Omid Amini, "Limit linear series over the moduli of stable curves"

A linear series of degree d and rank r, abbreviated g.r.d., on a smooth projective curve is the data of a line bundle of degree d on the curve and an (r+1) dimensional space of global sections of that line bundle.

The problem of defining a moduli space capturing all the possible limits of g.r.d.s on curves was posed by Eisenbud and Harris in the eighties in their study of the geometry of the general curve and the moduli space of curves. Eisenbud and Harris characterized themselves all the possible limits of linear series when smooth curves degenerate to a curve of compact type. Very little is known on this problem regarding other parts of the Deligne-Mumford compactification, essentially the case of

- canonical linear series (d=2g-2, r=g-1) with limit curve having two components (Esteves-Medeiros),

- limit curve being of pseudo-compact type (Osserman),

- sections of pluricanonical line bundles (r=0) with arbitrary limit curve (Bainbridge-Chen-Gendron-Grushevsky-Möller).

I will report on ongoing joint work with Esteves where we seek to construct a moduli space of limit linear series over the moduli space of stable curves. I will present the general idea of our approach, what we have done so far, and our recent discovery that relates this to hybrid curves and their moduli spaces (introduced in joint works with Nicolussi with the aim of describing analytic aspects of the asymptotic geometry of Riemann surfaces).

Francesca Carocci, "Rubber tori and the boundary of expanded stable maps"

Extending and generalising Jun Li’s original approach to define relative GW invariants, Ranganathan constructs moduli spaces of (log) expanded stable maps. These spaces parametrise  transverse stable maps to certain target expansions. In this talk, I will start by describing the geometry of the expansions that can appear as targets in the moduli space of expanded maps.  I will then explain the identification of maps in the boundary induced by the action of rubber tori on the “higher levels” of the expanded target; such action can be satisfyingly described at the tropical level. In particular, I will explain the difficulties in obtaining a recursive description for the boundary of the moduli space of  expanded maps. This is based on joint works with N.Nabijou.

Leo Herr, "Log Intersection Theory and the Log Product Formula"

 Kato: Log structures are “magic powder” that makes mildly singular spaces appear smooth. Log schemes literally lie between ordinary schemes and tropical geometry, and are related to Berkovich Spaces. Problems in Gromov-Witten Theory demand intersection theoretic machinery for slightly singular spaces. Log structures have solved similar problems in Hodge Theory, D-modules, Connections and Riemann-Hilbert Correspondences, Abelian Varieties (esp. Elliptic Curves), etc. How can they be used to define a reasonable intersection theory for curve counting on singular spaces? We'll give a product formula as proof-of-concept for a whole toolkit under development to tackle these types of problems.

David Holmes, "Measuring when a (log) line bundle on a curve is trivial"

Given a family of smooth proper curves C/S and a line bundle L on C, the set of points in S over which L is trivial (the 'double ramification cycle') plays a significant role in localisation calculations in enumerative geometry. What happens when the family C/S is allowed to be (semi/pre-)stable? One can try to extend the naive definition, but there is a combinatorial obstruction which makes the set of points on S only locally closed; not healthy if one wants to define an associated cycle class. Looking closely at some examples, we will see that this problem begs for a tropical/logarithmic solution, and how this leads to a natural logarithmic double ramification cycle, and a formula in terms of piecewise-polynomial functions on the moduli stack of tropical curves.

Gavril Farkas, "The birational geometry of M_g: new developments via non-abelian Brill-Noether theory and tropical geometry"

I will discuss how novel ideas from non-abelian Brill-Noether theory can be used to prove that the moduli space of genus 16 is uniruled and that the moduli space of Prym varieties of genus 13 is of general type. For the much studied question of determining the Kodaira dimension of moduli spaces, both these cases were long understood to be crucial in order to make further progress. I will also explain the use of tropical geometry in order to establish the Strong Maximal Rank Conjecture, necessary to carry out this program.

Sam Molcho, "Compactifying the Prym variety of a family of nodal curves"

Let C --> S, B-->S be families of nodal curves over a log smooth base S, smooth over the dense open subscheme U of S where the log structure is trivial. Suppose C --> B is a finite flat map over S. In this talk, I will explain how to extend the Prym variety Prym(C_U --> B_U) to a proper scheme over S. The construction comes from combining the semistable reduction theorem with the theory of compactified Jacobians, via the logarithmic Jacobian. This is joint work in progress with di Lorenzo-Gross-Horn-Ulirsch.

Johannes Schmitt, "A logarithmic approach to spaces of multi-scale differentials"

Strata of differentials are moduli spaces parameterizing a smooth curve together with marked points and a meromorphic differential form which has zeros and poles exactly at the markings, of prescribed multiplicities. For such a stratum, a smooth compactification (the moduli space of multi-scale differentials) was constructed by Bainbridge, Chen, Gendron, Grushevsky and Möller. Their definition is very hands-on and explicit, but with many moving parts: as the curve degenerates, one needs to remember a list of auxiliary data  encoded in so-called enhanced level graphs, twisted differentials and prong matchings) taken modulo an intricate equivalence relation (governed by the (simple) level rotation torus). In this talk I will recall their definition, and explain how logarithmic geometry can be used to give a second approach to defining the space of multi-scale differentials. This provides a conceptual interpretation for each of the objects mentioned before. If time permits I will also discuss applications to the geometry of the moduli space and conjectures about a Hodge Double ramification cycle. This is joint work with Chen, Grushevsky, Holmes and Möller.

Rosa Schwarz, "Piecewise polynomial functions and divisors"

Where in classic algebraic geometry we do instersection theory with homology or Chow groups, working with divisors, in log geometry we work with piecewise-polynomial functions: in simplest cases just the combinatorial data of a polynomial function on a toric fan. Although these concepts are not the same, there is a connection between the two. In work with David Holmes, we constructed a map from piecewise polynomial functions to the Chow group in order to prove properties of the double-double ramification cycle. In practice, this map means that there is a way to translate between a piecewise polynomial function on a log scheme to classes of divisors in Chow. By giving some key examples, including the practical example of the moduli space of stable 2-marked genus 1 curves, I would like to illustrate how you can get some intuition for these functions and the corresponding divisors, and give some explicit computations.

Qaasim Shafi, "Logarithmic Quasimaps"

Quasimap theory provides an alternative curve counting framework to Gromov-Witten theory by allowing the map from the curve to the target to acquire basepoints. It is related to Gromov-Witten theory via wall-crossing formulas and plays an important role in mirror symmetry. In recent years, there has been a growing interest in relative, or logarithmic Gromov-Witten theory, for example for its use in computing ordinary Gromov-Witten invariants via the degeneration formula. I will describe a construction of a theory of logarithmic quasimaps and explain some potential applications.

Filippo Viviani, "On the Universal Jacobian: algebraic, tropical and logarithmic aspects"

I will start by reviewing how to compactify the universal  Jacobian stack over the moduli stack of stable pointed curves. Then I will discuss  logarithmic universal Jacobians and their tropicalization morphism towards the corresponding tropical universal Jacobians.  Finally, I will discuss the connection with the Molcho-Wise's  logarithmic Picard and tropical Picard stacks. This is a joint work with M. Melo, S. Molcho, M. Ulirsch, J. Wise.