## Curves Seminar - Daniel Erman (Wisconsin-Madison)

### Wednesday, February 7th, 2018

**Time:** 3:30 p.m. **Place:** Jeffery Hall 319

**Speaker:** Daniel Erman (Wisconsin-Madison)

**Title:** Big polynomial rings and Stillman’s conjecture

**Abstract:** Ananyan–Hochster's recent proof of Stillman's conjecture is based on a key principle: if f_1,.., f_r are sufficiently general forms in a polynomial ring, then as the number of variables tends to infinity, they will behave increasingly like independent variables. We show that this principle becomes a theorem if ones passes to a limit of polynomial rings, using either the inverse limit or the ultraproduct. This yields the surprising fact that these limiting rings are themselves polynomial rings (in uncountably many variables). It also yields two new proofs of Stillman's conjecture. This is joint work with Steven Sam and Andrew Snowden.