## Northern California Symplectic Geometry Seminar: Planar Lagrangian graphs/The dimension problem in CR geometry

Seminar | March 13 | 2:30-5 p.m. | Stanford University Mathematics Department, 384H/383N

Peter Lambert Cole/Boris Kruglikov, University of Indiana/UiT - The Arctic University of Norway

Department of Mathematics

Peter Lambert-Cole: A foundational result in the study of contact geometry and Legendrian knots is Eliashberg and Fraser's classification of Legendrian unknots They showed that two homotopy-theoretic invariants -- the Thurston-Bennequin number and rotation number -- completely determine a Legendrian unknot up to isotopy. Legendrian spatial graphs are a natural generalization of Legendrian knots. We prove an analogous result for planar Legendrian graphs. Using convex surface theory, we prove that the rotation invariant and Legendrian ribbon are a complete set of invariants for planar Legendrian graphs. We apply this result to completely classify planar Legendrian embeddings of the Theta graph. Surprisingly, this classification shows that Legendrian graphs violate some proven and conjectured properties of Legendrian knots. This is joint work with Danielle O'Donnol.

3:30--4:00 PM: Tea break Boris Kruglikov: The dimension problem for CR-manifolds is to find the maximal finite symmetry dimension depending on CR-dimension $n$ and codimension m. This is difficult already in the hypersurface case ($m=1$). The simplest possible situation of 3-dimensional manifolds ($n=1$) was resolved only recently by I. Kossovsky and R. Shafikov (JDG-2016). Their proof is long and involved, I will explain a short proof found in collaboration with A. Isaev (PAMS-2016). Next, I will present the solution of the dimension problem in 5D ($n=2$) - another result of A. Isaev and the speaker. Note that if CR-structure is assumed Levi-nondegenerate then the maximal symmetry dimension (for any $n$) is known, due to E. Cartan, N. Tanaka, S. Chern and J. Moser. Even the next (submaximal) symmetry dimensions was established by the speaker (JGA–2015) for general n (the case $n=1$ is due to E. Cartan). The latter is an instance of the more general symmetry gap problem for geometric structures that I will also briefly discuss.