Number theory and Algebra Seminar at Asu (2024-25)

Title: The Analytic Class Number Formula

Abstract:  The class number is a useful invariant for a number field which gives a sense of how much unique factorization fails in that number field's ring of integers. We will be proving the analytic class number formula, which is a formula relating many invariants of a number field (including the class number) to the residue of its Dedekind zeta function at 1. This formula, used to show Dedekind zeta functions have simple poles at 1, is also used to prove Dirichlet's theorem on primes in arithmetic progressions.

Title: On a generalization of rational homology manifolds

Abstract:  A complex variety Z is called a rational homology manifold if the homology of the link of each singularity of Z is the same as that of a sphere. While smooth varieties are rationally smooth, there are several examples of singular varieties that satisfy this condition. These varieties exhibit interesting geometric properties, including Poincaré duality. We study a natural weakening of this notion, which gives rise to the notion of k-Hodge rational homology varieties. This notion captures the difference between higher Du Bois and higher rational singularities, two classes of singularities that have recently attracted significant interest. Work in collaboration with B. Dirks and S. Olano.

Title: A combinatorial method for the reduction number of an ideal

Abstract:  In the study of commutative rings, several algebraic properties are captured by numerical invariants which are defined in terms of ideals and their powers. Among these, of particular relevance are the reduction number and analytic spread of an ideal, which control the growth of the powers of the given ideal for large exponents. Unfortunately, these invariants are usually difficult to calculate for arbitrary ideals, and different methods might be required depending on the specific features of the class of ideals under examination. In this talk, I will discuss a combinatorial method to calculate the reduction number of an ideal, based on a homological characterization in terms of the regularity of a graded algebra. This is part of ongoing joint work with Louiza Fouli, Kriti Goel, Haydee Lindo, Kuei-Nuan Lin, Whitney Liske, Maral Mostafazadehfard and Gabriel Sosa.

Title: The number of equations defining an algebraic set, and connections with invariant theory

Abstract:  Given an algebraic set, i.e., the solution set of a family of polynomial equations, what is the least number of polynomials needed to define this set? The question is surprisingly difficult, with a rich history. We will discuss some of this history, with a special focus on examples coming from classical invariant theory. This includes results of Bruns, Bruns-Schwänzl, Barile, and our ongoing joint work with Jeffries, Pandey, and Walther.

Title: Generating Cohen-Macaulay module categories and Buchweitz-Orlov singularity categories

Abstract:  Cohen-Macaulay representation theory was born in the 1960s as a higher-dimensional version of representation theory of artin algebras. The purpose of this theory is, for a given Cohen-Macaulay ring R, to understand the structure of the category CM(R) of (maximal) Cohen-Macaulay R-modules. In this talk, we will first consider a certain question on how to generate the category CM(R) by using concrete examples. Next we will move on to thinking of a similar question for the singularity category D_{sg}(R) in the sense of Buchweitz and Orlov, and extend theorems of Ballard, Favero and Katzarkov.

Title: King's conjecture and the secondary fan

Abstract:  In 1997 King conjectured that each smooth projective toric variety has a "full strong exceptional collection of line bundles," or a collection which forms a sort of basis for all graded modules over its coordinate ring with respect to its irrelevant ideal.  The conjecture is false, but has motivated recent work in removing the dependence on irrelevant ideal by considering the secondary fan of the toric variety.  I will discuss a version of King's conjecture which my coauthors and I have proven in this case.

Title: A Fun Look at Ramification Theory in Number Fields

Abstract:  Join us as we take a light-hearted look at one of the core facets of algebraic number theory, ramification theory. We will be assuming general understanding of groups and rings from Algebra I (MAT-543), and provide the basics of field theory that those in Algebra II (MAT-544) might not have seen yet, enough so that the presentation will be enjoyable. We will then give a look into what we call “global theory,” and provide background information about number fields and what ramification theory is. Then, we will set out to prove a fun little fact about certain quadratic and cubic number fields.

In particular, given a number field of degree n over Q, we will call such a number field “pure” if it can be written in the form Q(sqrt(d)) for some integer d that is free of nth powers. Specifically, we will be answering a question about how a prime p ramifies in a pure degree p extension for the primes p=2 and p=3. Though this may seem like a small “sample,” you will see that the p=3 case is already extremely complicated, even with the restriction to a pure cubic number field.

