Title: A Morse $A_\infty$-model for the higher-dimensional Heegaard Floer homology of cotangent fibers
Abstract: Given a smooth closed $n$-manifold $M$ and a $\kappa$-tuple of basepoints ${\bf q} \subset M$, we define a Morse-type $A_\infty$-algebra called the based multiloop $A_\infty$-algebra and show the equivalence with the higher-dimensional Heegaard Floer $A_\infty$-algebra of $\kappa$ disjoint cotangent fibers of $T^*M$.
Title: Disk counting for tropical Lagrangians
Abstract: This is a continuation of work with Sushmita Venugopalan (Chennai). I will describe how to compute the potentials of the Lagrangian tori in del Pezzo surfaces (first computed by Pascaleff-Tonkonog), the potentials of tropical Lagrangians such as Manin collections of Lagrangian spheres, and the corresponding open-closed maps. The results work best for almost toric four-manifolds, but also work in some higher dimensional situations such as representation varieties.
Title: Floer homotopy as a bordism theory
Abstract: Floer envisioned that Morse theory should have an extension, beyond ordinary homology, to generalised homology theories such as stable homotopy or K-theory. Cohen, Jones, and Segal constructed a framework for defining such an extension, using as their main building block a formulation of the Pontryagin-Thom construction for manifolds with corners. I will describe an alternative approach which does not rely on such a construction, and which has the advantage both of giving a direct definition of generalised Floer homology groups, as well as providing a model for the category of spectra, in terms of objects that are completely natural in Floer theory.
Title: Symplectic rational homology ball fillings of Seifert fibered spaces
Abstract: Determining when a rational homology ball bounds a rational homology ball has a long history in topology and symplectic geometry. Answering such questions allows one to prove that the important 4-manifold construction of rational blowdown can be done in the symplectic category. In this talk, I will discuss what is known about symplectic rational homology ball fillings of small Seifert fibered spaces and give a complete classification of which contact structures on a small Seifert fibered space admit a symplectic rational homology ball filling when the $e_0$ invariant is less than -3. We will also show that the only spherical manifolds that bound rational homology balls are a small class of lens spaces and give evidence towards the Gompf conjecture that no Brieskorn homology sphere bounds a symplectic homology ball. This is joint work with Ozbagi and Tosun.
Title: Beyond reflexivity, symplectically
Abstract: Reflexive polytopes play an important role in mirror symmetry, as they define toric varieties within which mirror families of Calabi-Yau manifolds limit onto a union of toric varieties. A challenging problem in mirror symmetry is to construct dual Lagrangian fibrations on mirror Calabi-Yau manifolds. In this setting, three-dimensional polytopes yield Lagrangian-fibered K3 surfaces. Forgetting complex structures, one can construct Lagrangian-fibered symplectic K3 surfaces directly from the boundary of a reflexive polytope by recognizing the integral affine structures with nodes that are supported by the polytope. In this talk, I will explain a natural condition on a three-dimensional polytope for its boundary to support an integral affine structure with nodes and examine the implications of that condition if the polytope is Delzant. (Joint work with Liat Kessler.)
Title: Two or infinity
Abstract: We explain some of the main ideas behind a recent joint work, proving that every Reeb flow on a closed connected three-manifold has either two or infinitely many simple periodic orbits, as long as the associated contact structure has torsion Chern class. This resolves longstanding conjectures of Hofer-Wysocki-Zehnder (about Reeb flows on the three-sphere, giving the standard contact structure) and Alvarez-Paiva, Burns, Matveev and Long (about geodesic flows on Finsler surfaces).
Title: Symplectic excision
Abstract: Removing a properly embedded ray from a (noncompact) manifold does not affect the topology nor the diffeotype. What about the symplectic analogue of this fact? And can we go beyond rays? We use incomplete Hamiltonian flows to excise interesting subsets of positive codimension: the product of a ray with a Cantor set, a “box with a tail”, and - more generally - epigraphs of lower semicontinuous functions. This is joint work with Xiudi Tang, motivated by a question of Alan Weinstein.
Title: Universally counting curves in Calabi–Yau threefolds
Abstract: Enumerating curves in algebraic varieties traditionally involves choosing a compactification of the space of smooth embedded curves in the variety. There are many such compactifications, hence many different enumerative invariants. I will propose a “universal” (very tautological) enumerative invariant which takes values in a certain “Grothendieck group of 1-cycles”. It is often the case with such “universal” constructions that the resulting Grothendieck group is essentially uncomputable. But in this case, the cluster formalism of Ionel and Parker shows that, in the case of threefolds with nef anticanonical bundle, this Grothendieck group is freely generated by local curves. This reduces the MNOP conjecture (in the case of nef anticanonical bundle and primary insertions) to the case of local curves, where it is already known due to work of Bryan–Pandharipande and Okounkov–Pandharipande.