Abstract: I will start with a motivation of what algebraic (and model-theoretic) properties an algebraically closed field of characteristic 1 is expected to have. Then I will explain how these properties can be obtained by the well-known in model theory Hrushovski's construction and then formulate very precise axioms that such a field must satisfy. The axioms have a form of statements about existence of solutions to systems of equations in terms of a 'multi-dimansional' valuation theory and the validity of these statements is an open problem to be discussed.

Abstract: In the last five years Bridgeland stability has revolutionized our understanding of the geometry of moduli spaces of sheaves on surfaces, allowing us to compute ample and effective cones and describe different birational models. In this talk, I will survey some of my joint work with Daniele Arcara, Aaron Bertram, Jack Huizenga and Matthew Woolf on the birational geometry of moduli spaces of sheaves on the plane. I will describe the ample and effective cones of these moduli spaces, concentrating on Hilbert schemes of points and concrete examples.

Hurwitz spaces are moduli spaces for extensions of curves with prescribed ramification types. They arise naturally when enumerating extensions of global function fields and also in many other contexts. The classical Hurwitz space H_{m,n} may be viewed as a finite cover of the space of n-sets of points on P^1. We will show that this cover has a big monodromy group for n>4. This can be applied to study the statistics of extensions of F_q(t) with varying branching locus in the large q limit. Joint work with Chris Hall and Robert Guralnick.

The functoriality conjecture is a key ingredient in the theory of automorphic forms and the Langlands program. Given two reductive groups G and H, the principle of functoriality asserts that a map r:H^->G^ between their dual complex groups should naturally give rise to a map r*:Rep(H)->Rep(G) between their automorphic representations. In this talk, I will describe the idea of functoriality, its connection to L-functions and recent work on weak functorial lifts to the exceptional group of type G_2.

In his investigation of modular forms of half-integral weight, Shimura established, using Hecke theory, a family of relations between eigneforms of half-integral weight k+1/2 with a given level 4N and character chi and cusp forms of weight 2k and character chi^2. The level being subsequently determined by Niwa to be at most 2N.

Abstract: Let X be a regular scheme, projective and flat over Spec Z. We give a conjectural formula in terms of motivic cohomology, singular cohomology and de Rham cohomology for the special value of the zeta-function of X at any rational integer. We will explain how this reduces to the standard formula for the residue of the Dedekind zeta-function at s = 1. ‏האירוע הזה כולל שיחת וידאו ב-Google Hangouts.

Abstract: Calabi conjectured that the complex Monge-Ampère equation on compact Kaehler manifolds has a unique solution. This was solved by Yau in 1978. In this talk, we present a non-archimedean version on projective Berkovich spaces. In joint work with Burgos, Jell, Künnemann and Martin, we improve a result of Boucksom, Favre and Jonsson in the equicharacteristic 0 case. We give also a result in positive equicharacteristic using test ideals.

In this talk we present a database of rational elliptic curves with good reduction outside certain finite sets of primes, including the set {2, 3, 5, 7, 11}, and all sets whose product is at most 1000. In fact this is a biproduct of a larger project, in which we construct practical algorithms to solve S-unit, Mordell, cubic Thue, cubic Thue--Mahler, as well as generalized Ramanujan--Nagell equations, and to compute S-integral points on rational elliptic curves with given Mordell--Weil basis. Our algorithms rely on new height bounds, which we obtained using the

Abstract: In the talk I will discuss classical problems concerning the distribution of square-full numbers and their analogues over function fields. The results described are in the context of the ring Fq[T ] of polynomials over a finite field Fq of q elements, in the limit q → ∞. I will also present some recent generalization of these kind of classical problems. ‏האירוע הזה כולל שיחת וידאו ב-Google Hangouts.

Using the endoscopic classification of automorphic forms for unitary groups, I will prove conjecturally sharp upper bounds for the growth of Betti numbers in congruence towers of complex hyperbolic manifolds. This is joint work with Sug Woo Shin. ‏האירוע הזה כולל שיחת וידאו ב-Google Hangouts. הצטרף: https://plus.google.com/hangouts/_/calendar/ODdkc2JxNmlmbjNhZ2U0ODVvb3E3...

Let L(E/Q, s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the central value L(E, 1, χ) of twists of L(E/Q, s) by Dirichlet characters χ. We discuss the vanishing and non-vanishing frequencies of these values as χ ranges over characters of fixed order greater than 2. We also examine thee square-free part of the algebraic part of L(E/F, 1) for abelian fields F/Q when these values are non-zero.

Abstract: In modern algebraic geometry we encounter the notion of derived intersection of subschemes. This is a sophisticated way to encode what happens when two subschemes Y_1 and Y_2 of a given scheme X intersect non-transversely. The classical intersection multiplicity can be extracted from the derived intersection.

We prove cases of Rietsch mirror conjecture that the quantum connection for projective homogeneous varieties is isomorphic to the pushforward D-module attached to Berenstein-Kazhdan geometric crystals. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. In particular we link the purity of Berenstein-Kazhdan crystals to the Ramanujan property of certain Hecke eigensheaves. The isomorphism of D-modules comes from global rigidity results where a Hecke eigenform is determined by its local ramification. We reveal