Abstract: We will begin by reviewing the basic theory of modular forms and Eisenstein series and then present major results achieved in generalizing Ramanujan's congruences. We will end with an outline of some active open problems driving current progress in this area. 

Abstract: The concept of derived category was conceived by Grothendieck in the 1960s as a natural habitat for all sorts of left and right derived functors which had been inundating the literature after the arrival of homological algebra. Although these functors are indispensable today and have been of great technical aid, they lack a satisfactory general framework when viewed individually. Grothendieck being Grothendieck realised these left and right derived functors are shadows of a higher structure. Together with his student Verdier, he introduced derived categories and proved Serre’s duality theorem in greater generality showcasing its conceptual advantage. Initially derived categories were seen simply as an abstract framework, however the work of Mukai on derived categories of abelian varieties, Bondal-Orlov reconstruction theorem and Kontsevich’s Homological Mirror Symmetry conjecture have changed this perception. In this talk, we will begin with Serre duality and its derived-categorical incarnation, then move to Bondal-Orlov reconstruction theorem, as well as see a few of the derived invariant properties of smooth projective varieties. If time permits, we will also discuss the work of Mukai and Orlov on derived categories of abelian varieties. The goal of this talk is to provide a brief introduction to some of the research directions centred around derived categories, aimed at a broad audience.

Abstract: A classical theorem of Gelfand and Kazhdan states that a natural involution on the general linear group sends every irreducible representation to its dual. Motivated by a conjecture of Dipendra Prasad, we study an analogous question for finite reductive groups. Using Lusztig’s Jordan decomposition of characters, we relate this duality property to associated unipotent characters and their Frobenius eigenvalues. As a consequence, we obtain a large class of finite reductive groups for which every irreducible character is mapped to its dual under Prasad’s duality involution. This talk will focus on the main ideas and examples rather than technical details. 

Abstract: This introductory talk aims to discuss the linear algebraic group, root system, and root datum associated with a connected reductive linear algebraic group, with particular emphasis on the general linear group GLn. Further, the structure theory of connected reductive linear algebraic groups will be explained.

Prerequisite: First Course in Algebraic Geometry. 

Abstract:  A quiver is defined as a directed graph with an attitude towards representation theory. In this talk, I will introduce quiver, quiver representations, path algebra (which is an associative algebra), Morita equivalence and discuss a fundamental classification result due to Gabriel. If time permits, I will also discuss one possible answer to the question, “Why are quivers?” There are no prerequisites, and there will be many examples.  

Abstract:  In this expository talk, we will see some five notions of “canonical” Kähler metrics, three of which are classical and have been very well studied by the stalwarts of the field, while the other two were introduced quite recently (with their definitions being motivated by the classical ones) and have not received much attention. Canonical Kähler metrics are seen as “canonical” representatives of their respective Kähler classes in some nice ways, viz. They enjoy some nice properties with respect to their various notions of curvature. The classical notions of these metrics appear as the critical points (in fact minimizers) of some energy functionals defined on their Kähler classes, and are hence the solutions to some nice PDEs on the underlying complex manifold, while the newly introduced ones are simply defined as the solutions to some analogous PDEs. We will give a brief review of the key results about the existence and uniqueness of these metrics on various kinds of compact complex manifolds, including an overview of some algebro-geometric obstructions to the existence of these metrics (or equivalently to the solvability of the corresponding PDEs).

Prerequisites: Differential Geometry, Basics of Riemannian Geometry, Complex Analysis, Some basic knowledge of Complex Manifolds would be helpful, though we will be quickly recalling the required definitions.