Fields Workshop around Connes' Embedding Problem

May 16- 18, 2008
University of Ottawa



Hari Bercovici, Indiana University
Schubert calculus for the practical person

Schubert calculus predicts the existence of subspaces of Euclidean space which have intersections of given dimensions with several given spaces. The proofs of these results rely on cohomological calculations. We will see that in many important cases these subspaces can be calculated effectively, and they belong (generically) to the
lattice generated by the given spaces. This allows us to prove analogous results for projections in finite von Neumann algebras.

Nate Brown, Penn State University

Talk 1: Connes' Embedding Problem: An Introduction.

I'll discuss the problem, its history and several of the equivalent reformulations (mainly due to Kirchberg).

Talk 2: Metric spaces associated with embeddable factors.

I'll introduce metric spaces associated with an R^\omega embeddable factor. Some basic properties of these invariants will be discussed.

Ken Dykema, Texas A&M University
A linearization of Connes' embedding problem

Motivated by the results of H. Bercovici and W.S. Li, which state that (the analogues of) Horn's inequalities hold in R^\omega, we wondered whether it would be possible to construct a II_1-factor where one of Horn's inequalities failed to hold. In the other direction, we wondered whether it could be proven that all Horn inequalities hold in all II_1-factors, and, if so, whether this would imply a positive solution to Connes' embedding problem. The subject of this talk is a weaker version of this last implication, involving questions about eigenvalue data for sums of self-adjoint matrices, but with k-by-k Hermition matrix coefficients.
(joint work with Benoit Collins).

Gabor Elek, The Alfred Renyi Institute of Mathematics

Talk 1: Sofic groups and Connes' Embedding Problem

Sofic groups were introduced by Misha Gromov and Benjamin Weiss.
These groups are (roughly speaking) those groups that can be approximated by finite graphs. It turns out that if a finitely generated group is sofic then its von Neumann algebra is embeddable (in the sense of Connes). Interestingly, some other conjectures such as Gromov's Surjunctivity Conjecture, Kaplansky's Direct Finiteness Conjecture and Luck's Determinant Conjecture also hold for sofic groups. Amenable groups, residually finite groups, free and direct products of sofic groups are always sofic. Also, amenable extensions of sofic groups are sofic as well. In the moment no non-sofic group is known.

Talk 2:: Constant time algorithms and measurable equivalence relations.

Constant time algorithms form a new and interesting field of computer science. One has a very large but finite graph G and want to calculate some parameters of the graph. For example the size of the largest independent subset, or the spectral distribution function of the graph Laplace operator or the amount of edges must be discarded to obtain a 3-colorable graph. In some cases these problems are very (actually NP) hard. However, for certain graph classes (such as planar graphs or graphs with moderate growth) one can calculate these parameters with high probability and high accuracy, sampling only a constant (depending on the accuracy
parameter) amount of vertices. The goal of the talk is to illuminate the surprising connection between constant time algoritms and the main subjects of the conference: von Neumann algebras, hyperfiniteness, ultraproducts and approximability.

Ilijas Farah, York University
Incompleteness, independence, and absoluteness or: When to call a set theorist?

By G\"odel's icompleteness theorems in every strong enough mathematical theory one can formulate a statement that is independent: it can be neither proved nor refuted in this theory. In certain cases set-theoretic methods can be employed to _prove_ that a mathematical statement is independent. I will give several examples of such statements. I will also show why Connes's embedding problem is not likely to be independent.

Thierry Giordano (uOttawa)
Introduction to von Neumann algebras

This will be a review of the definition and basic properties of von Neumann algebras and factors.
This talk is intended for the audience who is not familiar with operator algebras, and especially for graduate students.

Igor Klep, University of Ljubljana
Sums of hermitian squares and the BMV conjecture
(based on joint work with Markus Schweighofer)

Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of the two letters is fixed is always a matrix with nonnegative trace. We show that this statement holds if the words are of length at most 13. In the proof a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming is established. As a by-product we obtain a family of examples of real polynomials in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators. Furthermore, it is not clear whether their trace is nonnegative when evaluated in type II_1 factors.

Wing Suet Li, Georgia Institute of Technology
Eigenvalue inequalities, in an embeddable factor and in other settings

Consider two self-adjoint operators $A,B:\sH\to\sH$ on a finite-dimensional Hilbert space. Let $\{\lda_j(A)\}$, $\{\lda_j(B)\}$, and $\{\lda_j(A+B)\}$ be sequences of eigenvalues of $A,B$, and $A+B$ counting multiplicity, arranged in decreasing order. In 1962, A. Horn conjectured that the relations of $\{\lda_j(A)\}$, $\{\lda_j(B)\}$, and $\{\lda_j(A+B)\}$ can be characterized by a set of inequalities defined inductively. This problem was eventually solved by A. Klyachko and Knutson-Tao in the late 1990s. In this talk we will show that these inequalities provide a characterization of the possible eigenvalues of the sum of two selfadjoint elements in a $\rm II_1$ factor which can be embedded in the ultrapower $\cal R^{\omega}$ of the hyperfinite $\rm II_1$ factor. Also, we will see that these (and related) inequalities can characterize the possible eigenvalues of the sum of two selfadjoint compact operators.

Vladimir Pestov (UOttawa)
Ultrafilters and ultraproducts for beginners

Abstact: I will define the concepts of a prefilter, filter, and ultrafilter, as well as their main properties, and discuss the concepts of an algebraic ultraproduct of a family of algebraic structures, followed by ultraproducts of metric and normed spaces, metric groups, and finally tracial ultraproducts of von Neumann factors, including a proof that R^\omega is a factor. The concepts of theory of von Neumann algebras will be defined as necessary (without proofs).

Markus Schweighofer, Université de Rennes 1
A purely algebraic formulation of Connes' embedding conjecture (joint work with Igor Klep)

We prove that Connes' embedding conjecture is equivalent to a purely algebraic statement about polynomials in cyclically permuting variables. The counterpart for commuting variables is Putinar's quadratic module representation of polynomials positive on a hypercube. From this result of Putinar, we derive in a surprising way a special case of our algebraic reformulation. The respective analog of our reformulation for noncommuting variables is also true by a recent result of Helton and McCullough. Moreover, it is now possible to investigate Connes' conjecture with convex optimization techniques.