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.