Numerous problems in extremal hypergraph theory ask to determine the maximal size of a k-uniform hypergraph on n vertices that does not contain an 'enlarged' copy H^+ of a fixed hypergraph H. These include well-known problems such as the Erdos-Sos 'forbidding one intersection' problem and the Frankl-Furedi 'special simplex' problem.

Abstract: We will prove a sharp average-case variant of a classical embedding theorem of John through the theory of nonlinear spectral gaps. We will use this theorem to provide a new answer to questions of Johnson and Lindenstrauss (1983) and Bourgain (1985) on metric dimension reduction, and explain how it leads to algorithms for approximate nearest neighbor search.

A finite graph is automatically also a metric space, but is there any interesting geometry to speak of? In this lecture I will try to convey the idea that indeed there is very interesting geometry to explore here. I will say something on the local side of this as well as on the global aspects. The k-local profile of a big graph G is the following distribution. You sample uniformly at random k vertices in G and observe the subgraph that they span. Question - which distributions can occur? We know some of the answer but by and large it is very open.

Consider a simplicial complex that allows for an embedding into R^d. How many faces of dimension d/2 or higher can it have? How dense can they be?
This basic question goes back to Descartes. Using it and other rather fundamental combinatorial problems, I will motivate and introduce a version of Grothendieck's "standard conjectures" beyond positivity (which will be explored in detail in the Sunday Seminar).
All notions used will be explained in the talk (I will make an effort to be very elementary)

Abstract:
The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold.
An alternative partition, based on the gradient field of the eigenfunction, is via the so called Neumann domains.
A Neumann domain of an eigenfunction is a connected component of the intersection between the stable
manifold of a certain minimum and the unstable manifold of a certain maximum.
We introduce this subject, discuss various properties of Neumann domains and
point out the similarities and differences between nodal domains and Neumann domains.

In the early 1970’s, Hindman proved a beautiful theorem in
additive Ramsey theory asserting that for any partition of the set of
natural numbers into finitely many cells, there exists some infinite set
such that all of its finite sums belong to a single cell.
In this talk, we shall address generalizations of this statement to the
realm of the uncountable. Among other things, we shall present a
negative partition relation for the real line which simultaneously
generalizes a recent theorem of Hindman, Leader and Strauss, and a

Abstract: I will discuss a collection of results about lattices and their subgroups in Euclidean space which are obtained using dynamics on homogeneous spaces. The ergodic theory of group actions on spaces obtained by quotienning a Lie group by a lattice (spaces of lattice-type) or on projective spaces are extensively studied and display distinct dynamical phenomena. Motivated by classical questions in Diophantine approximation we are led to study the ergodic theory of group actions on hybrid homogeneous spaces which are half projective and half of lattice type.

By a theorem of Borel and Harish-Chandra,
an arithmetic group in a semisimple Lie group is a lattice.
Conversely, by a celebrated theorem of Margulis,
in a higher rank semisimple Lie group G
any irreducible lattice is an arithmetic group.
The aim of this lecture is to survey an
arithmeticity criterium for discrete subgroups
which are not assumed to be lattices.
This criterium, obtained with Miquel,
generalizes works of Selberg and Hee Oh
and solves a conjecture of Margulis. It says:

A family of sets F is said to satisfy the (p,q)-property if among any p sets in F, some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any p > q > d there exists a constant c = c_d(p,q), such that any family of compact convex sets in R^d that satisfies the (p,q)-property, can be pierced by at most c points. Helly's Theorem is equivalent to the fact that c_d(p,p)=1 (p > d).

A tribute to the Legacy of Marina Ratner

For more information and registration click here. Read more about The 22nd Midrasha Mathematicae : Equidistribution, Invariant Measures and Applications

Speaker: Yuval Filmus, Technion
Title: Structure of (almost) low-degree Boolean functions
Abstract:
Boolean function analysis studies (mostly) Boolean functions on {0,1}^n.
Two basic concepts in the field are *degree* and *junta*.
A function has degree d if it can be written as a degree d polynomial.
A function is a d-junta if it depends on d coordinates.
Clearly, a d-junta has degree d.
What about the converse (for Boolean functions)?
What if the Boolean function is only *close* to degree d?