The seminar meets on selected Fridays 4:00-5:00 PM at the CUNY Graduate Center, room 6494. The CUNY Graduate Center is located at 365 Fifth Avenue, New York, NY 10016.
Organizers:
Laura Ghezzi, New York City College of Technology and the Graduate Center (CUNY), laura.ghezzi58@citytech.cuny.edu
Hans Schoutens, New York City College of Technology and the Graduate Center (CUNY), hschoutens@citytech.cuny.edu
Fei Ye, Queensborough Community College (CUNY), feye@qcc.cuny.edu
Spring 2024 dates: February 23, March 8, March 22, April 5, May 3
February 23
Speaker: Laura Ghezzi, New York City College of Technology and the Graduate Center, CUNY
Title: Comparing canonical and bicanonical degrees
Abstract: Let R be a Cohen-Macaulay local ring that has a canonical ideal C. We look at the properties of C as a way to refine our understanding of R; in particular we are interested in metrics that measure the deviation from R being Gorenstein. We define the bi-canonical degree of R as the multiplicity of C**/C, where C** is the bidual of C. First, we give an overview of the properties of the bi-canonical degree. We also discuss recent progress on the comparison conjecture, that compares the bi-canonical degree to the canonical degree, introduced in a previous paper. Numerical semigroups provide an important set-up for this comparison, in connection to recent work of Herzog and Kumashiro. The talk is based on joint work with S. Goto, J. Hong, R. Hutson and W. Vasconcelos and work in progress with J. Hong.
March 8
Speaker: Laura Cossu, University of Graz, Austria
Title: Factorization of Matrices over Integral Domains: Classical Problems and Recent Developments
Abstract: A classical open problem in ring theory is to characterize integral domains R such that every singular matrix over R can be expressed as a product of idempotent matrices. This problem’s significance lies in its connections to other major unsolved issues, including the classification of integral domains whose general linear groups are generated by elementary matrices and those satisfying weaker versions of the Euclidean algorithm. Notably, over a Bézout domain (i.e., a domain in which every finitely generated ideal is pricipal), a singular matrix can be decomposed into idempotent factors if and only if every invertible matrix can be decomposed into elementary matrices, and this holds true if and only if the domain admits a weak algorithm.
In this seminar, we will provide an overview of classical results concerning the idempotent factorization of matrices, along with recent developments in the field. In particular, we will briefly introduce a new approach to matrix factorization grounded in factorization theory.
March 22
Speaker: Bradley Dirks, Stony Brook University
Title: The Minimal Exponent of LCI Subvarieties
Abstract: Classification of singularities is an interesting problem in many areas of algebraic geometry, like the minimal model program. One classical approach is to assign to a singular subvariety a rational number, its log canonical threshold. For complex hypersurface singularities, this invariant has been refined by M. Saito to the minimal exponent. This invariant is related to Bernstein-Sato polynomials, Hodge ideals and higher du Bois/higher rational singularities of hypersurfaces.
In joint work with Qianyu Chen, Mircea Mustațǎ and Sebastián Olano, we defined the minimal exponent for LCI subvarieties of smooth complex varieties. I will describe what was done in the hypersurface case, give our definition in the LCI case, and explain the relation to local cohomology modules and (time permitting) the classification of singularities.
April 5 (4-5 PM virtual on zoom–see link below)
Speaker: Brian Laverty, West Virginia University
Title: Finite reducing dimensions and the derived depth formula
Abstract: In 1961, Maurice Auslander published a paper which introduced the depth formula:
$$depth_R(M) + depth_R(N) = depth(R) + depth_R(M \otimes_R N).$$
In his paper, Auslander showed that this formula holds over unramified regular local rings in which Tor^R_i(M, N) = 0 for all i > 0. Since then the results on the depth formula have expanded, and the depth formula itself has been generalized to the derived category. In 2020, Araya–O. Celikbas introduced reducing dimensions which further expanded the results on the depth formula. In this talk I will cover the definition of reducing dimension, some of its properties, and its results involving the depth formula.
Note: This talk will be virtual on zoom
https://us02web.zoom.us/j/81549606766?pwd=WGsrK3k4dVpjM0pGbDZzNGFPWFVNdz09
Meeting ID: 815 4960 6766
Passcode: 489570
May 3 (4-5 PM, GC room 6494)
Speaker: James Myer, CUNY Graduate Center
Title: (Toward) An Algorithm to (Explicitly) Produce a Regular Model of a Hyperelliptic Curve in (Bad) Characteristic (0, 2): A Criterion to Verify Regularity of the Normalization of a Candidate Model
Abstract: Given a hyperelliptic curve (defined over a “pleasant” field of characteristic 0 whose ring of integers is of mixed (bad) characteristic (0, 2)), we seek a regular model, i.e. a(n arithmetic) surface fibered over the (spectrum of the) ring of integers of the field whose generic fiber is the given curve, and with a special fiber: its avatar in characteristic 2.
A strategy is afforded within a paper of Dino Lorenzini & Qing Liu: there exists a (regular) model of the projective line whose normalization in the function field of the given hyperelliptic curve is its sought after regular model. So, we seek such a regular model of the projective line…
A candidate such model is gifted to us (explicitly) by work of Andrew Obus & Padmavathi Srinivasan.
We establish a stepping stone across the river toward an algorithm to (explicitly) produce a regular model of any hyperelliptic curve in mixed (bad) characteristic (0, 2): a criterion to verify the regularity of the normalization of a candidate model of a hyperelliptic curve (equivalently, the normalization of the candidate model of the projective line of Obus & Srinivasan in the function field of the hyperelliptic curve).
