This is based on joint work with Ely Kerman.
Symplectic Dynamics/Geometry Seminar
Suppose L is a link with n components and the rank of Kh(L;Z/2) is 2^n, we show that L can be obtained by disjoint unions and connected sums of Hopf links and unknots. This result gives a positive answer to a question asked by Batson-Seed, and generalizes the unlink detection theorem of Khovanov homology by Hedden-Ni and Batson-Seed. The proof relies on a new excision formula for the singular instanton Floer homology introduced by Kronheimer and Mrowka.
This is joint work with Yi Xie.
We prove several relations between spectrum and dynamics including wave trace expansion, sharp/improved Weyl laws, propagation of singularities and quantum ergodicity for the sub-Riemannian (sR) Laplacian in the four dimensional quasi-contact case. A key role in all results is played by the presence of abnormal geodesics and represents the first such appearance of these in sub-Riemannian spectral geometry.
I'll show a graphical user interface I wrote which explores the problem of inscribing rectangles in Jordan loops. The motivation behind this is the notorious Square Peg Conjecture of Toeplitz, from 1911.
I did not manage to solve this problem, but I did get the result that at most 4 points of any Jordan loop are vertices of inscribed rectangles. I will sketch a proof of this result, mostly through visual demos, and also I will explain two other theorems about inscribed rectangles which at least bear a resemblance to theorems in symplectic geometry.