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Aleksander Doan: Lectures

January 8, 2020
TITLE: On complexification and categorification

ABSTRACT: In this introductory talk, I will outline some basic ideas behind three related infinite-dimensional complex Morse theories, underlying Donaldson-Thomas invariants of Calabi-Yau three-folds, complex instanton Floer homology of 3-manifolds, and intersection Floer homology of hyperkahler manifolds. The second part of the talk will be devoted to the Fueter equation, an interesting generalization of the Dirac equation, whose solutions play an important role in all three theories.

September 13, 2019
TITLE: From Gauge Theory to Calibrated Geometry and Back

ABSTRACT: Twenty years ago, Donaldson and Thomas proposed to define invariants of special holonomy manifolds using the equations of gauge theory. In the first part of Doan’s talk, he will discuss the basic ideas behind this fascinating proposal and survey some recent advances made by the Simons Collaboration. He will then focus on Calabi-Yau manifolds, in particular on a joint project with Thomas Walpuski, whose goal is to define new invariants of Calabi-Yau three-folds using gauge theory and pseudo-holomorphic curves.

Slides of Lecture

January 09, 2018
TITLE: Multiple covers of associatives and ADHM monopoles

ABSTRACT: Continuing Thomas Walpuski’s talk, I will explain how to extend the definition of the putative enumerative invariant of G_2 manifolds to include contributions from multiple covers of associative submanifolds. The main idea, which is a special instance of the Haydys-Walpuski proposal, is to incorporate into the invariant solutions of the ADHM Seiberg-Witten equations on associatives.

September 12, 2017
TITLE: Fueter sections and wall-crossing in Seiberg-Witten theory

ABSTRACT: Fueter sections are solutions to a non-linear generalization of the Dirac equation on a Riemannian spin three-manifold. The goal of this talk, based on joint work in progress with Thomas Walpuski, is to explore the relationship between Fueter sections taking values in instanton moduli spaces and wall-crossing for solutions to the Seiberg-Witten equation with multiple spinors. Time permitting, I will explain how this discussion fits into the Donaldson-Segal program of counting G2-instantons.