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!!!). Come join us in LCB 222 on Mondays at 2pm-3pm!

Fall 2025

Date	Speaker	Title	Abstract
12/01	Ethan Morgan	TBD	TBD
11/24	Antonia Kekkou	TBD	TBD
11/17	Nawaj KC	TBD	TBD
11/03	Jorge Gaspar Lara	TBD	TBD
10/27	Gari Chua	TBD	TBD
10/20	Suchitra Pande	TBD	TBD
10/13	Rahul Ajit	TBD	TBD
09/29	Kashif Khan	TBD	TBD
09/22	Dimas Sanjoyo	TBD	TBD
09/15	Joseph Sullivan	TBD	TBD
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 Rodriquez 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