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Thomas Walpuski: Lectures

September 13, 2017
TITLE: The (1,k)-ADHM Seiberg­-Witten equation and k­-fold covers of associatives

ABSTRACT: The (1,k) ADHM Seiberg–Witten equations are a class of generalized Seiberg–Witten equations associated with the hyperkähler quotient appearing in the Atiyah, Drinfeld, Hitchin, and Manin’s construction of the framed moduli space of ASD instantons on R4. Formally, degenerating solutions of this equation are related to Fueter sections of bundles of symmetric products of k copies of R4. In this talk I will explain this relation in more detail and discuss why we believe these equations to be relevant to issues of multiply covered associatives. This is joint work in progress with Aleksander Doan.

September 7, 2016
TITLE: From G2 gauge theory to the ADHM Seiberg–Witten equation

ABSTRACT: The compactness question for G_2–instantons naturally leads one to consider the local problem around an associative submanifold. I will explain how, in a neighborhood of an associative submanifold, the G_2–instanton equation can be understood as a Seiberg–Witten type equation. A result of Haydys then motivates introducing a second Seiberg–Witten type equation associated with the ADHM construction. I will end the talk by discussing a compactness theorem for the simplest of the ADHM Seiberg–Witten equations. This is joint work with Andriy Haydys.