Three 20 minute research talks

Three 20 minute research talks

Simon Allais, ENS de Lyon, Orsola Capovilla-Searle, Duke University, Julian Chaidez, University Of California, Berkeley

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Three 20 min research talks: 

Simon Allais, ENS de Lyon

Generating functions in Hamiltonian dynamics and symplectic-contact rigidity

Generating functions of Hamiltonian diffeomorphisms are maps that can be seen as finite dimensional versions of the action functional. In various situations, classical Morse theory applied to them can retrieve the same information as the Floer theory. In this talk, I will introduce this tool and expose some old and new results of Hamiltonian dynamics and symplectic rigidity that can be retrieved and sometimes extended using elementary Morse theory and generating functions; among others, the recent theorem of Shelukhin about the Hofer-Zehnder conjecture in the special case of CP^d and a contact generalization of the symplectic camel theorem.

Orsola Capovilla-Searle, Duke University

Weinstein handle decompositions of complements of toric divisors in toric 4 manifolds

We consider toric 4 manifolds with certain toric divisors that have normal crossing singularities. The normal crossing singularities can be smoothed, changing the topology of the complement. In specific cases this complement has a Weinstein structure, and we develop an algorithm to construct a Weinstein handlebody diagram of the complement of the smoothed toric divisor. The algorithm we construct more generally gives a Weinstein handlebody diagram for Weinstein 4-manifolds constructed by attaching 2 handles to T*S for any surface S, where the 2 handles are attached along the conormal lift of curves on S. Joint work with Bahar Acu,  Agnes Gadbled, Aleksandra Marinkovic, Emmy Murphy, Laura Starkston and Angela Wu.

Julian Chaidez, Univeristy of California,  Berkeley

ECH Embedding Obstructions For Rational Surfaces

Is the Gromov width on toric varieties monotonic with respect to inclusions of moment polytopes? In this talk, I will prove a generalization in dimension 4: the "width" associated to a concave toric domain is monotonic with inclusion of momenty polygons. This is an application of some new algebro-geometric obstructions for embeddings of star-shaped domains into rational surfaces. This work is joint with Ben Wormleighton.