中科院数学与系统科学研究院
数学研究所
中科院华罗庚数学重点实验室
华罗庚青年数学论坛
学术报告
报告人： 郭昊 博士（Texas A&M University）
题 目：Positive scalar curvature, index theory, and quantitative K-theory
时 间：2021.05.17（星期一），09:00-11:00
地 点：腾讯会议：442 523 172
摘 要：In this talk, I will discuss my work on determining obstructions to and existence of metrics of positive scalar curvature (PSC) on manifolds. Almost since its inception, index theory has been successfully applied to the problem of determining (usually in the negative) when a given closed manifold admits a PSC metric, owing to the close relation between the square of the Dirac operator and the scalar curvature of the metric on a spin manifold. It turns out that by taking into account the action of fundamental group on the universal cover, much stronger obstructions can be obtained, which imply for instance that the n-dimensional torus does not admit a metric of PSC. After giving some background on the topic, I will discuss my work on obstructions to PSC coming from the indices of Callias-type operators in the non-compact (and more generally non-cocompact) setting. I will also discuss how more classical index obstructions can be refined using quantitative K-theory, and give an application of quantitative K-theoretic methods to Gromov's band width conjecture. I will also highlight some results on the existence of PSC metrics. This incorporates joint work with Peter Hochs, Mathai Varghese, Hang Wang, Zhizhang Xie, and Guoliang Yu.
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报告人： 郭昊 博士（Texas A&M University）
题 目：Higher invariants in the maximal setting and their applications
时 间：2021.05.20（星期四），09:00-11:00
地 点：腾讯会议：215 544 787
摘 要：This talk focuses on higher index invariants that take values in the K-theory of certain maximal completions of geometric operator algebras. One reason for considering such completions is that they have better functoriality properties than their more common "reduced" counterparts. I will first discuss my work with Guoliang Yu and Zhizhang Xie, in which we prove that such completions are well-defined even when the manifold is non-compact but satisfies reasonable geometric properties, as well as a functional calculus for elliptic operators in the maximal setting. This makes it possible to give a geometric proof of the fact that the maximal higher index is functorial with respect to maps between covering spaces, and furthermore extends naturally to a proof of functoriality for the higher rho invariant. This has applications to the computation of such invariants. I will then discuss my joint work with Peter Hochs and Mathai Varghese on the indices of Callias-type operators that are invariant under the action of a locally compact group. This work generalises the notion of the coarse index from the setting of discrete group actions to continuous group actions, and has an application to the geometric quantisation commutes with reduction problem.
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