Laura P. Schaposnik 1.1, Advanced topics in (mathematical) gauge theory / Higgs bundles (IAS | PCMI)
Channel: IAS | Park City Mathematics Institute | PCMI
Date: 15/07/2019

Laura P. Schaposnik, University of Illinois at Chicago
Higgs bundles provide a unifying structure for diverse branches of mathematics and physics. The Dolbeault Moduli space of Higgs bundles has a hyperkahler structure, and through diﬀerent complex structures it can be understood as diﬀerent moduli spaces: Via the non-abelian Hodge correspondence the moduli space is diﬀeomorphic as a real manifold to the De Rahm moduli space of ﬂat connections. Via the Riemann-Hilbert correspondence there is an analytic correspondence with the Betti moduli space of conjugacy clases of surface group representations.
We will begin by introducing Higgs bundles and their main properties (Lecture 1), and then we will discuss the Hitchin ﬁbration and its diﬀerent uses (Lecture 2). The second half of the course will be dedicated to studying diﬀerent types of subspaces (branes) of the moduli space of Higgs bundles, their apppearances in terms of ﬂat connections and representations (Lecture 3), as well as correspondences between them (Lecture 4).
This 4-lecture course is being presented at the 29th annual PCMI Summer Session taking place July 30–July 19 2019, at the Prospector Conference Center in Park City, Utah. The residential, three-week Summer Session is the flagship activity of the IAS | Park City Mathematics Institute (PCMI), an outreach program of the Institute for Advanced Study. The 2019 research theme is Quantum Field Theory and Manifold Invariants.

The PCMI Summer Session is an intensive program that includes several parallel sets of activities aimed at different groups of participants across the entire mathematics community. These activities include a program for mathematics researchers, eight mini-courses for graduate students on topics related to Quantum Field Theory and Manifold Invariants, two lecture series for undergraduate students, a program for faculty from predominantly undergraduate institutions, a workshop on equity and mathematics education for undergraduate faculty, and a professional development program for K–12 school teachers. All of PCMI’s groups meet simultaneously, pursuing individual courses of study and a meaningful amount of interaction both scientifically and socially.

For more information, visit https://pcmi.ias.edu