User:J D Bowen

Hi there! I'm a senior mathematics major at the University of San Francisco, currently applying to PhD programs around the US and Canada.

I'm particuarly interested in topology, and it is in this area I intend to contribute to Proofwiki.

Pages and Such
I've done a lot of topology, multivariate calculus, some number theory, differential geometry...  just a bunch of stuff.

Projects
I'm working on putting all of my lecture notes in topology up here, drawing heavily on 3 reference texts: Guillemin and Pollacks Differential Topology, Bredon's Topology and Geometry , and Hocking and Youngs Topology.

I hope to get cracking on knot theory soon-ish.

Pages I've Written
I keep this for my own reference, so I can keep track of what I should finish and what I've abandoned and what I would like to still do.

Complex Functions

Analytic Continuation of the Riemann Zeta Function

Trivial Zeroes of the Riemann Zeta Function

Equivalence of Riemann Zeta Function Definitions

Derivative of the Riemann Zeta Function

Inverse and Square of the Riemann Zeta Function

Gamma Difference Equation

Zeroes of the Gamma Function

Poles of the Gamma Function

Equivalence of Gamma Function Definitions

Analytic Function Theory

Zeroes of Analytic Function are Isolated

Uniqueness of Analytic Continuation

Cauchy Integral Theorem

Existence of Laurent Series

Equivalence of Definitions for Analytic Function

Fundamental Theorem of Algebra

Multivariate Calculus

Classical Stokes' Theorem

Divergence Theorem

Green's Theorem

Differential Geometry

General Stokes' Theorem

Implications of Stokes' Theorem

Existence of Local Coordinates

Manipulation of Exterior Derivative

Series

Manipulation of Absolutely Convergent Series

Products of Sums

Product Form of Sum on Completely Multiplicative Function

Ingham's Theorem on Convergent Dirichlet Series

Number Theory

Prime Number Theorem

Order of the Moebius Function

Sum of Moebius Function over Divisors

Order of the Divisor Function

Sum of Reciprocals is Divergent

Knot Theory

Alexander Polynomial is a Knot Invariant

Measure Theory

Banach-Tarski Paradox

Equidecomposable Nested Sets

Hausdorff Paradox

Equidecomposability is an Equivalence Relation

Equidecomposable Subsets

Existence of Non-Measurable Sets

Equidecomposability Unaffected by Union

Measure of Intervals is Length

Countable Sets Have Measure Zero

Measurable Image

Measurable Sets are an Algebra of Sets

Combinatorics

Five Color Theorem

Kuratowski’s Theorem

Topology

Weak Whitney Immersion Theorem

Classification of Compact One-Manifolds

Classification of Compact Two-Manifolds

Classification of Compact Three-Manifolds Supporting Zero-Curvature Geometry

Metric Induces a Topology

Heine-Borel Theorem

Rokhlin's Theorem on Bounded Manifolds and Induced Spin Structures

H-Cobordism Theorem

Homotopy Groups are Groups

Homology Groups are Groups

Rokhlin's Theorem

List of Fundamental Groups for 2-Manifolds

Fundamental Group is Independent of Base Point for Path-Connected Space

Homotopy Groups are Homeomorphism Invariants

Brouwer's Fixed Point Theorem

Extendability Theorem for Intersection Numbers

Homotopy is an Equivalence Relation

Preimage Theorem

Retraction Theorem

Sard's Theorem

Poincaré Conjecture

Algebra

Classification of Groups of Order up to 15

Cyclic Groups of Order pq

Dicyclic Group

Fundamental Theorem of Finite Abelian Groups

Integration Theory

Lebesgue Integral is Extension of Riemann Integral

Assorted and Uncategorized

Properties of Algebras of Sets

Ostrowski's Theorem

P-adic Metric is a Norm