Low-dimensional topology and geometry have many problems with an
easy formulation, but a hard solution. Despite our intuitive
feeling that these problems are "hard", lower or upper bounds on
algorithmic complexity are known only for some of them...
We present an overview of elementary methods to study extensions
of modular representations of various types of "groups". We shall
begin by discussing actions of an elementary
abelian pp-group, E=(Z/p)rE=(Z/p)r, on finite dimensional
vector spaces...
There are striking analogies between topology and arithmetic
algebraic geometry, which studies the behavior of solutions to
polynomial equations in arithmetic rings. One expression of these
analogies is through the theory of etale cohomology, which...
(joint work with Assaf Naor) A key problem in metric geometry
asks: given metric spaces X and Y, how well does X embed in Y? In
this talk, we will consider this problem for the case of the
Heisenberg group and explain its connections to geometric...
You can make a paper Moebius band by starting with a 1 by L
rectangle, giving it a twist, and then gluing the ends together.
The question is: How short can you make L and still succeed in
making the thing? This question goes back to B. Halpern and
C...
Matroids are combinatorial objects that model various types of
independence. They appear several fields mathematics, including
graph theory, combinatorial optimization, and algebraic geometry.
In this talk, I will introduce the theory of matroids...
Several well-known open questions, such as: "are all groups sofic
or hyperlinear?", have a common form: can all groups be
approximated by asymptotic homomorphisms into the symmetric groups
Sym(n) (in the sofic case) or the unitary groups U(n) (in...
Diagonalization is incredibly important in every field of
mathematics. I am a representation theorist, so I will start by
motivating the uses of diagonalization in representation theory.
Then comes a brief introduction to categorical representation...
Hilbert’s 23rd Problem is the last in his famous list of problems
and is of a different character than the others. The description is
several pages, and basically says that the calculus of variations
is a subject which needs development. We will...
