| Date: | August 19, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Yi Wang, Vanderbilt University |
| Title: | Helton-Howe trace formula on submodules |
| Abstract: | Helton-Howe trace formula gives an explicit description of the traces of antisymmetric sums of Toeplitz operators on the Bergman space. On a submodule of the Bergman module with relatively nice zero locus, previous results show that the cross commutators between the generalized Toeplitz operators with coordinate symbols belong to some Schatten class. A natural question is whether the same Helton-Howe trace formula remain true for such submodules. We answer this question in affirmative, using finite rank approximation and dilation techniques. |
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| Date: | August 26, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Giovanni Landi, Trieste University |
| Title: | Solutions to the quantum YB equation and related deformations |
| Abstract: | We present natural families of coordinate algebras of noncommutative Euclidean spaces and noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the quantum Yang–Baxter equation. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have spherical manifolds, and noncommutative quaternionic planes as well as noncommutative quaternionic tori. On these there is an action of the classical quaternionic torus SU(2)×SU(2) in parallel with the action of the torus U(1)×U(1) on a complex noncommutative torus. |
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| Date: | September 2, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Bruno de Mendonca Braga, University of Virginia |
| Title: | Coarse equivalences of metric spaces and outer automorphisms of Roe algebras |
| Abstract: | Given a metric space $X$, the Roe algebra of $X$, denoted by $\mathrm{C}^*(X)$, is a $ \mathrm{C} ^*$-algebra which encodes many of $X$'s large scale geometric properties. In this talk, I will discuss some uniform approximability results for maps between Roe algebras (we call those "coarse-like properties"). I will then talk about applications of these uniform approximability results to isomorphisms between Roe algebras. In particular, given a uniformly locally finite metric space $X$, we obtain that the canonical map from the group of coarse equivalences of $X$ modulo the relation of closeness to the group of outer automorphisms of $ \mathrm{C} ^*(X)$ is surjective if $X$ has property A. This is a joint work with Alessandro Vignati. |
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| Date: | September 9, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Anna Duwenig , University of Wollongong |
| Title: | Noncommutative Poincaré Duality of the Irrational Rotation Algebra |
| Abstract: | The irrational rotation algebra is known to be self-dual in a KK-theoretic sense. The required K-homology fundamental class was constructed by Connes out of the Dolbeault operator on the 2-torus, but there has not been an explicit description of the dual element. In this talk, I will geometrically construct that K-theory class by using a pair of transverse Kronecker flows on the 2-torus.This is based on joint work with Heath Emerson (University of Victoria). |
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| Date: | September 16, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Kristin Courtney, University of Münster (WWU) |
| Title: | C*-structure on images of completely positive order zero maps |
| Abstract: | A completely positive (cp) map is called order zero when it preserves orthogonality. Such maps enjoy a rich structure, which has made them a key component of completely positive approximations of nuclear C*-algebras. Motivated by generalized inductive limits arising from such cp approximations, we consider the structure of the image of a cp order zero map. It turns out that this is captured by a few key properties that one can ask of a self-adjoint subspace of a C*-algebra. We will discuss these properties and the implications for generalized inductive sequences. This is joint work with Wilhelm Winter. |
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| Date: | September 23, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Rudolf Zeidler, University of Göttingen |
| Title: | Scalar curvature comparison via the Dirac operator |
| Abstract: | In recent years, Gromov proposed studying the geometry of positive scalar curvature (abbreviated by "psc") via various metric inequalities. In particular, he proposed the following conjecture: Let $M$ be a closed manifold which does not admit a metric of psc. Then for any Riemannian metric on $V = M \times [-1,1]$ of scalar curvature $\geq n(n-1)$ the estimate $d(\partial_- V, \partial_+ V) \leq 2\pi/n$ holds, where $\partial_\pm V = M \times \{\pm 1\}$ and $n = \dim V$.
Previously, Rosenberg and Stolz conjectured similarly that if $M$ does not admit psc, then $M \times \mathbb{R}$ does not admit a complete metric of psc and $M \times \mathbb{R}^2$ does not admit a complete metric of uniformly psc.
In this talk, we will discuss a new geometric phenomenon consisting of a precise quantitative interplay between distance estimates and scalar curvature bounds which underlies these three conjectures. We will explain that this phenomenon arises if $M$ admits an obstruction to psc using the index theory of Dirac operators. |
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| Date: | September 30, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Simone Cecchini, University of Göttingen |
| Title: | A long neck principle for Riemannian spin manifolds with positive scalar curvature |
| Abstract: | We present results in index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary.
As a first application, we establish a ``long neck principle'' for a compact Riemannian spin n-manifold with boundary X, stating that if scal(X) ≥ n(n-1) and there is a nonzero degree map f into the n-sphere which is area decreasing, then the distance between the support of the differential of f and the boundary of X is at most π/n. This answers, in the spin setting, a question recently asked by Gromov.
As a second application, we consider a Riemannian manifold X obtained by removing a small n-ball from a closed spin n-manifold Y. We show that if scal(X) ≥ σ >0 and Y satisfies a certain condition expressed in terms of higher index theory, then the width of a geodesic collar neighborhood Is bounded from above from a constant depending on σ and n.
Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to N x [-1,1], with N a closed spin manifold with nonvanishing Rosenebrg index.
