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Thomas Walpuski: Lectures

January 9, 2018
TITLE: On counting associative submanifolds and Seiberg–Witten monopoles

ABSTRACT: From Nordström and Joyce we learn that counting associatives cannot lead to an invariant of G_2-manifolds. Classical surgery formulae for the Seiberg-Witten invariant SW(Y) of a 3-manifold Y show that the count of associative submanifolds weighted by SW is invariant under the transitions described Nordström and Joyce. Unfortunately, the Seiberg-Witten invariant SW(Y) is only defined if b_1(Y) > 1. In this talk I will explain how one might replace SW by a version of Kronheimer-Mrowka’s monopole homology and construct a Floer homology group which has a chance of being both defined and invariant. This is joint work with Aleksander Doan.

January 8, 2018
TITLE: Introduction to gauge theory on Riemannian manifolds with special holonomy

ABSTRACT: Riemannian manifolds with special holonomy typically allow for instanton-type equations implying the the Yang-Mills equation. I will recall the most important examples of these equations; discuss foundational results and examples; and finally mention a number of questions.

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.