Recently Nicolas and Serre have determined the structure of the
Hecke algebra acting on modular forms of level 1 modulo 2, and
Serre has conjectured the existence of a universal Galois
representation over this algebra. I'll explain the proof of...
One of the major insights of the ``fixed-parameter
tractability’’ (FPT) approach to algorithm design is that, for many
NP-hard problems, it is possible to efficiently *shrink* instances
which have some underlying simplicity. This preprocessing
can...
We prove a strong limitation on the ability of entangled provers
to collude in a multiplayer game. Our main result is the first
nontrivial lower bound on the class MIP* of languages having
multi-prover interactive proofs with entangled provers...
I will give an introduction to symplectic geometry and
Hamiltonian systems and then introduce an invariant called
symplectic cohomology. This has many applications in symplectic
geometry and has been used a lot especially in the last 5-10
years...
We study the hole probability of Gaussian entire functions. More
specifically, we work with entire functions given by a Taylor
series with i.i.d complex Gaussian random variables and arbitrary
non-random coefficients. A 'hole' is the event where the...
A twenty-year old conjecture by Manickam, Mikl\'os, and Singhi
asked whether for any integers $n, k$ satisfying $n \ge 4k$, every
set of $n$ real numbers with nonnegative sum always has at least
$\binom{n-1}{k-1}$ $k$-element subsets whose sum is...
I will give an introduction to the problem of parallel
repetition of two-prover games and its applications and related
results in theoretical computer science (the PCP theorem, hardness
of approximation), mathematics (the geometry of foams,
tiling...
