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 11/06/2020
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.