Abstract: Quiver Hecke algebras, or KLR algebras, have been a revolution in the representation theory of the symmetric group and more generally of Hecke-type algebras of a type A flavour. They opened the world of graded representation theory for these algebras, also including the Temperley--Lieb algebra and its one-boundary generalisation. All this is thanks to the Brundan-Kleshchev-Rouquier isomorphism relating KLR algebras and affine Hecke algebras of type A. In this talk I will explain a generalisation of some of these results for affine Hecke algebras of type B/C, involving the so-called orientifold KLR algebras. I will discuss connections with quantum groups and quantum symmetric pairs and hopefully have time to present a recent application to two-boundary Temperley--Lieb algebras.

Speaker(s):

Speaker(s):

Abstract: In the first part of the talk, I will introduce the BPS sheaf associated
with the preprojective algebra of a quiver. This is a perverse sheaf on
the moduli space of representations, endowed with a Lie algebra
structure, which encodes the Kac polynomials of the quiver. The Lie
algebra structure is described in terms…

A representable natural transformation u : U* —> U in the category Psh(C) of presheaves on a small category C is a “natural model" of dependent type theory. The type-forming operations may be described as algebraic structure on u, representing corresponding operations on the type-families classified by u. For example, the dependent product or “Pi-type” is an algebra structure for the polynomial endofunctor

P_u : Psh(C) —> Psh(C) .

Similar operations on u represent the other type-formers of unit type, dependent sums, and identity types. The latter are given by a newly discovered “path-type” structure, which relates such models to cubical (Quillen) model categories.

Abstract: In this talk, I will introduce the nilpotent cohomological Hall algebra COHA(S,Z) of coherent sheaves on a smooth quasi-projective complex surface S that are set-theoretically supported on a closed subscheme Z. This algebra can be viewed as the "largest" algebra of cohomological Hecke operators associated with modifications along a subscheme Z of S. When S is the minimal resolution of an ADE singularity and Z is the exceptional divisor, I will describe how to characterize COHA(S,Z) in terms of the Yangian of the corresponding affine ADE quiver Q (based on joint work with Emanuel Diaconescu, Mauro Porta, Oliver Schiffmann, and Eric Vasserot, arXiv:2603.03386). More generally, I will discuss nilpotent COHAs arising from Bridgeland stability conditions on the bounded derived category of nilpotent representations of the preprojective algebra of Q, following joint work with Olivier Schiffmann and Parth Shimpi (arXiv:2511.08576).

We show that various classical theorems of linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a general result that involves a tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing and interpreting the…

Abstract: TBA