| Date: | January 14, 2022 |
| Time: | 2:00pm |
| Location: | ZOOM |
| Speaker: | Gilles Pisier, Texas A&M University |
| Title: | The lifting property for C*-algebras |
| Abstract: | We give several characterizations of the lifting property (LP in short) using the maximal tensor product for C* -algebras. The class of algebras with LP includes all nuclear C*-algebras but also the full C*-algebras of free groups.
Meeting ID: 951 5490 4208
Passcode: 446996 |
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| Date: | January 21, 2022 |
| Time: | 08:00am |
| Location: | ZOOM |
| Speaker: | Misha Gromov |
| Title: | Invitation to Scalar Curvature |
| Abstract: | Abstract: There are three great domains in geometry, which lie on the boundary of "soft" and "rigid":
(1) low dimensional, especially 4-dimensional topology/geometry;
(2) symplectic topology/geometry;
(3) scalar curvature bounded from below.
I will try to elucidate in my lecture common features of these three and explain the specificity of the problems arising with the scalar curvature.
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| Date: | January 21, 2022 |
| Time: | 09:15am |
| Location: | ZOOM |
| Speaker: | Rudolph Zeidler |
| Title: | Scalar and mean curvature comparison via the Dirac operator |
| Abstract: | Abstract: I will explain a spinorial approach towards a comparison and rigidity principle involving scalar and mean curvature for certain warped products over intervals. This is motivated by recent scalar curvature comparison questions of Gromov, in particular distance estimates under lower scalar curvature bounds on Riemannian bands $M \times [-1,1]$ and Cecchini's long neck principle. I will also exhibit applications of these techniques in the context of the positive mass theorem with arbitrary ends. This talk is based on joint work with Simone Cecchini.
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| Date: | January 28, 2022 |
| Time: | 2:00pm |
| Location: | ZOOM |
| Speaker: | Yanli Song, Washington University in St. Louis |
| Title: | K-theory of the reduced C*-algebra of a real reductive Lie group |
| Abstract: | In 1987, Antony Wassermann announced a result of the structure of reduced C∗-algebra of a connected, linear real reductive group, up to Morita equivalence, and the verification of the Connes-Kasparov conjecture for these groups. In this talk, I will close a gap in the literature by providing the remaining details concerning the computation of the reduced C∗-algebra and discuss details of the C∗-algebraic Morita equivalence. In addition, I will also review the construction of the Connes-Kasparov morphism. The tool we used in the computation comes from David Vogan’s theory of minimal K-types. This is a joint work with Pierre Clare, Nigel Higson and Xiang Tang. |
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| Date: | February 11, 2022 |
| Time: | 2:00pm |
| Location: | ZOOM |
| Speaker: | Zhizhang Xie, Texas A&M University |
| Title: | On Gromov’s Dihedral extremality and rigidity conjectures |
| Abstract: | In this talk, I will present my recent joint work with Jinmin Wang and Guoliang Yu on Gromov’s dihedral extremality and rigidity conjectures. These two conjectures of Gromov concern comparisons of scalar curvature, mean curvature and dihedral angles for compact manifolds with corners, which can be viewed as scalar curvature analogues of the Alexandrov’s triangle comparisons for spaces whose sectional curvature is bounded below. Gromov’s conjectures have profound implications in geometry and mathematical physics such as the positive mass theorem. In our recent work, by using Dirac operator methods, we answer positively Gromov’s dihedral extremality conjecture for convex polyhedra in all dimensions, and Gromov’s dihedral rigidity conjecture for convex polyhedra in dimension three. |
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| Date: | February 18, 2022 |
| Time: | 08:00am |
| Location: | ZOOM |
| Speaker: | Shmuel Weinberger, University of Chicago |
| Title: | Some introductory remarks on the Novikov conjecture |
| Abstract: | I will explain a few simple ideas about the Novikov conjecture and related problems. |
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| Date: | February 18, 2022 |
| Time: | 09:15am |
| Location: | ZOOM |
| Speaker: | Guoliang Yu, Texas A&M University |
| Title: | The Novikov conjecture and scalar curvature |
| Abstract: | I will discuss some connections between the Novikov conjecture and scalar curvature. |
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| Date: | February 25, 2022 |
| Time: | 2:00pm |
| Location: | ZOOM |
| Speaker: | Ralph Kaufmann, Purdue University |
| Title: | Noncommutative geometry in Hochschild complexes |
| Abstract: | The fact that Hochschild cochain complexes are not commutative, but only commutative up to a controlled homotopy was a fundamental insight of Gerstenhaber.There is moreover a series of higher operations that can be neatly organized into an operadic structure, which is the content of Deligne’s conjecture.There is a whole package of operations on the chain level based on surfaces with extra structure, which we gave in 2006.In the case of a Frobenius algebra this yields a homotoy BV structure that essentially captures a circle action. In geometric situations this captures algebraic string topology operations, for instance an operation corresponding to the Goresky-Hingston coproduct.
