| Date: | January 22, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Emil Prodan, Yeshiva University |
| Title: | Non-Commutative Geometry and Materials Science |
| Abstract: | Starting with the pioneering works of Jean Bellissard in the 1980’s,
Non-Commutative Geometry has emerged as one of sharpest tools in the arsenal of a theoretical materials scientist. Perhaps for this audience, the most important questions are why Non-Commutative Geometry and how much of it?
Using well understood examples, where specially designed materials display extraordinary behaviours in extreme conditions, I will try to convince the audience that one has to walk the entire sequence: algebra of observables → K-theory →pairing with cyclic cohomology → local index formula, plus one additional step which I call “pushing into the Sobolev.” Furthermore, among such extraordinary behaviours is a certain relation between the dynamics of degrees of freedominside the bulk and at the boundary of a sample, dubbed the bulk-boundary correspondence principle. It is captured by a certain extension of C*-algebras and, as such, KK-theory offers a natural framework and supplies the necessary tools to investigate such phenomena. If the time permits, I will also discuss recent efforts trying to steer these tools from their usual use of explaining observed behaviours towards the discovery of new dynamical behaviours in materials science. |
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| Date: | January 29, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Pierre Albin, University of Illinois at Urbana-Champaign |
| Title: | The sub-Riemannian limit of a contact manifold |
| Abstract: | Contact manifolds, which arise naturally in mechanics, dynamics, and geometry, carry natural Riemannian and sub-Riemannian structures and it was shown by Gromov that the latter can be obtained as a limit of the former. Subsequently, Rumin found a complex of differential forms reflecting the contact structure that computes the singular cohomology of the manifold. He used this complex to describe the behavior of the spectra of the Riemannian Hodge Laplacians in the sub-Riemannian limit. As many of the eigenvalues diverge, a refined analysis is necessary to determine the behavior of global spectral invariants. I will report on joint work with Hadrian Quan in which we determine the global behavior of the spectrum by explaining the structure of the heat kernel along this limit in a uniform way. |
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| Date: | February 5, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Jens Hemelaer, University of Antwerp |
| Title: | Toposes in arithmetic noncommutative geometry |
| Abstract: | We give an introduction to topos theory from a geometric point of view, focusing on toposes that arise from a discrete group acting on a topological space. In particular, we will look at lattices over a global field, and see how the topos classifying them is related to the ring of finite adeles of the global field. In the case where the class group is trivial, this topos is equivalently described as a topos of presheaves on a monoid, leading to toposes that are analogous to (the underlying topos) of the Arithmetic Site of Connes and Consani. We then discuss how the different toposes are related to each other. Are there interesting geometric morphisms between them? When are these morphisms embeddings, or local homeomorphisms?
The talk is based on joint work in progress with Morgan Rogers and joint work in progress with Aurélien Sagnier. |
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| Date: | February 12, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Henry Yuen, Columbia University |
| Title: | Testing low-degree polynomials in the noncommutative setting |
| Abstract: |
In low-degree testing the following question is considered: given a multivariate function over a finite field, if a sufficiently large fraction of “local views” of the function are consistent with low-degree polynomials, does this imply that the function is _globally_ consistent with a single low-degree polynomial? Many low-degree testing theorems have been proved over the years, and have had important applications in theoretical computer science, including complexity theory and property testing.
Low-degree testing also plays an important role in the recent quantum complexity result MIP* = RE. Here, low-degree testing is considered in the _noncommutative_ setting: “local views” of a function are given via a sequence of measurements on a state, but the measurement operators do not necessarily commute with each other. Despite noncommutativity, there is still a sense in which local consistency with low-degree polynomials implies global consistency with low-degree polynomials.
