Schedule

Title. Character varieties and the moduli spaces of flat connections

Abstract. First, we will start by introducing the character variety of surface group representations as a tool to study the moduli space of flat (smooth) connections over a compact surface. We will give some examples for lower genus and how the character variety can be obtained as a quasi-Hamiltonian reduction. We will also show how to obtain examples of higher genus via doing fusions of examples of lower genus. Next, we will consider flat meromorphic connections. In particular, we will look at a Lie theoretic way to describe the Stokes data at the pole. This allows the moduli space of meromorphic connections to be obtained as a quasi-Hamiltonian reduction also. If time permits, we hope to end the talk with an example that relates the Coxeter plane with the singular directions of a pole of some special meromorphic connection. 

References

1. A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps, J. Diff. Geo., 48 (1998), 445--495. 

2. P. Boalch, Quasi-Hamiltonian geometry of meromorphic connections, Duke Math. J., 139 (2007), 369--405. 

Lecture 1. Fock-Goncharov moduli spaces of G-local systems on a surface abstract. 

Abstract. For a punctured oriented surface S and a split reductive algebraic group G, the G-character variety or the G-character stack for S is the moduli space of G-local systems on S. This space can be viewed as the space of group homomorphisms from the fundamental group of S to G, considered up to conjugation. Fock and Goncharov, and later Goncharov and Shen, considered certain modifications of this space, and proved that these have structures of cluster varieties. I will review these constructions. 

Lecture 2. On canonical bases of regular functions on Fock-Goncharov moduli spaces 

Abstract. The original form of Fock and Goncharov's duality conjectures predicts an existence of a canonical basis of regular functions on one version of the Fock-Goncharov moduli space associated to a surface S and an algebraic group G. This basis should be enumerated by the tropical integer points of the dual version of the Fock-Goncharov moduli space associated to S and the Langlands dual of G. I will briefly introduce the contents of this conjecture, and mention some recent achievements about it, including Gross, Hacking, Keel and Kontsevich's theta bases from the theory of mirror symmetry of log Calabi-Yau varieties.

Title: WKB analysis and Floer theory 

Abstract: I will give a brief introduction to WKB analysis and discuss some aspect from the point of view of Floer theory. This is based on my ongoing joint work with Kohei Iwaki and Tatsuki Kuwagaki.

Title: Hurwitz numbers, W-algebras, and topological recursion 

Abstract: Monotone Hurwitz numbers count the number of certain classes of brach covers of the sphere, and it is well known that the generating function of monotone Hurwitz numbers satisfies so-called Virasoro constraints. Algebraically, the Virasoro algebra admits a different name, the W-algebra of type A_1, and as the name suggests, there are other types of W-algebras (e.g. type A_r). Then, an interesting question arises: is there any higher rank analogue of monotone Hurwitz numbers corresponding to the W-algebra of type A_r? In these lectures, I will first give an overview of the story of monotone Hurwitz numbers, and then explore how they can be generalised into a higher rank setting. If time permits, I will also introduce the third character of the story, topological recursion, which serves as the "mirror dual" of Hurwitz numbers. These lectures are partly based on my recent work joint with Nitin Chidambaram and Maciej Dolega.

Title. Mirror symmetry among a noncommutative resolution of a chain of projective lines, a marked bordered surface and a generalized root system of multi-extended $A$ type.

Abstract. This lecture is based on the joint work in progress with Akishi Ikeda, Takumi Otani and Atsushi Takahashi. In this lecture, at first, I will explain how to derive a generalized root system $R$ of multi-extended $A$ type from the homological mirror symmetry between the derived category of left modules for a certain quiver algebra obtained as a localization of noncommutative resolution of a chain of projective lines and the partially wrapped Fukaya category $\W(S, \ell)$ of the corresponding marked bordered surface $S$ with the suitable choice of a line field $\ell$. At second, I will explain the Saito theory for the LG model associated to $S$ and a primitive form associated to $\ell$. The resulting Frobenius manifold can be considered as a mirror to "Gromov-Witten theory for a noncommutative manifold". Finally, if time permits, I will explain the (still conjectural) construction of a flat structure on the orbit space of the extended Weyl group for $R$ which should be isomorphic to the Frobenius manifold mentioned above, especially by focusing the structure of the discriminant and a choice of primitive direction.

Title: Set of full exceptional collections and mirror symmetry 

Abstract For ADE singularities, Deligne gave a characterization of sets of distinguished bases and a recursion relation for their cardinalities, and proved Looijenga's conjecture on their coincidence with the degrees of Lyashko-Looijenga map which capture topological information of bifurcation sets. Categorifying distinguished bases into full exceptional collections in derived directed Fukaya categories motivated by the idea of homological mirror symmetry, I shall explain how Deligne's recursion and comparison of degrees of Lyashko-Looijenga map with numbers of full exceptional collections  can naturally be generalized for the cases of Affine ADE and elliptic DE cases.

Title. 3-dimensional mirror symmetry

Abstract. In this lecture series, I will give a brief overview of the program of 3-dimensional mirror symmetry. Along the way, we will introduce basic ideas of geometric representation theory and topological quantum field theory.