| Date: | May 27, 2020 |
| Time: | 2:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Zhizhang Xie, Texas A&M University |
| Title: | Approximations of delocalized eta invariants by their finite analogues |
| Abstract: | Delocalized eta invariants for self-adjoint first order elliptic differential operators on closed manifolds were first introduced by Lott, as a natural extension of the classical eta invariant of Atiyah-Patodi-Singer. It is a fundamental invariant in the studies of higher index theory on manifolds with boundary, positive scalar curvature metrics on spin manfolds and rigidity problems in topology. The delocalized eta invariant, despite being defined in terms of an explicit integral formula, is difficult to compute in general, due to its non-local nature. In this talk, I will report on some recent results concerning when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces, where the latter are easier to compute. This talk is based on joint work with Jinmin Wang and Guoliang Yu. |
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| Date: | June 3, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Paolo Antonini, SISSA |
| Title: | The Baum–Connes conjecture localised at the unit element of a discrete group |
| Abstract: | For a discrete group Γ we construct a Baum–Connes map localised at the group unit element. This is an assembly map in KK–theory with real coefficients leading to a form of the Baum-Connes conjecture which is intermediate between the Baum–Connes conjecture and the Strong Novikov conjecture.
A second interesting feature of the localised assembly map is functoriality
with respect to group morphisms. We explain the construction and we show that the relation with the Novikov conjecture follows from a comparison at the level of KKR-theory of the classifying space for free and proper actions EΓ with the classifying space for proper actions EΓ.
Based on joint work with Sara Azzali and Georges Skandalis. |
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| Date: | June 10, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Jianchao Wu, Texas A&M University |
| Title: | The Novikov conjecture and C*-algebras of infinite dimensional nonpositively curved spaces |
| Abstract: | The rational strong Novikov conjecture is a prominent problem in noncommutative geometry. It implies deep conjectures in topology and differential geometry such as the (classical) Novikov conjecture on higher signatures and the Gromov-Lawson conjecture on positive scalar curvature. Using C*-algebraic and K-theoretic tools, we prove that this conjecture holds for any discrete group admitting an isometric and proper action on a (possibly infinite-dimensional) nonpositively curved space that we call an admissible Hilbert-Hadamard space, partially extending earlier results of Kasparov and Higson-Kasparov. In particular, our result can be applied to geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold, as they act on an infinite-dimensional symmetric space called the space of L^2-Riemannian metrics. A crucial ingredient of our proof is the construction of C*-algebras from infinite dimensional nonpositively curved spaces. This is joint work with Sherry Gong and Guoliang Yu. |
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| Date: | June 17, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Shintaro Nishikawa, Penn State University |
| Title: | Sp(n,1) admits a proper 1-cocycle for a uniformly bounded representation |
| Abstract: | We show that the simple rank one Lie group Sp(n ,1) for any n admits a proper 1-cocycle for a uniformly bounded Hilbert space representation: i.e. it admits a metrically proper affine action on a Hilbert space whose linear part is a uniformly bounded representation. Our construction is a simple modification of the one given by Pierre Julg but crucially uses results on uniformly bounded representations by Michael Cowling. An interesting new feature is that the properness of these cocycles follows from the non-continuity of a critical case of the Sobolev embedding. This work is inspired from Pierre Julg's work on the Baum-Connes conjecture for Sp(n,1). |
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| Date: | June 18, 2020 |
| Time: | 4:00pm |
| Location: | Zoom |
| Speaker: | Javier Alejandro Chavez-Dominguez, University of Oklahoma |
| Title: | Asymptotic dimension and coarse embeddings in the quantum setting |
| Abstract: | We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric quotients, direct sums, and quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of reflexive quantum expanders must have infinite asymptotic dimension. This is done by proving a vertex-isoperimetric inequality for quantum expanders, based upon a previously known edge-isoperimetric one due to Temme, Kastoryano, Ruskai, Wolf, and Verstraete. Joint work with Andrew Swift. |
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| Date: | June 24, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Carla Farsi, University of Colorado - Boulder |
| Title: | Proper Lie Groupoids and their structures |
| Abstract: | I will talk about two projects in their final phase of completion.
(Joint with Scull and Watts) After defining the orbit category for transitive proper Lie groupoids and equivariant CW-complexes, we define equivariant Bredon homology and cohomology theories for actions of transitive proper Lie groupoids by using similarities with the compact group action case. Our work can be seen as basic evidence for Morita equivalence invariance of general Bredon theories.
