Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars
| | Speaker: | 郭昊 博士，Texas A&M University
| | Inviter: | | Title:
| Positive scalar curvature, index theory, and quantitative K-theory | Time & Venue:
| 2021.05.17 09:00-11:00 腾讯会议：442 523 172 | Abstract:
| 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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