Organized by Semeon Artamonov, Nicolai Reshetikhin, Vera Serganova, and Alexander Shapiro.

Sep. 11 | Calvin McPhail-Snyder | Holonomy invariants from quantum groups |

Sep. 25 | Michael Shapiro | Noncommutative Pentagram map |

Oct. 17* | Milen Yakimov | Derived actions of groupoids of 2-Calabi-Yau categories |

Oct. 23 | Vladimir Fock | Riemann geometry without indices |

Nov. 6 | Peter Koroteev | The Quantum DELL System |

Nov. 19* | Benjamin Hoffman | String domains for coadjoint orbits |

* - special day/time

Geometric information about a topological space X can be described by a conjugacy class of representations of its fundamental group into a Lie group G, or equivalently by a gauge class of flat g-connections. In this talk I will discuss how to construct invariants of such spaces using quantum topology techniques, focusing in particular on U_{q}(sl_{2}) at a root of unity. If time permits I will also explain a connection to the twisted Reidemeister torsion.

A pentagram map is a discrete integrable transformation on the space of projective classes of (twisted) n-gons in projective plane. We will discuss a classical and non-commutative version of pentagram map and its integrability properties.

Starting with work of Seidel and Thomas, there has been a great interest in the construction of faithful actions of various classes of groups on derived categories (braid groups, fundamental groups of hyperplane arrangements, mapping class groups). We will describe a general construction of this sort in the setting of algebraic 2-Calabi-Yau triangulated categories. It is applicable to categories coming from algebraic geometry, cluster algebras and topology. To each algebraic 2-Calabi-Yau category, we associate a groupoid, defined in an intrinsic homological way, and then construct a representation of it by derived equivalences. In a certain general situation we prove that this action is faithful and that the green green groupoid is isomorphic to the Deligne groupoid of a hyperplane arrangement. This applies to the 2-Calabi-Yau categories arising from algebraic geometry. We will also illustrate this construction for categories coming from cluster algebras, where one gets categorical actions of braid groups. This is a joint work with Peter Jorgensen (Newcastle University).

The talk will be devoted to a well know subject - basic theorems of Riemann geometry - Bianchi identities, conformal invariance of the Weyl tensor, variation of the Hilbert functional and some others. We suggest a formalism to give a few line proofs of these statements based on formulation of Riemann geometry as a gauge theory in terms of Clifford forms as well as the action of a group sl(2) \times sl(2) thereon. Analogous formalism for Kähler geometry involves the action of the affine group \widehat{sl}(4). Joint work with P.Goussard.

We propose quantum Hamiltonians of the double elliptic many-body integrable system (DELL) and study its spectrum. These Hamiltonians are certain elliptic functions of coordinates and momenta. Our results provide quantization of the classical DELL system which was previously found in the string theory literature. The eigenfunction for the N-body model is conjectured to be a properly normalized equivariant elliptic genus of the affine Laumon space (in physics terms — an instanton partition functions of 6d SU(N) gauge theory with adjoint matter compactified on a torus with a codimension two defect). As a byproduct we discover new family of symmetric orthogonal polynomials which provide an elliptic generalization to Macdonald polynomials.

Partial tropicalizations are a kind of Poisson manifold built using techniques of Poisson-Lie theory and the geometric crystals of Berenstein-Kazhdan. They provide a bridge between linear Poisson manifolds and cones which parametrize the canonical bases of irreducible G-modules. I will talk about applications of partial tropicalization theory to questions in symplectic geometry. For each regular coadjoint orbit of a compact group, we construct an exhaustion by symplectic embeddings of toric domains. As a by-product we arrive at a conjectured formula for the Gromov width of coadjoint orbits. We prove similar results for multiplicity-free spaces. This is joint work with A. Alekseev, J. Lane, and Y. Li.