Functional Analysis Seminar
2018-2019
Time | Location | Seminar Organizers |
---|---|---|
Tuesday — 11:00am–12:00pm | AP&M 6402 (unless otherwise specified) | Adrian Ioana (aioana@ucsd.edu) Todd Kemp (tkemp@ucsd.edu) |
Fall 2018
Date | Speaker | Title + Abstract |
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October 9 |
Christopher Schafhauser
York University |
Subalgebras of AF-AlgebrasA long-standing open question, formalized by Blackadar and Kirchberg in the mid 90's, asks for an abstract characterization of C$^*$-subalgebras of AF-algebras. I will discuss some recent progress on this question: every separable, exact C$^*$-algebra which satisfies the UCT and admits a faithful, amenable trace embeds into an AF-algebra. Moreover, the AF-algebra may be chosen to be simple and unital with unique trace and the embedding may be taken to be trace-preserving. Modulo the UCT, this characterizes C$^*$-subalgebras of simple, unital AF-algebras. As an application, for any countable, discrete, amenable group $G$, the reduced C$^*$-algebra of $G$ embeds into a UHF-algebra. |
November 6 2:30pm in APM 6402 |
David Jekel
UCLA |
An Elementary Approach to Free Gibbs Laws Given by Convex PotentialsWe present an alternative approach to the theory of free Gibbs laws with convex potentials developed by Dabrowski, Guionnet, and Shlyakhtenko. Instead of solving SDE's, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N(\mathbb{C})_{sa}^m$ to prove the following. Suppose $\mu_N$ is a probability measure on on $M_N(\mathbb{C})_{sa}^m$ given by uniformly convex and semi-concave potentials $V_N$, and suppose that the sequence $DV_N$ is asymptotically approximable by trace polynomials in a certain sense. Then the moments of $\mu_N$ converge to a non-commutative law $\lambda$. Moreover, the free entropies $\chi(\lambda)$, $\underline{\chi}(\lambda)$, and $\chi^*(\lambda)$ agree and equal the limit of the normalized classical entropies of $\mu_N$. An upcoming paper will use the same techniques to obtain transport maps from $\lambda$ to a free semicircular family as the limit of transport maps for the matrix models $\mu_N$. |
November 8 1pm in APM 6402 |
Pooya Vahidi Ferdowsi
Caltech |
Classification of Choquet-Deny GroupsA countable discrete group is said to be Choquet-Deny if it has a trivial Poisson boundary for every non-degenerate probability measure on the group. In other words, a countable discrete group is Choquet-Deny if non-degenerate random walks on the group have trivial behavior at infinity. For example, all abelian groups are Choquet-Deny. It has been long known that all Choquet-Deny groups are amenable. I will present a recent result classifying countable discrete Choquet-Deny groups: a countable discrete group is Choquet-Deny if and only if none of its quotients have the infinite conjugacy class property. As a corollary, a finitely generated group is Choquet-Deny if and only if it is virtually nilpotent. This is a joint work with Joshua Frisch, Yair Hartman, and Omer Tamuz. |
November 13 |
Scott Atkinson
Vanderbilt University |
Tracial stability and graph productsA unital $C^\ast$-algebra $A$ is tracially stable if maps on $A$ that are approximately (in trace) unital $\ast$-homomorphisms can be approximated (in trace) by honest unital $\ast$-homomorphisms on $A$. Tracial stability is closed under free products and tensor products with abelian $C^\ast$-algebras. In this talk we expand these results to show that for a graph from a certain class, the corresponding graph product (a simultaneous generalization of free and tensor products) of abelian $C^\ast$-algebras is tracially stable. We will then discuss two applications of this result: a selective version of Lin’s Theorem and a characterization of the amenable traces on certain right-angled Artin groups. |
November 14 3pm in APM 6402 |
Anush Tserunyan
UIUC |
A pointwise ergodic theorem for quasi-pmp graphsWe prove a pointwise ergodic theorem for locally countable ergodic quasi-pmp (nonsingular) graphs, which gives an increasing sequence of Borel subgraphs with finite connected components, averages over which converge a.e. to the expectations of $L^1$-functions. This can be viewed as a random analogue of pointwise ergodic theorems for group actions: instead of taking a (deterministic) sequence of subsets of the group and using it at every point to compute the averages, we allow every point to coherently choose such a sequence at random with a strong condition that the sets in the sequence determine aconnected subgraph of the Schreier graph of the action. |
December 4 |
Daniel Hoff
UCLA |
