[QSMS Symplectic geometry seminar 2023-10-17, 18] New approaches to discovering symplectic non-convexity / Banach-Mazur distances in symplectic and contact geometry

by qsms posted Oct 10, 2023


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일정시작 2023-10-17
일정종료 2023-10-18
배경색상 #229954
반복주기 1
반복단위 1.일(단위)
  • Date:   2023-10-17 (Tue) 16:00 ~ 17:00  

                       2023-10-18 (Wed) 16:00 ~ 17:00  

  • Place:  129-406 (SNU)   
  • Speaker:  Jun Zhang (Institute of Geometry and Physics, University of Science and Technology of China)
  • TiTle & Abstract:

New approaches to discovering symplectic non-convexity  (Oct. 17th)

In this talk, we will provide new examples of star-shaped (toric) domains in \C^2 that are dynamically convex but not symplectically convex. Our examples are based on two approaches: one is from Chaidez-Edtmair’s criterion via Ruelle invariant and systolic ratio; the other is from the ECH capacities and an analog non-linear version of Banach-Mazur distance in symplectic geometry. In particular, from the second approach, we derive the first family of examples that can be numerically verified (instead of taking a certain limit from the first approach). We will also illustrate that the information given by these two approaches is in general independent of each other. This talk is based on joint work with Dardennes, Gutt, and Ramos.


Banach-Mazur distances in symplectic and contact geometry  (Oct. 18th)

In this talk, we will discuss how the metrical geometric concept, Banach-Mazur distance, naturally fits into the study of large-scale phenomena in symplectic and contact geometry. Here, a large-scale phenomenon means a high-dimensional Euclidean space can be quasi-isometrically embedded into a space of geometric objects which include certain domains in a symplectic or contact manifold, as well as geometric structures for instance contact 1-forms. The dimension of such a Euclidean space is called the rank of a quasi-flat. Successful applications of the Banach-Mazur distance are also based on various algebraic machinery: persistence module theory, shape invariant theory, etc. The main results of this talk are establishing a general principle that almost all the spaces of geometric objects in symplectic and contact geometry admit high-rank quasi-flats. This principle can be used to distinguish geometric objects in a quantitative way. This talk is based on joint works with Vukasin Stojisavljevic and Daniel Rosen, respectively. 





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