PCMI 2022 Undergraduate Faculty Program

The Undergraduate Faculty Program (UFP) at PCMI is aimed at faculty members from all types of colleges and universities with a strong interest in undergraduate teaching and research.  Some participants are looking to rekindle their engagement with mathematical research and to interact with the broader mathematical community, while others are looking for new teaching approaches.  The focus of the UFP varies from year to year, although its mathematical content is always aligned with the main research theme of PCMI that summer. In some years it is run as a small group seminar focused on some interesting parts of mathematics, in a setting which emphasizes participants working on ideas and problems together; in other years some part of it is also devoted to pedagogical issues or curriculum development.  The UFP typically has around 15 participants, which allows for close and informal interactions among the participants.

UFP participants also interact with the participants and lecturers in the other programs, and in particular may choose to attend some of the Undergraduate and Graduate Summer School lecture series.  Members from all parts of PCMI may take part in the Experimental Math Lab, 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.  Interaction among everyone at PCMI is fostered by various informal social activities open to all PCMI participants, as well as daily “cross-program activities’’ that include lectures and presentations on topics of general mathematical interest.

The 30th Annual PCMI Summer Session will be held July 17-August 6, 2022.

2022 Research Theme: Number Theory Informed by Computation.

The lecturer for the Undergraduate Faculty Program in 2022 will be Sinai Robins (University of Sao Paulo, Brazil), and here is a rough plan for what will be covered:

UFP TopicsWe will study the geometry of numbers, initiated by Minkowski, from a Fourier analytic perspective.   There are many number-theoretic consequences, arising from interactions between polytopes and lattices.   Using this geometric context, we will learn about various number-theoretic functions that include Dedekind sums and Gauss sums.    The Dedekind eta function also comes up, as well as theta functions attached to lattices.  Some open problems will be posed and discussed, arising naturally from these topics.