April 2026
Brauer Group
An introduction to Brauer groups, first for a field (i.e., Spec k for a geometer) and then for a scheme.
AlgebraAlgebraic Geometry
Read more →June 2025
Representation Theory
Basics on the representation of complex semisimple Lie algebras, with some examples worked out.
Algebra
Read more →April 2025
Beilinson's Resolution of the Diagonal
An explanation of Beilinson's famous 'Problems in Linear Algebra', in particular showing how his paper gives a full strong exceptional collection and a derived equivalence to quiver representations.
Algebraic GeometryCategory Theory
Read more →March 2025
Tilting
An introduction to the machinery of tilting sheaves for writing down derived equivalences. In particular, we compute the functors explicitly in some examples.
AlgebraCategory Theory
Read more →October 2024
Stability Conditions
An introduction to Bridgeland stability, which, granted, doesn't get much farther than discussing t-structures.
Algebraic GeometryCategory Theory
Read more →August 2024
Syzygies in Algebraic Geometry
An introduction to syzygies and minimal free resolutions in the context of algebraic geometry, with an emphasis on Castelnuovo-Mumford regularity.
Algebraic Geometry
Read more →April 2024
Variations of Hodge Structures
A note on how families of complex manifolds give rise to a 'variation of hodge structures'. Along the way, we compute the monodromy representation on cohomology for the Legendre family.
Algebraic Geometry
Read more →December 2023
De Rham’s Theorem
De Rham’s theorem provides an important connection between topology and smooth manifolds: de Rham cohomology is real singular cohomology, with the isomorphism given by integration.
Topology
Read more →August 2022
Covering Spaces
Covering spaces are a cool thing in topology. Finding covering spaces is like unwrapping a topological space, with intimate connections to the fundamental group.
Topology
Read more →July 2022
Nullstellensatz
Hilbert's Nullstellensatz bridges algebra and geometry. This article develops commutative algebra before arriving at a proof.
Algebra
Read more →December 2021
Sylow Theorems
The Sylow Theorems are fundamental to finite groups. We build toward their proof through group actions, existence, and conjugation.
Algebra
Read more →December 2021
Products and Coproducts
We get our feet wet in category theory through universal properties, examples, and the uniqueness of products and coproducts.
Category Theory
Read more →October 2021
FTFGMPID
The Fundamental Theorem of Finitely Generated Modules over a PID quickly gives the classification of finitely generated abelian groups and Jordan normal form.
Algebra
Read more →August 2020
Relations, Well-Foundedness, and Noetherian Induction
What exactly does induction require? We generalize the natural numbers with well-founded relations, and correspondingly generalize induction.
Foundations
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