Abstract. This is a joint work with Linhui Shen. A decorated surface is an oriented surface with punctures and a finite collection of special points on the boundary, considered modulo isotopy. Let G be a split adjoint group. We introduce a moduli space Loc(G,S) of G-local systems on a decorated surface S, which reduces to the character variety when S has no boundary, and quantize it.

Class field theory classifies abelian extensions of local and global fields in terms of groups constructed from the base. We shall survey the main results of class field theory for number fields and function fields alike. The goal of these introductory lectures is to prepare the ground for the study of explicit class field theory in the function field case, via Drinfeld modules. I will talk for the first 2 or 3 times.

There is a general slogan according to which the limit behaviour of a one-parameter family of complex algebraic varieties when the parameter t tends to zero should be (partially) encoded in the associated t-adic analytic space in the sense of Berkovich; this slogan has given rise to deep and fascinating conjecturs by Konsevich and Soibelman, as well as positive results by various authors (Berkovich, Nicaise, Boucksom, Jonsson...).

Abstract: We combine a technique of Steel with one due to Jensen and Steel to obtain a core model below singular cardinals kappa which are sufficiently closed under the beth function, assuming that there is no premouse of height kappa with unboundedly many Woodin cardinals. The motivation for isolating such core model is computing a lower bound for the strength of the theory: T = ''ZFC + there is a singular cardinal kappa such that the set of ordinals below kappa where GCH holds is stationary and co-stationary''.

Abstract: We combine a technique of Steel with one due to Jensen and Steel to obtain a core model below singular cardinals kappa which are sufficiently closed under the beth function, assuming that there is no premouse of height kappa with unboundedly many Woodin cardinals. The motivation for isolating such core model is computing a lower bound for the strength of the theory: T = ''ZFC + there is a singular cardinal kappa such that the set of ordinals below kappa where GCH holds is stationary and co-stationary''.

I will discuss results relating different partially wrapped Fukaya categories. These include a K\"unneth formula, a `stop removal' result relating partially wrapped Fukaya categories relative to different stops, and a gluing formula for wrapped Fukaya categories. The techniques also lead to generation results for Weinstein manifolds and for Lefschetz fibrations. The methods are mainly geometric, and the key underlying Floer theoretic fact is an exact triangle in the Fukaya category associated to Lagrangian surgery along a short Reeb chord at infinity.

Speaker: Kim Minki, Technion Title: The fractional Helly properties for families of non-empty sets Abstract: Let $F$ be a (possibly infinite) family of non-empty sets. The Helly number of $F$ is defined as the greatest integer $m = h(F)$ for which there exists a finite subfamily $F'$ of cardinality $m$ such that every proper subfamily of $F'$ is intersecing and $F'$ itself is not intersecting. For example, Helly's theorem asserts that the family of all convex sets in $d$-dimensional Euclidean space has Helly number $d+1$.

Speaker: Roy Meshulam (Technion) Title: Topology and combinatorics of the complex of flags Abstract:

Speaker: Eyal Karni, BIU Title: Combinatorial high dimensional expanders Abstract: An eps-expander is a graph G=(V,E) in which every set of vertices X where |X|<=|V|/2 satisfies |E(X,X^c)|>=eps*|X| . There are many edges that "go out" from any relevant set.

Speaker: Raphy Yuster, U. Haifa Title: On some Ramsey type problems in tournaments Abstract: I will talk about several Ramsey type problems in tournaments guaranteeing the existence of subgraphs with certain chromatic properties. Here are two such problems which attracted some attention recently: 1. Let g(n) be the smallest integer such that every tournament with more than g(n) vertices has an *acyclic subgraph* with chromatic number larger than n. It is straightforward that Omega(n) <= g(n) <= n^2. An improvement of both bounds is presented.

Speaker: Arindam Banerjee, RMVERI Title: Castelnuovo-Mumford Regularity, Combinatorics and Edge Ideals. Abstract: