# Faculty Activities

*Abstract:*

Let X be a random variable defined by X = sum_i a_i x_i where x_i are independent random variables uniformly distributed in {-1, 1}, and a_i are real numbers. We investigate the tail behavior of the variable X, and apply the results to study associated linear threshold functions f:{-1,1}^n→{0,1}, which are indicators of events of the form {sum a_i x_i > t} for real numbers t. A puzzle: Let a = max_i |a_i| and assume sum a_i^2 = 1. Is it true that Pr[|X| <= a] > a/10 ? Joint work with Nathan Keller.

*Abstract:*

Let M be an orientable hyperbolic surface without boundary and let c be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of c in the hyperbolic plane is shorter than c.

The talk will be presented for advanced undergraduate and beginning graduate students.

*Abstract:*

The profinite completion of a free profinite group on infinite set of generators is a profinite group of grater rank. However, it is still unknown whether it is a free profinite group too. I am going to present some partial results regarding to this question.

*Abstract:*

Let G be a group acting on a space X. A natural question to study is the asymptotical behavior of orbits when elements are chosen using some law.

We first look into a specific lattice in SL(2,R) acting on the real projective line when elements are chosen using a certain word norm, and show connections to the Minkowski question mark function, continued fractions and the stationary measure of a closely related random walk.

We then study the random walk on the real plane, generated by action of subgroups in SL(2,R). By applying carefully chosen scaling suggested by Maucourant and using a recent result of Benoist-Quint we reach a reasonable candidate for solution.

*Announcement:*

You are cordially invited to:

** the 33 ^{rd} **

**Elisha Netanyahu Memorial Lecture**

**Professor ****Claire Voisin**

**College de France, France**

**Title**: *Hodge structures in algebraic geometry*

**Abstract**:

Hodge structures of weight 1 appear implicitly in the study of Riemann surfaces and their Jacobians, which are associated complex tori. In higher weight (degree, dimension), the theory of Hodge structures has been developed by Griffiths starting from the Hodge decomposition theorem.

I will explain what a Hodge structure is and how it can be used to study the geometry of algebraic varieties (or Kähler manifolds).

**Reception will be held at 16:30 in the Faculty Lounge Amado Mathematics Building, 8th Floor.**

*Abstract:*

After summarizing 1D periodic Jacobi matrices, I will define periodic Jacobi matrices on infinite trees. I'll discuss the few known results and some interesting examples and then discuss lots and lots of interesting conjectures. This is joint work mainly with Nir Avni and Jonathan Breuer but also with Jacob Christensen, Gil Kilai and Maxim Zinchenko.

It is on the spectral theory of a class of operators on trees, for which there has been literature on the random case even in the theoretical physics literature but I am not aware of any application to anything close to real physics so this is probably better as a math talk but I leave it to you to sort it out if you are interested. I don’t care at all which it is called or even if it is jointly sponsored.