CARS
Community Algebra Reading Seminar
This semester, CARS was organized by Jake Kettinger and David Lieberman
Shah Zamir
A Problem of Hilbert
April 27In 1900 David Hilbert published a list of 23 open problems. The 14th problem concerns the finite generation of the ring of invariants of a group acting on a finitely generated K-algebra. In 1959 Nagata provided a negative answer to Hilbert's 14th problem. In this talk, I intend to sketch a proof of this counter example. Our proof will follow a simplified version by R. Steinberg. I only assume basic knowledge of algebraic objects such as polynomial rings, algebraic varieties, and groups
Nawaj KC
Jacobian Criterion
April 20Oscar Zariski proved that a point in a variety is smooth if, and only if, its coordinate ring localized at the point is regular. This theorem yields the very well known Jacobian Criterion which can compute all the singular points of a variety. We will discuss these ideas in the talk.
Matthew Bachmann
Čech Cohomology from Another Perspective
April 13Last week Jordan introduced Čech cohomology for sheaves of topological spaces. In today’s talk we will translate the theory introduced last week into a more ring theoretic perspective. We will first discuss how we can associate a topological space to a commutative ring and then analyze what the Čech construction says about this topological space. A somewhat surprising and amazing fact is that the Čech cohomology applied to rings will recover certain local cohomology modules.
Of course, this is only cool if you think local cohomology is useful; so, the talk will conclude with a brief introduction to local cohomology and some of the initial applications of the theory.
Jordan Barrett
Čech Cohomology
April 6In this talk, I will give an introduction to Sheaf theory and the Čech Cohomology of topological spaces. Also I will do a sick af kick flip.
David Lieberman
Toric Rings: How to Turn a Cone Into a Coordinate Ring
March 16Toric rings and their varieties were first introduced in the 1970's, and are a rich source of interesting finitely generated algebras that are not boring polynomial-esque rings. In this talk, we will go through the construction of these objects that takes us from a polyhedral cone in affine space to the coordinate ring of some variety. No advanced algebra knowledge should be needed for this talk, but the audience will need to be well versed in being a good pal and not judging my ability to draw cones in R3.
Jake Kettinger
Unexpected Curves
March 23Boo! I'm sorry, I didn't mean to be so... unexpected. Anyway. I'm going to talk about unexpected curves in algebraic geometry. But what does it mean for a curve to be unexpected? Where can we find unexpected curves and what are the different ways we can construct them? All these will be answered! And then we will talk about unexpectedness in higher characteristics and dimensions, and how they relate to other geometric constructions such as quasi-elliptic fibrations and geproci sets.
Above is a demonstration of the unique (up to isomorphism) unexpected quartic in characteristic 0. Points 1, 2, 3, 4, and 10 are click-and-draggable, and point 10 will always be a triple point no matter where you drag it! (Although it may not always appear that way because non-real slopes don't render in Desmos.) You can read about its construction and uniqueness in the 2019 paper by Farnik, Galuppi, Sodomaco, and Trok.
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Spring Break 💐 🌻 🌼 🌷
March 16A time of rest! A time of joy! A time of best! A time of rhyme! This week we all take a nice break from all this commotion that surrounds us. We will return next week!
Julie Geraci and Dakota White
Gröbner Bases: The Sequel
March 9This week we till show applications of Gröbner Bases through Macaulay2 and Graph Colorings!
Julie Geraci and Dakota White
Gröbner Bases: Who is she?
March 2Gröbner bases are powerful and magical objects in commutative algebra. They help us answer questions like the ideal membership problem or the parametrization problem. They even work their little magic in programs like Macaulay2, where you may ask an innocent question like "I=J?" Could you believe that all of this comes from an abstraction of the division algorithm?
In this two-part series, we will dive into the world of Gröbner bases. In this week's talk, we will motivate and define Gröbner bases as well as give some neat facts and algorithms about Gröbner bases that one may not learn in your typical commutative algebra course.
Next time, we will show the power of Gröbner bases in M2 and applications to other areas of mathematics.
Open Discussion
Community Day!
February 23Today in CARS we will use the time to check in with each other as an algebra community instead of having the usual talk. We want to open back up the discussion about our group culture, and hear how everyone is feeling as a member of the algebra group.
We hope to continue working on creating an inclusive space for all graduate students, and everyone's input is greatly appreciated.
David Lieberman
D-modules and Bernstein's Inequality or: How I Learned to Stop Worrying and Love the Lower Bound on the Dimension of a D-Module
February 16As algebraists, we are (almost exclusively) concerned with the study of modules. We often ask questions about modules along the lines of "Is this module big? How big is it? Can we wrangle this thing in a nice way or is it too big to bend it into submission with our will power alone?" Answers to this question can involve looking at dimension, generating sets, and length, then determining how "bad" it can get.
There is another question we can ask though: given a ring R how small can an R-module be? Is there a lower bound on the dimension of an R-module? What even is the dimension of an R-module? In this talk we will answer these questions when the ring in question is the ring of differential operators over a polynomial ring. The only prior knowledge I will assume is 818 algebra, partial derivatives, and the product rule. Proper hydration and general curiosity are also encouraged, but not necessary. If time permits, we may also introduce the notion of holonomic modules.
Zach Nason
A Counterexample Concerning the Existence of Height One Prime Ideals in the Intersection of Height Two Prime Ideals
February 9In the early 1970s, Irving Kaplansky conjectured that in a Noetherian domain, the intersection of any two height two prime ideals must contain a height one prime ideal. Although this conjecture is true in many Noetherian domains, it is false in general, as demonstrated by Stephen McAdam in 1974. In my talk today, I will construct this counterexample, and will build up several important results necessary for this construction. In addition, I will prove that this conjecture is true in R[x], where R is a Noetherian domain.
Nawaj KC
Serre's Intersection Formula
February 2Given subvarieties X and Y in an affine n-space which meet only at the origin, we want to define their intersection multiplicity, a rather innocent and natural definition to desire. Over the plane, it suffices to simply define it as the length of Rm/(I, J) where I, J are the defining ideals of these varieties and Rm is the polynomial ring localized at the origin. However, in higher dimensions, we see clear evidence that this formula is simply inadequate and that it is just not precise enough. In today's talk, we present Serre's fix to this. Some basics of commutative algebra at the 901/902 level will be assumed.
Andrew Soto
A Rigidity Theorem for Ext
January 26The goal of this talk is to present the following theorem: if R is an unramified hypersurface, if M and N are finitely generated R modules, and if ExtRn(M, N) = 0 for some n ≤ grade M, then ExtRn(M, N) = 0 for i ≤ n. A corollary of this says that ExtRi(M, M) ≠ 0 for i ≤ grade M. This gives a partial answer to a question of Jorgensen: if (R, m, k) is a complete intersection and if M is a nonzero finitely generated module of finite projective dimension, then must ExtRn(M, M) be nonzero for 0 ≤ n ≤ pdR(M)?