pcmi undergraduate faculty program (2021)
The 2021 PCMI Undergraduate Faculty Program is allied with the motivic homotopy Graduate Summer School and Undergraduate Summer School. We will discuss Milnor forms (aka motivic or $\mathbb{A}^1$-Milnor numbers), which generalize classical Milnor numbers. Participants should finish the week prepared to conduct independent and student research on this topic. We will also discuss 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 will adhere).
schedule
All times US Mountain Daylight (UTC-06:00). All events will occur virtually at pcmi2021.sococo.com on the Undergraduate Faculty Program floor.
| 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 will be structured:
Tech check-in
Optional. Stop by to test you technology setup and familiarize yourself with our meeting space at pcmi2021.sococo.com.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/Khimshiashvilli 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 sococo Gathering Space to socialize with participants.Discussion / problem session
These 60-minute sessions will 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 will 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 will help guide our discussions, so make sure you get a copy!Peer collaboration
Optional. Those interested in and available to continue mathematical and mentoring discussions will be divided into teams that can meet during these times.resources
The following notes, papers, and books might be useful.
- Lecture 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 $\mathbb{A}^1$-enumerative geometry of the $\mathbb{A}^1$-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 $\mathbb{A}^1$-Milnor numbers with Macaulay2 – S. Pauli
- The class of Eisenbud–Khimshiashvili–Levine is the local $\mathbb{A}^1$-Brouwer degree - J.L. Kass & K. Wickelgren
- $\mathbb{A}^1$-Milnor numbers (Oberwolfach report) - K. Wickelgren
- An arithmetic enrichment of Bézout’s Theorem - S. McKean
- $\mathbb{A}^1$-Milnor numbers - M.U. Hafeez (Reed thesis)