Organized by Semeon Artamonov and Nicolai Reshetikhin
*-special day/time
Carlsson and Mellit introduced the Dyck path algebra and its polynomial representation, which was used to prove some important conjectures in algebraic combinatorics. I will define this algebra and construct its action on the equivariant K-theory of certain smooth strata in the flag Hilbert schemes of points on the plane. In this presentation, the fixed points of torus action correspond to generalized Macdonald polynomials and the the matrix elements of the operators have explicit combinatorial presentation. The talk is based on a joint work with Erik Carlsson and Anton Mellit.
We propose a categorification of link invariants in Euclidean 3d space associated to a semi-simple Lie algebra, based on category of A-branes in finite dimensional Landau-Ginzburg (LG) models. The category of A-branes in such abstract LG model was constructed recently by Gaiotto, Moore and Witten; its mathematical counterpart is a version of Fukaya-Seidel category. The specific Landau-Ginzburg model needed is derived from string theory. We explain the relation to some other approaches to the same problem.
Khovanov and Rozansky defined a link invariant called triply graded homology. It is conjectured by Gorsky, Negut and Rasmussen that this invariant can be expressed geometrically by a functor from complexes of Soergel bimodules to the derived category of coherent sheaves on the dg flag Hilbert scheme followed by taking cohomology. A functor with similar properties has been constructed by Oblomkov and Rozansky using matrix factorizations and it is believed that this functor solves the conjecture. The aim of this joint work in progress with Roman Bezrukavnikov is to relate the two constructions using previous work of Arkhipov and Kanstrup.
In the category of tangles objects are even natural numbers and morphisms between n and m are (n,m)-tangles up to isotopy. Weak triangulated represenations of this category are usually constructed in terms of generators and relations: we associate a functor to each tangle generator (a "cup", a "cap", or a "crossing"), and prove that certain compositions of these functors are isomorphic. It turns out that if we impose additionally a certain skein relation, which is not intrinsic to the tangle category but requires a triangulated representation, all tangle relations follow from those that only involve cups and caps. We are going to discuss the analogue of this result for the category of sl_n webs. This talk is based on joint work with Timothy Logvinenko
The notion of topological field theory was formalized by Michael Atiyah; it is a purely mathematical notion inspired by physics. In particular, such a theory gives invariants of closed d-manifolds.
Examples of 3-dimensional topological field theories have been well studied, most notably Reshetikhin-Turaev and Turaev-Viro theories. However, in dimension 4, situation is much less understood.
In this talk, we give an overview of one construction of a 4-dimensional topological field theory based on the notion of pre-modular category; in particular, we give computation of invariants which such a theory would associate to some 2-dimensional surfaces.
Galois algebras is an important class of algebras with invariant skew group structure that allow an effective study of their Gelfand-Tsetlin representations. We will give an overview of recent advances based on joint results with D.Grantcharov, E.Ramirez, P.Zadunaisky and J.Zhang.
Matrix Factorizations were introduced by Eisenbud to study minimal resolutions of Cohen-Macaulay modules. The notion was rediscovered from a physics perspective, where such factorizations appeared as boundary conditions for topological quantum field theory, and led to the (curved) deformation theory of the category of coherent sheaves on complex manifolds. An important stability result here is the Knoerrer periodicity theorem, the invariance of the MF category under Cartesian crossings with non degenerate quadratic functions. I will describe a generalization of this to Morse-Bott functions. The answer involves the full Gerstenhaber structure on the Hochschild complex of a manifold, instead of the more commonly used Lie structure.
The pentagram map was introduced by Richard Schwartz in 1992, and is now one of the most renowned discrete integrable systems which has deep connections with such topics as cluster algebras, dimer models etc. In this talk I will present a geometric construction which identifies the pentagram map, as well as its various multidimensional generalisations, with refactorization type mappings in Poisson-Lie groups.
Virtual knot theory is a generalization of classical knot theory that studies stabilized knots and links in thickened surfaces. Two knots (links) in thickened surfaces are said to be stably equivalent if they can be obtained one from another by a finite sequence of ambient isotopies along with surgeries on their complements (that can change of genus of the embedding surface). There is a diagrammatic theory that captures stable equivalence. One adds virtual crossings (neither over nor under) and rules for handling them that generalize the Reidemeister moves. Then virtual knots can be studied using strictly planar diagrams. This means that one has access to both a rich background of combinatorial topology and the three dimensional topology of the thickened surfaces. This talk will discuss the basic definitions for virtual knot theory and the construction of a number of invariants of interest, including the Jones polynomial, the arrow polynomial and the affine index polynomial, Khovanov homology and relations with virtual knot cobordism. We will discuss how quantum link invariants extend to virtual knot theory and we will attempt to discuss how virtual knot theory could or should be related to physics and quantum information theory.
We will discuss the Arkhipov's twisting functor associated to a positive root of a complex simple finite-dimensional Lie algebra. By applying this twisting functor for a non-simple root on generalized Verma modules we obtain the so called partial Gelfand-Tsetlin modules, which are weight modules outside the category $\mathcal{O}$. The talk is based on joint results with Vyacheslav Futorny and Luis Enrique Ramirez.