| Date: | September 5, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Michael Brannan, Texas A&M University |
| Title: | Quantum permutations and their matrix models |
| Abstract: | A quantum permutation matrix is an N x N matrix P whose entries are orthogonal projections on some common Hilbert space H with the property that the rows and columns of P sum to the identity operator on H. In the special case where H is the one dimensional Hilbert space, a quantum permutation matrix simply corresponds to an ordinary permutation matrix, and in this case can be thought of as describing a symmetry of an N point set. In this talk I will explain how arbitrary quantum permutation matrices describe the ``quantum symmetries'' of an N point set. Putting all of these quantum permutation matrices together in a cohesive way yields the structure of a quantum group, which is commonly called the Quantum Permutation Group on N letters. Unlike the classical permutation groups, quantum permutation groups turn out to highly infinite and noncommutative objects -- in many ways they behave algebraically like the C*- and von Neumann algebras associated to nonabelian free groups. Despite the inherent infiniteness of quantum permutation groups, I will show how these objects can nonetheless be well-approximated by finite-dimensional structures. In particular, these objects turn out to be residually finite as discrete quantum groups, and this residual finiteness can in fact be achieved using certain very simple finite-dimensional matrix models which I will describe. (Joint work with Alexandru Chirvasitu and Amaury Freslon.) |
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| Date: | September 12, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Hao Guo, Texas A&M University |
| Title: | Index of Equivariant Callias-Type Operators |
| Abstract: | I will discuss a class of Dirac-type operators, called equivariant Callias-type operators, on manifolds equipped with a Lie group action, where the orbit space is non-compact. It turns out that these operators are Fredholm with an index in the K-theory of the group C*-algebra and can be constructed by adding an ordinary (non-Fredholm) Dirac operator to an element of the K-theory of the equivariant Higson corona of the manifold. One can apply the index theory of such operators to prove an obstruction theorem for invariant metrics of positive scalar curvature. |
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| Date: | September 26, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Li Gao, Texas A&M University |
| Title: | Pseudo-differential Operators and the local index formula on Moyal Planes |
| Abstract: | Moyal planes are noncommutative deformation of Euclidean spaces given by canonical commutation relations (CCR). They are prototypes of noncommutative noncompact manifolds. In the recent work of Gonzalez-Perez, Junge and Parcet, the Pseudo-differential operators on Moyal planes were studied. In this talk, I will talk about the application of Pseudo-differential operators of Moyal plane to the local index formula. We will consider the setting that for a Moyal plane, its covariant derivatives also satisfy CCR relations. This is a joint work with Marius Junge and Edward McDonald. |
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| Date: | October 3, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Alan Reid, Rice University |
| Title: | (Joint seminar with Groups and Dynamics Seminar) Distinguishing certain triangle groups by their finite quotients |
| Abstract: | We prove that some arithmetic Fuchsian triangle groups are profinitely rigid in the sense that they are determined by their set of finite quotients amongst all finitely generated residually finite groups. These include the (2,3,8) triangle group.
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| Date: | October 10, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Dean Baskin, Texas A&M University |
| Title: | Diffraction for the Dirac equation with Coulomb-like potentials |
| Abstract: | The Dirac equation describes the relativistic evolution of
electrons and positrons. We consider the (time-dependent!) Dirac equation
in three spatial dimensions coupled to a potential with Coulomb
singularities. We show that singularities of the solutions are typically
diffracted by the singularities of the potential and compute the symbol of
the diffracted wave.
In this talk I will 1) describe what I mean by diffraction in a simpler
setting, 2) describe our results for the Dirac equation, and 3) ask the
audience for help understanding these results in a more geometric way.
This talk is based on joint work with Oran Gannot and Jared Wunsch. |
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| Date: | October 12, 2018 |
| Time: | 4:00pm |
| Location: | BLOC 220 |
| Speaker: | Rufus Willett, University of Hawaii |
| Title: | Representation stability and topology (Joint with Linear Analysis Seminar) |
| Abstract: | Let G be a discrete group with a fixed finite generating set
S. A map from G into some (finite dimensional) unitary group U(n) is an
epsilon-representation if it is a group homomorphism up to epsilon error
(for the operator norm) on the finite set S. Thus a
quasi-representation is a close to being a representation in some sense.
The group G is stable if every epsilon representation is close to an
actual representation, in a precise sense. For example, free groups are
fairly obviously stable. However, a famous result of Voiculescu shows
that the rank two free abelian group is not stable. In his thesis,
Loring gave this a topological interpretation: it turns out that
Voiculescu’s result is more-or-less equivalent to Bott periodicity.
I’ll try to explain all this, and how topological information can be
used to produce many other examples of non-stable groups. |
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| Date: | October 31, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Zhenhua Wang, University of Houston |
| Title: | Noncommutative topology and operator *-algebras |
| Abstract: | An operator $*$-algebra is an operator algebra with a completely isometric conjugate linear involution. In this talk, we will talk about general theory of operator $*$-algebras such as characterizations of operator $*$-algebras, the relationship to their $C^*$-covers and real positivity. In the second part of my talk, noncommutative topology in the involutive setting will be discussed. This is joint work with David Blecher. |
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| Date: | November 7, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Clement Dell'Aiera, University of Hawaii |
| Title: | Dynamical Property T |
| Abstract: | We will present a notion of topological property T for group actions
which generalizes Kazhdan's property T for groups and geometric property
T of Willett and Yu. This is work in progress with Rufus Willett. |
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| Date: | November 14, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Robin Deeley, University of Colorado Boulder |
| Title: | Relative constructions in K-homology and KK-bordism |
| Abstract: | The Baum-Douglas model for K-homology provides a geometric counterpart to the analytic construction of Kasparov. In the framework of index theory, the former is more related to the topological index, while the latter is more related to the analytic index. I will discuss various relative constructions in geometric (i.e., Baum-Douglas) K-homology and each case the associated index theoretic invariant. If time permits, some more analytic constructions involving Hilsum's notion of KK-bordism will also be discussed. The results in this talk are (in part) joint work with Magnus Goffeng and Bram Mesland. |
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| Date: | November 28, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Jinmin Wang, Texas A&M University |
| Title: | Higher rho invariant and pairings with higher cyclic cocyles |
| Abstract: | In this talk, I will introduce a pairing map between K-theory of obstruction algebra and higher cyclic cocyles for the group algebras of hyperbolic groups. The main focus of this pairing is to establish a connection between two different approaches to the index theory of manifolds with boundaries. This is joint work with Xiaoman Chen, Zhizhang Xie and Guoliang Yu. |
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| Date: | December 5, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Jeffrey Kuan, Texas A&M University |
| Title: | Noncommutative random walks on the dual of gl_n |
| Abstract: | We construct two noncommutative versions of random walks on the dual of gl_n. The observables are elements of the universal enveloping algebra U(gl_n) [in the first construction] and of the quantum group U_q(gl_n) [in the second construction]. We take asymptotics to find two different generalizations of the two--dimensional Gaussian free field. |
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