July 21 - 25 School

Schedule

• Roman Bezrukavnikov (MIT, HSE International Laboratory of Representation Theory and Mathematical Physics)

  Categorification and dualities in representation theory

  Abstract: I will start with discussion of Lusztig's theory of character sheaves and D-modules which will be presented as an example of categorification.

  I will then talk about an approach to the affine analogue of this theory. Time permitting I will touch upon a (partly conjectural) relation to abelian subcategories in the derived categories of coherent sheaves.

• Tom Braden (University of Massachusetts, Amherst)

  Recent developments in modular sheaf theory

  Abstract: Constructible sheaves on flag varieties, nilpotent cones, and related spaces have long played a central role in geometric approaches to representation theory. However, until recently they have almost always been taken with coefficients in a field of characteristic zero, in part because powerful tools from Hodge theory and D-modules are not available for sheaves with other coefficients. This mini-course will survey some of the recent applications of constructible sheaves with positive characteristic coefficients to modular representation theory, with emphasis on explicit examples and methods for computation.

• Ivan Losev (Northeastern University)

  Quantized quiver varieties

  Abstract: Nakajima quiver varieties are remarkable symplectic algebraic varieties of interest for Geometric Representation theory, Algebraic Geometry and Mathematical Physics. Their quantizations are interesting associative algebras. I will discuss the representation theory of these quantizations and emphasize connections to the geometry.

• David Nadler (UC Berkeley)

  Elliptic character sheaves

  Abstract: We will discuss approaches to character sheaves for loop groups via Geometric Langlands of the two-torus. For the spectral approach (appearing in preprints 1312.7164, 1312.7164), we will explain basic aspects of the microlocal geometry of coherent sheaves. For the automorphic approach (initial steps appearing in 1302.7053), we will focus on approaches to an elliptic horocycle transform. The results intertwine various strands joint with D. Ben-Zvi (Texas), P. Li (Berkeley), A. Preygel (Berkeley).

• Geordie Williamson (Max-Planck-Institut für Mathematik)

  Hodge theory and Soergel bimodules

  Abstract: De Cataldo and Migliorini have given us a beautiful proof of the decomposition theorem using only classical Hodge theory. I will explain their proof and how one can adapt it to certain settings (Soergel bimodules) where the underlying space (and hence classical Hodge theory) might be missing. The positivity of the Hodge-Riemann relations is provided by simple positivity properties in Coxeter systems. Most of what I will discuss is joint work with Ben Elias.

• Andrei Okounkov (Columbia University, HSE International Laboratory of Representation Theory and Mathematical Physics)

  Representation theory and enumerative geometry

• Ivan Danilenko (Independent U. Moscow)

  Khovanov-Rozansky Homologies and Cabling

• Tanmay Deshpande (IPMU)

  Geometric idempotents and character sheaves on algebraic groups

• Akishi Ikeda (U. Tokyo)

  Stability conditions on $N$-Calabi-Yau categories associated to $A_n$- quivers

• Tina Kantrup (Aahus)

  Quasi-coherent Hecke category and Demazure Descent

• Yoshiyuki Kimura (RIMS)

  Graded quiver varieties and quantum cluster algebras

• Nicolas Libedinsky

  Representations of algebraic groups, Soergel bimodules approach.

• Masaki Mori (U. Tokyo)

  Iwahori-Hecke algebras in non-integral rank

• Masanari Okumura (U. Tokyo)

  Chiral Equivariant Cohomology and Representations of Lie Algebroids

• Laura Rider (MIT)

  Geometric Satake Equivalence and Modular Representation Theory

• Yosuke Saito (Tohoku U.)

  Elliptic Ding-Iohara-Miki Algebra and Related Topics

• Oleksandr Tsymbaliuk (MIT)

  The affine Yangian of $\mathfrak{gl}_1$ and quantum toroidal algebras

• Jon Xu (U. Melbourne)

  Schubert varieties and finite geometry