In the last forty years, the study of singularity formation has
mostly concerned model problems and focusing non linearities. In
this second lecture, we will try to give a unified overview on the
known mechanisms of singularity formation, with in...
Given i.i.d. samples drawn from an unknown distribution over a
large domain [N], approximating several basic quantities, such as
the distribution's support size and its Shannon Entropy, requires
at least roughly (N / \log N) samples [Valiant and...
The 4th Clay Millenium problem has a very simple formulation:
may viscous incompressible fluids form a singularity in finite
time? The answer is no in dimension two as proved by Leray in 1932,
but the three dimensional problem is out of reach. More...
I will discuss some recent progress in analytic number theory
for polynomials over finite fields, giving strong new estimates for
the number of primes in arithmetic progressions, as well as for
sums of some arithmetic functions in arithmetic...
In the past decade convex integration has been established
as a powerful and versatile technique for the construction of weak
solutions of various nonlinear systems of partial differential
equations arising in fluid dynamics, including the Euler...
