| Date: | August 28, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Sherry Gong, UCLA |
| Title: | On the Kronheimer-Mrowka concordance invariant |
| Abstract: | We will talk about Kronheimer and Mrowka’s knot concordance invariant, $s^\sharp$. We compute the invariant for various knots. Our computations reveal some unexpected phenomena, including that $s^\sharp$ differs from Rasmussen's invariant $s$, and that it is not additive under connected sums. We also generalize the definition of $s^\sharp$ to links by giving a new characterization of the invariant in terms of immersed cobordisms.
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| Date: | September 13, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 220 |
| Speaker: | Caterina Consani, Johns Hopkins University |
| Title: | [Colloquium] Under and above Spec(Z) |
| Abstract: | The talk will describe some recent geometric constructions related to the algebraic spectrum of the integers and the adele class space of the rationals which are part of a joint work with A. Connes. |
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| Date: | October 2, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Clément Dell'Aiera, University of Hawai'i |
| Title: | Exhaustivity of representations at infinity of Roe algebras |
| Abstract: | Exhaustive representations were introduced by Nistor and Prudhon, motivated by characterizations of Fredholm operators and spectral theory of N-body Hamiltonians. Coarse geometry deals with the geometry at infinity of metric spaces with bounded geometry via the Roe algebra. In particular, Spakula and Willlett show how to construct a family of representations of the Roe algebra which can be thought of "the representations at infinity". We will be studying when this family is exhaustive. This is joint work with Yu Qiao. |
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| Date: | October 9, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Chunlan Jiang, Hebei Normal University |
| Title: | Similarity invariants of essentially normal Cowen-Douglas operators and Chern polynomials |
| Abstract: | In this talk, I will discuss our resent work on a class of essentially normal operators by using the geometry method from the Cowen-Douglas theory and a Brown-Douglas-Fillmore theorem in the Cowen-Douglas theory. More precisely, the Chern polynomials and the second fundamental forms are
the similarity invariants (in the sense of Herrero) of this class of essentially normal operators. |
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| Date: | October 16, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Jianchao Wu, Texas A&M University |
| Title: | The K-theory of C*-algebras associated to certain infinite dimensional spaces |
| Abstract: | Noncommutative geometry provides a potent approach to the study of the algebraic geometry (e.g., K-theory) of infinite dimensional manifolds. In this talk, I will outline the construction of C*-algebras associated to Hilbert-Hadamard spaces, understood as a kind of (typically infinite dimensional) nonpositively curved manifolds. Under mild assumptions, these C*-algebras retain a remnant of Bott periodicity, which we exploit to prove the Novikov conjecture of geometrically discrete groups of diffeomorphisms. This is joint work with Sherry Gong and Guoliang Yu. |
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| Date: | October 23, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Rufus Willett, University of Hawai'i |
| Title: | Decompositions and K-theory |
| Abstract: | We introduce a notion of 'local decomposability' for a C*-algebra, inspired by the theory of nuclear dimension (due to Winter and Zacharias) and of dynamical complexity (due to Guentner, Yu and the speaker). We derive the existence of a sort-of controlled Mayer-Vietoris sequence in this setting, inspired by work of Oyono-Oyono and Yu in controlled K-theory (although not using that language), and give applications to Baum-Connes theory and to the Künneth formula.
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| Date: | October 30, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Quanlei Fang, The City University of New York |
| Title: | Revisiting Arveson’s Dirac operator of a commuting tuple |
| Abstract: | About twenty years ago, Arveson introduced an abstract Dirac operator based on Taylor spectrum and functional calculus. He showed that every Dirac operator is associated with a commuting tuple. The Dirac operator of a commuting tuple has inspired several interesting problems in multivariable operator theory. In this talk, we will revisit the Dirac operator and discuss some related problems. |
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| Date: | November 6, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Jintao Deng, Texas A&M University |
| Title: | The Novikov conjecture and group extensions |
| Abstract: | The Novikov conjecture is an important problem in higher dimensional topology. It claims that the higher signatures of a compact smooth manifold are invariant under orientation preserving homotopy equivalences. The Novikov conjecture is a consequence of the strong Novikov conjecture in the computation of the K-theory of group C*-algebras. In this talk, I will talk about the Novikov conjecture for groups which are extensions of coarsely embeddable groups. |
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| Date: | November 13, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Tatiana Nagnibeda, University of Geneva |
| Title: | Various types of spectra and spectral measures on Schreier and Cayley graphs |
| Abstract: | We will be interested in the Laplacian on graphs associated
with finitely generated groups: Cayley graphs and more generally
Schreier graphs corresponding to some natural group actions. The
spectrum of such an operator is a compact subset of the closed interval
[-1,1], but not much more can be said about it in general. We will
discuss various techniques that allow to construct examples with
different types of spectra: connected, union of two intervals, totally
disconnected…, and how this depends on the choice of the generating set
in the group. Types of spectral measures that can arise in these
examples will also be discussed. |
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| Date: | November 13, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Paul Schupp, University of Illinois at Urbana Champaign |
| Title: | Closures of Turing Degrees |
| Abstract: | This talk is on aspect of my general project with Carl
Jockusch on “the coarsification of computability theory”, that is,
bringing the asymptotic-generic point of view of geometric group theory
into the theory of computability. Classically, computability theory
studies Turing degrees, that is, equivalence classes of subsets of N
which are computationally equivalent. Coarse computability studies how
closely arbitrary subsets of N can be approximated by computable sets.
