The interface between water and vacuum (governed by the "water
wave equation"), and the interface between oil and water in sand
(governed by the "Muskat equation") can develop singularities in
finite time. Joint work with A. Castro, D. Cordoba, F...
A basic fact of algebraic graph theory is that the number of
connected components in an undirected graph is equal to the
multiplicity of the eigenvalue zero in the Laplacian matrix of the
graph. In particular, the graph is disconnected if and only...
The problem of combinatorial auctions is one of the basic
questions in algorithmic mechanism design: how can we allocate/sell
m items to n agents with private valuations of different
combinations of items, so that the agents are motivated to...
In 1964 Arnold constructed an example of instabilities for
nearly integrable systems and conjectured that generically this
phenomenon takes place.
There has been big progress attacking this conjecture in the past
decade. Jointly with Ke Zhang we...
I will talk about two natural notions of convergence for
sequences of graphs of bounded degree and their connection to
groups and group actions. The first is Benjamini-Schramm
convergence, which is strongly related to parameter testing. The
second...
We describe some results concerning the number of connected
components of nodal lines of high frequency Maass forms on the
modular surface. Based on heuristics connecting these to a critical
percolation model, Bogomolny and Schmit have conjectured...
