Noncommutative Geometry Seminar
Fall 2022
Date: | August 15, 2022 |
Time: | 09:00am |
Location: | BLOC 306 |
Speaker: | YMNCGA |
Title: | Young Mathematicians in Noncommutative Geometry and Analysis workshop |
Date: | August 16, 2022 |
Time: | 09:00am |
Location: | BLOC 306 |
Speaker: | YMNCGA |
Title: | Young Mathematicians in Noncommutative Geometry and Analysis workshop |
Date: | August 17, 2022 |
Time: | 09:00am |
Location: | BLOC 306 |
Speaker: | YMNCGA |
Title: | Young Mathematicians in Noncommutative Geometry and Analysis workshop |
Date: | August 18, 2022 |
Time: | 09:00am |
Location: | BLOC 306 |
Speaker: | YMNCGA |
Title: | Young Mathematicians in Noncommutative Geometry and Analysis workshop |
Date: | September 16, 2022 |
Time: | 1:00pm |
Location: | ZOOM |
Speaker: | Omar Mohsen , Paris-Saclay University |
Title: | Characterization of Maximally Hypoelliptic Differential Operators Using Symbols, and Index Theory |
Abstract: | In this talk we will give an introduction to maximally hypoelliptic differential operators. This is a class of differential operators generalizing elliptic operators and includes operators like Hormander’s sum of squares. We will present our work where we define a principal symbol and show that maximally hypoellipticity is equivalent to invertibility of our principal symbol generalizing the classical regularity theorem for elliptic operators. We will also give a topological index formula for maximally hypoelliptic differential operators using our symbol. Explicit examples of index computations will be included at the end. This talk is based on joint work with Androulidakis and Yuncken. |
Date: | September 28, 2022 |
Time: | 2:00pm |
Location: | BLOC 302 |
Speaker: | Rudolf Zeidler, University of Munster |
Title: | Nonnegative scalar curvature on manifolds with at least two ends |
Abstract: | I will present an obstruction to positive scalar curvature (psc) on complete manifolds with at least two ends based on the existence of incompressible hypersurfaces that do not admit psc. This result mixes an analytic technique based on $\mu$-bubbles, an augmentation of the classical minimal hypersurface obstructions to psc, with a topological argument based on positive scalar curvature surgery. Due to the latter a surprising (but necessary!) spin condition enters our result even though our methods are not based on the Dirac operator. Concretely, let $M$ be an orientable connected $n$-dimensional manifold with $n\in\{6,7\}$ and $Y\subset M$ a two-sided closed connected incompressible hypersurface that does not admit a metric of psc. Suppose that the universal covers of $M$ and $Y$ are either both spin or both non-spin. Then $M$ does not admit a complete metric of psc. As a consequence, our result answers questions of Rosenberg-Stolz and Gromov up to dimension $7$. Joint work with Simone Cecchini and Daniel Räde. |
Date: | September 29, 2022 |
Time: | 11:00am |
Location: | BLOC 302 |
Speaker: | Christopher Wulff, University of Goettingen |
Title: | Generalized asymptotic algebras and E-theory for non-separable C*-algebras |
Abstract: | Many common ad hoc definitions of bivariant K-theory for non-separable C*-algebras have some kind of drawback, usually that one cannot expect the long exact sequences to hold in full generality. I report on my current project to define E-theory for non-separable C*-algebras without such disadvantages via a generalized notion of asymptotic algebras. The intended model is appropriate to define cycles in situations where an approximation procedure is not done over a real parameter but over more complex directed sets. I will also pose the question whether the equivariance of bivariant K-theory can be generalized in a potentially very useful way. |
Date: | October 5, 2022 |
Time: | 2:00pm |
Location: | BLOC 302 |
Speaker: | Ryo Toyota, TAMU |
Title: | Controlled K-theory and K-homology |
Abstract: | I will introduce a new perspective of K-homology of spaces. This work is motivated by a paper of Guoliang Yu, where he showed that the K-theory of the localization algebra is isomorphic to K-homology for finite simplicial complexes. The localization algebra consists of functions from [1,\infty) to Roe algebra whose propagations go to 0. "The reason" we get K-homology is that by focusing operators whose propagation is small, we can recover some local information on spaces we lost by taking Roe algebras. Here we discuss how we can recover K-homology by focusing on operators whose propagation is smaller than a certain threshold r instead of thinking of operator valued functions. I will report what we can prove and what should be true. |
Date: | October 12, 2022 |
Time: | 2:00pm |
Location: | BLOC 302 |
Speaker: | Zhaoting Wei, Texas A&M University-Commerce |
Title: | Grothendieck-Riemann-Roch theorem and index theorem |
Abstract: | It is well-known that the Hirzebruch–Riemann–Roch theorem in algebraic geometry is a special case of the Atiyah-Singer index theorem. In this talk I will present a proof of the Grothendieck-Riemann-Roch theorem as a special case of the family version of the Atiyah-Singer index theorem. In more details, we first give a Chern-Weil construction of characteristics forms of coherent sheaves in terms of antiholomorphic flat superconnections, and then give a heat-kernel proof of Grothendieck-Riemann-Roch theorem. This is a joint work with J.M. Bismut and S. Shen. ZOOM link: https://tamu.zoom.us/j/98547610481 |
Date: | November 9, 2022 |
Time: | 2:00pm |
Location: | BLOC 302 |
Speaker: | Dan Lee, Queens College CUNY |
Title: | The equality case of the spacetime positive mass theorem |
Abstract: | The spacetime positive mass theorem states that asymptotically flat initial data sets satisfying the dominant energy condition (a physical condition expressing nonnegativity of matter sources) must have “nonnegative mass” in the sense that the ADM energy-momentum vector (E,P) must be “future causal,” that is, E \ge |P|. This result goes back to Witten in the spin case and Schoen-Yau and Eichmair-Huang-Lee-Schoen for manifolds with dimension less than 8. It was always conjectured that the equality E=|P| should only be possible for initial data sets arising from slices of Minkowski space, but it is surprisingly tricky to prove. A rigorous proof in the spin case was not discovered until 15 years after Witten’s proof, by Beig-Chrusciel (n=3) and Chrusciel-Maerten (n>3). Recently, in joint work with Lan-Hsuan Huang, we built on some insights of Beig-Chrusciel to find a proof that depends only upon knowing that the inequality E \ge |P| holds for all nearby initial data sets that also satisfy the hypotheses of the spacetime positive mass theorem. Or in other words, our proof characterizing the equality case does not depend on *how* one proves the inequality. |
Date: | December 7, 2022 |
Time: | 2:00pm |
Location: | BLOC 302 |
Speaker: | Sven Hirsch |
Title: | On a generalized Geroch Conjecture |
Abstract: | The Theorem of Bonnet-Myers implies that manifolds with topology M^{n-1} x S^1 do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus T^n does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so called m-intermediate curvature), and use stable weighted slicings to show that for n <= 7 the manifolds N^n = M^{n-m} x T^m do not admit a metric of positive m-intermediate curvature. This is joint work with Simon Brendle and Florian Johne. |