bikes

Welcome to BIKES!

B ridges

I nto

K nowledge,

E .g., the knowledge of commutative algebra

S eminar

It's a well-known fact that BIKES are better than CARS (We're looking at you, Nebraska!!!). For the Fall 2026 semester, we'll meet at 12:15 in FASB 101.

Fall 2026

Date	Speaker	Title	Abstract
12/10	Kolton O'Neal	F-stuff 3
12/03	Ethan Morgan	F-stuff 2
11/19	Gari Chua	F-stuff 1
11/12	Brendan Murphy	Reading group 2 part 1
11/05	Kashif Khan	Reading group 2 part 3
10/29	Avery Bainbridge	Reading group 2 part 2
10/22	Vince Dillon	Reading group 2 part 1
10/08	Joseph Sullivan	Derived categories 3
10/01	Dimas Sanjoyo (or maybe Ethan)	Derived categories 2
09/24	Jorge Gaspar Lara	Derived categories 1
09/17	Somebody, presumably
09/10	Brendan Murphy	Primary invariants for conjugation by GLₙ(𝔽ₚ) on Mₙ(𝔽ₚ)	Over an infinite field, the invariant ring k[Mₙ]^GLₙ(k) is easy to describe: it is a polynomial ring on the coefficients c₁, c₂, …, cₙ of the characteristic polynomial of the generic matrix. Aryaman and before him Larry Smith worked out a description of primary invariants over a finite field in the case n=2, curiously finding that in addition to the coefficients det, tr we obtain the Steenrod power 𝒫¹(det). In this talk we present (as far as time allows) a proof that the n(n+1)/2 invariants c₁, c₂, …, cₙ, 𝒫¹(c₂), …, 𝒫¹(cₙ,), ..., 𝒫ⁿ⁻¹(cₙ) form regular sequence, i.e. they get us just over halfway to a homogeneous system of parameters/set of primary invariants for 𝔽ₚ[Mₙ]^GLₙ(𝔽ₚ). This research was AI-assisted, and the main result has been formalized in Lean.
09/03	Aryaman Maithani	On the Cohen–Macaulayness of abelian extensions of regular local rings	Let R be a regular local ring with quotient field K. Let L be a Galois extension of K and S the integral closure of R in L. Assume that the Galois group is abelian. Is S Cohen–Macaulay? Roberts showed the answer is affirmative when the order of the group is invertible in the residue field of R, and Koh gave a counterexample in mixed characteristic. Recently, in joint work with Singh and Sridhar, we give equicharacteristic counterexamples. We shall discuss all these results, being rather elementary constructions and proofs!

