# Seminars & Events for Joint PU/IAS Symplectic Geometry Seminar

##### Mirror symmetry for Gromov-Witten invariants of a quintic threefold

The mirror symmetry principle of string theory provides closed formulas for GW-invariants, with special attention devoted to a quintic threefold, $Q3$. The genus $0$ mirror prediction for $Q3$ was verified 12 years ago by using the Atiyah-Bott localization theorem. In this talk, I will outline how the analoguos genus 1 localization problem is solved by making use of a number of its relations with the genus $0$ localization problem. This approach confirms the 1993 BCOV mirror symmetry prediction for genus $1$ GW-invariants of $Q3$. It also produces mirror formulas for genus $1$ GW-invariants of a degree $n$ hypersurface in $P^{n-1} (Q3$ is $n=5)$, confirming a recent prediction of Klemm-Pandharipande for a sextic fourfold ($n=6$) and producing a puzzling combinatorial identity related to unbranched covers of tori ($n=3$).

##### Area-dependence in gauged Gromov-Witten theory

I will describe joint work with E. Gonzalez, in which we study the dependence of the moduli space of gauged pseudoholomorphic maps from a surface to a target X as the area form on the surface is varied. As an application, we get some version of the "abelianization" conjecture of Bertram et al relating Gromov Witten theory of symplectic quotients by a group and its maximal torus. This is part of a larger project which aims to develop functoriality of Gromov-Witten invariants of quotients, joint with Ziltener, Ma'u, and Ott.

##### (Conjectural) triply graded link homology groups of the Hopf link and Hilbert schemes of points on the plane

Gukov et al. suggested triply graded link homology groups via refined BPS counting on the deformed conifold. Through large N duality they identify their Poincaré polynomials as refined topological vertices. I further apply the geometric engineering to interpret them as holomorphic Euler characteristics of natural vector bundles over Hilbert schemes of points on the affine plane. Then they perfectly make sense mathematically. This work is very preliminary, but I hope it could be developed further.

##### Special Lagrangian fibrations, instanton corrections and mirror symmetry

We study the extension of mirror symmetry to the case of Kahler manifolds which are not Calabi-Yau: the mirror is then a Landau-Ginzburg model, i.e. a noncompact manifold equipped with a holomorphic function called superpotential. The Strominger-Yau-Zaslow conjecture can be extended to this setting by considering special Lagrangian torus fibrations in the complement of an anticanonical divisor, and constructing the superpotential as a weighted count of holomorphic discs. In particular we show how "instanton corrections" arise in this setting from wall-crossing discontinuities in the holomorphic disc counts. Various explicit examples in complex dimension 2 will be considered.

##### The action gap and periodic orbits of Hamiltonian systems

The action and index spectra of a Hamiltonian diffeomorphism and their behavior under iterations carry important information about the periodic orbits of the diffeomorphism. In a recent joint work with Ginzburg, we proved that for a certain sequence of iterations of a Hamiltonian diffeomorphism, the minimal action-index gap is bounded from above when the ambient manifold is closed and symplectically aspherical. This theorem implies the Conley conjecture asserting that such a diffeomorphism has simple periodic orbits of arbitrarily large period. The proof uses the facts, also established in the same work, that an isolated fixed point remains isolated for admissible iterations and that the local Floer homology groups for all such iterations are isomorphic to each other up to a shift of degree.

##### Mirror symmetry of Fano toric A-model and Landau-Ginzburg B-model

In this talk, I will introduce the notion of weakly unobstructed Lagrangian submanifolds and balanced Lagrangian submanifolds. I will explain construction of certain potential function constructed out of study of deformation theory of Floer cohomology and explain its relationship to the earlier work of Givental which advocates that quantum cohomology ring is isomorphic to the Jacobian ring of Landau-Ginzburg superpotential. I will explain these result in the context of mirror symmetry between Fano toric A-model and Landau-Ginzburg B-model. If time permits, I will indicate how this study can be related to construction of Entov-Polterovich's symplectic quasi-states on toric manifolds.

##### Numerical Methods in Calabi-Yau Compactications of String Theory

After a brief introduction to N=1 compatifications in String Theory, it will become clear why one needs to know explicit solutions to important PDEs, such as the Kaehler-Einstein metrics. This fact motivates the use of numerical methods to approximate solutions to such PDEs. Instead of using relaxation methods/finite differences I will explain how to use geometric quantization combined with many powerful results in complex analysis (Yau's theorem, DUY, balanced metrics...) to approximate transcendental objects by algebraic-geometric ones. I will finish by showing several examples of these techniques.

##### Algebraic properties of quantum homology

In this talk we discuss certain algebraic properties of the quantum homology algebra of toric Fano manifolds. In particular, we describe an easily-verified sufficient condition for the semi-simplicity of the quantum homology. Moreover, we provide some examples of monotone toric Fano manifolds for which the quantum homology is not semi-simple.

This is joint work with Ilya Tyomkin.