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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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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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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I will discuss some connections between the Novikov conjecture and scalar curvature.
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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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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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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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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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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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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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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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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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