UCR | Commutative Algebra Seminar

# Commutative Algebra Seminar

### University of California, Riverside

This was a virtual seminar in Commutative Algebra, primarily aimed at graduate students who do not yet have a strong background on the subject. Rather than having traditional research talks, our main goal was to introduce the audience to the field; the talks will survey results, techniques and ideas in current commutative algebra research.

## Spring 2021

• May 28, 2021
Eloísa Grifo (UCR)
Symbolic powers II

Abstract: Symbolic powers live somewhere between algebra and geometry, and give us the polynomials vanishing to higher order on a given set of points, or more generally any variety. We will introduce some currently active open problems regarding symbolic powers, and discuss some recent developments.

• May 21, 2021
Patricio Gallardo (UCR)
Point and Line to Plane

Abstract: I will discuss new and old perspectives on both the ideals associated with n points in the complex plane and the compact spaces parametrizing families of them. The talk is based on ongoing work with Benjamin Schmidt.

• May 14, 2021
José González (UCR)
Separating points, tangent vectors and their higher order analogs on toric varieties

Abstract: We will review how we get a map from an algebraic variety to projective space and will recall some algebraic properties which guarantee that the map has desirable geometric properties; like being defined on the whole variety, separating points, separating tangent vectors, etc. We will briefly introduce a class of varieties called toric varieties and explain how the algebraic properties mentioned above can be characterized in concrete combinatorial terms in the case of toric varieties.

• May 7, 2021
Josh Pollitz (University of Utah)
The derived category of a complete intersection ring

Abstract: The goal of this talk is to discuss how ring theoretic information is captured in terms of derived categories and illustrate applications from this perspective. The starting point is the celebrated Auslander-Buchsbaum-Serre theorem which characterize the least singular rings, namely, regular local rings. It can be phrased in the following way: A local ring is regular if and only if each object of its bounded derived category is a small object; this solved the localization problem for regular rings. In this talk I will review this theorem and discuss a recent derived category characterization of the rings which are in many ways the closest to being regular, namely, complete intersection rings. I will discuss an application of this characterization, and time permitting even more recent developments with various collaborators.

• April 30, 2021
Eloísa Grifo (UCR)
Characteristic p or: How I learned to stop worrying and love Frobenius

Abstract: In characteristic p, we gain a very simple but very powerful tool: the Frobenius map. We will talk about some of the basic ideas in characteristic p commutative algebra, how one classifies singularities (say, how bad a ring can be) using Frobenius, and how all this can sometimes answer questions in characteristic 0 as well.

• April 23, 2021
Lance Miller (Arkansas)
Title: Measuring singularities

Abstract: Local rings which are regular, i.e., their dimension and embedding dimension agree, are among the nicest rings to consider. They are the algebraic analogue of a "smooth point" and behave like the local rings of functions of a point of a manifold. In this talk, we'll discuss different ways to measure the failure of a ring to be regular. Along the way, we will discuss some classic interpretations in terms of differential forms and time permitting some very recent ways as well. The talk will utilize both algebraic (specifically commutative algebra) and geometric (specifically algebraic geometry) language to highlight the analogies that are going on, however no prior knowledge in these subjects will be required to enjoy the talk.

• April 16, 2021
Francesca Gandini (Kalamazoo College)
Polymatroids, Subspace Arrangements, and Representations

Abstract: Given a hyperplane arrangement, or more generally a subspace arrangement, we can associate to it a combinatorial object, a (poly)matroid, which has applications in many different fields of mathematics. We will discuss some results relating the algebraic and combinatorial aspects of hyperplane arrangements, including some that hint at a possible connection between the representation theory and the combinatorics of these objects... which is yet to be explored!

• April 9, 2021
Nora E. Youngs (Colby College)
Title: Using algebraic geometry to understand the neural code

Abstract: One major problem in neuroscience is to understand how the brain uses neural activity to form representations of the external world. It is known that combinatorial information in the firing patterns of neurons often reflects important features of the stimuli that generated them. How can we efficiently extract such information? This talk will introduce a method from algebraic geometry that we can use to encode and extract combinatorial structure from neural codes. We will discuss how this structure can be used to infer features of the underlying stimulus space, and some of the issues that arise under this approach.

