The seminar meets on selected Fridays 4:00-5:00 PM at the CUNY Graduate Center, room 4214.03 (math thesis room). 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 and the Graduate Center (CUNY), feye@qcc.cuny.edu
Fall 2024 dates: September 20, October 18, November 1, November 15, November 22, December 6
September 20 GC 4214.03
Speaker: Nathan Chen, Harvard University
Title: On measures of irrationality and rational group actions
Abstract: In this talk, we will survey measures of irrationality, with an emphasis on recent results for surfaces. We will then explore some of the variational properties of the degree of irrationality, in the presence of a rational group action. This is joint work with L. Esser.
October 18 GC 4214.03
Speaker: Anh Duc Vo, Harvard University
Title: Du Bois complexes and singularities
Abstract: In this talk, I will discuss the notion of Du Bois complexes and provide an overview of classical notions such as rational and Du Bois singularities. I will then discuss their recent extensions to k-rational and k-Du Bois singularities, both for local complete intersections (LCIs) and non-LCI varieties. Additionally, I will discuss results on the injectivity and vanishing properties of Du Bois complexes in the context of these generalizations. This talk is based on joint work with Mihnea Popa, Wanchun Shen, and Sridhar Venkatesh.
November 1 on zoom
Speaker: Shiji Lyu, University of Illinois at Chicago
Title: Approximation of schemes over complete local rings
Abstract: Let k be a field and R be the power series ring k[[T_1,…,T_n]]. Finite type schemes over R were used in, for example, Hironaka’s resolution of singularities and the works of de Fernex-Ein-Mustață on ACC of log canonical thresholds. In this talk, we discuss a systematic way of approximating finite type schemes over R using schemes essentially of finite type over k, preserving various types of singularities and homological properties. This allows us to extend known results and constructions for varieties to finite type schemes over R, including formulas for multiplier ideals, deformation of singularities, and big Cohen-Macaulay algebras. This is joint work in preparation with Shizhang Li and Bogdan Zavyalov.
Note: This talk will be virtual on zoom
https://us02web.zoom.us/j/81549606766?pwd=WGsrK3k4dVpjM0pGbDZzNGFPWFVNdz09
Meeting ID: 815 4960 6766
Passcode: 489570
November 15 GC 4214.03
Speaker: Ben Blum-Smith, Johns Hopkins University
Title: Degree bounds for rational invariants (v2.0)
Abstract: Degree bounds have a long history in invariant theory. The Noether bound on the degrees of algebra generators for a ring of invariants is over a century old, and there is a vast literature sharpening and generalizing it. In the last two decades, there has also been an active program on degree bounds for invariants which are able to distinguish orbits as well as algebra generators can (known as separating invariants).
In this talk I make the case that generators for the field of rational invariants represent an exciting avenue for research on degree bounds as well. Even the case of G=Z/pZ, uninteresting from the point of view of generating and separating invariants, has a story to tell for rational invariants. I present new upper bounds. The methods involve the classical Minkowski geometry of numbers, and more recent work on explicit computation of invariant fields.
This talk is based on joint work with Harm Derksen, Thays Garcia, Rawin Hidalgo, Consuelo Rodriguez, Sylvan Crane, Karla Guzman, Alexis Menenses, and Maxine Song-Hurewitz.
(The “v2.0″ in the title is because I gave a talk 2 years ago in the same research program, but this talk will not assume any prior familiarity with the program.)
November 22 GC 4419 (note the room change for this talk)
Speaker: Aycin Iplikci Arodirik, Ohio State University
Title: A way to view quasi-coherent sheaves of ideals as ideals of a ring
Abstract: A blow-up $X$ of an integral domain along an ideal is a projective scheme. In a naive sense, we want to consider this projective scheme as an affine scheme. We construct an associated ring $X^*$ and present a way to view the quasi-coherent sheaves of ideals belonging to the scheme $X$ as ideals of the ring $X^*$. The construction is based on the Nagata extension, which is a particular localization of polynomial rings of a given ring. First, we extend the notion of homogeneous ideals from polynomial rings to Nagata extension rings and then to $X^*$. In this extended sense, homogeneous ideals will be relevant to our purpose and we can establish a one-to-one correspondence between the set of relevant ideals of $X^*$ and the set of quasi-coherent sheaves of ideals on $X$. No knowledge of algebraic geometry is required to follow this talk, as our goal is obtaining insight into the projective schemes that correspond to blow-ups from a ring theory perspective.
December 6 GC 4214.03
Speaker: James Myer, CUNY Graduate Center
Title: (Toward) A(nother) Algorithm for Resolution of Singularities (for Curves) in Any Characteristic
Abstract: This is joint work with Raymond van Bommel & Mike Montoro. Hironaka proved in 1964 that we can resolve singularities in characteristic 0, and there have since been substantial efforts to simplify the algorithm, e.g. by Kollár. The state of the art is an algorithm of Abramovich, Temkin, and Wlodarczyk, and independently, McQuillan, wherein the category of schemes is enlarged to that of (Deligne-Mumford) stacks. To this day, no one knows whether we can resolve the singularities of an algebraic variety of any dimension defined over a field of positive characteristic. So, curiosity led the three of us to investigate what exactly goes awry in the case of positive characteristic, and we aimed to adjust the work of Abramovich et al. to handle the case of positive characteristic. For simplicity, we only studied curves. We hope this method will blaze a trail toward varieties of higher dimension, but this requires a closer look at the invariant in this case of positive characteristic. Even if this algorithm cannot handle any new cases, there is still a good chance that the algorithm will be more efficient than existing methods to resolve singularities. We’ll define a new invariant via Newton polygons, learn a little about stacky blowups, play with some examples, and see the proof that we can resolve the singularities of any (generalized cusp) in one fell swoop!
