Let F be a non-Archimedean local field. In the representation theory of GL_n(F), one of the basic problems is to characterize its irreducible representations up to isomorphism. There are many invariants (e.g., epsilon factors, L-functions, gamma factors, depth, etc) that we can attach to a representation of GL_n(F). Roughly, the local converse problem is to find the smallest subcollection of twisted local \gamma-factors which classifies the irreducible admissible representations of GL_n(F) up to isomorphism.

First we am going to recall first basic facts about vector bundles on smooth projective curves. Then we will talk about moduli "spaces" of vector bundles on curves. If time permits, we will also talk about related "spaces" like Hecke stacks and moduli "spaces" of shtukas. Key words: Riemann-Roth theorem for curves, vector bundles on curves, degree.

Abstract: The goal of this (and the next) talk is to introduce automorphic L-functions for GL(n) and other split groups, and to discuss some of their properties and some conjectures. Key words: L-functions, Langlands dual group, modular forms

Abstract: The starting point of the geometric approach to the theory of automorphic forms over function fields is a beautiful observation of Weil asserting that there is a natural bijection between the two-sided quotient GL(n,F)\GL(n,A)/GL(n,O) and the set of isomorphism classes rank n vector bundles on a curve. The goal of my talk will be to explain this result and to give some applications. Key words: adeles and ideles in the function field case, algebraic curves, line and vector bundles on curves, Picard group, Riemann-Roch theorem.

Probabilistic limit theorems for (distance expanding and hyperbolic) dynamical systems is a well studied topic. In this talk I will present conditions guaranteeing that a local central limit theorem holds true for certain families of distance expanding random dynamical systems. If time permits, I will also discuss a version of the Berry-Esseen theorem. Joint work with Yuri Kifer.

Randomisations, coheir sequences and NSOP1 [Joint with A Chernikov and N Ramsey] Recall that if T is a theory, then its Keisler randomisation, T^R, is the theory of spaces of random variables which take values in a model of T . It was show some time ago that if T has IP (e.g., simple unstable), then T^R has TP2, and in particular not simple. In Eilat I announced the following result [with Chernikov and Ramsey] : A. If T is NSOP1, then its randomisation T^R is NSOP1

Speaker: Zlil Sela
Title: Basic conjectures and preliminary results in non-commutative algebraic geometry

 Abstract: Algebraic geometry studies the structure of varieties over
 fields and commutative rings. Starting in the 1960's ring theorists
 (Cohn, Bergman and others) have tried to study the structure of varieties
 over some non-commutative rings (notably free associative algebras).

 The lack of unique factorization that they tackled and studied in detail,

Speaker: Zlil Sela
Title: Basic conjectures and preliminary results in non-commutative algebraic geometry

 Abstract: Algebraic geometry studies the structure of varieties over
 fields and commutative rings. Starting in the 1960's ring theorists
 (Cohn, Bergman and others) have tried to study the structure of varieties
 over some non-commutative rings (notably free associative algebras).

 The lack of unique factorization that they tackled and studied in detail,
 and the pathologies that they were aware of, prevented any attempt

Abstract: Let G be a finite abelian group. We say that a given subset of G is uniform if all of its Fourier coefficients are small. We'll show that uniform sets are common and explore some of their nice additive properties.

כותרת: מה הקשר בין גרפים מרחיבים לבין מידות אינוריאנטיות על S^n? תקציר: מידת לבג על הספירה S^n היא סיגמא-אדיטיבית ואינוריאנטית לסיבובים, וזה מאפיין אותה לחלוטין מבין המידות (המנורמלות) על S^n המוגדרות על כל הקבוצות המדידות-לבג. בעית בנך-רוזייביץ', מתחילת שנות ה-20, שואלת: מה לגבי מידות סוף-אדיטיביות? האם מידת לבג היא עדיין היחידה? בשנת 1923, בנך נתן תשובה שלילית למקרה של המעגל S^1. מה לגבי n\geq2? טרסקי עשה את הצעד הראשון לקראת הפתרון (בעזרת פרדוקס בנך-טרסקי!).

Speaker: Michael Chapman Title: Pell's Equation Abstract:Think about the following problems: Find all natural numbers that are both squares and triangular. Definition of Triangular Number Find all non-negative integers a such that a+1 and 3a+1 are both perfect squares. Also, prove that if you take the ascending sequence (a_n)_{n=1}^\infty of such numbers, the number a_n \cdot a_{n+1}+1 is also a perfect square. Prove there are infinitely many triplets of consecutive positive integers, all of them are sums of two squares. For example 8=2^2 +2^2,9=3^2 +0^2,10=3^2+1^2.