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Texas A&M University
Mathematics

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.