Combinatorial group theory began with Dehn's study of surface
groups, where he used arguments from hyperbolic geometry to solve the
word/conjugacy problems. In 1984, Cannon generalized those ideas to all
"hyperbolic groups", where he was able to give a solution to the
word/conjugacy problem, and to show that their growth function satisfies
a finite linear recursion. The key observation that led to his
discoveries is that the global geometry of a hyperbolic group is determined locally:
first, one discovers the local picture of G, then the recursive structure
Combinatorial group theory began with Dehn's study of surface
groups, where he used arguments from hyperbolic geometry to solve the
word/conjugacy problems. In 1984, Cannon generalized those ideas to all
"hyperbolic groups", where he was able to give a solution to the
word/conjugacy problem, and to show that their growth function satisfies
a finite linear recursion. The key observation that led to his
discoveries is that the global geometry of a hyperbolic group is determined locally:
first, one discovers the local picture of G, then the recursive structure
I'll tell a couple of anecdotes related to imaginary quadratic fields
(e.g. primes in the sequence n^2+n+41), and then open a new story --
local CFT and the explicit construction of K^ab due to Lubin-Tate.
I will give an introduction to sheaves and microlocal sheaves, as pioneered by Kashiwara-Schapira. The goal will be to explain recent work with Shende establishing that microlocal sheaves on a Weinstein manifold are a symplectic invariant.
Abstract: Any birational geometer would agree that the best algorithm
for resolution of singularities should run by defining a simple invariant of
the singularity and iteratively blowing up its maximality locus.
The only problem is that already the famous example of Whitney umbrella
shows that this is impossible, and all methods following Hironaka had
to use some history and resulted in more complicated algorithms.
Nevertheless, in a recent work with Abramovich and Wlodarczyk we did
construct such an algorithm, and an independent description of a similar
Abstract:
Paul L\'evy's classical arcsine law states that the occupation time ratio of one-dimensional Brownian motion for the positive side is arcsine-distributed. The arcsine law has been generalized to a variety of classes of stochastic processes and dynamical systems.
