The Toda lattice is one of the earliest examples of non-linear
completely integrable systems. Under a large deformation, the
Hamiltonian flow can be seen to converge to a billiard flow in a
simplex. In the 1970s, action-angle coordinates were...
Symmetric power functoriality is one of the basic cases of
Langlands' functoriality conjectures and is the route to the proof
of the Sato-Tate conjecture (concerning the distribution of the
modulo p point counts of an elliptic curve over Q, as the...
A large toolbox of numerical schemes for dispersive equations
has been established, based on different discretization techniques
such as discretizing the variation-of-constants formula (e.g.,
exponential integrators) or splitting the full equation...
This talk is based on a joint work with Steve Lester.
We review the Gauss circle problem, and Hardy's conjecture
regarding the order of magnitude of the remainder term. It is
attempted to rigorously formulate the folklore heuristics behind
Hardy's...
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...