The idea of coarse computabilty leads to a natural definition of the
closure of a Turing degree in the space S of coarse similarity classes
of subsets of N with the Besicovich metric. It turns out that S is an
interesting space. We will discuss interactions of the topology of S and
properties of Turing degrees. |
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| Date: | November 13, 2019 |
| Time: | 4:00pm |
| Location: | BLOC 220 |
| Speaker: | Tullio Ceccherini-Silberstein, University of Sannio |
| Title: | Hecke algebras of multiplicity-free induced representations |
| Abstract: | Given a finite group G and a subgroup K, one says that (G,K) is a
Gelfand pair provided the associated permutation representation
(\lambda, L(G/K)) is multiplicity-free (that is, decomposes into
pairwise non-equivalent irreducible subrepresentations). This condition
is equivalent to the algebra End_G(L(G/K)) of interwining operators
being commutative. Observe that \lambda is nothing but the induced
representation Ind_K^G \iota_K of the trival representation \iota_K of
K. In [CS-S-T] we consider triples (G,K,\theta), where \theta is, more
generally, an irreducible K-representation and introduce a Hecke-type
algebra H(G,K,\theta) - analogous to End_G(L(G/K)) - and show that that
Ind_K^G\theta is multiplicity-free if and only if H (G,K,\theta) is
commutative. We apply our results in the context of the representation
theory of GL_2(q), the general linear group of a field with q elements.
[CS-S-T] Harmonic analysis and spherical functions for multiplicity-free
induced representations of finite groups. Springer (to appear) arXiv:
1811.09526.
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| Date: | November 20, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Peter Hochs, University of Adelaide |
| Title: | A localised equivariant index for proper actions and an APS index theorem. |
| Abstract: | Roe defined a localised version of the coarse index of an elliptic operator that is invertible outside a subset Z of the manifold M it is defined on. An equivariant version of this index was defined for proper and free actions by discrete groups by Xie and Yu. With Guo and Mathai, we extended this to proper actions by any locally compact group G. If Z/G is compact, then this index takes values in the K-theory of the group C* algebra of G, and generalises the Baum-Connes analytic assembly map. It also generalises an equivariant index of Callias-type operators constructed earlier by Guo. Another special case is an equivariant index for proper, cocompact actions on manifolds with boundary, generalising the Atiyah-Patodi-Singer (APS) index and its equivariant version. With Bai-Ling Wang and Hang Wang, we obtained an equivariant APS index theorem in this context. Using a version for maximal group C*-algebras and Roe algebras, we obtain a link with an index on invariant sections defined earlier with Mathai. |
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| Date: | November 22, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 624 |
| Speaker: | Ilya Kachkovskiy, Michigan State University |
| Title: | Almost commuting matrices |
| Abstract: | Suppose that $X$ and $Y$ are two self-adjoint matrices with the commutator $[X,Y]$ of small operator norm. One would expect that $X$ and $Y$ are close to a pair of commuting matrices. Can one provide a distance estimate which only depends on $\|[X,Y]\|$ and not on the dimension? This question was asked by Paul Halmos in 1976 and answered positively by Huaxin Lin in 1993 by indirect C*-algebraic methods, which did not provide any explicit bounds. It was conjectured by Davidson and Szarek that the distance estimate would be of the form $C\|[X,Y]\|^{1/2}$. In the talk, I will explain some background on this and related problems, and the main ideas of the proof of this conjecture, obtained jointly with Yuri Safarov. If time permits, I will discuss some current work in progress. |
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| Date: | November 25, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Ilan Hirshberg, Ben-Gurion University |
| Title: | Mean cohomological independence dimension and radius of comparison. |
| Abstract: | I will report on joint work in progress with N. Christopher Phillips.
In 2010, Giol and Kerr published a construction of a minimal dynamical system whose associated crossed product has positive radius of comparison. Subsequently, Phillips and Toms conjectured that the radius of comparison of a crossed product should be roughly half the mean dimension of the underlying system. Upper bounds were obtained by Phillips, Hines-Phillips-Toms and very recently by Niu, however there were no results concerning lower bounds aside for the examples of Giol and Kerr. In the non-dynamical context, work of Elliott and Niu suggests that the right notion of dimension to consider is cohomological dimension, rather than covering dimension (notions which coincide for CW complexes). Motivated by this insight, we introduce an invariant which we call "mean cohomological independence dimension" (more precisely, a sequence of such invariants), for actions of countable amenable groups on compact metric spaces, which are related to mean dimension, and obtain lower bounds for the radius of comparison for crossed products in terms of this invariant. |
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| Date: | December 4, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Mizanur Rahaman, Institute for Quantum Computing / University of Waterloo |
| Title: | Bisynchronous Games and Factorizable Maps |
| Abstract: | In the theory of non-local games, the graph isomorphism game stands out to be an intriguing one. Specially when the algebra of this game is considered. This is because this game establishes a close connection between the algebra of the game and the theory of quantum permutation groups. It turns out that the graph isomorphism game is an example of a bisynchronous game. In this talk, I will introduce these games and the corresponding correlations arising from the perfect strategies for such games. Moreover, when the number of inputs is equal to the number of outputs, each bisynchronous correlation gives rise to a completely positive map which will be shown to be factorizable in the sense of Haagerup-Musat. This is a joint work with Vern Paulsen. |
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| Date: | December 11, 2019 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Chris Phillips, University of Oregon |
| Title: | Simplicity of reduced Banach algebras of free groups and their relatives |
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