In newer work with M. Rivera and Zhengfang Wang, we extend these operations to Hochschild chain complexes and the Tate Hochschild complex,which connects the chain and cochain complexes and plays a role in singularity theory.
There is a particular dualization which dualizes a higher multiplication found in this complex toa double Poisson bracket. These operations are natural from the point of view of surfaces and also allow for animation in the sense of Nest and Tsygan. |
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| Date: | March 4, 2022 |
| Time: | 08:00am |
| Location: | ZOOM |
| Speaker: | Bernhard Hanke, Augsburg University |
| Title: | Surgery, bordism and scalar curvature |
| Abstract: | One of the most influential results in scalar curvature geometry, due to Gromov-Lawson and Schoen-Yau, is the construction of metrics with positive scalar curvature by surgery. Combined with powerful tools from geometric topology, this has strong implications for the classification of such metrics. We will give an overview of the method and point out some recent developments. |
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| Date: | March 4, 2022 |
| Time: | 08:00am |
| Location: | ZOOM |
| Speaker: | Johannes Ebert, University of Münster |
| Title: | Rigidity theorems for the diffeomorphism action on spaces of positive scalar curvature |
| Abstract: | The diffeomorphism group, Diff(M), of a closed manifold acts on the space, R+(M), of positive scalar curvature metrics. For a basepoint, g, we obtain an orbit map
σg : Diff(M) → R+(M)
which induces a map on homotopy groups
(σg)∗ : π∗(Diff(M)) → π∗( R+(M)).
The rigidity theorems from the title assert that suitable versions of the map (σg)∗ factors through certain bordism groups. A special case of our main result asserts that (σg)∗ has finite image if M is simply connected, stably parallelizable, and of dimension at least 6. The results of this talk are from joint work of the speaker with Oscar Randal–Williams. |
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| Date: | March 11, 2022 |
| Time: | 2:00pm |
| Location: | ZOOM |
| Speaker: | Rufus Willett, University of Hawai’i |
| Title: | Decomposable C*-algebras and the UCT |
| Abstract: | A C*-algebra satisfies the UCT if it is K-theoretically the same as a commutative C*-algebra, in some sense. Whether or not all (separable) nuclear -algebras satisfy the UCT is an important open problem; in particular, it is the last remaining ingredient needed to prove the ‘best possible’ classification result for simple nuclear C*-algebras in the sense of the Elliott classification program.
We introduce a notion of a ‘decomposition’ of a C*-algebra over a class of C*-algebras. Roughly, this means that there are almost central elements of the C*-algebra that cut it into two pieces from the class, with well-behaved intersection. Our main result shows that the class of nuclear C*-algebras that satisfy the UCT is closed under decomposability.
Decomposability introduces a natural ‘complexity hierarchy’ on the class of -algebras: one starts with finite-dimensional C*-algebras, and the ‘complexity rank’ of a C*-algebra is roughly the number of decompositions one needs to get to down to the finite-dimensional level. There are interesting examples: we show that all UCT Kirchberg (i.e. purely infinite, separable, simple, unital, nuclear) C*-algebras have complexity rank one or two, and characterize when each of these cases occur. The UCT for all nuclear -algebras thus becomes equivalent to the statement that all Kirchberg algebras have finite complexity rank.
