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).
Title: Recent progress on the Eisenbud-Huneke-Ulrich conjecture
Abstract: Over a polynomial ring, an ideal I is said to be linearly presented if its first syzygy module is generated by linear relations. More generally, the resolution of I is linear for p steps if this linearity continues through the first p steps. In this talk, we explore some fundamental properties of ideals with partially linear minimal free resolution, such as asymptotic behavior of powers, Castelnuovo-Mumford regularity, and lower bounds on the number of minimal generators. In particular, we prove the Eisenbud-Ulrich conjecture on powers of linearly presented ideals and establish a linear effective bound toward the more general Eisenbud-Huneke-Ulrich conjecture.
Sep 30: Eduardo Fernández (Georgia Tech).
Title: Overtwisted disks detect tight contact structures.
Abstract: Overtwisted contact structures were introduced by Eliashberg in his seminal 1989 article, and their classification is governed by an h-principle. Nevertheless, in this talk I will introduce the subclass of rigid overtwisted contact structures in dimension three, characterized by the existence of more than one overtwisted disk up to contact isotopy. As its name suggests, these overtwisted contact structures exhibit a genuine form of geometric rigidity. The motivation of this notion comes from work of Vogel and Chekanov, which established the existence of a unique rigid overtwisted contact structure on the 3-sphere.
I will discuss the nature of rigid overtwisted 3-manifolds, explain how they force the existence of tight contact structures, and construct new examples of rigid overtwisted contact structures. This talk is based on ongoing joint work with John Etnyre and Gordana Matić.
Oct 07:
Oct 14: Sean Eli (Georgia Tech).
Title: Exotic R^4's and Heegaard Floer homology
Abstract: We discuss an explicit construction of exotic R^4's: smooth 4-manifolds homeomorphic but not diffeomorphic to R^4. In the 90's De Michelis and Freedman asked how the choices in this construction affect the resulting smooth manifolds. This problem has seen some progress but is still not well understood. We show how to tell these spaces apart using Heegaard Floer homology, and give an explicit infinite family of exotic R^4's. We also discuss related phenomena and open problems. This is joint work with Jen Hom, Tye Lidman, and Shunyu Wan.
Oct 21: Debaditya Raychaudhury (New Mexico).
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: