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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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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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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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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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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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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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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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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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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