PCMI 2027 Undergraduate Summer School
The Undergraduate Summer School (USS) at PCMI provides a unique opportunity for undergraduate students to learn some fascinating mathematical ideas in a setting that allows them to interact with mathematicians at all levels. The program itself is typically centered around lecture series delivered by leading experts on topics related to the main research theme of PCMI that summer. These lectures generally present material not usually part of an undergraduate curriculum, allowing students to become familiar with key ideas and techniques in the field, and often leading toward further research. The program is structured so that students at different levels will have many opportunities to learn new things.
The USS is not like a typical REU in several ways. The focus of the USS is more on the specialized lecture series and most importantly, the ability to interact informally with the many graduate students and researchers attending other parts of PCMI. Students can get to know mathematicians who have pursued a wide variety of career paths, and they can get a sense of which of these paths may be most appealing to them. Many USS participants report making connections that strongly influence their choice of graduate school. Interactions are fostered by the various informal social activities open to all PCMI participants, as well as daily "cross-program activities," which include lectures and presentations on topics of general mathematical interest. USS attendees participate in the Experimental Math Lab (XML) held daily at 9:40am, in which small groups of participants with close mentorship from a more senior mathematician investigate open-ended problems and report on their findings at the end of the three-week Summer Session.
The USS is open to undergraduate students at all levels, including those who have just completed their undergraduate studies. Participants are expected to be in residence for the entire three weeks of PCMI.
The PCMI Summer Session will be held June 27-July 17, 2027.
Research Theme: Patterns, Primes and Dynamics
In 2027 there will be one daily lecture series at 1:00pm, given by Kyle Pratt (Brigham Young University), and the morning sessions will involve Experimental Mathematics Lab (XML) component with open-ended problems and computational work. Here is an outline of Prof. Pratt's lectures:
Title: The primes and the Riemann zeta function
Course description: This course introduces analytic number theory through the relationship between the prime numbers and the Riemann zeta function. We will explore how the zeros of the zeta function govern the distribution of primes and what they reveal about prime numbers in short intervals. Along the way, we will study Dirichlet polynomials—finite sums that model the zeta function—and see how estimates for their average size lead to information about zeros and primes. Examples, selected proofs, and problems will emphasize the ideas behind the methods. Time permitting, we will also explore recent breakthroughs on the prime numbers and the Riemann zeta function.
Recommended prerequisites: Experience reading and writing mathematical proofs. Single-variable calculus, including infinite series and improper integrals. A first course in real analysis is recommended. Some familiarity with complex numbers is expected. Some exposure to complex analysis would be helpful but is not required; the specific complex-analytic tools used in the course will be introduced as needed. No previous course in number theory is assumed.
All admitted students will be expected to contribute toward an inclusive and collegial atmosphere in this program. USS students come from many different backgrounds, and with different strengths, and we ask that all USS students commit to working together positively and noncompetitively.
