Contact person: Dietmar Bisch

Contact person: Larry Schumaker

Contact person: Guoliang Yu

Contact person: Gieri Simonett

Contact person: Mike Plummer

Contact person: Paul Edelman

Contact person: Dietmar Bisch

I will describe the ideas and methods underlying coarse geometry. The
relation with curvature is given by a geometric differential operator
(the
Dirac operator). The index theory for this operator gives various
obstructions to positive scalar curvature. I will present some of
these
obstructions on open manifolds and draw conclusions for general closed
manifolds.

Contact person: Guoliang Yu

Kazhdan property T originated from the representation
theory of Lie groups. Shortly after its introduction it was
used by Margulis to construct an explicit example of a
family of expanders. Unfortunately, the expanding constant
of this family was unknown, because all proofs that a group
has a property T were not quantitative, and the expanding
constants of this family of expanders was unknown.

In a resent paper, A. Lubotzky and I. Pak showed that
Kazhdan property T of the group SL_n(Z) implies that the
working time of the product replacement algorithm on
k-generated abelian groups is logarithmic in the size of
the groups, but its dependence on k was unknown.

A recent result by Y. Shalom gave an explicit bound of the
Kazhdan constant for the group SL_n(Z), which lead to
quantitative bounds for the constants in these two
combinatorial constructions.

Contact person: Mark Sapir

The fundamental group F(M)
of a type II₁ factor M was introduced by
Murray and von Neumann in 1943 in connection with their notion of
continuous dimension. It measures the extent to which “amplifications”
of M are isomorphic to M (e.g., if the algebra of
2×2 matrices over M is isomorhic to M then 2 is in F(M)}.
It is a puzzling and still poorly understood invariant.

We will present results providing the first examples of factors M
with trivial fundamental group. Thus, if G is the arithmetic group
Z² \rtimes SL(2, Z) and M=L(G) is the associated
group von Neumann algebra then F(M)={1}. The proof uses in a
crucial way the “weak amenability” of \Gamma=SL(2, Z)
(i.e. Haagerup’s approximation property or equivalently Gromov’s
a-T-menability) and the relative property (T) of Kazhdan-Margulis
of the inclusion Z² \subset Z² \rtimes \Gamma.
The combination of these two properties makes it possible to prove
a unique decomposition of M as a cross-product
M = L^\infty(T²) \rtimes \Gamma, thus allowing us to define
l²-Betti number invariants b_n(M) from the
l²-Betti numbers b_n(R), defined by Gaboriau
in 2001, of the equivalence relation R induced by \Gamma on
T².

Contact persons: Dietmar Bisch and Gennadi
Kasparov

I originated this theory in 1981–84 (with the considerable help of my
then graduate student David Hobby).
In this talk, I will tell the story of how a long-running fascination
with one little problem and several big problems, combined with
stubbornness and luck, led to some big results.

Contact person: Dietmar Bisch

Operator space theory is a natural non-commutative quantization of
functional analysis (Banach space theory). In this talk, I will first
discuss some fundamental results in operator spaces, and then
discuss some interesting applications to operator algebras and
non-commutative harmonic analysis.

Contact person: Guoliang Yu and Dechao
Zheng

Contact persons: Dietmar Bisch and Bruce
Hughes

Contact person: Mark Sapir

Abstract:
The notion of skew-symmetric bicharacter on a Hopf algebra is dual to
that of
R-matrix widely used in mathematics and beyond. It appears in the theory of
quantum groups, Yang – Baxter equations, etc. While R-matrices work in the
case of finite-dimensional Hopf algebras, bicharacters are more universal
and work fine in the case of infinite dimensions. One of the basic examples
of bicharacters are the commutation factors on abelian groups used, in
particular, to define so called color Lie superalgebras, a notion generated
in physics few decades ago. In general, bicharacters on a commutative and
cocommutative Hopf algebra H allow one to define Lie structures on the
algebras with the coaction H. On the other hand, Hopf algebras whose
structure includes a fixed skew-symmetric bicharacter form an important
class of cotriangular Hopf algebras, dual to the triangular ones
introduced by Drinfeld. If we want to classify such Lie structures or such Hopf
algebras, we may apply so called Scheunert’s trick, saying that any
skew-symmetric
bicharacter b(g,h) on a finitely generated abelian group G can be written in
the form b(g,h)=a(g,h)[s(g,h)/s(h,g)], where a(g,h) is either trivial or the
bicharacter defining ordinary Lie superalgebras and s(g,h) is a not
necessarily skew-symmetric bicharacter. In particular, if we deform the
product in a G-graded algebra using s(g,h), then the color commutator defined
by b(g,h) becomes either an ordinary Lie bracket or an ordinary superbracket.
The goal of this talk is to report on most recent result in this area,
including the extension of Scheunert’s trick to arbitrary cocommutative Hopf
algebras of characteristic different from 2 and the classification of
finite-dimensional algebras, which are commutative under a suitable
generalized Lie bracket.

Contact person: Mark Sapir