Spring 2026

Date	Speaker	Title	Abstract
04/16	Jorge Gaspar Lara	Trivial Extensions 3
04/09	Ethan Morgan	Resolutions Over Trivial Extensions	In this talk, we will continue to discuss trivial extensions of rings. First, we will finish characterizing projective and injective modules over a trivial extension. With this in hand, we will discuss a process to construct projective resolutions of these modules.
04/02	Jorge Gaspar Lara	Homological Algebra over Trivial Extensions	Let R be a commutative Noetherian ring. Sometimes when studying a R-module M, things might be less than ideal. In this talk we will discuss a construction known idealization of M (or trivial extension of R by M) that helps us study properties of M through the ideal lens. We will learn about what the category of modules of the trivial extension looks like and start taking our initial steps to do homological algebra in this category.
03/19	Kashif Khan	ASL	In this talk, we give the definition of Hodge algebra and ASLs (Not American Sign Language) with the motivation coming from determinantal rings. Then, we explain how to determine properties (such as Cohen Macaulyness) about an ASL by going to its discrete Hodge algebra. If time permits, we give an alternate calculation of the dimension of determinantal rings and a rough outline of the proof to show they are CM.
03/05	Ethan Morgan	More Properties of Determinantal Rings	In this talk, I'll wrap up the proof that determinantal rings of the form k[X]/I_2(X) are Cohen-Macaulay. After that, we'll move on to an overview of one way to generalize these methods to determine the same properties we've done in the 2x2 case for k[X]/I_t(X) with arbitrary t.
02/26	Kolton O'Neal	Intro to Determinantal Rings	Let X be a matrix of indeterminates. We consider rings of the form k[X]/I_t(X). They turn out to be fairly nice rings (in particular, they are Cohen-Macaulay). This first in a series of talks about determinantal rings will introduce these rings, talk about some related open problems, and show they are integral domains. The proof of this fact is surprisingly long.
02/19	Joseph Sullivan	Tilting Quivers	While Rickard's theorem (from Kashif's talk) characterizes all derived equivalences between certain algebras in terms of tilting complexes, we will specialize to an easier setting by defining tilting objects. After going through a cute proof for how tilting objects in particular give derived equivalences, we will find several tilting objects explicitly in the context of quiver representations (continuing Dimas's noncommutative trend). I hope to draw several pictures.
02/12	Dimas Sanjoyo	Motivating Rickard’s Morita Theory	It turns out that one of the motivations for the Derived Morita Theory that Kashif spoke about last week is a problem from modular representation theory. Today, I will be giving a brief tour of the field and a bird’s eye view on Broué’s Abelian Defect Conjecture.
02/05	Kashif Khan	Derived Morita Theory	Classical Morita Theory gives conditions for when the module categories of rings are equivalent. An analogous question can be asked about whether two rings have equivalent derived categories. In this talk, we answer this question by going through Jeremy Rickard's paper ‘Morita Theory for Derived Categories,’ where we are introduced to the notion of a Tilting module which generalizes the finitely generated projective generators in classical Morita Theory. We also talk about ‘Morita invariants’ for the derived category.
01/29	Gari Chua	Multiplicities, Closures, and Interpolation	The Hilbert-Samuel and Hilbert-Kunz multiplicities are invariants of classical interest, and both generalize the notion of the multiplicity of a plane curve at a point. On the other hand, certain ideal closures, such as integral closure and tight closure, have seen a lot of interest in defining singularities. These multiplicities and closures are related in the following sense: for nice enough local rings, two ideals have the same Hilbert-Samuel (resp. Hilbert-Kunz) multiplicity if and only if they have the same integral (resp. tight) closure. All these notions can be defined in characteristic p, with tight closure contained in integral closure and Hilbert-Kunz multiplicity smaller than Hilbert-Samuel multiplicity. In this talk, we explore Taylor's theory of s-multiplicities and s-closures, which provide a way to continuously interpolate between the two closures and the two multiplicities. We will see that this theory gives rise to certain distinct multiplicities and closures. Finally, we state certain results and conjectures regarding these interpolations, such as a version of the Briancon-Skoda Theorem.
01/15	Aryaman Maithani	On invariant rings of permutation groups	Consider the action of a subgroup G of S_n acting on the polynomial ring S = k[x_1, ..., x_n] by permuting the variables. Noting that this action does not really refer to the base field k, one may ask if the ring of invariants S^G is also suitably independent of k. For example, the properties of being Cohen–Macaulay, Gorenstein, a UFD, F-regular, etc… <br> One may also ask whether suitable numerical invariants are also independent: the a-invariant, the Hilbert series of S^G, the Hilbert series of H^n(S)^G and H^n(S^G). If you would like the answers to these questions, this talk is for you.

Fall 2025

Date	Speaker	Title	Abstract
12/01	Nawaj KC	Local cohomology over polynomial Z-algebras	In this talk, I will sketch the proof of the following theorem: If R is a polynomial ring over the integers Z, then the local cohomology modules of R supported at any ideal has finitely many associated primes. The proof is quite amazing, so please tune in.
11/24	Antonia Kekkou	Associated Primes of Local Cohomology Modules over a Polynomial Ring with Complex Coefficients	In the second talk of this series, we’ll show that the local cohomology modules of a polynomial ring over the complex numbers, for any ideal, have finitely many associated prime ideals. To show this, we’ll introduce the theory of D-modules.
11/17	Ethan Morgan	Associated Primes of Local Cohomology Modules	In this talk, I'll provide some background information relevant to Antonia and Nawaj's upcoming talks. We will do a quick refresher course on what local cohomology is, then talk about associated primes of local cohomology modules and some conjectures/results related to cases where there are finitely many. We'll then take a look at an example to show that local cohomology modules can have infinitely many associated primes.
11/03	Gari Chua	Proof of the Central Result of Huneke-Lyubeznik	In this talk, we present the proof of the central result of Huneke-Lyubeznik, which states that if R is the homomorphic image of a Gorenstein local ring, with absolute integral closure R^+, then the natural induced map on local cohomology is the zero map for all degrees smaller than dim R. This is an extension of a result of Hochster-Huneke that proves the result for regular rings. This result has some consequences that help us understand the absolute integral closure better. Of note is the implication that R^+ is a big Cohen-Macaulay algebra for R. If time permits, we can also reformulate this in terms of dualizing complexes.
10/27	Jorge Gaspar Lara	Local Cohomology and Equational Lemma	In the second entry on our series of talks we will briefly introduce local cohomology. I will do some basic computations of it as well as define the Čech complex. After this, we will turn our attention to the Equational Lemma, which will be a key ingredient for the proof of our Main Theorem on the next talk!
10/20	Suchitra Pande	Cohen-Macaulay rings and splinters via local cohomology	This is a first in a series of three talks in which we will understand a fundamental theorem about the structure of excellent local domains in prime characteristics due to Hochster and Huneke. This theorem states that the absolute integral closure of such a ring is a Big Cohen-Macaulay algebra. In the first talk, we review some preliminary material about local cohomology and Cohen-Macaulay rings and understand the precise statement of the main theorem. As quick applications, we will see how this theorem can be used to prove some well-known results - for example the Hochster-Roberts theorem and the Cohen-Macaulayness of splinters. Time permitting, we will also discuss how the theorem of Hochster and Huneke can be used to prove that regular rings are splinters in positive characteristics.
10/02	Desiree Laurel Martin	A Complete Classification of DG Trees and Cycles	Let G be a tree and I(G) the edge ideal of G. In this talk we explore the relationship between the Taylor resolution of an edge ideal and its minimal free resolution using Morse theory. We then ask when the minimal free resolutions of these edge ideals inherit the differential graded (dg) algebra structure from the Taylor resolution. This is joint work with High Geller and Henry Potts-Rubin.
09/29	Kashif Khan	The same De Rham duality statement for D-modules	In continuing the attempt to compromise on a common interest between Dimas and myself, we provide and define the necessary conditions needed for the De Rham duality for D-modules. Recall we have R is a complete local ring (i.e. R=k[[x_1,...,x_n]]). We first go back to Matlis Duality and provide the framework for how the Matlis Dual is a D module for non-finitely generated R-modules that are also D-modules. Then we'll give examples of D-modules and their Matlis duals that arise naturally. Lastly, we go over the ingredients for the De Rham duality statement, such as why holonomic modules appear. But most importantly, we celebrate two birthdays!
09/26	Manav Batavia	Algebras with a straightening law in invariant theory	Algebras with a straightening law (ASLs) provide a striking illustration of how combinatorial methods can simplify ring-theoretic computations. In this talk, I will introduce the concept of ASLs, outline their key structural properties, and present natural examples arising as invariant rings of specific group actions.
09/22	Dimas Sanjoyo	A De Rham Duality Statement for D-modules	In attempting to compromise on a common interest between Kashif and myself, we stumbled upon a paper of Switala with the following result: if R is a complete regular local ring containing k, and D = D(R,k) is the ring of k-linear differential operators on R, then the Matlis dual of M of any left D-module M can be given the structure of a left D-module. Further, if M is a holonomic D-module, then the de Rham cohomology spaces of the Matlis dual of M are k-dual to those of M. In this talk, I will present the first result, and some basic definitions and results about holonomic D-modules.
09/15	Joseph Sullivan	Koszul Rings	Last week Kolton defined for us (in particular) the Koszul complex for a regular sequence. We're going to reinterpret a Koszul complex for the graded polynomial ring/symmetric algebra as a duality with the exterior algebra. More generally, we will define this Koszul complex for "quadratic algebras", and when it happens to be exact, we will get Koszul duality, which relates to certain Ext algebras.
09/08	Kolton O'Neal	The Tate Resolution Over a Complete Intersection Ring	Given a Noetherian commutative ring R and an ideal I in R, the Koszul complex is a differential graded (DG) R-algebra. If the generators for I form a regular sequence, the Koszul complex is also a free resolution for R/I. But if they don't form a regular sequence, Tate provided a general construction in 1957 that always produces a DG R-algebra that is also a free resolution for R/I. In this talk, we define free resolutions, the Koszul complex, and DG algebras. We then describe Tate's construction. Finally, we state a nice result from Tate's paper about certain rings for which this construction terminates after only 2 steps. In general, Tate's construction may take infinitely many steps. If time allows, I'll discuss an outline of a proof of this theorem in a special case.
08/25	Aryaman Maithani	Splittings in Classical Invariant Theory	In this talk, we shall talk about the classical group actions of the classical groups — the general linear, special linear, orthogonal, and symplectic groups. Each of these is a subgroup G of GL_t and acts naturally on a ring like R = K[Y_{t x n}] via matrix multiplication, and the ring of invariants R^G is what shall be looking at. In characteristic zero, the inclusion R^G ↪ R possesses an appropriate splitting, and we shall describe methods of computing this over an arbitrary field of characteristic zero. Lastly, we will contrast this with positive characteristic and note a curious phenomenon that every characteristic zero splitting must have every prime appearing in the denominator.

Fall 2024

Date	Speaker	Title
11/06	Ethan Morgan	A Few Cute Proofs with Spectral Sequences
09/04	Anne Fayolle	F this and F that!
08/26	Aryaman Maithani	Very bad no-good evil rings

Spring 2024

Date	Speaker	Title
04/12	Ethan Morgan	Introduction to Determinantal Rings
04/05	Peter McDonald	The McKay Correspondence: History and Generalizations
03/29	Debbie Wooton	Minimal free resolutions of edge ideals of graphs
03/22	Brendan Murphy	Seeing Spec R inside of the derived category
03/01	Rahul Ajit	Jet Bundles or Bundles of Principal Parts
02/23	Joseph Sullivan	Introduction to Gröbner bases
02/16	Aryaman Maithani	Linear quotients of connected ideals of graphs

Fall 2023

Date	Speaker	Title
12/01	Jorge Gaspar Lara	Fusion invariant representations
10/27	Anne Fayolle	On tame ramification and compatible ideals
10/20	Rahul Ajit	Hilbert-Kunz multiplicity, F-signature & a conjecture of Monsky
10/06	Jenna Judd	Galois descent, cohomology, and conjugacy
09/29	Aryaman Maithani	Invariant Theory of Commutative Rings
09/22	Antonia Kekkou	Introduction to Local Duality
09/15	Peter McDonald	Singularities in characteristic zero: multiplier ideals, trace ideals, and closure operations
09/08	Brendan Murphy	The Lean Theorem Prover and Formal Mathematics
09/01	Trung Chau	Morse resolutions

Spring 2023

Date	Speaker	Title
04/07	Anne Fayolle	An introduction to compatible ideals
03/31	Trung Chau	Principal radical systems
03/17	Trevor	F-Pure Threshold and Semi-quasi Homogeneous Singularities
02/24	Brendan Murphy	The Lean Theorem Prover and Formal Mathematics
02/17	Peter McDonald	The Test Ideal and The Multiplier Ideal Are Morally Equivalent
02/10	Seungsu Lee	F-Splittings and Strongly F-Regular Rings
01/20	Kriti	Minkowski Inequality for Multiplicity of Ideals

Fall 2022

Date	Speaker	Title
10/20	Daniel McCormick	The Improveder New Intersection Theorem
09/29	Brendan Murphy	Intersection Theorems in Local Algebra (part 1)
09/22	Trung Chau	Big Cohen-Macaulay Modules and Homological Conjectures (Theorems)
09/25	Antonia Kekkou	Examples and results on bounds on Loewy length via derived categories
09/08	Josh Pollitz	Bounds on Loewy length via derived categories

Spring 2022

Date	Speaker	Title
04/07	Daniel McCormick	Simplicial Rings and André-Quillen (Co)homology
03/31	Antonia Kekkou	The Dold-Kan Correspondence
03/24	Peter McDonald	An introduction to simplicial objects
03/17	Trung Chau	Morse resolutions
03/03	Trung Chau	Cellular resolutions and hull resolutions
02/24	Vaibhav Pandey	Local cohomology modules of a polynomial ring over the integers have finitely many associated primes
02/17	Seungsu Lee	Discreteness and rationality of F-jumping numbers of hypersurfaces on regular rings
02/10	Eamon Quinlan-Gallego	Unit F-modules and local cohomology
02/03	Peter McDonald	Splinters and derived functors
01/27	Hanlin Cai	Splinters and derived splinters