In this case, we show that if scal(V) ≥ n(n-1), then the distance between the boundary components of V is at most 2π/n. This last constant is sharp by an argument due to Gromov. |
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| Date: | October 7, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Bogdan Nica, IUPUI |
| Title: | On norms of averaging operators on geometric groups |
| Abstract: | Given a finite subset S of an infinite discrete group, consider the operator \lambda(S)=\sum_{g\in S} \lambda(g). I will discuss the problem of estimating the operator norm of \lambda(S) in certain `geometric' situations. |
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| Date: | October 14, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Paolo Piazza, Università di Roma La Sapienza |
| Title: | Higher genera and C*-indices on G-proper manifolds |
| Abstract: | Higher general for a G-proper manifold without boundary can be defined in analogy with Galois coverings and they are, by definition, geometric objects. To understand their stability properties we need to connect them to higher C^*-indices of suitable Dirac operators. This is possible but under additional assumptions on the group G, for example G semisimple and connected and more generally G satisfying the Rapid Decay condition and G/K of nonpositive sectional curvature. I will begin my talk by explaining these results. I will then move to manifolds with boundary and explain how it is possible to define higher genera in this more complicated situation. Crucial to the analysis is a higher C^*-index theorem of Atiyah-Patodi-Singer type. All these results, the last very recent, are in collaboration with Hessel Posthuma. |
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| Date: | October 21, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Jonathan Block, University of Pennsylvania |
| Title: | Singular foliations and characteristic classes |
| Abstract: | We revisit the residue theorem of Baum and Bott computing characteristic classes for certain objects in terms of residues calculated along the singularities of a foliation using techniques from higher homotopy structures. |
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| Date: | October 28, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Matthew Lorentz, University of Hawai‘i at Mānoa |
| Title: | The Hochschild cohomology of uniform Roe algebras |
| Abstract: | Recently Rufus Willett and I showed that all bounded derivations on Uniform Roe Algebras associated to a bounded geometry metric space X are inner in our paper “Bounded Derivations on Uniform Roe Algebras”. This is equivalent to the first Hochschild cohomology group $H^1(C^*_u(X), C^*_u(X))$ vanishing. It is then natural to ask if all the higher groups $H^n(C^*_u(X), C^*_u(X))$ vanish. To investigate the continuous cohomology of a Uniform Roe Algebra we employ the technique of “reduction of cocycles” where we modify a given cocycle by a coboundary to obtain certain properties. I will discuss this procedure and give examples of calculating the higher cohomology groups. |
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| Date: | November 4, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Rudy Rodsphon, Northeastern University |
| Title: | A KK-theoretical perspective on quantization commutes with reduction |
| Abstract: | We propose a reframing of Paradan--Vergne's approach to the quantization commutes with reduction problem in KK-theory, more especially the index theoretic part that leads to their "Witten non-abelian localization formula". While our method is similar to theirs at least in spirit, interesting conceptual simplifications occur, and it makes the relationship to
Ma--Tian--Zhang's analytic methods quite apparent.
Time permitting, I'll also sketch another possible way to derive this localization formula, which is purely functorial. |
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| Date: | November 11, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Nigel Higson, Penn State University |
| Title: | The Oka principle and Novodvorskii’s theorem |
| Abstract: | In the early days of Banach algebra K-theory, Novodvorskii proved that the Gelfand transform for any commutative Banach induces an isomorphism in Banach algebra K-theory. This is a version of the Oka principle in several complex variables, which identifies equivalence classes of structures, including vector bundles, in the holomorphic and continuous categories in a variety of contexts. Since the Oka principle has long been proposed as a mechanism to understand and indeed prove the Baum-Connes conjecture, Novodvorskii’s theorem continues to be of interest in noncommutative geometry. I shall give a more or less self-contained proof of Novodvorskii’s theorem, along with a rough sketch of possible future extensions into the noncommutative realm. This is joint work with Jacob Bradd. |
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| Date: | December 2, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Sheagan John, Texas A&M University |
| Title: | Pairing of Secondary Higher Invariants and Cyclic Cohomology for Virtually Nilpotent Groups |
| Abstract: | We prove that if G is a virtually nilpotent group, then each delocalized cyclic cocycle on the group algebra has a representative of polynomial growth. For each delocalized cyclic cocyle we thus define a higher analogue of Lott’s delocalized eta invariant and prove its convergence for invertible differential operators. We also use a determinant map construction of Xie and Yu to prove that if G is of polynomial growth then there is a well defined pairing between delocalized cyclic cocyles and K-theory classes of C*-algebraic secondary higher invariants. When this K-theory class is that of a higher rho invariant of an invertible differential operator we show this pairing is precisely the aforementioned higher analogue of Lott’s delocalized eta invariant. |
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| Date: | December 9, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Iakovos Androulidakis, University of Athens |
| Title: | The Heisenberg calculus of a singular Lie filtration |
| Abstract: | In this talk we will report on progress towards addressing the question of Fredholmness for general linear differential operators, including ones with singularities. To this end, we combine singular foliations theory with adiabatic groupoid methods developed in recent generalisations of the Heisenberg calculus, in order to build a pseudodifferential calculus whose principal symbols are defined on a family of nilpotent Lie algebras with non-constant dimension. We are able to state the Rockland condition for appropriate representations of these algebras. For differential operators which satisfy this condition, we are able to construct a parametrix in our calculus. This is work in progress together with Erik van Erp, Omar Mohsen and Robert Yuncken. |
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| Date: | December 16, 2020 |
| Time: | 1:00pm |
| Location: | Zoom 942810031 |
| Speaker: | Zhaoting Wei, Texas A&M University-Commerce |
| Title: | Determinant line bundles and cohesive modules |
| Abstract: | An interesting property of coherent sheaves on complex manifolds is that they have well-defined holomorphic determinant line bundles. In this talk I will show the attempt to construct such determinant line bundles in the framework of cohesive modules, introduced by Jonathan Block. I will also discuss some potential applications. |
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