This is a 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 is to introduce the audience to the field; the talks will survey results, techniques and ideas in current commutative algebra research.

In Winter 2020, all talks are happening on zoom; if you'd like to attend, please contact us.

- January 22, 2021

Ethan Kowalenko (UCR)

Title: 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 29, 2021

Jenny Kenkel (University of Michigan)

Title TBA

Abstract: TBA - February 5, 2021

Alessandra Costantini (UCR)

Rees algebras and free resolutions

Abstract: TBA - February 12, 2021

Alessandra Costantini (UCR)

Rees algebras and multiplicity

Abstract: TBA - February 19, 2021

Eloísa Grifo (UCR)

Symbolic powers

Abstract: TBA - February 26, 2021

Javier Gonzalez-Anaya (UCR)

Title TBA

Abstract: TBA - March 5, 2021

Francesca Gandini (Kalamazoo College)

Title TBA

Abstract: TBA

- 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, itâ€™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.

- 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

Jack Jeffries (University of Nebraska-Lincoln)

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

talk notes (see also notes from a related previous talk)

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

- commalg, a website with conference listing and updates from the commutative algebra community
- Fellowship of the Ring, a worldwide virtual commutative algebra seminar hosted by MSRI
- Commutative and Homological Algebra Market Presentations, a virtual commutative algebra seminar with talks by graduate students and young researchers
- Macaulay2, a commutative algebra software
- Macaulay2 online, a way to run Macaulay2 without installing it
- Recent arXiv papers in Commutative Algebra
- Virtual seminar on recent connections between algebra, geometry, and combinatorics