Abstract: Gross-Hacking-Keel constructed scattering diagrams for log Calabi-Yau surfaces. The scattering diagram is the key ingredient for the construction of the mirror family of the log Calabi-Yau surfaces. In this talk, I will explain the background on scattering diagrams and some ideas from algebraic geometry to help construct the scattering diagram from Floer theory. As corollaries, this leads to the folklore conjecture that open Gromov-Witten invariants coincide with certain log Gromov-Witten invariants, 21 dics in cubic surfaces as an open analogue 27 lines in cubic surfaces and a version of mirror symmetry of cluster varieties. This is a joint work with Bardwell-Evans, Cheung and Hong.

The absolute Galois group of the field of rational numbers and the Grothendieck-Teichmueller (GT) group introduced by V. Drinfeld in 1990 are among the most mysterious objects in mathematics. In my talk, I will introduce (the gentle version) of the Grothendieck-Teichmueller group. I will also introduce the groupoid GTSh of GT-shadows that can be used to study this group. I will explain how the groupoid GTSh acts on child's drawings, describe properties of this action and show some interesting examples. My talk is based on papers in preparation with Jacob Guynee, Jessica Radford and Jingfeng Xia.

The moduli space M_{g,n} of smooth algebraic curves of genus g with n distinct marked points is not compact. However, it admits many compactifications that are themselves moduli spaces, and it remains an outstanding problem in algebraic geometry to classify these modular compactifications. An important family of examples is that of the moduli spaces of "weighted pointed" curves constructed by Hassett, in which a vector of real numbers determines which of the n marked points are permitted to come together. In this talk, I will present joint work with Vance Blankers that constructs modular compactifications of M_{g,n} using a simplicial complex rather than vector of weights as an input. Not only do the resulting "simplicial" moduli spaces generalize Hassett's, but they also classify the modular compactifications coming from colliding markings. If time permits, we will also discuss how this idea can be combined with an earlier classification result to produce a classification of modular compactifications of M_{1,n} by Gorenstein curves admitting collisions.

Manin’s Conjecture predicts the asymptotic formula for the counting function of rational points over number fields or global function fields. In the late 80’s, Batyrev developed a heuristic argument for Manin’s Conjecture over global function fields, and the assumptions underlying Batyrev’s heuristics are refined and formulated as Geometric Manin’s Conjecture. Geometric Manin’s Conjecture is a set of conjectures regarding properties of the space of sections of Fano fibrations, and it consists of three conjectures: (i) Pathological components are controlled by Fujita invariants; (ii) For each nef algebraic class, a non-pathological component which should be counted in Manin’s Conjecture is unique (This component is called as Manin components); (iii) Manin components exhibit homological or motivic stability. In this talk we discuss our proofs of GMC (i) over complex numbers using theory of foliations and the minimal model program. Using this result, we prove that these pathological components are coming from a bounded family of accumulating varieties. This is joint work in progress with Brian Lehmann and Eric Riedl.

Gendron and Tahar studied the number of meromorphic differentials on rational curves with a unique zero, fixed orders of poles, and given residue conditions by using flat geometry. I will talk about generalizing these results to arbitrary residue conditions by using intersection theory on compactified strata of abelian differentials.

In this continuation of my Spring 2022 talk, I will remind you of a notion of Kollár stability for families of projective varieties fibered over one-dimensional bases, a common generalization of Tate's minimal models for elliptic curves and Kollár's stability for hypersurfaces over PIDs. Having discussed some successful applications in my previous talk, I will focus on several open problems in the field.

Abstract: An open question of David Prill from the 1970's, popularized in ACGH, asks whether there is a cover of curves $f: X \to Y$ with $g(Y) \geq 2$ so that every fiber moves in a pencil. We discuss joint work with Daniel Litt, answering this question. In order to do so, we describe its connection to a conjecture in geometric topology, the Putman-Wieland conjecture, by studying the derivative of an associated period map.

Abstract: Punctured logarithmic R-maps are objects that naturally appear in the boundary of logarithmic compactifications of gauged linear sigma models. They are combinations of stable maps and (roots of) canonical divisors of the domain curves twisted using the logarithmic geometry. Their moduli stacks admit virtual cycles. The corresponding invariants give rise to explicit correction terms to the quantum Lefschetz principles of Gromov-Witten theory in arbitrary genus via explicit formulas. This leads to many recent progresses in Gromov-Witten theory. For example, in the quintic three-folds case, the number of invariants needed for these correction terms are $\lfloor (2g-2)/5 \rfloor + 1$. This matches the number of free parameters of the BCOV B-model theory from physics. Similar results apply to other Calabi-Yau three-fold complete intersections.

This talk is based on collaborations with Dan Abramovich, Mark Gross and Bernd Siebert on the logarithmic Gromov-Witten theory, and collaborations with Felix Janda, Yongbin Ruan and Rachel Webb on gauged linear sigma models.