I have given several talks for the Mirror Symmetry seminar I run.

Why Counting Curves on toric surfaces isn't scary GAeL 2022

The logarithmic Hilbert scheme and its Tropicalisation Frankfurt Algebraic Geometry 05/2021

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.

Compactifying with logarithmic geometry Imperial Junior Geometry Seminar 04/02/2021

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.

Tropical and algebraic curve counting Kings/UCL Junior Geometry Seminar 28/10/2021

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.

Passing between obstruction theories [Internal Seminar] 16/08/2021

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.

Donaldson--Thomas theory Imperial Junior Geometry Seminar 11/06/2021

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.