In this talk, I will give an introduction to low-degree testing and discuss its analysis. This is based on joint work with Ji, Natarajan, Vidick, and Wright. (https://arxiv.org/abs/2009.12982 ) |
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| Date: | February 19, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Eckhard Meinrenken, University of Toronto |
| Title: | Differential Geometry of Weightings |
| Abstract: | The idea of assigning weights to local coordinate functions appears in many areas of mathematics, such as singularity theory, microlocal analysis, sub-Riemannian geometry, or the theory of hypo-elliptic operators, under various terminologies. In this talk, I will describe some differential-geometric aspects of weightings along submanifolds. This includes a coordinate-free definition, and the construction of weighted normal bundles and weighted deformation spaces. As an application, I will discuss the osculating tangent bundle for Lie filtrations, and the corresponding tangent groupoid of Choi-Ponge, van Erp-Yuncken, and Haj-Higson. (Based on joint work with Yiannis Loizides.) |
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| Date: | February 26, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Alcides Buss, Universidade Federal de Sanata Catarina |
| Title: | Amenable actions of groups on C*-algebras |
| Abstract: | In this lecture I will explain recent developments in the theory of amenability for actions of groups on C*-algebras and Fell bundles, based on joint works with Siegfried Echterhoff, Rufus Willett, Fernando Abadie and Damian Ferraro. Our main results prove that essentially all known notions of amenability are equivalent. We also extend Matsumura’s theorem to actions of exact locally compact groups on commutative C*-algebras and give a counter-example for the weak containment problem for actions on noncommutative C*-algebras. |
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| Date: | March 5, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Claire Debord, Paris 7 |
| Title: | Cyclic subsets, groupoids and VB-groupoids |
| Abstract: | In this talk, after recalling basic facts on groupoids, we will see that the structure of a groupoid G is entirely determined by giving invariant subsets by cyclic permutations of the Cartesian product G^k for k=0,…,3 and satisfying some additional properties. We will then focus our attention on a special type of groupoids, namely VB-groupoids and see how this point of view allows to find very simply the construction of the dual groupoid of a VB-groupoid. In particular, we recover the famous Weinstein’s cotangent groupoid. Finally, we will construct a Fourier transform in this situation which induces an isomorphism between the C*-algebra of a VB-groupoid E and the C*-algebra of the VB-groupoid dual E*. This is a work in progress with Georges Skandalis. |
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| Date: | March 12, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Asghar Ghorbanpour, University of Western Ontario |
| Title: | Scalar curvature of functional metrics on noncommutative tori |
| Abstract: | In recent years, the investigation of geometries of noncommutative tori has attracted attention from both mathematics and physics. In this talk, I will report on recent development in the study of different geometries on noncommutative tori and discuss a new family of metrics, called functional metrics, on noncommutative tori and the study of spectral invariants of geometric operators related to these metrics. My aim would be to show how some computational ideas gave us advantages, at least symbolically, in computing the invariants, among others the scalar curvature of these metrics. The talk is based on joint work with M. Khalkhali: arXiv:1811.04004 [math.QA]. |
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| Date: | March 19, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Pere Ara, Universitat Autónoma de Barcelona |
| Title: | Crossed products and the Atiyah problem |
| Abstract: | It was shown by Austin in 2013 that the question by Atiyah about the rationality of $\ell^2$-Betti numbers has a negative answer. However the problem of determining the exact set of real numbers appearing as $\ell^2$-Betti numbers from a given group $G$ is widely open. In particular, this is an open question for the lamplighter group $L=\mathbb Z_2 \wr \mathbb Z$, which was the first known counterexample to the Strong Atiyah Conjecture, stating that all $\ell^2$-Betti numbers arising from a group $G$ belong to the subgroup $\sum_{H\le G, H \text{ finite}} \frac{1}{|H|} \mathbb Z$ of $\mathbb R$.
I will review some recent progress on this question, obtained in joint work with Joan Claramunt and Ken Goodearl. I will recall some of the basic techniques in our approach, which involve the consideration of Sylvester matrix rank functions on certain crossed products, and their associated $*$-regular envelopes. |
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| Date: | March 26, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Dan Voiculescu, University of California, Berkeley |
| Title: | Around the Quasicentral Modulus |
| Abstract: | The quasicentral modulus is a numerical invariant associated with a n-tuple of Hilbert space operators and a normed ideal of compact operators. It plays a key role in perturbations of n-tuples of operators and invariance of absolutely continuous spectra results. The talk will be about new results on the quasicentral modulus and commutants mod normed ideals. This will include exact formulas for the quasicentral modulus in fractional dimension and for hybrid perturbations. I will then talk about commutants mod normed ideals for compact differentiable manifolds with boundary and the K-theory exact sequence for their Calkin algebras for connected sums of manifolds. |
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| Date: | April 2, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Henrique Bursztyn, Instituto Nacional de Matematica Pura e Aplicada |
| Title: | Relating Morita equivalence in algebra and geometry via deformation quantization |