This is based on joint work with Arturo Jaime, and with Guoliang Yu. |
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| Date: | April 1, 2022 |
| Time: | 08:00am |
| Location: | ZOOM |
| Speaker: | Zhizhang Xie, Texas A&M University |
| Title: | Comparisons of scalar curvature, mean curvature and dihedral angle, and their applications |
| Abstract: | In this talk, I will review Gromov’s dihedral extremality and rigidity conjectures regarding comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. These conjectures have profound implications in geometry and mathematical physics such as the positive mass theorem. I will explain the recent work on positive solutions to these conjectures, and some related applications (such as a positive solution to the Stoker conjecture). The talk is based on my joint works with Jinmin Wang and Guoliang Yu. |
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| Date: | April 1, 2022 |
| Time: | 09:15am |
| Location: | ZOOM |
| Speaker: | Jinmin Wang, Texas A&M University |
| Title: | Gromov's dihedral rigidity conjecture and index theory on manifolds with corners |
| Abstract: | In this talk, I will explain the key ideas of our recent work on positive solutions to Gromov's dihedral extremality and rigidity conjectures. One of the main ingredients is a new index theory on manifolds with corners (more generally, manifolds with polytope singularities), which is of independent interest on its own. Our approach is based on the analysis of differential operators arising from conical metrics. The comparison of dihedral angles enters into the study of these differential operators in an essential way. This is based on my joint works with Zhizhang Xie and Guoliang Yu. |
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| Date: | April 8, 2022 |
| Time: | 08:00am |
| Location: | ZOOM |
| Speaker: | Weiping Zhang, Nankai University |
| Title: | Deformations of Dirac operators |
| Abstract: | Deformations of Dirac operators have played important roles in various aspects in geometry and topology. In this expository talk I will discuss some of these applications. |
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| Date: | April 8, 2022 |
| Time: | 09:15am |
| Location: | ZOOM |
| Speaker: | Guangxiang Su, Chern Institute of Mathematics |
| Title: | Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds |
| Abstract: | Let $(M,g^{TM})$ be a noncompact complete Riemannian manifold of dimension $n$, and $F\subseteq TM$ be an integrable subbundle of $TM$. Let $g^F=g^{TM}|_{F}$ be the restricted metric on $F$ and $k^F$ be the associated leafwise scalar curvature. Let $f:M\to S^n(1)$ be a smooth area decreasing map along $F$, which is locally constant near infinity and of non-zero degree. We show that if $k^F> {\rm rk}(F)({\rm rk}(F)-1)$ on the support of ${\rm d}f$, and either $TM$ or $F$ is spin, then $\inf (k^F)<0$. As a consequence, we prove Gromov's sharp foliated $\otimes_\varepsilon$-twisting conjecture. Using the same method, we also extend two famous non-existence results due to Gromov and Lawson about $\Lambda^2$-enlargeable metrics (and/or manifolds) to the foliated case. This is a joint work with Xiangsheng Wang and Weiping Zhang. |
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| Date: | April 22, 2022 |
| Time: | 08:00am |
| Location: | ZOOM |
| Speaker: | Yuguang Shi, Peking University |
| Title: | Quasi-local mass and geometry of scalar curvature |
| Abstract: | Abstract: Quasi-local mass is a basic notion in General Relativity. Geometrically, it can be regarded as a geometric quantity of a boundary of a 3-dimensional compact Riemannian manifold. Usually, it is in terms of area and mean curvature of the boundary. It is interesting to see that some of quasi-local masses, like Brown-York mass, have deep relation with Gromov’s fill-in problem of metrics with scalar curvature bounded below. In this talk, I will discuss these relations. This talk is based on some of my recent joint works with J.Chen, P.Liu, W.L. Wang , G.D.Wei and J. Zhu etc. |
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| Date: | April 22, 2022 |
| Time: | 09:15am |
| Location: | ZOOM |
| Speaker: | Jintian Zhu, Peking University |
| Title: | Incompressible hypersurface, positive scalar curvature and positive mass theorem |
| Abstract: | In this talk, I will introduce a positive mass theorem for asymptotically flat manifolds with fibers (like ALF and ALG manifolds) under an additional but necessary incompressible condition. I will also make a discussion on its connection with surgery theory as well as quasi-local mass and present some new results in these fields. This talk is based on my recent work joint with J. Chen, P. Liu and Y. Shi. |
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| Date: | April 29, 2022 |
| Time: | 08:00am |
| Location: | ZOOM |
| Speaker: | Christian Bär |
| Title: | Boundary value problems for Dirac operators |
| Abstract: | This introduction to boundary value problems for Dirac operators will not focus on analytic technicalities but rather provide a working knowledge to anyone who wants to apply the theory, i.e. in the study of positive scalar curvature. We will systematically study "elliptic boundary conditions" and discuss the following topics:
* typical examples of such boundary conditions
* regularity of the solutions up to the boundary
* Fredholm property and index computation
* geometric applications |
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| Date: | April 29, 2022 |
| Time: | 09:15am |
| Location: | ZOOM |
| Speaker: | Simone Cecchini |
| Title: | Distance estimates in the spin setting and the positive mass theorem |
| Abstract: | The positive mass theorem states that a complete asymptotically Euclidean manifold of nonnegative scalar curvature has nonnegative ADM mass. It relates quantities that are defined using geometric information localized in the Euclidean ends (the ADM mass) with global geometric information on the ambient manifold (the nonnegativity of the scalar curvature). It is natural to ask whether the positive mass theorem can be ``localized’’, that is, whether the nonnegativity of the ADM mass of a single asymptotically Euclidean end can be deduced by the nonnegativity of the scalar curvature in a suitable neighborhood of E.