(Joint with Seaton) After defining groupoid Euler characteristics for cocompact proper Lie groupoids we prove that they can be realized as the usual Euler characteristic of groupoid inertias spaces. We prove that these Euler Characteristics are Morita invariant and extend those defined for orbifolds and G-spaces where G is a compact Lie group. |
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| Date: | July 8, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Aurélien Sagnier, John Hopkins University |
| Title: | Towards arithmetic sites at some places |
| Abstract: | Adèle class spaces for number fields are, as A. Connes stressed it in 1996, very important for the spectral interpretation of zeroes of Hecke $L$-functions (and so in particular for the Riemann zeta function or for the Dedekind zeta function of a number field). The semiringed topos $\left(̂\widehat{\N^{\times}},(\Z∪{−∞},\max,+)\right)$ called by A. Connes and C. Consani the arithmetic site and introduced by them in 2014 provides a algebro-geometric background for the adèle class space of $\Q$. Thanks to this algebro-geometric backbground, one could hope in the long term to transfer and adapt to the context of number fields (and in A. Connes' and C. Consani's case $\Q$) ideas coming from Weil's proof of the analogue of the Riemann hypothesis in the function field case.
In my PhD thesis, I introduced for the number field $\Q(\imath)$ a semiringed topos $\left(\widehat{\mathcal{\Z[\imath]}},(\Z[\imath]^{\text{conv}},\text{Conv}(\cup),+)\right)$ similar to the one introduced by A. Connes and C. Consani and which is linked to the adèle class space of $\Q(\imath)$ and consequently linked to the Dedekind zeta function of $\Q(\imath)$ and a family of Hecke $L$-functions. However the question of naturality of the choice of $\Z[\imath]^{\text{conv}}$ as structure sheaf remained unanswered.
In this lecture, I will show that $\Z[\imath]^{\text{conv}}$ is the solution to an universal problem coming from the hyperaddition on $\Z[\imath]/\{\pm 1,\pm\imath\}$ and so that this choice of structure sheaf was natural after all. This universal problem coming from an hyperaddition will help us to get hints on how to define arithmetic sites for other number fields and for example for $\Q(\sqrt{2})$. |
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| Date: | July 15, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Masoud Khalkhali, University of Western Ontario |
| Title: | Newton divided differences, higher curved quantum tori, and scalar curvature |
| Abstract: | Two major problems in the study of curvature invariants for curved NC tori consists of extending curvature computations to higher dimensions in a uniform manner, and computing curvature invariants for non-conformally flat geometries. In sharp distinction to the classical situation where uniform algebraic formulas exist in all dimensions and for general metrics, no such formulas exist in NC settings yet. Both problems need new ideas and are quite hard in general. In this talk I will first discuss both problems and then I shall present a solution to both problems under some restrictions on the nature of the metrics on higher NC tori. We introduce an extension of conformally metrics, called functional metrics and associate a Laplace type operator to these metrics. Using Newton divided difference, we show that the first two terms of the heat trace for these metrics can be computed in a uniform manner in all dimensions. This gives universal explicit formulas for scalar curvature. Based on joint work with Asghar Ghorbanpour (available in arXiv). |
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| Date: | July 22, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Raphael Ponge, Sichuan University |
| Title: | Analysis on curved noncommutative tori |
| Abstract: | Noncommutative tori are important examples of noncommutative spaces. Following seminal work by Connes-Tretkoff, Connes-Moscovici, Fathizadeh-Khalkhali, and others a differential geometric apparatus on NC tori is currently being built. So far the main focus has been mostly on conformal deformation of the (flat) Euclidean metric or product of such metrics.
This talk will report on ongoing work to deal with general Riemannian metrics on NC tori (in the sense of Jonathan Rosenberg). Results include local and microlocal Weyl laws, Gauss-Bonnet theorems metrics, and local index formulas. |
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| Date: | July 29, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Markus Pflaum, University of Colorado - Boulder |
| Title: | Localization in Hochschild homology and an application to the Hochschild homology of convolution algebras of circle actions |
| Abstract: | Localization in Hochschild homology is a strong tool which allows to
sheafify the Hochschild chain complex of function algebras over
singular spaces or convolution algebras. We explain this idea
in the case of a proper Lie groupoid and how it leads to reduce
the computation of Hochschild homology groups of smooth convolution
algebras to those of stalks of the associated sheaf complex.
We then specialize to the case of convolution algebras of compact Lie
group actions on manifolds. We explain the ansatz by Brylinski to
compute the corresponding Hochschild homology groups and verify
Brylinski's conjecture that it is given by basic relative forms
on the associated inertia space for the case of smooth circle actions.
This is joint work with Hessel Posthuma and Xiang Tang. |
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| Date: | August 5, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| 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: | August 12, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Jonathan Belcher, University of Colorado Boulder |
| Title: | Bridge Cohomology - A Generalization of Hochschild and Cyclic Cohomologies |
| Abstract: | In this presentation we extend the geometric correlation between Hochschild (respectively cyclic) cohomology of the algebra of smooth functions on a manifold with its de Rham homology towards a similar correlation for a manifold with boundary. The natural tool that arises will be called bridge cohomology since it provides a series of steps between the cyclic and Hochschild chain complexes. Here we develop the full cohomology theory, ending with corresponding Gysin--Connes sequences that relate this cohomology with its Hochschild and cyclic counterparts. We also provide an excision theorem for bridge cohomology. Finally, we give a geometric example and extend the theorems by Hochschild--Kostant--Rosenberg and Connes to manifolds with boundary. |
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