professor Manoj Kumar Yadav (HRI, Pragayraj)
Chair:Prof. Anupam Singh
Time -4 to 5 pm
Title: Construction of skew braces
Abstract: I am planning to present a mostly pedagogical treatment of a very recently introduced concept called skew left brace. It was invented in connection with solutions of the Yang-Baxter equation. I'll indicate this connection and show how to construct a sequence of skew left braces from a given one. To satisfy curiosity, if any, an algebraic structure $(G, +, *)$ is said to be a skew left brace if $(G, +)$ and $(G, *)$ are groups and + and * speak to each other as follows: $a*(b+c) = a*b -a +a*c$ for all $a, b, c \in G$.
13/10/2022: Dr. Vinay Kumaraswamy (TIFR)
Time-2 - 3 pm
Chair: Dr. Kaneenika Sinha
Title: Counting rational points on cubic hypersurfaces
Abstract: Let X be the cubic hypersurface cut out by a non-degenerate rational cubic form in n variables. If n is at least 14, then Heath-Brown has shown that the Hasse principle holds for X. By a result of Kollar, this implies that there are infinitely many rational points on X. In this talk, I will discuss the problem of obtaining quantitative lower bounds for the number of rational points of bounded height on X.
21/10/2022: Dr. Prafullkumar Tale (IISER, Pune)
Chair:-Dr. Soumen Maity
Time -4 to 5 pm
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 have been developed to cope with this hardness, focusing on parameterized complexity. We will see the applicability of these frameworks to various types of graph problems.
27/10/2022: Dr. Anna Seigal (Harvard University)
Chair: Professor Rama Mishra
Time -4 to 5 pm
Title: Ranks of tensors and connections to data analysis
Abstract: Linear algebra is the foundation to methods for finding structure in matrix data. There are many challenges in extending this to the multi-linear setting of tensors. I will discuss the comparison of rank and symmetric rank for a tensor, including recent progress in finding tensors whose rank and symmetric rank differ (in joint work with Kexin Wang, building on work of Yaroslav Shitov). We will see connections to classical algebraic geometry, via cubic surfaces, and small open problems of a combinatorial nature.
Sumit Chandra Mishra (IISER, Mohali)
Chair:Professor Anupam Singh.
Date:9/11/22
TItle: A Ruled Residue Theorem for function fields of elliptic 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? In this talk, we will consider these questions for the function field of elliptic curves. This is a joint work with Karim J. Becher and Parul Gupta.
Professor Ritabrata Munshi (ISI, Kolkata)
Date-11/11/2022
Venue-ZOOM
Time-3-4 pm.
Title: Bounds for L-functions
Abstract: This will be a survey talk on the subconvexity problem. After a brief historical introduction, we will discuss the several aspects of the problem and the different techniques that are used to tackle them. At the end we will mention some recent results.
Professor Nitin Nitsure (TIFR).
Date-18/11/22
Time 4-5 pm.
Speaker: Nitin Nitsure.
Title: Curvature and torsion for parallel connections: a new approach.
Abstract: In geometry, we often want to compare two small figures in space, which may be far away from each other. In spaces such as the Euclidean space, where there is a large symmetry group, this is done via group action.But most spaces are not very symmetric, that is, they do not have enough automorphisms. Then a comparison between distant figures involves `parallel translating' one of them to the other along a curve. This uses a differential geometric device called a connection on the tangent bundle. The departure of the geometry from the Euclidean model shows itself in the presence of what are called as the curvature and a torsion tensor fields. Invented by Gauss and Riemann, these objects are of fundamental importance in both mathematics and physics. In this talk, I will begin with a general introduction to these ideas. Then I will explain a simple new way to determine the curvature and torsion tensors, via quadrilateral gaps. This research was instigated by questions posed to me by Professor Rajaram Nityananda, a renowned physicist.
Reference: Nitin Nitsure `Curvature, torsion and the quadrilateral gaps' Proc. Indian Acad. Sci, 2021.
DOI: 10.1007/s12044-020-00596-2
arXiv:1910.06615
- Chandranandan Gangopadhyay (IISER, Pune)
Date-25/11/22.
