I will discuss a proof of the existence of infinitely many
solutions for the singular Yamabe problem in spheres using
bifurcation theory and the spectral theory of hyperbolic
surfaces.
The Langlands and Fontaine–Mazur conjectures in number theory
describe when an automorphic representation f arises geometrically,
meaning that there is a smooth projective variety X, or more
generally a Chow motive M in the cohomology of X, such...
On an elliptic curve $y^2=x^3+ax+b$, the points with coordinates
$(x,y)$ in a given number field form a finitely generated abelian
group. One natural question is how the rank of this group changes
when changing the number field. For the simplest...
I will describe joint work with Boris Solomyak, in which we show
that the stationary (Furstenberg) measure on the projective line
associated to 2x2 random matrix products has the "correct"
dimension (entropy / Lyapunov exponent) provided that the...
An important question in hydrodynamic turbulence concerns the
scaling proprties in the inertial range. Many years of experimental
and computational work suggests---some would say, convincingly
shows---that anomalous scaling prevails. If so, this...
Deligne's "Weil II" paper includes a far-reaching conjecture to
the effect that for a smooth variety on a finite field of
characteristic p, for any prime l distinct from p, l-adic
representations of the etale fundamental group do not occur
in...
Worst-case analysis of algorithms has been the central method of
theoretical computer science for the last 60 years, leading to
great discoveries in algorithm design and a beautiful theory of
computational hardness. However, worst-case analysis...
At leading order, the fluctuations around the typical dynamics are
described by the second cumulant. They actually satisfy a
stochastic PDE with time-space white noise. Can we say more using
higher order cumulants?
The problem of finding metrics with constant Q-curvature in a
prescribed conformal class is an important fourth-order cousin of
the Yamabe problem. In this talk, I will explain how certain
variational bifurcation techniques used to prove non...
Although the distribution of hard spheres remains essentially
chaotic in this regime, collisions give birth to small
correlations. The structure of these dynamical correlations is
amazing, going through all scales. How combinatorial techniques
can...