Title: The slope of the v-function and the Waldschmidt constant

Abstract:  In this talk, we discuss the asymptotic behaviour of the v-number of a Noetherian graded filtration I = {I_k} of a Noetherian N-graded domain R. Previous work by Ficcara and Sgroi showed that the v-number of the ideals of filtration, denoted as v(I_k) exhibits periodic linearity for sufficiently large values of k. Building on this foundation, we describe that the limit of the ratio v(I_k )/k as k approaches infinity not only exists but is also equal to the limit of the ratio of the minimum degree of non-zero elements of the ideal to k. This implies that all these linear functions associated with the v-numbers share the same slope. In particular, for a Noetherian symbolic filtration, the slope of these linear functions corresponds to the Waldschmidt constant of the ideal. In the last, we discuss the comparison of v-number and regularity of symbolic power of certain classes of monomial ideals.

Title: The integral Chow rings of the stacks of hyperelliptic Weierstrass points

Abstract:  In this talk, we will discuss the computation of the integral Chow rings of the stacks parametrizing smooth hyperelliptic curves with marked Weierstrass points. Aside from their intrinsic interest, products of these stacks naturally arise as boundary strata of proper moduli stacks, such as stack of stable hyperelliptic curves and some compactified Hurwitz stacks. We prove that the integral Chow ring of each of these stacks of hyperelliptic Weierstrass points is generated as an algebra by any of the psi-classes and that all relations live in degree one. We will also discuss how computation can be carried out when working with rational coefficients. The talk is mainly based on joint work with Dan Edidin.

Title: Implicitization of tensor product surfaces via syzygies

Abstract:  It is a classical problem in algebraic geometry to find the implicit equations of the image or graph of a rational map between projective spaces. This so-called implicitization problem has been studied to great length by geometers, algebraists and, more recently, the geometric modelling community, for its applications to computer-aided design (CAD). As one such application, we consider tensor product surfaces and the implicitization of these varieties, using the syzygies of their defining ideals. We discuss previous and ongoing work using these methods, when these syzygies have low degree. 

Title: Higher singularities for determinantal ideals

Abstract:  Higher Du Bois and higher rational singularities are newly-defined properties of complete intersections that can be detected using local cohomology, refining the classical notions of Du Bois and rational singularities. We will discuss the commutative algebra behind these properties, and a number of challenges that arise while attempting to generalize them outside the complete intersection case. Via the example of determinantal ideals, we will theorize about what a "correct" definition could be.

Title: Epsilon Multiplicity is a Limit of Amao Multiplicities

Abstract:  In a 2014 paper, Cutkosky proved a volume equals multiplicity formula for the multiplicity of an m_R-primary ideal. We will discuss a generalization of this result to the epsilon multiplicity.

Title: Cartier algebras through the lens of p-families

Abstract:  A Cartier subalgebra of a prime characteristic commutative ring R is an associated non-commutative ring of operators on R that play nicely with the Frobenius map. When R is regular, its Cartier subalgebras correspond exactly with sequences of ideals called F-graded systems. One special subclass of F-graded system is called a p-family; these appear in numerical applications such as the Hilbert-Kunz multiplicity and the F-signature. In this talk, I will discuss how to characterize some properties of a Cartier subalgebra in terms of its F-graded system. I will further present a way to construct, for an arbitrary F-graded system, a closely related p-family with especially nice properties.

Title: Asymptotic colengths for families of ideals.

Abstract:  In this talk we discuss the existence and the importance of asymptotic colengths for families of m-primary ideals in a Noetherian local ring (R,m).  We explore various families such as weakly graded families, weakly p-families and weakly inverse p-families. If time permits we discuss the new analytic method to prove the existence of limits. This talk is based on a joint work with Cheng Meng.

Title: Explicit composition identities for higher composition laws

Abstract:  In 2001, Manjul Bhargava gave a new proof of Gauss composition of binary quadratic forms by using 2x2x2 integer cubes. Moreover, he showed that there are five higher composition laws which are related to quadratic rings similar to the case of binary quadratic forms. The proof of these higher composition laws relies on bijections between certain orbits of the spaces on which the composition is defined under some natural group action and certain suitable (tuples of) ideal classes of quadratic rings. In my PhD thesis, we formulated the higher composition laws in a manner similar to Gauss' formulation of composition of binary quadratic forms. More precisely, we provided explicit composition identities for the higher composition laws in the quadratic case. In this talk, I will briefly outline Bhargava's work on composition laws in the quadratic case, (and the cubic case if time permits), and describe our results on the composition identities.