Title: Picard Groups of some Quot Schemes
Abstract: Let E be a vector bundle on a smooth complex projective curve C of genus at least 1. Fix integers k, d > 1. Let Q denote the Quot scheme of quotients of E of rank k and degree d. For large d, it was proved by Popa and Roth that Q is irreducible and generically smooth. In this talk we will show that for large d, Q is in fact locally factorial. As a consequence, we compute the Picard group of Q for large d. This is a joint work with Ronnie Sebastian.
Information about the seminars can be accessed here:
9. Professor Haruzo Hida (02/12/22).
Title: Adjoint L-value formula and its relation to Tate conjecture,
Abstract: For a Hecke eigenform $f$, we present an adjoint L-value formula relative to each quaternion algebra $D$ over ${\mathhbb Q}$ with reduced norm $N$ (which is a natural generalization of Siegel's mass formula). A key to prove the formula is the theta correspondence for the quadratic ${\mathhbb Q}$-space $(D,N)$. Under the $R=T$-theorem, $p$-part of the Bloch-Kato conjecture is known; so, the formula is an adjoint Selmer class number formula. We also describe how to relate the formula to a consequence of the Tate conjecture for quaternionic Shimura varieties.
10. Speaker: Gurleen Kaur, IISER Mohali
Title: Central units of integral group rings
Abstract:
In this talk, we are concerned with the rank of Z(U(ZG)) and large subgroups of Z (U (ZG)), i.e., subgroups of finite index in Z (U (ZG)). Both of these prob- lems have been the center of attraction for several decades starting with the work of Higman and require a deep understanding of the structure of G and that of the rational group algebra QG. In a joint work with 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. We have also seen the vast- ness of the class of generalized strongly monomial groups. We proved that every supermonomial group is generalized strongly monomial. We have seen the impact of this study on the group of central units of an integral group ring. We proved that the generalized Bass units generate a subgroup of finite index in the center Z(U(ZG)) of the unit group U(ZG) for generalized strongly monomial group G. Furthermore, for a generalized strongly monomial group G, the rank of Z(U(ZG)) is determined. The formula so obtained is in terms of generalized strong Shoda pairs of G. This generalizes the corresponding work done by Jespers, Olteanu, del Rio and Van Gelder.
Date: 6 December 22, at 4pm
Venue: Madhav hall
10. Professor Jacques Tilouine (09/12/22)
TITLE: Results and Conjectures on Integral period relations
Abstracts:
in a series of works with E. Urban, we investigate integral period relations for an automorphic
form and its image by functoriality, in some cases. In the case of $GL_2$ and for quadratic base change, our results are rather complete. In the case of $GL_2$ and for $Symm^2$, we can formulate a conjecture in which a new phenomenon seems to appear. A potential proof for one divisibility predicted by this conjecture is a joint work in progress.
11. Jotsaroop Kaur, IISER Mohali (16/12/2022)
Title: Equiconvergence for perturbed Jacobi polynomial expansions
Abstract: We show asymptotic expansions of the eigenfunctions of certain perturbations of the Jacobi operator in a bounded interval, deducing equiconvergence results between expansions with respect to the associated orthonormal basis and expansions with respect to the cosine basis. Several results for pointwise convergence then follow. This is a joint work with Giacomo Gigante (University of Bergamo, Italy).
12. Jaya Iyer, IMSC (20/12/2022).
Title: Lefschetz theorems in ALgebraic Geometry
Abstract: We will discuss Chow Lefschetz questions and provide certain
explicit examples which fulfil injectivity of homological Chow
Lefschetz morphisms.
13. Arpan Kabiraj (IIT, Pallakad) 29/12/2022
Title: Center of skein algebras associated to loops on surfaces
Abstract: We will discuss a method (using hyperbolic geometry) to compute the center of various skein algebras introduced by Turaev for the quantization of Poisson algebras of loops introduced by Goldman and Wolpert in 80’s. We will also discuss how these methods can be used to compute the center of homotopy skein algebra introduced by Hoste and Przytycki.
14. Professor Somnath Jha, IIT, K(30/12/22).
Title: Selmer group and ideal class group.
Abstract: The relation between isogeny induced Selmer group of an elliptic curve and ideal class group has been extensively studied; from the work of Cassels to the recent works of Bhargava et. al. and Chao-Li. In this talk, we will explore the relation between 3-isogeny induced Selmer group of an elliptic curve and the (3 part of) the ideal class group, over certain number fields. We will apply this to a classical Diophantine problem related to rational cube sum. This is a joint work with D. Majumdar and P. Shingavekar.