Blog
Valuation rings and specialization
July 23, 2020
Four equivalent definitions of valuation rings in algebraic geometry and their applications to scheme theory.
Semistable Abelian Varieties and unipotent action
July 21, 2020
Exploring semi-stable reduction and inertia group actions on Tate modules.
The reduction of base change theorems to nice cases
May 2, 2020
How to reduce proper base change theorems to simpler cases using Chow's lemma.
Why did Iwasawa work so hard?
April 28, 2020
Understanding the necessity of control theorems in Iwasawa theory.
Solving non linear differential equations using combinatorics!
December 9, 2019
Using Cayley's formula and labeled tree counting to solve differential equations.
Obstructions to Weil Cohomology coming from supersingular elliptic curves
September 25, 2019
Why a cohomology theory with Q_p coefficients cannot exist in characteristic p.
Constructible sets, openness of flat maps and generic freeness
August 22, 2019
Three interconnected results in algebraic geometry over fields.
Irreducibility of Galois representations attached to Elliptic Curves
January 11, 2018
Properties of Galois representations from elliptic curves over number fields.
Noether Normalization, Spreading out and the Nullstellensatz
November 22, 2017
Three geometric proofs of Hilbert's weak Nullstellensatz.
Weak Mordell-Weil and various approaches to it
November 19, 2017
Multiple approaches to proving the Weak Mordell-Weil theorem.
Weil pairing and Galois descent
November 16, 2017
An elegant connection between the Weil pairing and Galois descent theory.
Some notes of mine
October 29, 2017
Notes on modular forms, complex multiplication, and class groups in Z_p extensions.
Formal Summation and Dirichlet L-functions
October 15, 2017
Computing L-function values at negative integers through formal summation.
The Groupoid Cardinality of Finite Semi-Simple Algebras
January 4, 2017
Semi-simple finite algebras of order p^m have groupoid cardinality exactly 1.
A nice proof that the irreducible characters form a basis for class functions
December 27, 2016
A fundamental result in representation theory using ring-theoretic methods.
Schur's Lemma and the Schur Orthogonality Relations
December 21, 2016
A conceptual approach to the Schur Orthogonality Relations.
Congruent Numbers and Elliptic Curves
December 6, 2016
The deep connection between congruent numbers and elliptic curves.
Descent on Vector Spaces and Cohomology
November 26, 2016
Descent theory, vector spaces, and their relationship to Galois cohomology.
The Tumor Suppressing Theory of Aging
August 8, 2014
Aging as a consequence of the evolutionary tension between cancer suppression and cellular growth.
Joscha Bach
August 7, 2014
Exploring Joscha Bach's ideas on intelligence, consciousness, and civilization.
Technology as a value system
August 7, 2014
How technologies inherently carry embedded value systems that shape society.