Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

Rohil Prasad (Princeton), The smooth closing lemma for area-preserving surface diffeomorphisms

In this talk, I will discuss recent joint work with D. Cristofaro-Gardiner and B. Zhang showing that a generic area-preserving diffeomorphism of a closed surface has a dense set of periodic points. This follows from a result called a “smooth closing lemma” for area-preserving surface diffeomorphisms; this answers in the affirmative Smale’s 10th problem in the setting of area-preserving surface diffeomorphisms. The proof uses quantitative analysis of spectral invariants from periodic Floer homology via various estimates in Seiberg-Witten theory.

Alex Pieloch (Columbia), Sections and unirulings of families over the projective line

We will discuss the existence of rational (multi)sections and unirulings for projective families $f : X \to P^1$ with at most two singular fibres. Specifically, we will discuss two ingredients for constructing the above rational curves. The first is local symplectic cohomology groups associated to compact subsets of convex symplectic domains. The second is a degeneration to the normal cone argument that allows one to produce closed curves in $X$ from open curves (which are produced using local symplectic cohomology) in the complement of $X$ by a singular fibre.

Chi Hong Chow (Chinese University of Hong Kong), Hofer geometry of coadjoint orbits and Peterson's theorem

We will discuss a complete computation of Savelyev's homomorphism associated to any coadjoint orbit of a compact Lie group $G$, where the domain is restricted to the based loop homology of $G$. This gives at the same time some applications to the Hamiltonian groups of these spaces and a geometric proof of an unpublished theorem of Peterson. This theorem tells us explicitly how the multiplicative structure constants of the based loop homology of $G$ determine those of the quantum cohomology of its coadjoint orbits.