Abstract: Modular forms are historically the first example of automorphic
forms, and are still studied today as they have many applications. In
this talk I want to introduce modular forms, give some examples, and, if
time permits, explain the connection to elliptic curves, objects we
already met in the first lecture.

Title: An improved bound for Hadwiger’s covering problem via thin shell inequalities for the convolution square.
Abstract: A long-standing open problem, known as Hadwiger’s covering problem, asks what is the smallest natural number N(n) such that every convex body in {\mathbb R}^n can be covered by a union of the interiors of at most N(n) of its translates. Despite continuous efforts, the best-known general upper bound for this number remain as it was more than half a decade ago, and is of the order of \binom{2n}{n}n\ln n.

We present an approach to computing the scaling limits of Christoffel-Darboux kernels using transfer matrix evolution.

Speaker: Ehud de Shalit
Title:Periods and Hodge theory, complex and p-adic
P-adic theory
p-adic Hodge theory started with Tate in the 1960's, and witnessed two periods of great expansion: one by Fontaine and his school, the other one more recently, by Scholze and his collaborators. It has found spectacular applications in arithmetic algebraic geometry. Scholze got the Fields medal last August for this work. By analogy with the first lecture we shall try to explain the main ideas, without going into details

Speaker: Ehud de Shalit
Title:Periods and Hodge theory, complex and p-adic
Complex theory
This will be a survey of complex Hodge theory, with applications to period maps and moduli and with an emphasis on the example of abelian varieties. It will cover classical results of Riemann, Siegel, Hodge and Griffiths

This is a new seminar, whose official name is "Topics in number theory
and algebraic geometry". At least in the beginning the goal
of the seminar will be to give a (relatively) gentle introduction to
various topics, which should be accessible to beginning but motivated graduate students.
The seminar has a number in the shnaton (80942), so graduate students
can get a credit for it.
First talk: The goal of this organizational/introductory talk will be to
describe areas of mathematics, connected to Langlands program.

Abstract:
The Chabauty method is a remarkable tool which employs p-adic analitic methods (in particular Colman integration.) To study rational points on curves. However the method can be applied only when the genus of the curve in question is larger than its Mordell-Weil rank. Kim developed a sophisticated "nonableian" generalisation.
We shall present the classical methid, and give an approachable introduction to Kim's method.
I'm basically going to follow http://math.mit.edu/nt/old/stage_s18.html

The Whitehead conjecture asks whether a subcomplex of an aspherical 2-complex is always
aspherical. This question is open since 1941. Howie has shown that the existence of a finite counterexample implies (up to the Andrews-Curtis conjecture) the existence of a counterexample within the class of labelled oriented trees. Labelled oriented trees are algebraic generalisations of Wirtinger presentations of knot groups.

Title: Szego theorem for measures on the real line: optimal results and
applications.
Abstract: Measures on the unit circle for which the logarithmic integral
converges can be characterized in many different ways: e.g., through
their Schur parameters or through the predictability of the future from
the past in Gaussian stationary stochastic process. In this talk, we
consider measures on the real line for which logarithmic integral exists
and give their complete characterization in terms of the Hamiltonian in

Speaker: Bruno Benedetti, U. Miami
Title: Local constructions of manifolds
Abstract: Starting with a tree of tetrahedra, say you are allowed to
recursively glue together some two boundary triangles that have nonempty
intersection. You may perform this type of move as many times you want.
Let us call "Mogami manifolds" the triangulated 3-manifolds (with or
without boundary) that can be obtained this way. Mogami, a quantum
physicist, conjectured in 1995 that all triangulated 3-balls are Mogami.

Speaker: Tammy Ziegler, HU
Title: Extending weakly polynomial functions from high rank varieties
Abstract: Let k be a field, V a k-vector space, X in V a subset. Say that f: X —> k is weakly polynomial of degree a if its restriction to any isotropic subspace is a polynomial degree of a. We show that if X is a high rank variety then any weakly polynomial function of degree a is the restriction to X of a polynomial of degree a on V. Joint work with D. Kazhdan.

Speaker: Ohad Klein, BIU
Title: Biased halfspaces, noise sensitivity, and local Chernoff inequalities
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 in R, the reals. Assume Var(X)=1=sum a_{i}^2. We investigate the tail behavior of the variable X, and apply the results to study halfspace functions f:{-1,1}^{n}-->{-1,1} defined by f(x)=1 (\sum_i a_i x_i > t) for some t in R.
A puzzle: Let a = max_{i} |a_{i}|. Is it true that Pr[|X| \leq a] \geq a/10?

Speaker: Spencer Backman, HU
Title: Cone valuations, Gram's relation, and flag-angles

Speaker: Igor Pak, UCLA
Title: Counting linear extensions.
Abstract:

Speaker: Noam Lifshitz, BIU
Title: Sharp thresholds for sparse functions with applications to extremal combinatorics.
Abstract:
The sharp threshold phenomenon is a central topic of research in the analysis of Boolean functions. Here, one aims to give sufficient conditions for a monotone Boolean function $f$ to satisfy $\mu_{p}(f)=o(\mu_{q}(f))$, where $q = p + o(p)$, and $\mu_{p}(f)$ is the probability that $f=1$ on an input with independent coordinates, each taking the value $1$ with probability $p$.