For recordings of virtual talks see
Fall 2023, Spring 2024 and Fall 2024 talks
October 13, 2023
Speaker: Andrew Stout, BMCC (CUNY)
Title: Jet Schemes: From Local to Global Deformations
Abstract: We consider jet schemes of flat deformations over simple linear fat points. It is shown that, for locally complete intersection varieties, these spaces can be viewed as global flat deformations over affine n space of the classical jet scheme of order n. This has natural implications for defining a motivic volume and developing an analogue of the motivic Milnor fiber. These results were obtained by further developing previous work done by the speaker on Auto Arc Spaces (a type of generalized jet scheme). Auto Arc Spaces were originally introduced into the literature by Prof. Hans Schoutens.
Preprint: https://arxiv.org/abs/2309.14656
October 27, 2023
Speaker: Han-Bom Moon, Fordham University
Title: Derived category of moduli space of vector bundles
Abstract: The derived category of moduli spaces of vector bundles on a curve is expected to be decomposed into the derived categories of symmetric products of the base curve. I will explain the current status of knowledge and how one can show that the derived category of the symmetric product of the base curve can be embedded into the derived category of the moduli space. This is joint work with Kyoung-Seog Lee.
November 10, 2023
Speaker: Valeriy Sergeev
Title: F-Rational Rings and Rational Singularities
Abstract: Rational singularities are an important class of singularities, closely related to many areas of algebraic geometry, including geometric invariant theory and the minimal model program. Rational singularities are defined using desingularizations. Despite this fact, rational singularities are closely related to singularities defined in terms of Frobenius map in prime characteristic, where existence of desingularizations is still an open question. We will discus results of Smith and Schoutens that explore this relationship.
December 1, 2023
Speaker: Uyen (Enni) Le, West Virginia University
Title: Remarks on a conjecture of Huneke and Wiegand
Abstract: This talk centers around the Huneke-Wiegand Conjecture, which has been a long-standing problem. Unlike vector spaces over a field, not all modules over a ring have a basis. Using an operation known as the tensor product, the Huneke-Wiegand Conjecture seeks conditions on a module M with rank over a one-dimensional local ring such that M has a basis (i.e., M is free).
Building upon the foundational work of Huneke and R. Wiegand in 1994, and O. Celikbas in 2020, we provide affirmative results for this conjecture in specific cases. In particular, we establish results for 2-periodic modules over local rings (joint work with Olgur Celikbas, Hiroki Matsui, and Arash Sadeghi), as well as 4-periodic modules over Gorenstein rings.
Note: This talk will be virtual on zoom
https://us02web.zoom.us/j/81549606766?pwd=WGsrK3k4dVpjM0pGbDZzNGFPWFVNdz09
Meeting ID: 815 4960 6766
Passcode: 489570
February 23, 2024
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, 2024
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, 2024
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, 2024 (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, 2024 (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).
September 20, 2024 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, 2024 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, 2024 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, 2024 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, 2024 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, 2024 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!
