Research Interests

my Publications/ Errata

my Seminar Slides

other Handouts

Open Problems

video

G. Eric Moorhouse:

Handouts

Disclaimer: These notes have been prepared for distribution to students in my classes, and participants in local seminars, following a lower standard for proofreading than my publications. They vary in intended level from elementary to graduate student level. Also expect to find casual remarks, inside jokes, homework problems etc., intended for my students. Misprints are par for the course, but you can let me know when you find mistakes (my email address) and if I have time, I will post corrections.

General Advice for Students

  1. So You Want to Give a Math Talk. UW Math REU, June 22, 2010
  2. Some very basic pointers on summation notation

Proofs

  1. On learning to read and write proofs. Three examples of proofs
  2. Mathematical Induction

Algebra

  1. The field of rational numbers: an exercise in the use of equivalence classes
  2. A linear algebra refresher for later courses
  3. A very quick summary of what eigenvalues and eigenvectors are
  4. Numbers and Polynomials. Introducing polynomials as formal objects
  5. Division Algorithm for Polynomials; Euclidean Algorithm for polynomials, computation examples in Maple
  6. Extended Euclidean Algorithm for polynomials
  7. Newton and Lagrange Interpolation of Polynomials
  8. Equiangular Lines in ℝn: the absolute bound via the dimension of the space of d-homogeneous polynomials in n variables
  9. Basic notation, terminology and examples of ring theory
  10. Polynomials and power series: the formal viewpoint
  11. Homomorphisms, ideals and quotients in ring theory
  12. Introduction to complex numbers and the Fundamental Theorem of Algebra
  13. Introduction to fields
  14. Extensions of fields with applications to straightedge-and-compass constructibility
  15. Some consequences of field characteristic
  16. Free groups
  17. Algebraic closures of fields with an appendix on Zorn's Lemma
  18. Abstract Algebra I
    Used as a text for our first-semester graduate abstract algebra course. Groups, rings and fields, culminating in the Fundamental Theorem of Galois Theory.
  19. Representation Theory
    Ordinary representation theory of finite groups. Notes for one of the units in our graduate-level group theory course, culminating in the theorems of Frobenius (existence of Frobenius kernels) and Burnside (the pαqβ Theorem).
  20. Introduction to Cohomology beginning with algebra background
  21. Universal Coefficient Theorem for homology, beginning with algebra background
  22. The skinny on group cohomology

Combinatorics via Generating Functions

  1. Funny Dice: an introduction to generating functions. Some helpful Maple code is included
  2. Fibonacci numbers I and II: our next examples of generating functions
  3. Catalan numbers
  4. Counting walks
  5. Partition numbers and Pentagonal numbers
  6. Bell and Stirling numbers
  7. Counting necklaces via generating functions (see Bona's book)
  8. Selected applications of infinite series: generating functions and signal processing

Number Theory

  1. Review: basic definitions and properties of the integers
  2. Classifying Pythagorean Triples. A first proof in our course; I prefer this treatment to that in the Silverman textbook
  3. Using continued fractions to recognize rationals and quadratic irrationals from their decimal approximations
  4. CFRAC Factorization Method: An introduction to modern methods of integer factorization, with a small example implemented in Maple
  5. Dirichlet Series. Dirichlet's proof that there are infinitely many primes of the form 4k+1.
  6. Needles and Numbers: from the Buffon Needle problem to special values ζ(2n) of the Riemann zeta function
  7. The Basel Problem.  Here we give a rigorous elementary proof that ζ(2) = π2/6.
  8. Quadratic Reciprocity
  9. Writing a number as a sum of two squares with Maple implementation
  10. Factorization in rings of algebraic integers
  11. Transcendence of e and of π
  12. Functional equation for the Riemann zeta function using the adèles to motivate the Euler factors
  13. A Taste of Galois Theory
  14. Computations in the p-adics
  15. Introduction to modern cryptography
  16. Modular exponentiation of large integers using Maple. Background for RSA and Diffie-Hellman
  17. Maple implementation of RSA encryption
  18. A first (very rough) working version of Cyclotomic Fields with Applications.  Lecture notes for Fall 2018 course

Information Theory/Coding Theory/Entropy

  1. Data Compression using Huffman codes
  2. Entropy
  3. Error-Correcting Codes using a systematic version of the [7,4,3] binary Hamming code as an example
  4. The [7,4,3] binary Hamming code again: Alternative version with syndromes using linear algebra for encoding/decoding
  5. A sheet of qualitative examples comparing entropy of various information sources
  6. Reed-Solomon Codes
  7. Bell's Theorem via the CHSH game

Geometry

  1. A quick Overview of Geometry
  2. Affine Planes
  3. An application of finite geometry to experimental design
  4. Projective Planes
  5. On the role of the axiomatic method in plane geometry
  6. Inversive Planes
  7. Some straightedge and compass constructions, with Steiner's porism
  8. The real hyperbolic plane
  9. Incidence Geometry. Course notes for a Fall 2007 graduate course. Current revision: August, 2017
  10. The E8 Root Lattice and Conway's Ovoids

Exams

  1. Algebra Qualifying Exam, January 2005 (for our graduate students)
    With solutions.
  2. December 2005 Putnam Exam with my solutions
  3. December 2006 Putnam Exam with my solutions

Logic/Set Theory/Topology

Warning: More pictures ahead of old white males wearing neckties

  1. My favourite problem. Only after thinking seriously about it, look at the comments, solution and generalization.
  2. Knot Theory and Alexander polynomials
  3. Cardinality from the Bernstein-Cantor-Schröder Theorem
  4. Introducing the Infinite: Why the distinction between countable and uncountable is of practical importance
  5. Infinite Dimensional Vector Spaces
  6. Transfinite Induction.   Notes for UW seminar on Jan 20, 2009
  7. Separation properties in topology inclyding Urysohn's Lemma
  8. Ultrafilters and Tychonoff's Theorem
  9. Alternative proof of Tychonoff's Theorem by transfinite induction
  10. Ultraproducts
  11. The Tychonoff Plank
  12. Stone-Čech Compactification and Hindman's Theorem
  13. Construction of the Non-Standard Reals using ultrafilters
  14. The Hardin-Taylor Theorem on predicting the future
  15. Notes on Baire Category with applications (existence of an every-where-continuous-but-nowhere-differentiable function; also infinite permutation groups)
  16. Applications of logic to field theory
  17. Notes on Cherlin's proof that a generalised quadrangle with five points per line is finite. Includes background on order indiscernibles. Prepared for discussion with my friend Stan

 Under Construction

/ revised March, 2017