Dmitri Pavlov's invited talks
- 2011-5-3:
- Jones index via a symmetric monoidal bicategory of von Neumann algebras.
Notre Dame Topology Seminar.
Abstract: I will describe a new symmetric monoidal structure
on the bicategory of von Neumann algebras, bimodules and intertwiners,
which is motivated by conformal and Euclidean field theories.
I will then demonstrate how the bicategorical formalism of
shadows of 1-morphisms and traces of 2-morphisms
developed by Ponto and Shulman yields the Jones index
in a purely categorical way.
- 2010-12-1:
- Bivariant 0|1-dimensional field theories and de Rham homology and cohomology.
University of Utrecht talk organized by André Henriques.
Abstract: This talk is an introduction to bivariant field theories in the Stolz-Teichner program.
I will discuss the easiest non-trivial case, namely
0-dimensional bivariant field theories with one supersymmetry.
It turns out that the resulting bi-cycles are combinations
of currents and forms, as in the de Rham homology and cohomology.
Finally, I will give hints as to what these simple field theories
might teach us about higher dimensional ones, in particular K-homology
and KK-theory.
- 2010-8-6:
- 2|1-dimensional Euclidean field theories and noncommutative L^p-spaces.
FRG Workshop on mathematical 2D-field theory and the algebraic topology of closed manifolds
at Stony Brook University.
Abstract: A conjecture by Stolz and Teichner states that concordance classes
of 2|1-dimensional Euclidean field theories are in bijective correspondence with
cohomology classes of the cohomology theory TMF (topological modular forms).
Here a field theory is a functor from the bicategory of 2|1-dimensional Euclidean bordisms
to the bicategory of von Neumann algebras, L^p-bimodules, and their morphisms.
A significant amount of labor is required to make the definitions of
the two bicategories mentioned above precise.
Most of the talk will be devoted to a rigorous definition of the algebraic bicategory
of von Neumann algebras, L^p-bimodules, and their morphisms, which involves proving
several theorems about noncommutative L^p-spaces.
If time permits, I will also explain how the study of 2|1-dimensional Euclidean field theories
naturally leads to consider such interesting structures as one-parameter semigroups of bimodules
and two-parameter semigroups of bimodule endomorphisms further parametrized
by the moduli space of elliptic curves.
- 2009-10-20:
- Tensor products of noncommutative L_p-spaces and equivalences of categories of L_p-modules. Oberseminar C*-Algebren at the University of Münster by Joachim Cuntz and Siegfried Echterhoff.
Abstract:
In the first part of this talk I will introduce Haagerup's
theory of noncommutative $L_p$-spaces using the nice algebraic formalism
of modular algebras by Yamagami.
(Here $L_p=L^{1/p}$, in particular, $L_0=L^\infty$ and $L_{1/2}=L^2$.)
Then I will discuss some interesting properties of the resulting $L_p$-spaces,
in particular I will prove the following theorem:
$L_p(M)\otimes_M L_q(M)=L_{p+q}(M)$ for an arbitrary von Neumann algebra~$M$
and arbitrary complex $p$ and $q$ with nonnegative real parts.
Equality here means isometric isomorphism of $M$-$M$-bimodules.
In the second part of the talk I will describe $L_p$-modules by Junge and Sherman,
which are the noncommutative analogs of modules of $p$-sections of bundles
of Hilbert spaces over a measurable space.
The special cases $p=0$ and $p=1/2$ correspond to the well-known cases
of Hilbert W*-modules and Connes' correspondences.
I will prove that W*-categories of $L_p$-modules for all values
of~$p$ are equivalent to each other.
After that I will explain how Connes' fusion (and its generalized version), which originally had
very technical definition, can be described easily in this algebraic formalism.