Research
My research interests span a broad spectrum of mathematics generally classified as geometry, including
- General topology
- Differential topology
- Differential geometry
- Algebraic geometry
- Manifold theory in the categories of smooth, analytic, and complex manifolds
Reprints and preprints
- Harmonic Volume Can Be Computed as an Iterated Integral, by William M. Faucette, Canadian Mathematical Bulletin, vol. 35, no. 3, 1992
- Harmonic Volume, Symmetric Products, and the Abel-Jacobi Map, by William M. Faucette, Transactions of the American Mathematical Society, vol. 335, no. 1, January 1993
- Geometric Interpretation of the Reduction of the General Quartic by Galois Theory, by William M. Faucette, American Mathematical Monthly, vol. 103, no. 1, January 1996
- The Generalized Torelli Problem: Reconstructing a Curve and its Linear Series From its Canonical Map and Theta Geometry, by William M. Faucette, completed
- Circling up the Wagons: Unifying Mathematics for the Calculus Student, by William M. Faucette, completed
- Divisibility Rules for 7 and 13, by William M. Faucette, completed
- How Not To Prove Fermat's Last Theorem, by William M. Faucette, completed
- The Miracle Substitution: How and Why It Works, by William M. Faucette, completed
- Trisecting an Angle . . . By Cheating, by William M. Faucette and Wendy C. Davidson, completed
- Generalized Geometric Series, The Ratio Comparison Test and Raabe's Test, by William M. Faucette, accepted by The Pentagon
- Pascal's Theorem in Degenerate Cases, by William M. Faucette, completed
- Around the Cubic Curve in Fifty Minutes, by William M. Faucette, completed
- A Poor Man's Derivation of the Double Angle Formula for Sine, by William M. Faucette, completed
- Ceva's Therem and Its Applications, by William M. Faucette, completed
- The Nine Point Circle, by William M. Faucette, completed
- The Euler Line of a Triangle, by William M. Faucette, completed
Presentations
- Math Makes the World(s) Go 'Round: A Mathematical Derivation of Kepler's Laws of Planetary Motion
- How Many Ways Can 945 be Written as the Difference of Squares: An Introduction to the Mathematical Way of Thinking
Lectures on Public Key Cryptography
- Cryptography: Public Key vs. Private Key Cryptosystems
- Public Key Cryptography: The RSA Cryptosystem
- Public Key Cryptography: Elliptic Curve Cryptography
Presentations on Algebraic Number Theory
- An Application of Quotient Rings to Number Theory
- The Chinese Remainder Theorem and Its Generalizations
- Exploring xn+yn=zn
Presentations on Hodge Theory
- Lecture 1: Calculus on Smooth Manifolds
- Lecture 2: The Hodge Theory of a Smooth, Oriented, Compact Riemannian Manifold
- Lecture 3: Complex Manifolds
- Lecture 4: Hermitian Linear Algebra
- Lecture 5: The Hodge Theory of Hermitian Manifolds
- Lecture 6: Kähler Manifolds
- Lecture 7: The Hard Lefschetz Theorem and the Hodge-Riemann Bilinear Relations
- Lecture 8: Mixed Hodge Structures
Presentations on Multivariable Calculus based on Michael Spivak's Calculus on Manifolds
- Multivariable Differentiation
- The Inverse Function Theorem
- The Implicit Function Theorem
Presentations on Commutative Algebra
- Presentation: Factorization of Ideals
Presentations based on Commutative Algebra in Algebraic Geometry, by David Eisenbud
Presentations based on Commutative Algebra, by Michael F. Atiyah & Ian G. Macdonald
- Lecture 1: Rings and Ideals
- Lecture 2: Modules
- Lecture 3: Rings and Modules of Fractions
- Lecture 4: Primary Decomposition
- Lecture 5: Integral Dependence and Valuations
- Lecture 6: Chain Conditions
- Lecture 7: Noetherian Rings
- Lecture 8: Artin Rings
- Lecture 9: Discrete Valuation Rings and Dedekind Domains
- Lecture 10: Completions
- Lecture 11: Dimension Theory
Presentations based on Algebraic Curves, by William Fulton
- Lecture 1: Algebraic Preliminaries
- Lecture 2: Affine Space and Algebraic Sets
- Lecture 3: The Ideal of a Set of Points
- Lecture 4: The Hilbert Basis Theorem
- Lecture 5: Irreducible Components of an Algebraic Set
- Lecture 6: Algebraic Subsets of the Plane
- Lecture 7: Hilbert's Nullstellensatz
- Lecture 8: Modules and Finiteness Conditions
- Lecture 9: Integral Elements
- Lecture 10: Field Extensions
Notes & Solutions
Notes from A Course in Algebraic Number Theory, by Robert B. Ash
- Notes to Chapter 1
- Notes to Chapter 2
- Notes to Chapter 3
- Notes to Chapter 4
- Notes to Chapter 5
- Notes to Chapter 6
- Notes to Chapter 7
- Notes to Chapter 8
- Notes to Chapter 9
Notes to Basic Algebraic Geometry I, by Igor R. Shafarevich
- Chapter 1, Section 1
- Chapter 1, Section 1
- Chapter 1, Section 2
- Chapter 1, Section 3
- Chapter 1, Section 4
- Chapter 1, Section 5
Notes to Basic Algebraic Geometry II, by Igor R. Shafarevich
Solutions
- Solutions to Commutative Algebra, by Atiyah and Macdonald
- Solutions to Algebraic Curves, by William Fulton