## Winter 2021 schedule

• March 5, 2021
Javier Gonzalez-Anaya (UCR)
Finite generation of symbolic Rees algebras from a geometric perspective II

Abstract: Determining the finite generation of symbolic Rees algebras is a major industry in commutative algebra. In this talk, we will discuss the history of the problem in the case of monomial-curve ideals in three variables, and how geometric methods can help tackle it . Broadly speaking, the problem is equivalent to the finite generation of the so-called Cox ring of a specific class of varieties. By a result of Cutkosky, the Cox ring for these varieties is f.g. iff they contain two disjoint curves in them. We will outline how toric geometry allows us to translate this existence problem to studying lattice points of triangles in the plane. We'll conclude the talk with a summary of what is known as well as open problems.

## Past talks

• February 19, 2021
Eloísa Grifo (UCR)
Symbolic powers

Abstract: Many geometric shapes we might want to study are cut out by polynomial equations; these are called varieties. There is an algebra-geometry correspondence assigning each variety X to the ideal of all the polynomials that vanish at all the points in X. But how do we measure that vanishing? What are the polynomials that vanish on X to order n? What is the smallest degree of such a polynomial? These and many other questions are answered by studying symbolic powers, which are ubiquitous in commutative algebra. In this talk, we will introduce symbolic powers and briefly discuss some of the open problems about them.

• February 26, 2021
Javier Gonzalez-Anaya (UCR)
Finite generation of symbolic Rees algebras from a geometric perspective I

Abstract: Determining the finite generation of symbolic Rees algebras is a major industry in commutative algebra. In this talk, we will discuss the history of the problem in the case of monomial-curve ideals in three variables, and how geometric methods can help tackle it . Broadly speaking, the problem is equivalent to the finite generation of the so-called Cox ring of a specific class of varieties. By a result of Cutkosky, the Cox ring for these varieties is f.g. iff they contain two disjoint curves in them. We will outline how toric geometry allows us to translate this existence problem to studying lattice points of triangles in the plane. We'll conclude the talk with a summary of what is known as well as open problems.

• February 12, 2021
Alessandra Costantini (UCR)
Rees algebras and multiplicity
talk notes

Abstract: In this talk I will discuss the Cohen-Macaulay property of Rees algebras. The problem is geometrically motivated by the study of singularities. When blowing up a variety at a singular point, one does not necessarily get a non-singular variety, but at least would hope that the nature of the singularity does not get worse in the process. In particular, since Cohen-Macaulay singularities are very well-behaved, one would wish that the blowup of a Cohen-Macaulay singularity is still Cohen-Macaulay. This is not always true, however some sufficient conditions can be determined by studying the multiplicity of the ring.

• February 5, 2021
Alessandra Costantini (UCR)
Rees algebras and free resolutions
talk notes

Abstract: Rees algebras appear in algebraic geometry as homogeneous coordinate rings of the blow up of a variety or a scheme at a singular point. In order to understand the algebraic structure of Rees algebras, free resolutions represent a very useful tool. In this talk I will explain how one can use free resolutions to describe the Rees algebra in terms of generators and relations.

• January 29, 2021
Jenny Kenkel (University of Michigan)
FI-algebras

Abstract: Consider a polynomial ring with infinitely many variables. "No," you say, "I shan't! That's too many variables" and you are correct. The study of FI-algebras allows us to add notions like finite generation and noetherianity to algebras that are, strictly speaking, way too big.

