Talks

Dan Abramovich - Punctured logarithmic maps and punctured logarithmic invariants, slides

My intention is to set the stage for the two talks by Yixian Wu and Bernd Siebert following.

Andrea Brini - Stable maps to Looijenga pairs, slides

Let X be a smooth complex projective surface and D a reduced, singular anticanonical divisor in X whose irreducible components are all nef and smooth.

I will describe a series of correspondences relating five different classes of enumerative invariants specified by the geometry of (X,D): the log Gromov-Witten theory of X relative to D, the twisted Gromov-Witten theory of X by the sum of the dual line bundles to the irreducible components of D, the open Gromov-Witten theory of toric Lagrangians in a toric Calabi--Yau 3-fold determined by (X,D), the Donaldson-Thomas theory of a symmetric quiver specified by (X,D), and a class of BPS invariants considered in different contexts by Klemm-Pandharipande, Ionel-Parker, and Labastida-Marino-Ooguri-Vafa. I will also show how the problem of computing these invariants turns out to be closed-form solvable.

This is joint work with P. Bousseau (ETH Zurich) and M. van Garrel (Warwick).

Liana Heuberger - Finding Q-Fano threefolds using Laurent inversion, slides

I will illustrate how by using the above technique one can build high codimension Q-Fano threefolds from the Graded Ring Database. I will report on my latest progress yielding a table with constructions of these objects, either as complete intersections or Pfaffian varieties inside toric ambient spaces.

Al Kasprzyk - Experimenting with Fanosearch: exploring the landscape of Fano classification, slides

Our understanding of Fano classification via Mirror Symmetry has grown considerably over the past decade. Although still very much conjectural, we now have systematic ways to begin exploring what this classification might eventually look like. I will describe recent work, joint with Giuseppe Pitton, Liana Heuberger, and others, which builds upon the existing classifications of Fano polytopes in dimensions 3 and 4 and begins to systematically construct examples against which our conjectures and intuition can be tested. This is very-much work-in-progress.

Sean Keel - Analytic Disks, the Secondary Fan, and moduli of log Fano varieties

I'll discuss partial results, joint with Hacking and Yu, towards the following conjecture: Let Q be a connected component of the moduli space of tuples (X,E_1,...,E_n,Z) with E_i smooth divisors on smooth projective X, with E := E_1 + ... + E_n snc and anti-canonical, with a zero stratum, and Z an ample effective divisor not passing through a zero stratum of E. Note this last condition is equivalent to (X,E + e Z) being stable for e >0 sufficiently small, so Q lives in the moduli space SP of stable pairs. Conj: There is a complete toric variety TV with a finite map TV --> SP with image the closure of Q.

Cristina Manolache - Genus one stable maps: a 3-term obstruction theory, slides

Morphisms with perfect obstruction theories are the starting point of modern enumerative geometry. These are unfortunately rare. For example, given an embedding of a smooth variety in a projective space, the induced embedding of the moduli spaces of genus one stable maps is a map which carries a 3-term obstruction theory. I will show that this map gives rise to a construction very similar to the cosection localized construction in the sense of Kiem and Li and that this construction has good properties.

Andrea Petracci + Helge Ruddat - On deformations and smoothings of toric varieties, slides

Toric geometry provides a class of varieties for which geometric properties are read off from combinatorics. In this talk I will discuss some properties of the deformation theory of toric affine varieties and of toric Fano varieties.

This talk is based on collaborations with some subsets of {Alessio Corti, Simon Felten, Matej Filip, Paul Hacking, Anne-Sophie Kaloghiros, Helge Ruddat}.

Helge Ruddat - Advances in smoothing toroidal crossing varieties, slides

I am presenting the conceptual framework for dealing with fairly complicated log singularities in order to be able to use a Bogomolov-Tian-Todorov type theorem to produce smoothings. Previously, mostly log singularities with toric local models have been discussed in the literature. In a joint work with Corti-Felten-Petracci, we go away from that restriction in order to be able to enable general Fano smoothings and in order to prove a conjecture by Corti-Filip-Petracci about the smoothing components of affine non-isolated toric singularities.

Bernd Siebert - Canonical wall structures via punctured Gromov-Witten theory, slides

I will sketch a construction of consistent wall structures for (generalized) logarithmic Calabi-Yau pairs (X,D). This shows compatibility of the intrinsic mirror construction jointly with Mark Gross in the preprint "Intrinsic mirror symmetry" from last year, which is based on the direct definition of structure coefficients of the coordinate ring, and earlier constructions based on wall structures and rings of generalized theta functions.

The proof has been made possible by recent advances in punctured Gromov-Witten theory and gluing formulas, presented in the two previous talks by Dan Abramovich and Yixian Wu. This is joint work with Mark Gross.

Yixian Wu - Splitting of Logarithmic Gromov-Witten Invariants with Toric Gluing Strata, slides

Based on the punctured logarithmic Gromov-Witten theory built by Abramovich, Chen, Gross and Siebert, I will present a numerical splitting formula of log GW-invariants assuming that the gluing stratas in the target are toric. In the formula, the degenerate log GW-invariants are written as the sum of the products of invariants from splitted domain curves. As an application, I will sketch an alternative proof of the curve counting formula under toric degeneration of Nishinou and Siebert.