Ruzsa asked whether there exist Fourier-uniform subsets of ℤ/Nℤ
with very few 4-term arithmetic progressions (4-AP). The standard
pedagogical example of a Fourier uniform set with a "wrong" density
of 4-APs actually has 4-AP density much higher than...
In the last few years, expanders have been used in fast graph
algorithms in different models, including static, dynamic, and
distributed algorithms. I will survey these applications of
expanders, explain the expander-related tools behind this...
In a vertex expanding graph, every small subset of vertices
neighbors many different vertices. Random graphs are near-optimal
vertex expanders; however, it has proven difficult to create
families of deterministic near-optimal vertex expanders, as...
A symplectic embedding of a disjoint union of domains into a
symplectic manifold M is said to be of Kahler type (respectively
tame) if it is holomorphic with respect to some (not a priori
fixed) integrable complex structure on M which is compatible...
The quantum unique ergodicity conjecture of Rudnick and Sarnak
concerns the mass equidistribution in the large eigenvalue limit of
Laplacian eigenfunctions on negatively curved manifolds. This
conjecture has been resolved by Lindenstrauss when this...
We discuss a one-phase degenerate free boundary problem which
arises from the minimization of the so-called Alt-Phillips
functional. We establish partial regularity results for the free
boundary and discuss the rigidity of global minimizers when...
I will discuss pointwise ergodic theory as it developed out of
Bourgain's work in the 80s, leading up to my work with Mirek and
Tao on bilinear ergodic averages.
Let C be a class of metric spaces. We consider the following
computational metric embedding problem: given a vector x in R^{n
choose 2} representing pairwise distances between n points, change
the minimum number of entries of x to ensure that the...
A finite set system is union-closed if for every pair of sets in
the system their union is also in the system. Frankl in 1979
conjectured that for any such system there exists an element which
is contained in ½ of the sets in that system (the only...
Given a convex billiard table, one defines the set swept by
locally maximizing orbits for convex billiard. This is a remarkable
closed invariant set which does not depend (under certain
assumptions) on the choice of the generating function. I...
