Day 1
2:30pm - 3:30pm
Speaker: Daniel Perales (Texas A&M)
Abstract: Finite free additive and multiplicative convolutions are binary operations of polynomials that behave well with respect to the roots. These operations have gained interest in recent years due to its interpretation as expected characteristic polynomials of random matrix operations and their connection to free probability, geometry of polynomials, representation theory and combinatorics.
We will study in detail the basic properties of these convolutions of polynomials, survey the results in the area, and mention some interesting open problems.
Day 2
10:30am - 11:30am
Speaker: Camila Sehnem (Waterloo)
Abstract: A discrete group has the unique trace property if its reduced group C*-algebra has a unique trace. Examples constructed by Le Boudec show that a group G with the unique trace property need not be C*-simple. In this case the reduced group C*-algebra of G admits noncanonical pseudo-expectations, i.e. equivariant unital completely positive maps into the algebra of continuous functions on the Furstenberg boundary. I will discuss pseudo-expectations in the setting of noncommutative C*-dynamical systems and their relation to the ideal intersection property. Featuring joint work with M. Kennedy and L. Kroell.
2pm - 3pm
Speaker: Roy Araiza (UIUC)
Abstract: During this lecture I will present some recent results in the Grothendieck programme. In particular, after discussing historical work and some milestones in the programme, I will state a counterexample to a matricial Grothendieck theorem, answering a question of Blecher, Pisier, and Shlyakhtenko.
Day 3
12pm - 1pm
Speaker: Jorge Vargas (Caltech)
Abstract: Friedman's celebrated 2004 result states that, as the number of vertices goes to infinity, random d-regular graphs are (with high probability) nearly optimal expanders, meaning that the top non-trivial eigenvalue of their (random) adjacency matrix converges in probability to 2 sqrt(d-1). Since expanders are of great interest in mathematics and computer science, Friedman's paper (which is ~100 pages long) has attracted a lot of attention in the last two decades and more efficient proofs of his result (which yield vast generalizations) have been found. However, all the approaches to Friedman's theorem and its extensions relied on very delicate and sophisticated combinatorial considerations, making it hard to apply those ideas to other settings of interest.
In this talk I will discuss a fundamentally new (analytic) approach to Friedman's theorem which yields an elementary proof that can be written in just a few pages. Our approach also allows us to establish strong convergence (i.e. sharp norm estimates) for much more general models of tuples of random matrices (random regular graphs corresponding to the particular case of adding independent random permutations). These results can be used to show that certain infinite objects admit very strong finite dimensional approximations, which has important implications in operator algebras, spectral geometry, and differential geometry.
This is joint work with Chi-Fang Chen, Joel Tropp, and Ramon van Handel.
3pm - 4pm
Speaker: Haonan Zhang (University of South Carolina)
Abstract: It is well-known that the heat semigroup on the discrete hypercube is contractive over $L_p$'s for $p>1$. A question of Mendel and Naor concerns a stronger contraction property of the heat semigroup over the tail subspaces, known as the heat-smoothing conjecture. In this talk, I will discuss a similar problem on free group von Neumann algebras with some sharp estimates.