Rigid Components of s-Malleable DeformationsIn the theory of von Neumann algebras, fundamental unsolved problems going back to the 1930s have seen remarkable progress in the last two decades due to Sorin Popa's breakthrough deformation/rigidity theory. Popa's discovery hinges on the fact that, just as stirring a soup allows one to locate its most rigid (and desirable) hidden components, ``deformability" of an algebra $M$ allows one to precisely locate ``rigid" subalgebras known to exist only via a supposed isomorphism $M \cong N$. This talk will focus on joint work with Rolando de Santiago, Ben Hayes, and Thomas Sinclair, which shows that any diffuse subalgebra which is rigid with respect to a mixing $s$-malleable deformation is in fact contained in subalgebra which is uniquely maximal with respect to that rigidity. In particular, an algebra generated by a family of rigid subalgebras which intersect diffusely must itself be rigid with respect to that deformation. The case where this family has two members answers a question of Jesse Peterson asked at the American Institute of Mathematics (AIM), but the result is most striking when the family is infinite. |
"Winter" 2019
Date | Speaker | Title + Abstract |
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January 15 Joint with Geometry |
Artem Pulemotov
University of Queensland |
The Ricci iteration on homogeneous spacesThe Ricci iteration is a discrete analogue of the Ricci flow. Introduced in 2007, it has been studied extensively on Kähler manifolds, providing a new approach to uniformisation. In the talk, we will define the Ricci iteration on compact homogeneous spaces and discuss a number of existence, convergence and relative compactness results. This is largely based on joint work with Timothy Buttsworth (Queensland), Yanir Rubinstein (Maryland) and Wolfgang Ziller (Penn). |
February 5 Joint with Probability |
Brian Hall
University of Notre Dame |
Eigenvalues of random matrices in the general linear groupI will consider random matrices in the general linear group GL(N;C) distributed according to a heat kernel measure. This may also be described as the distribution of Brownian motion in GL(N;C) starting at the identity. Numerically, the eigenvalues appear to cluster into a certain domain $\Sigma_t$ as $N$ tends to infinity. A natural candidate for the limiting eigenvalue distribution is the “Brown measure” of the limiting object, which is Biane’s "free multiplicative Brownian motion." I will describe recent work with Driver and Kemp in which we compute this Brown measure. The talk will be self contained and will have lots of pictures. |
February 8 Note unusual day |
Ben Hayes
University of Virginia |
Quotients of Bernoulli shifts associated to operators with an $\ell^{2}$-inverseLet G be a countable, discrete, group and f an element of the integral group ring over G. It is well known how to associate to f an action of G on a compact, metrizable, abelian group. It turns out to be particularly interesting to consider those f with an $\ell^{2}$-inveres: i.e. a vector $\xi\in \ell^{2}(G)$ so that $f*\xi=\delta_{1}$. Many nice ergodic theoretic properties of the corresponding action have been established in this context. I will give certain examples of f,G for which we can say that this action is a quotient of a Bernoulli shift. When G is amenable, this implies that it *is* a Bernoulli shift. |
February 19 |
Daniel Drimbe
University of Regina |
On the tensor product decomposition of II$_1$ factors arising from groups and group actionsIn a joint work with D. Hoff and A. Ioana, we have discovered the following product rigidity phenomenon: if $\Gamma$ is an icc group measure equivalent to a product of non-elementary hyperbolic groups, then any tensor product decomposition of the II$_1$ factor $L(\Gamma)$ arises only from the canonical direct product decomposition of $\Gamma$. Subsequently, I. Chifan, R. de Santiago and W. Sucpikarnin classified all the tensor product decompositions for group von Neumann algebras arising from a large class of amalgamated free products. In this talk we will give an overview of these results and discuss about a similar rigidity phenomenon that appears in the context of von Neumann algebras arising from actions. More precisely, we prove that if $\Gamma$ is a product of certain groups and $\Gamma\curvearrowright (X,\mu)$ is an arbitrary free ergodic measure preserving action, then we show that any tensor product decomposition of the II$_1$ factor $L^\infty(X)\rtimes\Gamma$ arises only from the canonical direct product decomposition of the underlying action $\Gamma\curvearrowright X.$ |
February 26 |
Sayan Das
University of Iowa |
On the generalized Neshveyev-Stormer conjectureThe study of group actions on probability measure spaces plays a central role in modern mathematics. The (generalized) Neshveyev-Stormer conjecture states that the group action on a probability measure space can be completely understood by studying the inclusion of the group von Neumann algebra inside the group measure space construction. In my talk I shall show that the Neshveyev-Stormer conjecture is true for a large class of actions. This talk is based on a joint work with Ionut Chifan. |
Spring 2019
Date | Speaker | Title + Abstract |
---|---|---|
April 4 Joint with Probability |
Mark Meckes
Case Western University |
TBA |
April 9 |
Rolando de Santiago
UCLA |
TBA |
April 23 |
Todor Rsankov
Universite Paris Diderot |
TBA |
April 30 |
Amine Marrakchi
RIMS, Kyoto |
TBA |
May 7 |
Matthew Wiersma
University of Alberta |
TBA |