• January 22, 2021
Ethan Kowalenko (UCR)
Simplicial polytopes and commutative algebra

Abstract: In the intersection of commutative algebra, combinatorics, and algebraic geometry, there lives a beautiful theorem called the g-theorem. This theorem asserts that some (finite) sequence of numbers has a specific interpretation coming from simplicial polytopes (combinatorics) if and only if it has a specific interpretation coming from (graded) commutative algebra. The first proof of the forward implication of this theorem, due to Stanley, used the cohomology of toric varieties (algebraic geometry). A later proof, due to McMullen, dodged the algebraic geometry, allowing us to think of this theorem as purely algebro-combinatorial. In this talk, I will describe the g-theorem, and illustrate what it means in examples, using only combinatorics and commutative algebra. If I have time, I will describe some generalizations and conjectures for more algebro-combinatorial gadgets.

• January 15, 2021
Branden Stone (Assured Information Security, Inc)
Life as a government contractor

Abstract: Before leaving academia I was always intrigued at what life would be like in industry. It turns out, it’s pretty fun! In this talk I will answer questions that I wish I knew as a graduate student. E.g. What is my day like? What type of problems do you work on? Is it selling out to work on government contracts? What could a commutative algebraist do? The answers may surprise you.

• January 8, 2021
Eloísa Grifo (UCR)
Using computer software for commutative algebra research
M2 code

Abstract: Computer (algebra) software plays a very important role in modern mathematics research, and in particular in commutative algebra. We will talk briefly about why and how such software can be used in research, and introduce Macaulay2. We will see Macaulay2 in action, and give an overview of where and how one could learn more about using it.

## Fall 2020 schedule

• December 4, 2020
Benjamin Briggs (University of Utah)
The friendship between commutative algebra and rational homotopy theory
talk notes

Abstract: Starting in the 80s several mathematicians (especially Avramov and Roos) started noticing that some of the things that happen in rational homotopy theory mysteriously also happen in commutative algebra. As more connections were uncovered, eventually the two fields made contact, and even held joint conferences, leading to a lot of process on both sides. The connection between the two areas became known as "the looking glass", and ideas and results have now been passed back and forth through it for decades.
I'll try to describe some of the main similarities between rational homotopy theory and commutative algebra, and I'll introduce an object called the homotopy Lie algebra, which exists on both sides of the looking glass. It will be as accessible as possible!

• November 20, 2020
Rings of invariants

Abstract: In this talk, we will introduce rings of invariants from a commutative algebra perspective. This is a classical topic with connections to the origins of commutative algebra.

• November 6, 2020
Alessandra Costantini (UCR)
Combinatorial Commutative Algebra II
talk notes

Abstract: This is the second of three talks aiming to understand how to use combinatorial properties to understand free resolutions and Betti numbers of ideals in a polynomial ring. In the first part of the talk I will explain how to use combinatorics to understand free resolutions of squarefree monomial ideals, and of monomial ideals that are not square free. If time allows, I will also explain how you can construct ideals generated by monomials or by squarefree monomials starting from any homogeneous ideal.

• October 30, 2020
Alessandra Costantini (UCR)
Combinatorial Commutative Algebra I

Abstract: This is the first of three talks aiming to understand how to use combinatorial properties to understand free resolutions and Betti numbers of ideals in a polynomial ring. I will describe free resolutions of ideals generated by monomials, and in particular by squarefree monomials. I will also explain how you can construct ideals generated by monomials or by squarefree monomials starting from any homogeneous ideal.

• October 23, 2020
Adam Boocher (University of San Diego)
What are Betti Numbers?
talk notes part 1 and part 2

Abstract: In this talk I'll give lots of examples illustrating what betti numbers say about ideals in a polynomial ring. We will also talk about conjectures for lower bounds for betti numbers including recent work with Derrick Wigglesworth.

• October 16, 2020
Eloísa Grifo (UCR)
Betti numbers and free resolutions
talk notes

Abstract: A (minimal) free resolution of a finitely generated module M describes its generators, relations, the relations among the relations, etc, and contains lots of geometric and algebraic information about a module. In this talk, we will give a friendly introduction to free resolutions and construct several examples.

• October 9, 2020
Eloísa Grifo (UCR)
What is Commutative Algebra and what will this seminar be about?
talk notes

• October 2, 2020
Organizational meeting