Related papers: Modular generalized Springer correspondence: an ov…
These are the lecture notes from the 26th Winter School "Geometry and Physics", Czech Republic, Srni, January 14 - 21, 2006. These lectures are an introduction into the realm of generalized geometry based on the tangent plus the cotangent…
This talk is dedicated to various aspects of Mirror Symmetry. It summarizes some of the mathematical developments that took place since M. Kontsevich's report at the Z\"urich ICM and provides an extensive, although not exhaustive,…
In this lecture note, we give a basic introduction to the rapidly developing concepts of generalized symmetries, from the perspectives of both high energy physics and condensed matter physics. In particular, we emphasize on the (invertible)…
We study de Rham character sheaves on a commutative connected algebraic group $G$, defined as multiplicative line bundles with integrable connection. We construct a group algebraic space $G^\flat$ representing their moduli problem on…
These notes are an extended version of the authors' lectures at the 2013 CMI Workshop "Mixed Hodge Modules and Their Applications". We give an overview, with an emphasis on applications, of recent developments on the interaction between…
I review recent developments of "non-semisimple" modular tensor categories in the sense of Lyubashenko and the categorical Verlinde formula for such categories (this is a proceedings article for Meeting for Study of Number theory, Hopf…
Let $(G,G')$ be a reductive dual pair of a symplectic group and an orthogonal group over a finite field of odd characteristic. The Howe correspondence establishes a correspondence between a subset of irreducible characters of $G$ and a…
Based upon new global class field concepts leading to Langlands two-dimensional global correspondences,a modular representation of cusp forms is proposed in terms of global elliptic (bisemi)modules which are (truncated) Fourier series over…
The general structure of the representation theory of a $Z_2$-graded coalgebra is discussed. The result contains the structure of Fourier analysis on compact supergroups and quantisations thereof as a special case. The general linear…
Comments on the results presented at the Conference "Frontiers Beyond the Standard Model," FTPI, Oct. 2012. This summary traces a historical perspective. v2: A reference corrected and a footnote added; v3: a few grammar mistakes and typos…
We introduce and motivate -- based on ongoing joint work with Germ\'an Stefanich -- the notion of potent categorical representations of a complex reductive group $G$, specifically a conjectural Langlands correspondence identifying potent…
This article is based on my lecture notes from summer schools at the Universities of Utah (June 2007) and Warwick (September 2007). We provide an introduction to explicit methods in the study of moduli spaces of quiver representations and…
These Lectures are based on a course on noncommutative geometry given by the author in 2003 at the University of Chicago. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild…
This note is devoted to some questions about the representation theory over the finite field $\mathbb{F}_2$ of the general linear groups $\mathbb{GL_n(F_2)}$ and Poincar\'e series of unstable modules. The first draft was describing two…
This article is based on lecture notes prepared for the August 2006 Cologne Summer School. The first part contains background material and references for beginners. The second (and main) part is a survey of the current status in the theory…
The Springer modules have a combinatorial property called ``coincidence of dimensions,'' i.e., the Springer modules are naturally decomposed into submodules with common dimensions. Morita and Nakajima proved the property by giving modules…
The article surveys published and not yet published results about moduli spaces of algebraic surfaces.
These notes are an extension of the rough notes provided for my four lecture graduate level course on "Quadratic Forms and Automorphic Forms" at the March 2009 Arizona Winter School on Quadratic Forms. They are meant to give a survey of…
We describe the Springer correspondence explicitly for exceptional Lie algebras of type $G_2$ and $F_4$ and their duals in bad characteristics, i.e. in characteristics 2 and 3.
We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for a certain type of Courant algebroids, called…