program Description

The main activity of the Graduate Summer School will be a set of several interwoven minicourses offered by leading experts in the field, designed to introduce students to exciting, current research in mathematics. These minicourses will not duplicate standard courses available elsewhere. 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. Teaching assistants will be available for each minicourse.

The 2026 Research Topic is Knotted Surfaces in Four-Manifolds: 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 aim of the planned summer graduate program at PCMI is to provide in-depth introduction to several preeminent themes and methods in extremal and probabilistic combinatorics, with particular emphasis on strong connections of these fields with other areas of mathematics such as analysis, geometry, number theory, statistical physics, and theoretical computer science. 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.

APPLICATION PROCEDURE

1. Complete a Standard Coversheet for MathPrograms.Org. Your Standard Coversheet should include the name/s and email address/es of your reference writer/s. One letter of reference is REQUIRED (see below); you may submit up to two references in support of your application.

2. Provide one Reference Letter (REQUIRED). On the application, fill in the checkbox for your reference writer from the name you entered on your Standard Coversheet, then click on the green arrow next to the reference writer’s name so the writer will receive your reference request by email. When notified, your writers will then receive login info and upload instructions to come to the MathPrograms.org site to upload their letters. If your reference writer does not have access to a computer, he or she may submit the reference letter by mail no later than January 15, 2026.

3. Provide your Curriculum Vitae/Resume (REQUIRED): Applicants must upload their CV to their online application.

4. Provide your publication list.(OPTIONAL)