PCMI 2019 Graduate Summer School

The Gradate Summer School at PCMI consists of a series of several interwoven minicourses on different aspects of the main research theme of that summer.  These courses are taught by leading experts in the field, chosen not only for their stature in the field but their pedagogical abilities. Each minicourse comprises three to five lectures.  These minicourses vary in level of preparation needed, and the schedule is structured so there are good opportunities for students just entering the field as well as courses suitable for more advanced students.  Each course is accompanied by a daily problem session, structured to help students develop facility with the material.

The GSS takes place within the broader structure of PCMI, so there are many researchers at all levels in the field in attendance, as well as participants in the other PCMI programs.  This provides an outstanding way for graduate students to get to know leaders in their field and to interact with them in a leisurely way.  There are also numerous group activities which allow participants in the GSS to interact with people in other groups, including formal and informal social events, and the PCMI Experimental Math Lab, which brings together small groups of participants to work on accessible and open-ended problems.   Graduate students have many opportunities to get good advice about career paths after they complete their PhDs, and can meet mathematicians who are working at a wide variety of institutions, from top research centers to undergraduate-focused colleges.

There are three graduate minicourses scheduled each day (Wednesday afternoons are free) and problem sessions accompanying each minicourse. Participants may attend talks from the other programs as they see fit.

The 2019 Program: Quantum Field Theory and Manifold Invariants

Quantum field theory, a branch of high energy theoretical physics, has enjoyed a vigorous interaction with many parts of mathematics over the past several decades.  Indeed, that interaction was the subject of the first PCMI program in 1991.  The 2019 Graduate Summer School will focus on invariants of smooth manifolds which arose in a broad sense from quantum field theory. There will be two introductory minicourses, one on mathematical gauge theory and the other on knot invariants.  More advanced minicourses will cover such topics as Heegaard-Floer homology, knots and BPS states, characteristic classes for manifold bundles, BPS states and spectral networks, Higgs bundles, and applications of Floer homology.  The goal of the program is to bring together students and researchers at all levels interested in these areas to share exciting recent developments in these subjects, stimulate further interactions, and inspire a new generation to pursue research at the interface of geometry and physics.

Student preparation: We seek motivated students interested in manifold invariants and in mathematical aspects of quantum field theory. Though familiarity with some of the topics listed above would be helpful, the formal general prerequisites are graduate-level courses in the basics of manifold theory, topology, and analysis.  No formal training in physics is required. Additional prerequisites for individual courses are listed with the course descriptions.

The 29th Annual PCMI Summer Session will be held June 30 - July 20, 2019.