I will present the following localized version of the positive mass theorem in the spin setting. Let E be an asymptotically Euclidean end in a connected Riemannian spin manifold (M,g). If E has negative ADM-mass, then there exists a constant R > 0, depending only on the geometry of E, such that M must either become incomplete or have a point of negative scalar curvature in the R-neighborhood around E in M. This gives a quantitative answer, for spin manifolds, to Schoen and Yau's question on the positive mass theorem with arbitrary ends. Similar results have recently been obtained by Lesourd, Unger, and Yau without the spin condition in dimensions <8 assuming Schwarzschild asymptotics on the end E. I will also present explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end E. The bounds obtained are reminiscent of Gromov's metric inequalities with scalar curvature. This is joint work with Rudolf Zeidler. |
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| Date: | April 29, 2022 |
| Time: | 2:00pm |
| Location: | ZOOM |
| Speaker: | Anna Marie Bohmann, Vanderbilt |
| Title: | Assembly in the Algebraic K-theory of Lawvere Theories |
| Abstract: | Lawvere’s algebraic theories are an elegant and flexible way of encoding algebraic structures, ranging from group actions on sets to modules over rings and beyond. We discuss a construction of the algebraic K-theory of such theories that generalizes the algebraic K-theory of a ring and show that this construction allows us to build Loday assembly-style maps. This is joint work with Markus Szymik. |
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| Date: | May 6, 2022 |
| Time: | 2:00pm |
| Location: | ZOOM |
| Speaker: | Yves Andre, Institut de Mathématiques de Jussieu |
| Title: | A Remark on the Tate Conjecture |
| Abstract: | The Tate conjecture has two parts: i) Tate classes are generated by algebraic classes, ii) semisimplicity of Galois representations coming from pure motives. In a recent note with the same title, B. Moonen proved that i) implies ii) in characteristic 0. I’ll recast his result in the framework of observability theory, and discuss the case of positive characteristic. |
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| Date: | May 13, 2022 |
| Time: | 08:00am |
| Location: | ZOOM |
| Speaker: | Bernd Ammann, University of Regensburg |
| Title: | Yamabe constants, Yamabe invariants and Gromov-Lawson surgeries |
| Abstract: | In this talk I want to study the (conformal) Yamabe constant of a closed Riemannian (resp. conformal) manifold and how it is affected by Gromov-Lawson type surgeries. This yields information about Yamabe invariants and their bordism invariance. So far the talk gives an overview over older results of mine in joint work with M. Dahl, N. Große, E. Humbert, and N. Otoba. A further consequence is that many results about the space of metrics with positive scalar curvature may be generalized to spaces of metrics with Yamabe constant above $t>0$. In particular we will present the following Chernysh-Walsh type result which is work in progress: if $N^n$ arises from $M^n$ by a surgery of dimension $k\in\{2,3,\ldots,n-3\}$, then a Gromov-Lawson type surgery construction defines a homotopy equivalence from the space of metrics on $M$ with Yamabe constant above $t\in (0,\Lambda_{n,k})$ to the corresponding space on $N$. |
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| Date: | May 13, 2022 |
| Time: | 09:15am |
| Location: | ZOOM |
| Speaker: | Claude LeBrun, Stony Brook University |
| Title: | Yamabe Invariants, Weyl Curvature, and the Differential Topology of 4-Manifolds |
| Abstract: | The behavior of the Yamabe invariant, as defined in Bernd Ammann’s previous lecture, differs strangely in dimension 4 from what is seen in any other dimension. These peculiarities not only manifest themselves in the context of the usual scalar curvature, but also occur in connection with certain curvature quantities that are built out of the scalar and Weyl curvatures. In this lecture, I will explain how the Seiberg-Witten equations not only allow one compute the Yamabe invariant for many interesting 4-manifolds, but also give rise to other curvature inequalities.
I will then point out applications of these results to the theory of Einstein manifolds, while also highlighting related open questions that have so far proved impervious to these techniques. |
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