et K be a convex body in Rn. In some cases (say when K is a
cube), we can tile Rn using translates of K. However, in general
(say when K is a ball) this is impossible. Nevertheless, we show
that one can always cover space "smoothly" using translates...
Following Birkhoff's proof of the Pointwise Ergodic Theorem, it
has been studied whether convergence still holds along various
subsequences. In 2020, Bergelson and Richter showed that under the
additional assumption of unique ergodicity, pointwise...
Can you hear the shape of LQG? We obtain a Weyl law for the
eigenvalues of Liouville Brownian motion: the n-th eigenvalue grows
linearly with n, with the proportionality constant given by the
Liouville area of the domain (times a certain...
Graph Crossing Number is a fundamental and extensively studied
problem with wide ranging applications. In this problem, the goal
is to draw an input graph G in the plane so as to minimize the
number of crossings between the images of its edges. The...
Many cohomology theories in algebraic geometry, such as
crystalline and syntomic cohomology, are not homotopy invariant.
This is a shame, because it means that the stable motivic homotopy
theory of Morel--Voevodsky cannot be employed in studying
Given a flow on a manifold, how to perturb it in order to create
a periodic orbit passing through a given region? While the first
results in this direction were obtained in the 1960-ies, various
facets of this question remain largely open. I will...
Determining the complexity of matrix multiplication is a
fundamental problem of theoretical computer science. It is
popularly conjectured that matrices can be multiplied in
nearly-linear time. If true, this conjecture would yield
Phenomena involving interactions of waves happen at different
scales and in different media: from gravitational waves to the
waves on the surface of the ocean, from our milk and coffee in the
morning to infinitesimal particles that behave like wave...
I will start by introducing and motivating the (two-component)
Coulomb gas on the d-dimensional lattice Zd. I will then present
some puzzling properties of the fluctuations of this Coulomb gas.
The connection of this model with integer-valued fields...