Titles

Huai-Liang Chang, Geometry and Applications of Mixed Spin P-fields

This is a series of lectures introducing Mixed-Spin-P fields. When it comes to the higher genus GW invariants of the quintic hypersurface of P4, one needs to study those degenerate stable maps which contract positive genus to points. These maps are sometimes called ghost maps. Their contribution to GW invariant are known to be controlled by P fields.

It turns out the GW invariants can be equated as enumerating stable maps to P4 with P fields. A natural question is then how to capture these P fields in enumerative meaning, so as to determine GW invariants. One way is to stratify and refine the moduli to locate P fields's contribution. Another (strange) way is to enlarge the moduli to see what kind of final objects can a P field be deformed to, and then capture it using calculation methods.

Using the philosophy of Landau Ginzburg theories wall crossing, we now know that such final object is a ``five-Spin bundle on curve (with five sections)". The extended moduli parameterizing ``Mixed-Spin-P fields" (MSP) provides many structures of all genus quintic GW invariants, and determines lower genus ones (for genus one and two for example).

Indeed, this provides the first proof of the Yamaguchi-Yau finite generation conjecture of quintic CY 3 folds. Namely the GW generating function Fg lies in a special polynomial ring of five generators. Secondly, the propagator and Feynman graph sums of BCOV are provided by MSP's R matrices and stable graph sums. These will allow MSP to write Fg as an explicit solution of Holomorphic Anomaly Equations (HAE), as that in BCOV's paper.

Jonathan Wise, Logarithmic Curves: Moduli and Enumerative Geometry

Logarithmic geometry is a convenient language for discussing degenerate objects from algebraic geometry, and for translating between algebraic geometry and tropical geometry. Logarithmic curves turn out to be essentially the same as Deligne-Mumford stable curves, and logarithmic geometry gives us new insight about the Deligne-Mumford moduli space, as well as related spaces like stable maps and Picard groups. I will discuss how these spaces are constructed and how they relate to their tropical counterparts, with emphasis on logarithmic Gromov-Witten theory.

Luca Battistella, Applications of Gathmann's algorithm for relative invariants and quantum Lefschetz

In joint work with N. Nabijou, we introduce the notion of genus zero relative quasimaps to a toric target X with respect to a smooth (but not necessarily toric) hyperplane section Y. Increasing the tangency requirement at the markings gives a series of nested moduli spaces, whose virtual classes are related by a simple formula, originally derived by Gathmann in the setting of stable maps. This formula results in a recursive algorithm which allows us to compute the restricted invariants of Y from those of X, and prove a version of the quantum Lefschetz theorem for quasimaps, recovering a result of Ciocan-Fontanine and Kim. Time permitting, I will hint at how to extend these arguments to the Vakil-Zinger desingularisation in genus one, which is a joint work in progress with Nabijou and D. Ranganathan.

Francesca Carocci, Reduced vs Cuspidal invariants for the quintic 3-fold

Moduli spaces of stable maps of genus g>0 are highly singular and with many irreducible components which affect the enumerative meaning of the invariants arising from them. We will discuss two possible approaches to deal with the so called "degenerate contributions" in genus 1: Li-Zinger reduced invariants and Viscardi-Smyth cuspidal invariants. These two approaches, in the case of the quintic 3-fold, give the same invariants. We will give an idea of how to prove the previous statement based on Chang-Li's approach to the proof of the Li-Zinger formula. This is a joint project with L. Battistella and C. Manolache.

Jinwon Choi, Towards logarithmic theory of the BPS invariants

We propose a new curve counting invariant from log geometry of del Pezzo surfaces, which is motivated by numerical property of the genus zero local BPS invariants. We present a rigorous definition of the log BPS invariant via Gromov-Witten theory and formulate conjectures which relates the local and log BPS invariants. This is joint work with Michel van Garrel, Sheldon Katz and Nobuyoshi Takahashi.

Alessio Corti, Smoothing singular toric Fano 3-folds

TBA

Barbara Fantechi, TBA

TBA

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 assume some familiarity with log geometry from Jonathan Wise's series of lectures.

Martijn Kool, Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau fourfolds

The Hilbert scheme of n points on a Calabi-Yau fourfold carries a virtual cycle of degree n constructed by Borisov-Joyce and Cao-Leung. I present a conjectural formula for the invariants obtained by capping the virtual cycle with the top Chern class of a tautological bundle. I discuss evidence for this conjecture by considering (1) small numbers of points and (2) toric fourfolds. For affine four-space, this implies a conjecture about solid partitions. The latter is a specialization of a recent (more general) conjecture on solid partitions derived in physics by N. Nekrasov. Joint work with Y. Cao.

Étienne Mann, Gromov-Witten invariants via derived algebraic geometry

Let X be a smooth projective variety. Using the idea of brane actions discovered by Toën, we construct a lax associative action of the operad of stable curves of genus zero on the variety X seen as an object in correspondences in derived stacks. This action encodes the Gromov-Witten theory of X in purely geometrical terms and induces an action on the derived category Qcoh(X) which allows us to recover the Quantum K-theory of Givental-Lee. This is a joint work with Marco Robalo.

Navid Nabijou, Towards a Recursive Formula for Log Gromov-Witten Invariants in Genus Zero

Given a smooth projective variety X containing an snc divisor D, the log Gromov-Witten invariants of (X,D) give virtual counts of stable maps to X with specified orders of tangency to (the components of) D. They generalise the more classical "relative Gromov-Witten invariants" (defined for D smooth) and are a central object in the Gross-Siebert program.

In this talk, we describe work in progress towards obtaining a recursive formula for computing the invariants of (X,D), in genus zero. This is a generalisation of Gathmann's recursion formula for relative invariants, but some new and surprising phenomena appear in the snc setting.

Along the way, we will get some hands-on experience working with moduli spaces of log stable maps. In particular, we will show how in certain situations such a space may be viewed as a "desingularisation of the main component" of another, more naively defined moduli space of log stable maps.

Helge Ruddat, How tropical curves give a swift proof of the degeneration formula

I am reporting on my joint work with Kim and Lho where we give a short proof for the classical Li-Ruan/Jun Li degeneration formula using log stable maps. This is a pretty interplay of tropical and algebraic geometry.

Richard Thomas, Refined Vafa-Witten invariants for projective surfaces

I'll describe how to define Vafa-Witten invariants -- and refined Vafa-Witten invariants -- of projective surfaces. Then I'll explain how to calculate some parts of the theory via degeneracy loci and "Carlsson-Okounkov operators". The talk describes different projects joint with Yuuji Tanaka, Davesh Maulik and Amin Gholampour.