Number theory and Algebra Seminar at Asu

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!