PCMI 2027 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.
Research topic: Patterns, Primes and Dynamics
Organizers: Nikos Frantzikinakis, University of Crete; Bryna Kra, Northwestern University; Sarah Peluse, Stanford University; Florian Richter, EPFL; and Joni Teräväinen, University of Cambridge.
Clay Scholars: Vitaly Bergelson, The Ohio State University, and Peter Sarnak, Princeton University and Institute for Advanced Study
Which arithmetic patterns must appear in a dense sets of integers? in the primes? in finite colorings of the integers? What forms of randomness emerge in the distribution of primes? What about in the behavior of multiplicative functions? These questions connect additive combinatorics, analytic number theory, and ergodic theory, and the interactions among these fields have transformed our understanding of order and randomness.
The 2027 Graduate Summer School at PCMI will introduce central themes and methods in these fields through nine closely interconnected minicourses, ranging from classical foundations to recent breakthroughs and open problems. Particular emphasis will be placed on how ideas from one area illuminate questions in another.
Topics will include arithmetic patterns, Gowers norms and inverse theorems, structure and recurrence in dynamical systems, multiplicative functions and the Sarnak and Chowla conjectures, primes in short intervals, infinite additive configurations, and pointwise ergodic theorems along polynomials and primes.
The program welcomes graduate students beginning to explore these subjects as well as those seeking to broaden their research perspective. Concurrently, we will have lectures aimed at more senior researchers, as well as activities for undergraduate students.
The PCMI Summer Session will be held June 27-July 17, 2027.
