The seminar meets on Wednesdays at 11 am in Gordon Palmer Hall 302.
Schedules from past semesters: https://sites.ua.edu/btosun/seminars/.
Schedule for Fall 2026.
Aug 19:
Aug 26: Joe Breen (Alabama).
Title: Equality in the slice-Bennequin inequality
Abstract: There is a well-known inequality in topology known as the “slice-Bennequin inequality,” which can be thought of as a local, relative, inequality version of the adjunction formula in algebraic geometry. I will describe forthcoming work (joint with A. Zupan) on understanding when equality holds, which is an open problem in general.
Sep 2: Thomas Polstra (Alabama).
Title: (S_2)-ifications and a Uniform Briançon-Skoda Theorem for Non-Reduced Rings
Abstract: The Uniform Briançon–Skoda Theorem states that for a large class of commutative Noetherian reduced rings, there exists a natural number B such that for every ideal $I\subseteq R$ and every natural number n, if $\overline{I^{n}}$ denotes the integral closure of I^{n}, then $\overline{I^{n+B}} \subseteq I^n$. In the classical Briançon–Skoda Theorem, one can take B = d - 1 when R is regular, where d is the global dimension of R.
The ideal $\overline{I^n}$ can be realized as the n-th graded piece of the Rees or extended Rees algebra of I. When R is not reduced, the normalization of a Rees algebra fails to be Noetherian, and there exist ideals I such that $\overline{I^t} \not\subseteq I^2$ for all $t \geq 0$.
To adapt the significance of the Uniform Briançon–Skoda Theorem to the non-reduced setting, we instead consider graded algebras obtained as the (S_2)-ification of the (extended) Rees algebra of I. We establish uniform behavior in the (S_2)-ification process for (extended) Rees algebras and prove a uniformity result for non-reduced rings analogous to the Uniform Briançon–Skoda Theorem for reduced rings. Our approach makes essential use of homological methods, in particular the identification of the (S_2)-ification as the endomorphism ring of a finitely generated module that localizes generically to a canonical module.
This talk is based on joint work with Daniel Katz.
Sep 9:
Sep 16:
Sep 23: Fuxiang Yang (Notre Dame).
Sep 30: Eduardo Fernández (Georgia Tech).
Oct 07:
Oct 14: Sean Eli (Georgia Tech).
Oct 21:
Oct 28: Vera Vertesi (IAS/ University of Vienna).
Nov 04: Anirban Bhaduri (South Carolina).
Nov 11: Uly Alvarez (Alabama).
Nov 18:
Nov 25: Thanksgiving Break.
Dec 02: