park city mathematics institute · 2021

pcmi undergraduate faculty program

Milnor forms, and the theory and practice of leading undergraduate research.

The 2021 PCMI Undergraduate Faculty Program was allied with the motivic homotopy Graduate Summer School and Undergraduate Summer School. We discussed Milnor forms (aka motivic or $A^{1}$ -Milnor numbers), which generalize classical Milnor numbers. Participants finished the week prepared to conduct independent and student research on this topic. We also discussed the theory and practice of leading undergraduate research projects.

Below you can find the UFP schedule (including descriptions of activities) and a list of resources for the program. Scroll to the bottom of this page to see PCMI’s code of conduct, to which we adhered.

schedule

_{All times US Mountain Daylight (UTC−06:00). All events occurred virtually on the Undergraduate Faculty Program floor of the program’s Sococo space.}

	Mon 2 Aug	Tue 3 Aug	Wed 4 Aug	Thu 5 Aug	Fri 6 Aug
9:30–10	Tech check-in
10–10:25	Welcome & introductions
10:30–12	Lecture: Milnor numbers	Lecture: Quadratic forms	Lecture: Motivic degree	Lecture: Milnor forms	Lecture: Open problems
12–1	Lunch	Lunch	Lunch	Lunch	Lunch
1–2	Discussion / problem session	Discussion / problem session	Discussion / problem session	Discussion / problem session	Discussion / problem session
2:15–3:15	UR discussion: Selecting problems	UR discussion: Program design	UR discussion: Recruiting	UR discussion: Funding	UR discussion: Communication
3:30–5	Peer collaboration	Peer collaboration	Peer collaboration	Peer collaboration	Peer collaboration

Click on the items below to see how each activity was structured:

Tech check-in

Optional. Stop by to test your technology setup and familiarize yourself with our meeting space.

Welcome & introductions

Brief opening remarks and a chance to introduce yourself and meet the other participants.

Lecture

Series of 90-minute interactive lectures developing the classical theory of Milnor numbers (of curve and hypersurface singularities), quadratic forms and the Grothendieck–Witt ring, (local) motivic degree and the Eisenbud–Levine/Khimshiashvili form, Milnor forms, and open problems focused on the behavior of Milnor forms under blowup. The lectures should prepare participants to engage with exercises and ultimately explore open problems individually or through undergraduate research programs.

Lunch

Feel free to log off and have lunch on your own, or stop by the Gathering Space to socialize with participants.

Discussion / problem session

These 60-minute sessions permit a flexible mix of discussion with the lecturer and individual and group work on exercises.

Undergraduate research (UR) discussion

Facilitated group discussions about how to lead and mentor undergraduate research. Topics covered include selection of research problems, program design and group dynamics, funding and recruiting, and communication of results. Some required readings from A mathematician’s practical guide to mentoring undergraduate research help guide our discussions.

Peer collaboration

Optional. Those interested in and available to continue mathematical and mentoring discussions were divided into teams that could meet during these times.

resources

The following notes, papers, and books might be useful.

Lecture notes:
- Milnor forms of algebraic singularities — K. Ormsby
- Handwritten notes
Required texts:
- A mathematician’s practical guide to mentoring undergraduate research — M. Dorff, A. Henrich, & L. Pudwell
Recommended texts:
- Singular points of plane curves — C.T.C. Wall
- Singular points of complex hypersurfaces — J. Milnor
- Applications to $A^{1}$ -enumerative geometry of the $A^{1}$ -degree — S. Pauli & K. Wickelgren
Supplemental texts:
- A singular mathematical promenade — É. Ghys
- Plane algebraic curves — E. Brieskorn & H. Knörrer
- Introduction to plane algebraic curves — E. Kunz
- Introduction to quadratic forms over fields — T.Y. Lam
- An algebraic formula for the degree of a $C^{\infty}$ map germ — D. Eisenbud & H.I. Levine
- Computing $A^{1}$ -Milnor numbers with Macaulay2 — S. Pauli
- The class of Eisenbud–Khimshiashvili–Levine is the local $A^{1}$ -Brouwer degree — J.L. Kass & K. Wickelgren
- $A^{1}$ -Milnor numbers (Oberwolfach report) — K. Wickelgren
- An arithmetic enrichment of Bézout’s Theorem — S. McKean
- $A^{1}$ -Milnor numbers — M.U. Hafeez (Reed thesis)