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Siqi He: Lectures

09/14/2021: The branched deformations of special Lagrangian submanifolds
01/10/2020: A compactness result for the Hitchin-Simpson equations
01/11/2018: The extended Bogomolny equations and generalized Nahm pole solutions

September 14, 2021
TITLE: The branched deformations of special Lagrangian submanifolds

ABSTRACT: Special Lagrangian submanifolds are a distinguished class of real minimal submanifolds defined in a Calabi-Yau manifold, which is calibrated by the real part of the holomorphic volume form. Given a compact, smooth special Lagrangian submanifold, Mclean proved that the space of nearby special Lagrangian submanifolds of it could be parametrized by the harmonic 1-forms. In this talk, we will discuss some recent progress on generalizing Mclean’s result to the branched deformations. We will describe how to use multi-valued harmonic functions to construct branched nearby deformations.

January 10, 2020
TITLE: The counting problem of the Kapustin-Witten equations

ABSTRACT: I will discuss Witten’s program to define the Jones polynomial for knots inside a 3-manifold using gauge theory. In this program, the coefficients of the Jones polynomial will come from counting solutions to the Kapustin-Witten equations. In this talk, I will explain the difficulties in defining this invariant. In addition, I will discuss the non-compactness behavior of the Kapustin-Witten equations and the relationship with Fueter sections.

January 11, 2018
TITLE: The extended Bogomolny equations and generalized Nahm pole solutions

ABSTRACT: We will discuss Witten’s gauge theory approach to Jones polynomial and Khovanov homology by counting solutions to some gauge theory equations with singular boundary conditions. When we reduce these equations to 3-dimensional, we call them the extended Bogomolny equations. We develop a Donaldson-Uhlenbeck-Yau type correspondence for the moduli space of the extended Bogomolny equations on Riemann surface Σ times R^+ with Generalized Nahm pole singularity at Σ × {0} with the stable SL(2,R) Higgs bundle. This is joint work with Rafe Mazzeo.