Gave talk 4 in a student run seminar based off this program following this paper. My slides.

The Hilbert scheme of a projective variety X is the moduli space of closed subschemes of X. In this talk we will discuss a version of the Hilbert scheme for a pair (X,D) with D a (reasonable) divisor on X. This logarithmic Hilbert scheme is a special case of the logarithmic Quot scheme. As motivation, note hard algebraic geometry problems can be studied by degenerating to simpler situations. Logarithmic geometry provides a suite of tools to study such degenerations. For example, these techniques have been applied to study (logarithmic) GromovWitten theory and more recently (logarithmic) Donaldson-Thomas theory. The logarithmic Hilbert scheme of curves on a threefold is the moduli space studied in logarithmic Donaldson-Thomas theory. A long-term hope is to study other moduli spaces of coherent sheaves with techniques from logarithmic geometry.

The goal of this talk is to appreciate a striking result originally due to Mikhalkin. The question is easy to state: how many degree d curves pass through 3d-1 points in the complex plane. Mikhalkin’s answer illustrates a much--celebrated link between the geometry of polynomials (algebraic geometry) and the geometry of cone complexes (tropical geometry). Cone complexes, unlike algebraic curves, can be drawn on the blackboard: tropical geometry thus gives a powerful tool for visualising problems. Time permitting, I will sketch a modern proof of Mikhalkin’s result.

Let X be a scheme equipped with a SNC divisor D. I will sketch the construction of a proper moduli space of subschemes Z in expansions of X such that Z satisfies an appropriate transversality condition to D. The central insight is a tropical understanding for when Z is flat over the Artin fan of X, and how this behaves in families. I will present the example of the logarithmic linear system, a toric modification of projective space. The associated fan is closely related to the geometry of tropical curves and secondary polytopes.

Fix X a toric surface and $\beta$ the homology class of a curve on X. The moduli space of curves of class $\beta$ in X is a projective space. In this talk, we explore the logarithmic version of this statement. Logarithmic geometry is a way of imposing boundary conditions in algebraic geometry. Our moduli space illustrates the link between logarithmic geometry and the geometry of cone complexes - so-called tropical geometry. Along the way, we will give one answer to the question, "what is a tropical curve and why should one care?".

The first character in this talk is an irreducible closed subvariety Z in the n dimensional torus (C^*)^n. We describe a class of toric varieties {X_\sigma} associated to Z. We encounter tools for understanding the chow class of Z in such toric varieties. Our approach will allow us to “discover” tropical geometry.

The second character in this talk is a closed subvariety Z’ in a logarithmic scheme. The simplest example of a logarithmic scheme is a toric variety. We meet analogues of phenomena in the toric situation. Our story touches on the technology of Artin fans, piecewise polynomials and the logarithmic Chow group.

In enumerative geometry we often try to count curves, but computations can be hard. Logarithmic geometry, and its cousin tropical geometry, provide insights and new approaches to computing these curve counts. Gromov--Witten invariants are closely related to these curve counts. I will start by explaining a result of Mikhalkin reducing the Gromov--Witten theory of toric surfaces to a combinatorial (perhaps you might say "tropical") problem. I will go on to discuss a new construction ("The Logarithmic Linear System") which seems a promising route to a new proof of Mikhalkin theorem.

Wise defined obstruction theories in a general setting allowing him to compare widely used definitions. I will recall the definitions of obstruction theories given by Li/Tian and Behrend/Fantechi. I will go on to explain the definition given by Wise here and explain the relationships he presents with Li/Tian and Behrend/Fantechi obstruction theories.

Notes available here.

Note: Obstruction theories are used to define virtual fundamental classes in enumerative geometry. An excellent exposition for an obstruction theory in the sense of Behrend/Fantechi can be found here.

Abstract: We will start this talk with an introduction to Donaldson–Thomas (DT) theory. DT theory counts ideal sheaves to answer questions starting “how many curves on the 3 dimensional variety X…”. We will look at a linear system as a simple example of sheaf counting. We will then discuss the general DT construction, its relationship to Gromov–Witten theory and a handful of interesting results.

Subject to time, we explain why a logarithmic Donaldson–Thomas theory should prove useful. We discuss the logarithmic linear system as a simple example of logarithmic Donaldson–Thomas spaces. The example will show how solving a tropical moduli problem solves an algebraic moduli problem for free.