This Summer CARS is organized by Julianne Faur, Kesavan Mohana Sundaram and Ryan Watson
Kesavan Mohana Sundaram
Measuring Singularities with Frobenius
July 14, 16 and 18
In a polynomial ring over a perfect field of characteristic p or a F-finite regular local ring, one can define a numerical invariant called F-pure threshold. In the first talk, we will define F-pure threshold and go over computing some examples. We will also see its comparison with multiplicity. In the second talk, we will be defining test ideals and F-jumping numbers. In the last talk, we get to see an interpretation of F-threshold and test ideals using differential operators.
Zach Nason
Journeying towards Jensen's Theorem
July 21, 23 and 25
If you've taken Eloísa's homological algebra class, you may remember that she proved that over a noetherian local ring a finitely generated flat module is free. It's not possible to drop the "finitely generated" part, as there are flat (even projective) modules over noetherian local rings that aren't free. However, it is possible (although extremely difficult) to show that all flat modules have finite projective dimension over a noetherian local ring - this result is called Jensen's theorem. I definitely won't be able to prove Jensen's theorem in a week, but I'm planning to build up tools that will help prepare you to understand the proof, and that are interesting in their own right. Depending on time, I'll go through some important facts and characterizations of flat modules and will prove Lazard's theorem. I'll also define pure modules and prove some results about then, and if all works out, I may be able to sketch the proof that all flat modules with countably generated relations have finite projective dimension.
Anna Brosowsky
Three numerical invariants: The Hilbert-Samuel multiplicity, the Hilbert-Kunz multiplicity & F-signature
July 28, 30 and August 1
When studying singularities (or equivalently, when studying local rings) we are often interested in classifying "how bad" a singularity is, or "how far" away from regular a local ring is. One way to approach this is via numerical invariants, and to show that if the invariant falls within a certain range, the ring satisfies a corresponding nice property. In this talk series, we will consider three related invariants:
- the Hilbert-Samuel multiplicity, a characteristic-free numerical invariant, as well as its connection to integral closure.
- the Hilbert-Kunz multiplicity, an analogous positive-characteristic invariant, as well as its connection to tight closure.
- the F-signature, a positive-characteristic invariant, as well as its connection to strong F-regularity (and again to tight closure).
Ben Huenemann
The Buchsbaum-Eisenbud Theorem
July 31
With vector-spaces, we can tell whether a complex is exact by looking at the ranks of the R-modules and differentials. The Buchsbaum-Eisenbud Theorem gives us a similar mechanism for finite complexes of free R-modules. We will be going over the motivation for this theorem and its proof as covered in Keller Vanderbogert's notes on free resolutions.
Ryan Watson
An Introduction to Tensor Triangulated Geometry
August 4, 6 and 8
In this series of talks I'll go over some of the basics of the relatively new field of tensor triangulated geometry. Tensor triangulated geometry is a branch of math that took off in 2005 with Paul Balmer's paper "The spectrum of ideals in tensor triangulated categories." This branch of math allows one to give a theory of support analogous to the one in commutative algebra to certain categories, namely tensor triangulated categories. Many areas of math, including commutative algebra, naturally come equipped with tensor triangulated categories, so this theory lets one transfer ideas from one area of math to another. An example of such (which we won't see in these talks) is applying the idea of gluing in algebraic geometry to modular representation theory. In these talks I will go over what a tensor triangulated category is, how one defines a theory of support over said categories, and how this relates to commutative algebra. No prior knowledge of triangulated categories is necessary, just a basic understanding of category theory (functors, natural transformations, etc.) and some homological algebra.
Cleve Young
A p-Derivation Characterization of Quadratic Reciprocity
August 5
The Law of Quadratic Reciprocity is a classical result in Number Theory with its first proofs being published in 1801 and having over 240 published proofs of the result in the centuries since. By comparison p-derivations are a recent construction with a range of applications in number theory and arithmetic geometry. In this talk we discuss a new result using p-derivations, when p3(mod 4), which affords a complete characterization of the Law of Quadratic Reciprocity via p-derivations.
Victor Daniel Mendoza Rubio
Some homological dimensions and related problems
August 7
In the literature, there are homological invariants that extend or refine the classical notion of projective dimension. In this talk, I will introduce some of them, such as Cohen–Macaulay dimension, Gorenstein dimension, and complete intersection dimension. In addition, I will tell you a few related research topics and open problems.
Uyen Tran
The Hilbert-Burch Theorem and the Peskine-Szpiro Acyclicity Lemma
August 11
This talk will build upon Ben's discussion of the Buchsbaum-Eisenbud Theorem. We will begin by proving the Hilbert-Burch Theorem, a key application of the Buchsbaum-Eisenbud Theorem, then discuss the Peskine-Szpiro Acyclicity Lemma, a useful tool for establishing the acyclicity of complexes.
Nicole Xie
An introduction to resultants
August 15
In Elimination Theory, Gröbner bases and techniques from linear algebra are the common and well-studied tools to understand, when does a system of equations admit a nontrivial solution. In this talk, we will look at an alternate approach, which is given by resultants. The resultant of two polynomials is implemented in many computer algebra systems. In this talk, we will see some properties and applications of them. We will also explore, its generalization to a system of polynomials in several variables, some examples and applications, along with comparisons with Gröbner bases and a connection to Algebraic Geometry.