Title: Multidegrees of Binomial Edge Ideal

Abstract:  A binomial edge ideal is a type of ideal that we can generate using a simple, connected graph. We prove how to calculate the multidegree of a binomial edge ideal based on combinatorial properties of the underlying graph. In particular, we study the collection of subsets of vertices whose prime ideals have minimum codimension. We provide results which assist in determining these subsets, then find these collections for star, horned complete, barbell, cycle, wheel, and friendship graphs, and use the main result of the paper to obtain the multidegrees of their binomial edge ideals.

Title: Zeta Values and Integral Representations in Characteristic p

Abstract:  Having an integral representation for a function means being able to write that function as a complex line integral of some (hopefully simple) function. In number theory, integral representations allow us to prove important properties of zeta functions and L-functions, such as functional equations and analytic continuations. My work focuses on properties of zeta values, L-functions and multiple zeta values in the characteristic-p function field setting. Because we're working in characteristic p, meaningful measures and integration theory are difficult to describe, so in general we don't have integral representations of these functions. I will describe some of my recent work which seeks to give an algebraic alternative to integral representations for these special values. This allows us to prove new properties for these values, such as describing new linear relations between them.

Title: Symbolic Powers of Invariant Ideals

Abstract:  The symbolic powers first appear in Krul's principal ideal theorem.  Though defined algebraically, symbolic powers connect geometry and algebra. It reveals many geometric properties as well. One of the properties is finding minimal degree hypersurfaces vanishing at a given variety with a certain multiplicity. Even though they are extremely useful, it is difficult to find a tangible description of the symbolic powers. They are known only in a few cases.  

In this talk, I will describe symbolic powers and different open problems related to them. I will also describe some numerical invariants connected with symbolic powers. I will give a tangible description of the symbolic powers of invariant ideals arising via young diagrams. I will discuss the results from my joint with Sudipta Das and Alexandra Seceleanu. 

Title: What is a Rees algebra?

Abstract:  In this talk, we introduce the notion of the Rees algebra of an ideal, and the various ways this ring can be realized. In short, the Rees ring encapsulates the data of every power of an ideal, all within a single algebra. Hence, for algebraists, this object is invaluable within the study of multiplicities, reductions, and integral closures, to name a few. For geometers, the Rees ring serves as the coordinate ring of the graph of a rational map between varieties. As these constructions are parametric in nature, one question is how to achieve the implicitization of these algebras, namely how to obtain a tangible description of the Rees ring as a quotient of a polynomial ring. We discuss various results in this direction, as well as their applications. We end with some open questions, particularly in the much less explored realm of Rees rings of modules.

Title: Valuations, integral dependence, and multiplicities

Abstract:   The theory of integral closure of ideals, originating in the early twentieth century with work of Krull, Zariski, Rees, and others, remains a vibrant area of research in commutative algebra. This theory's significance stems from its connections with valuations, which enable a wide range of applications in other mathematical fields such as combinatorics, algebraic geometry, and number theory. Asymptotic properties of integral closures can be understood through multiplicities, which have their origins in intersection theory, developed by Hilbert, Noether, Van der Waerden, and others. In the 1950s, advancements by Samuel, Serre, and Rees brought multiplicities to the forefront of commutative algebra, leading to extensive further development in various directions, including recent works. This talk will provide an overview of the history and key results in these areas, highlighting some open problems along the way.

Title: An Enhancement to the Galois Correspondence in Abelian Extensions of Q

Abstract: Algebra is a study of mathematics geared towards structure and classification. When you think of the three core facets of modern abstract algebra, group, ring and field theory, each can seem somewhat unrelated past a surface level.

For instance, most introductory courses on group theory focus on finite groups, symmetry, and finding subgroups of given groups. Most courses on ring theory focus on infinite rings, ideals, and subrings. Field theory on the other hand has a different focus entirely. It’s mainly about field extensions, rather than subfields. When you learn each of these theories, they certainly have common points, but each feels unique in their focus. So what happens when two of these completely different branches of algebra collide?

Group theory and field theory are connected in a way seemingly unimaginable at first, and this beautiful correspon- dence is the core of Galois theory. However, it does not end at the Galois theory you would learn in a typical abstract algebra course. In certain abelian Galois extensions, there is an enhancement to the normal correspondence. The aim of this talk is to explore that enhanced correspondence.

We want this talk to be accessible to anyone, and as such, will be spending some time in the beginning explaining background information for those with minimal algebra experience. Please feel free to ask any questions you have. Also, if you have an idea of where the talk might be going, please speak up and join in!