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Titles and Abstracts

Pierrick Bousseau: The quantum tropical vertex

Gross-Pandharipande-Siebert have shown that 2-dimensional scattering diagrams compute some genus zero Gromov-Witten invariants of log Calabi-Yau surfaces. I will explain that the q-refined 2-dimensional scattering diagrams compute some higher genus Gromov-Witten invariants of log Calabi-Yau surfaces. This result can be motivated by the expected relation, going back to Witten, between open A-model and Chern-Simons theory.

Mark Gross: Gluing log Gromov-Witten invariants: a progress report

I will talk about joint work with Abramovich, Chen, and Siebert aiming at generalising the Li-Ruan and Jun Li degeneration formulas for Gromov-Witten invariants. The goal is to understand how to use degenerations of smooth varieties into, say, arbitrary normal crossings varieties in order to calculate Gromov-Witten invariants of the smooth variety. Li-Ruan solved this problem in the case that a smooth variety degenerates into a union of two smooth varieties, but log Gromov-Witten theory aims to deal with much more general degenerations. I will attempt to give this talk without needing a log geometric background.

Yang-Hui He: Learning Algebraic Geometry: Lessons from the String Landscape

We propose a paradigm to machine-learn the ever-expanding databases which have emerged in mathematical physics, algebraic geometry and particle phenomenology, as diverse as the statistics of string vacua and the classification of varieties.

As concrete examples, we establish multi-layer neural networks as both classifiers and predictors and train them with a host of available data ranging from Calabi-Yau manifolds to quiver representations for gauge theories, achieving impressive precision in a matter of minutes. This paradigm should prove useful in various investigations in landscapes in physics as well as pure mathematics.

Paul Johnson: Topology of Hilbert schemes of points on orbifold surfaces

Intuitively, the Hilbert Scheme of $n$ points on a smooth surface S parameterises sets of n unordered points on $S$, "remembering" extra information if the points collide. For a fixed $S$ and varying n, it turns out there is a lot of structure hidden in the topology of Hilbert Schemes -- Goettsche gave a product formula for them, which was later shown to be a shadow of a Heisenberg algebra action. It is natural to ask what happens for orbifold surfaces. When $G$ is a finite subgroup of $SL_2$, Hilbert schemes of points on $\mathbb C^2/G$ are important players in Nakajima's construction of representations of quantum groups, and on the other hand are connected to the classical combinatorial notion of cores and quotients of partitions.

However, when $G$ is not a subgroup of $SL_2$, much less is known, and we start to remedy this. For $G$ a finite cycle group *not* in $SL_2$, we state conjectural analogs of Goettsche's product formula and of the Heisenberg action, and prove a few results using a variation of cores and quotients of partitions, showing connections to the minimal and maximal resolutions of $\mathbb C^2/G$.

Tyler Kelly: Open Mirror Symmetry for the Landau-Ginzburg Model $x^r$

I will describe open B-model invariants in the context of Saito-Givental theory that mirror open r-spin invariants constructed by Buryak, Clader, and Tessler. This is joint work with Mark Gross and Ran Tessler.

Clélia Pech: Quantum Cohomology for Horospherical Varieties

Non-homogeneous horospherical varieties have been classified by Pasquier and include the well known odd symplectic Grassmannians. In this talk I will explain how to study their quantum cohomology, with a view towards Dubrovin’s conjecture. In particular, I will describe the cohomology groups of these varieties as well as a Chevalley formula, and prove that many Gromov-Witten invariants are enumerative. The consequence is that we can prove in many cases that the quantum cohomology is semisimple. I will also give a presentation of the quantum cohomology ring for odd symplectic Grassmannians. Finally, I will explain mirror constructions in two cases. This is joint work with R. Gonzales, N. Perrin, and A. Samokhin.

Yongbin Ruan I: Structure of higher genus Gromov-Witten invariants of quintic 3-fold I, background and early attempts

The computation of Gromov-Witten invariants of compact Calabi-Yau 3-folds such as quintic 3-fold has been a central problems in geometry and physics. Unfortunately, the higher genus computation turns out to be a very difficult problem. Via B-model and mirror symmetry, physicists have proposed a zoo of conjectures regarding the structure of theory as well as explicit computation. In this talk, we will review these conjectures and some of early attempts centering around the analytic continuation of Gromov-Witten theory.

Yongbin Ruan II: Structure of higher genus Gromov-Witten invariants of quintic 3-fold II, recent breakthrough

The effort to study the analytic continuation of Gromov-Witten theory leads to the invention of FJRW-theory and more recent mathematical theory of gauged linear sigma model (GLSM). One consequence of these new theories is the appearance of an alternative definition of Gromov-Witten theory. The recent breakthrough starts from a logarithmic compactification of relevant GLSM. The localization formula of log GLSM moduli space immediately reduces Gromov-Witten invariants of quintic 3-fold to finitely many unknown "effective invariants" and a twisted theory. An "advanced Givental theory" leads to the computation of generating function in a closed form and the solutions of ALL the B-model conjectures. This is a joint work with Qile Chen, Felix Janda and Shuai Guo.

Yongbin Ruan III: Calabi-Yau 3-fold, Reid's fantasy and Gromov-Witten invariants

Calabi-Yau manifolds always occupy an important place in the classification of algebraic varieties, In dimension one and two, there are only one family of Calabi-Yau manifolds. In early 90's, physicists surprised mathematician by discovering millions of different families of Calabi-Yau 3-folds and there is a "mirror symmetry" among them. To put an order to such a chaotic situation, Miles Reid proposed to connect ALL the Calabi-Yau 3-folds by algebraic surgeries such as flop and extremal transition. The hope is that one can prove mirror symmetry of Calabi-Yau 3-folds by proving mirror symmetry among surgeries. His proposal was known as Reid's fantasy. Flop was well-understood. Mark Gross soon classified so called primitive extremal transitions. In 96's, Anmin Li and I started a program to calculate the change of Gromov-Witten invariants under these surgeries. We did the case of flop and conifold transition and failed for other types of transitions. After twenty years, Rongxiao Mi made some breakthrough recently using the language of quantum D-module (motivated by early works of Lee-Lin-Wang).