PCMI 2026 Graduate Summer School
The Graduate 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 2026 Program: Knotted Surfaces in Four-Manifolds
Organizers: R. İnanç Baykur, University of Massachusetts Amherst; Kyle Hayden, Rutgers University – Newark; András Stipsicz, Rényi Institute of Mathematics; Gordana Matic, University of Georgia; Masaki Taniguchi, Kyoto University; and Ian Zemke, University of Oregon.
There are many complex mysteries in the theory of 4-manifolds. Adding a knotted surface to a 4-manifold gives additional structure that makes some questions easier, while other questions become even more difficult. Just as in the case of closed 4-manifolds the topological and smooth theories have very different characters.
The 2026 Graduate portion of PCMI will consider ways to construct and represent interesting surface embeddings in four-dimensinal spaces. The topological theory will be addressed along with invariants that distinguish topologically equivalent smooth embeddings. Complex, and symplectic notions will also be covered. Techniques ranging from gauge theory, to Floer theory, to mapping class groups will be considered as well.
The core of the program will be nine graduate mini-courses taught by a diverse group of leading researchers in the field renowned for their clear and engaging lecturing styles. In parallel, we plan thematic workshops aimed at more senior researchers as well as activities for undergraduate students.
The PCMI Summer Session will be held June 28-July 18, 2026.
