Upper Midwest Commutative Algebra Colloquium

Saturday, November 14, 2015
at the University of Wisconsin

Plenary speakers:
Srikanth Iyengar (Utah)
Kevin Tucker (UIC)

Saturday, April 23, 2016
at the University of Minnesota

Speakers:
Graham Leuschke (Syracuse)
Brooke Ullery (Utah)
Christine Berkesch Zamaere (Minnesota)
Megan Maguire (Wisconsin)
Principal organizers:
Christine Berkesch Zamaere (Minnesota)
Daniel Erman (Wisconsin)

Organizing committee:
Gennady Lyubeznik (Minnesota)
Steven Sam (Wisconsin)

This fantastic image was designed by Aych and you can buy a T-shirt!

Goals of workshop

This conference is being created with two main goals in mind:

To showcase the depth and breadth of research connected to commutative algebra;
To build a regional community among faculty and students across algebraic disciplines and between our two institutions.

We will especially seek out speakers from both inside and outside of commutative algebra whose work touches on multiple algebraic topics. We hope to promote promote a broad view of commutative algebra, inspiring conversations and collaborations between different areas of algebra.

Registration and funding for April 23 Workshop in Minnesota

Thanks to the NSF, there is some funding available! For Minnesota/Wisconsin participants, this can include reimbursements for driving, plus lodging. We also have some funding to cover outside participants interested in attending in the conference. Funding for outside participants will focus on graduate students and junior mathematicians who will particularly benefit from the conference. We encourage you to apply or contact the organizers if you have any questions! Registration and funding form

Child care

If you are in need of child care during the workshop, please contact one of the organizers and we will help with arrangements.

Schedule for November 14 Workshop in Wisconsin

Date and time	Location	Speaker	Title
Friday, November 13
7:00-9:00	The Side Door (240 W Gilman St)		Happy hour
Saturday, November 14
8:00-9:00	9th Floor Van Vleck Hall		Welcome with coffee and discussion
9:00-10:00	B239 Van Vleck Hall	Srikanth Iyengar (Utah)	Commutative algebra and modular representations of finite groups
10:00-10:45	9th Floor Van Vleck Hall		Coffee and discussion
10:45-11:45	B239 Van Vleck Hall	Kevin Tucker (UIC)	Limit F-signature functions
11:45-1:15			Lunch
1:15-1:45	9th floor Van Vleck Hall		Coffee and discussion
1:45-2:45	B239 Van Vleck Hall	Daniel Erman (Wisconsin)	Splendid Resolutions

Srikanth Iyengar
Title: Commutative algebra and modular representations of finite groups
Abstract: The goal of this talk will be to describe a bridge between the modular representation theory of finite groups and modules over polynomial rings. This has given us new insights and results concerning modular representations, and has also lead to unexpected results in commutative algebra. The talk with be based on joint work with Avramov, Benson, Carlson, Buchweitz, Krause, Claudia Miller, and Julia Pevtsova.

Kevin Tucker
Title: Limit F-signature functions
Abstract: In the study of commutative rings with prime characteristic p > 0, a number of so-called Frobenius invariants defined via the p-th power map are used in the study of singularities. The F-signature function is a real-valued convex function on the unit interval that interpolates between a number of other Frobenius invariants, including the Hilbert-Kunz multiplicity and F-pure threshold. In many examples, limiting values as p ? ? of the Frobenius invariants have been observed. Moreover, in some cases this leads to connections with geometric or analytic invariants used in complex algebraic geometry. In this talk, I will give an overview of recent progress in computing limit F-signature functions. Notably, I will relate the F-signature function of sums of squares with the Euler polynomials (joint work with S. Shideler).

Christine Berkesch Zamaere
Title: Finiteness properties of local cohomology modules
Abstract: We introduce the main ideas and sketch the proof of the main result in Lyubeznik's 1993 Inventiones article in which local cohomology modules over a regular ring containing a field of characteristic zero are viewed as D-modules.

Schedule for April 23 Workshop in Minnesota

Date and time	Location	Speaker	Title
Friday, April 22
8:00-10:00	Applebee's		Happy hour
Saturday, April 23
8:00-9:00	Keller 3-176		Welcome with coffee and discussion
9:00-10:00	Keller 3-180	Graham Leuschke	A generalization of Knorrer periodicity, with applications to noncommutative hypersurfaces
10:00-10:45	Keller 3-176		Coffee and discussion
10:45-11:45	Keller 3-180	Brooke Ullery	Measures of irrationality for hypersurfaces of large degree
11:45-1:00	Keller 3-180		Organized lunch banquet
1:00-2:00	Keller 3-180	Megan Maguire	Stable and Unstable Homology of Configuration Spaces
1:45-2:45	Keller 3-180	Christine Berkesch Zamere (Minnesota)	Finiteness properties of local cohomology modules

Brooke Ullery
Title: Measures of irrationality for hypersurfaces of large degree
Abstract: The gonality of a projective algebraic curve is the smallest degree of a map from the curve the projective line. There are a few different definitions that attempt to generalize the notion of gonality to higher dimensional varieties. The intuition is that the higher these numbers, the further the variety is from being rational. I will discuss some of these definitions, and present joint work with Lawrence Ein and Rob Lazarsfeld. Our main result is that if X is an n-dimensional hypersurface of degree d at least 5/2 n, then any dominant rational map from X to P^n must have degree at least d-1.

Graham Leuschke
Title: A generalization of Knorrer periodicity, with applications to noncommutative hypersurfaces
Abstract: Kn\"orrer periodicity asserts a tight relationship between MCM modules over a hypersurface ring S/(f) (AKA matrix factorizations of f) and the double branched cover S[z]/(f+z^2). I will describe a generalization of Kn\"orrer periodicity, relating matrix factorizations f and matrix factorizations of f+z^n. For each k =1, \dots, n$ we obtain a covariantly finite subcategory of matrix factorizations of f+z^n. The case of k=1 and an ADE singularity f leads to a class of noncommutative rings over which every minimal projective resolution is eventually periodic of period 2, suggesting that they might be called noncommutative hypersurfaces. This talk will discuss joint work with Alex Dugas (U. Pacific).

Megan Maguire
Title: Stable and Unstable Homology of Configuration Spaces
Abstract: In it's weakest form, we say that a family of topological spaces is homologically stable if for fixed i the ith homology groups of X_n and X_{n+1} are isomorphic for n sufficiently large. Notions of homological stability have been investigated for a wide range of topological families, including Hurwitz spaces, moduli of curves, and configuration spaces. Arnol'd first proved integral homological stability for the unordered configuration spaces of R^2. This was extended to open (connected, orientable, finite type) manifolds by McDuff and Segal (independently), and recently Church (via the method of representation stability) and Randal-WIlliams independently proved rational homological stability for all (connected, orientable, finite type) manifolds. Using the tools of Totaro, we compute the Betti numbers, both stable and unstable, of the unordered configuration spaces of some example spaces, such as a genus 1 Riemann surface and CP^3, and prove a vanishing theorem about the unstable homology a la Church, Farb, and Putman (joint with Melanie Wood).

Christine Berkesch Zamaere
Title: Finiteness properties of local cohomology modules
Abstract: We introduce the main ideas and sketch the proof of the main result in Lyubeznik's 1993 Inventiones article in which local cohomology modules over a regular ring containing a field of characteristic zero are viewed as D-modules.