PCMI 2021 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.                                                                     

The UFP normally runs as a three-week summer program parallel to the other PCMI programs, but because of COVID-19, this summer’s UFP will be a held as a single one-week online program.  It is scheduled for the week after the Undergraduate Summer School (USS) to make it easier for UFP members to attend the USS lectures and interact with USS participants if so desired.  As in normal years, UFP members will be encouraged to interact with all the participants and lecturers in the other programs during the full three-week program, and in particular may choose to virtually attend some of the other lecture series. 

Members from all parts of PCMI may also take part in the Experimental Math Lab (EML) 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 virtual session of the Undergraduate Faculty Program will be held August 2 - 6, 2021.

The lecturer for the Undergraduate Faculty Program in 2021 will be Kyle Ormsby (Reed College), and here is a rough plan for what will be covered:

Motivic degree and Milnor numbers

This short course will take a computational dive into the world of quadratic enumerative geometry, focusing on motivic Milnor numbers. Given an isolated complex hypersurface singularity (a simultaneous solution of f = grad(f) = 0), the classical Milnor number can be computed as the degree of the normalized gradient on a small sphere centered at the singularity.  Motivic (or A^1-) geometry generalizes this invariant to singularities over other fields by counting degree in the Grothendieck-Witt ring of quadratic forms.  We will review the classical story, generalize to arbitrary fields, and produce numerous hand- and computer-aided computations.  By the end of the week, we'll be on the cusp of some open problems; versal deformation of this course into nodes of original research (conducted independently or with students) is possible and encouraged.

Prerequisites: basic algebraic theory of quadratic forms, including definitions and computations of Grothedieck-Witt rings; some commutative algebra (Gröbner bases, zero-dimensional Gorenstein rings) may be useful but could be picked up en passant. Participation in the USS would provide ample preparation in quadratic forms. The USS will be held virtually July 11-31, 2021.