| Abstract: | The notion of Morita equivalence, native to ring theory, plays an important role in noncommutative geometry and has a geometric version in Poisson geometry (closely related to Morita equivalence of Lie groupoids). I will present some key aspects of Morita theory of Poisson structures and explain a concrete link with Morita equivalence of algebras via deformation quantization. |
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| Date: | April 9, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Jean-Eric Pin, Université Paris Denis Diderot et CNRS |
| Title: | A noncommutative extension of Mahler’s interpolation theorem |
| Abstract: | I will report on a result recently obtained with Christophe Reutenauer. Let p be a prime number. Mahler’s theorem on interpolation series is a celebrated
result of p-adic analysis. In its simplest form, it states that a function from N to Z is uniformly continuous for the p-adic metric d_p if and only if it can be uniformly approximated by polynomial functions. We prove a noncommutative generalization of this result for functions from a free monoid A* to a free group F(B) (or more generally to a residually p-finite group), where d_p is replaced by the pro-p metric. One of the challenges is to find a suitable definition of polynomial functions in this
noncommutative setting. |
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| Date: | April 16, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Yang Liu, SISSA |
| Title: | Hypergeometric Functions and Heat Coefficients on Noncommutative Tori |
| Abstract: | As the counterpart of conformal geometry, modular geometry on noncommutative manifolds explores the basic notions such as metric and curvature in Riemannian geometry (e.g. noncommutative tori) in a purely spectral framework. It was initiated by Connes-Tretkoff’s Gauss-Bonnet theorem on noncommutative two tori. Another milestone is the construction of modular Gaussian curvature due to Connes-Moscovici, which is derived from variation of the second heat coefficient of some Laplacian type operator. In this talk, I would like to report a few observations on the general structures of those heat coefficients. The word “modular” refers to the new ingredient of the coefficients, arising from the interaction between modular automorphisms associated to the volume state and the underlying smooth structure of the noncommutative manifolds. More precisely, one has to upgrade coefficients of local differential expressions from scalars to so-called rearrangement operators that fix various issues caused by the noncommutativity between metric coordinates and their derivatives. Like the notion of genus to a characteristic class, the spectral functions behind the rearrangement operators turn out to be intriguing. That is where hypergeometric functions come into play. The main result is the explicit formula of the second heat coefficient of a more general Laplacian type operator (beyond conformal perturbations studied in the literature). The talk is based on my recent preprint arxiv:2004.05714. |
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| Date: | April 23, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Ieke Moerdijk, Utrecht University |
| Title: | An Introduction to Dendroidal Topology |
| Abstract: | Simplicial sets form a classical and well known tool for modelling topological spaces, and more recently topologically enriched categories and infinity categories. I will present an extension of the category of simplicial sets, called “dendroidal sets”, and explain how these can model topological operads and their algebras. The definition is based on a simple category of trees, and the goal of the talk will be to give a leisurely introduction to dendroidal sets and some of their uses. Applications include an efficient infinite loop space machine, the analysis of derived mapping spaces of E_n operads, and Koszul duality, for example.
The talk will be based on work with or by many people, among whom Boavida, Cisinski, Goeppl, Heuts, Hinich, Weiss, and others. |
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| Date: | April 30, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Anna Skripka, University of New Mexico |
| Title: | Approximation of functions of unbounded operators |
| Abstract: | We will discuss recent advancements in approximation of functional calculus arising in problems of mathematical physics and noncommutative geometry. Our goal is to establish nice bounds and representations for (semi-)norms of approximation errors. Challenges arise, in particular, from noncommutativity of a perturbation and initial unbounded operator as well as noncompactness of the perturbation. Successful resolution rests on multilinear operator integration, a powerful technical method with a long history in noncommutative analysis. |
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| Date: | May 7, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Zhizhang Xie, Texas A&M University |
| Title: | A relative index theorem for incomplete manifolds and Gromov’s conjectures on positive scalar curvature |
| Abstract: | In this talk, I will speak about my recent work on a relative index theorem for incomplete manifolds. As applications, this new relative index theorem gives positive solutions to some conjectures and open questions of Gromov on positive scalar curvature. |
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| Date: | May 14, 2021 |
| Time: | 1:00pm |
| Location: | Zoom 951 5490 42 |
| Speaker: | Nikita Sopenko, Caltech |
| Title: | Invertible states of quantum lattice spin systems and their invariants |
| Abstract: | Recently methods of quantum statistical mechanics have been fruitfully applied to the study of phases of quantum lattice systems with interacting gapped Hamiltonians at zero temperature. I will discuss some of these developments focusing on the class of systems known as “invertible”. After introducing the corresponding class of quantum states, I will show how to define H^{d+1}(G,U(1)) valued indices for equivalence classes of such states in the presence of a symmetry group G for d=1 and d=2 lattices. If time permits, I will also discuss invariant for families of such states known as higher Berry curvatures. |
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