Fall 2021

Date	Speaker	Title
12/09	Trung Chau	Scarf complex and Lyubeznik resolution
12/02	Sandra Rodriguez Villalobos	Simplicial resolution
11/18	Annie + Daniel	Taylor resolution
11/04	Peter McDonald	D(R) and Ho(Mod-HR) are the same!
10/28	Hanlin Cai	Stable homotopy category
10/21	Daniel McCormick	Singularity category over a hypersurface ring
10/07	Jack	Derived categories and the singularity category
09/30	Trung Chau	A quick look at the singularity category
09/23	Devlin Mallory	Differential operators in characteristic p
09/16	Vaibhav Pandey	Introduction to differential operators
09/09	Seungsu Lee	F-purity and F-regularity

Spring 2021

Date	Speaker	Title
04/09	Vaibhav Pandey	How bad are pinched Veronese rings?
04/02	Isaac Martin	F-signature in affine semigroup rings
03/26	Josh Pollitz	Golod rings and DGAs
03/19	Daniel McCormick	Golod rings and DGAs
03/12	Ben	Golod rings and DGAs
02/26	Hanlin Cai	Complete and torsion modules
02/19	Peter McDonald	Complete and torsion modules
02/12	Sam	Peskine-Szpiro
02/05	Trung Chau	Peskine-Szpiro

Fall 2020

Date	Speaker	Title
12/01	Daniel McCormick	Rational homotopy theory
11/24	Peter McDonald	Rational homotopy theory
11/17	Josh Pollitz	Rational homotopy theory
11/10	Vaibhav Pandey	D-modules and differential operators
11/03	Trung Chau	D-modules and differential operators
10/27	Sandra Rodriguez Villalobos	D-modules and differential operators
10/20	Daniel McCormick	Hochschild cohomology and group cohomology
10/13	Jian	Kahler differentials and the Hochschild cohomology ring
10/06	Janina Letz	Introduction to group cohomology, Hochschild cohomology, and Hopf algebras
09/29	Seungsu Lee	Geometric aspects of F-splitting
09/22	Ruyi	Kunz's Theorem
09/15	Isaac Martin	Introduction to F-singularities
09/08	Aeolus	Shallow root systems in desert landscapes
09/01	Marcus Robinson	Universally dendroidal quasi-images

Summer 2020

Date	Speaker	Title
		For the summer, BIKES was rebranded as UNICYCLES (Utah's New International Commutative Algebra You-Can-Learn-Electronically Seminar
08/03	Sudipta	Quantifying singularities using differential signature
07/27	Shalom	Valuation domains and the Frobenius endomorphism
07/13	JJ	Tilting equivalence: the key ingredient to perfectoid spaces
06/29	Sandra Rodriguez Villalobos	Some invariants in prime characteristic
06/22	Jian	Classifying thick subcategories of the stable category of maximal Cohen-Macaulay modules over a hypersurface
06/15	Jenny Kenkel	The depth of powers of an ideal
06/01	Vaibhav Pandey	Finiteness properites of local cohomology modules: an extended example
05/25	Peter McDonald	Computing Rf_*F for coherent sheaves on projective space

Spring 2020

Date	Speaker	Title
04/23	Janina Letz	The dimension of an intersection in an ambient hypersurface
04/16	Jenny Kenkel	FI-modules
04/09	Isaac	A remark concerning multiplicities
04/02	Peter McDonald	Stability of associated primes
03/26	JJ	Homology of Noetherian and local rings
03/05	Vaibhav Pandey	Cohomological dimension of ideals defining Veronese subrings
02/27	Daniel	Total Betti numbers of moduels of finite projective dimension
02/20	Jian	Support varieties and cohomology over CIs
02/13	Janina Letz	Support varieties and cohomology over CIs
02/06	Wei	Support varieties and cohomology over CIs
01/30	Eloisa Grifo	An introduction to characteristic p
01/23	Ben Briggs	Hochschild cohomology
01/16	Thomas Polstra	Basic elements theorem
01/09	Josh Pollitz	Cohomology operators