Current Courses

• Mathematics 4440/5440. Coding and Cryptography. Fall 2026.
  Gives an introduction, with proofs, to the algebra and number theory used in coding and cryptography. Basic problems of coding and cryptography are discussed; prepares students for the more advanced ECEN 5032 and 5682. Prereq., MATH 3130. Recommended prereqs., MATH 3110 and 3140.
• Mathematics 5810. Special Topics in Mathematics: Math and AI. Fall 2026.
  How language models work and what AI means for mathematics: transformers and their training, vibecoding, Lean, AI-assisted research, and ethics — with a build-your-own-LLM project. Co-taught with Agnès Beaudry and Jonathan Wise.
  • Course Website

Summer Schools

• The two summer schools below led to Course Notes, available on the arXiv: An illustrated introduction to the arithmetic of Apollonian circle packings, continued fractions, and other thin orbits.
• Graduate Summer School, NUS IMS, June 3-7, 2024.
  Integral packings and number theory.
  • Summer School Website
  • Lecture Notes
• Graduate Summer School, CIRM, July 2023.
  Number theory as a door to geometry, dynamics and illustration.
  • Summer School Website
  • Lecture Notes

Past Courses at CU Boulder

• Mathematics 8114. Topics in Number Theory: Computational Mathematics. Fall 2025.

  • Monday, Wednesday and Friday, 10:10 – 11:00 am, Duane Physics G2B60.
  • Course Website
  • First Worksheet

• Mathematics 6180. Algebraic Number Theory. Spring 2025.
  Introduces number fields and completions, norms, discriminants and differents, finiteness of the ideal class group, Dirichlet’s unit theorem, decomposition of prime ideals in extension fields, decomposition, and ramification groups. Prereqs., MATH 6110 and 6140. Undergraduates must have approval of the instructor.

  • Monday, Wednesday and Friday, 12:20 am – 1:10 pm, please check room online.
  • Homework List
  • In-Class Notes
  • Course Info/Announcement
  • Required CU Syllabus Statements 
  • Algebraic Number Theory (by Matthew Baker) — these are the notes we base the course on.
  • My own corrections to Baker’s notes — probably now incorporated into Baker’s most recent version, so out of date.
  • Number Theory Motivating Questions/Answers 
  • Quadratic Forms, Lattices, Ideals

• Mathematics 4440/5440. Cryptography and Coding Theory. Fall 2024.
  Gives an introduction, with proofs, to the algebra and number theory used in coding and cryptography. Basic problems of coding and cryptography are discussed; prepares students for the more advanced ECEN 5032 and 5682. Prereq., MATH 3130. Recommended prereqs., MATH 3110 and 3140.

  • Monday, Wednesday and Friday, 10:10 – 11:00 am, Duane Physics G2B60.
  • Course website

• Mathematics 6110. Introduction to Number Theory, Fall 2024
  Examines divisibility properties of integers, congruencies [sic], diophantine equations, arithmetic functions, quadratic residues, distribution of primes, and algebraic number fields. Prereq., MATH 3140. Undergraduates must have approval of the instructor.

  • Monday, Wednesday, and Friday, 2:30 – 3:20 pm, Duane Physics G1B25
  • Getting started with Sage in Number Theory
  • Course Announcement

• Mathematics 8114. Topics in Algebra:  Mathematical Cryptography. Spring 2024.
  Please see the Course Announcement.

  • Monday, Wednesday and Friday, 11:15 am – 12:05 pm, Math 220.
  • Course Announcement.
  • Daily Resources & Exercises.

• Mathematics 6180. Algebraic Number Theory. Spring 2023.
  Introduces number fields and completions, norms, discriminants and differents, finiteness of the ideal class group, Dirichlet’s unit theorem, decomposition of prime ideals in extension fields, decomposition, and ramification groups. Prereqs., MATH 6110 and 6140. Undergraduates must have approval of the instructor.

  • Monday, Wednesday and Friday, 11:15 am – 12:05 pm, Muenzinger Psyc & Biopsych E118.
  • Homework List
  • In-Class Notes
  • Course Info/Announcement
  • Required CU Syllabus Statements 
  • Algebraic Number Theory (by Matthew Baker) — these are the notes we base the course on.
  • My own corrections to Baker’s notes — probably now incorporated into Baker’s most recent version, so out of date.
  • Number Theory Motivating Questions/Answers 
  • Quadratic Forms, Lattices, Ideals

• Mathematics 2001.  Introduction to Discrete Mathematics.  Spring 2023.
  Introduces the ideas of rigor and proof through an examination of basic set theory, quantification theory, elementary counting, discrete probability, and additional topics. Prereq., MATH 1300 or APPM 1350.

  • Section 4: Monday, Wednesday and Friday, 1:25-2:15 pm, Clare Small Arts and Sciences 302.
  • Course website

• Mathematics 4440/5440. Cryptography and Coding Theory. Fall 2022.
  Gives an introduction, with proofs, to the algebra and number theory used in coding and cryptography. Basic problems of coding and cryptography are discussed; prepares students for the more advanced ECEN 5032 and 5682. Prereq., MATH 3130. Recommended prereqs., MATH 3110 and 3140.

  • Monday, Wednesday and Friday, 10:10 – 11:00 am, Duane Physics G2B60.
  • Course website

• Mathematics 6180. Algebraic Number Theory. Spring 2021.
  Introduces number fields and completions, norms, discriminants and differents, finiteness of the ideal class group, Dirichlet’s unit theorem, decomposition of prime ideals in extension fields, decomposition, and ramification groups. Prereqs., MATH 6110 and 6140. Undergraduates must have approval of the instructor.

  • Monday, Wednesday and Friday, 11:30 – 12:20 am, Remotely.
  • Homework List
  • Course Info/Announcement
  • Required CU Syllabus Statements 
  • Algebraic Number Theory (by Matthew Baker) — these are the notes we base the course on.
  • My own corrections to Baker’s notes — based on my reading last time I taught it.
  • Number Theory Motivating Questions/Answers

• Mathematics 4440/5440. Cryptography and Coding Theory. Fall 2020.
  Gives an introduction, with proofs, to the algebra and number theory used in coding and cryptography. Basic problems of coding and cryptography are discussed; prepares students for the more advanced ECEN 5032 and 5682. Prereq., MATH 3130. Recommended prereqs., MATH 3110 and 3140.

  • Monday, Wednesday and Friday, 1:50 – 2:40 pm, Remotely.
  • Course website

• Mathematics 2001.  Introduction to Discrete Mathematics.  Fall 2020.
  Introduces the ideas of rigor and proof through an examination of basic set theory, quantification theory, elementary counting, discrete probability, and additional topics. Prereq., MATH 1300 or APPM 1350.

  • Section 5: Monday, Wednesday and Friday, 3:00 – 3:50 pm, Remote.
  • Course website

• Mathematics 2001.  Introduction to Discrete Mathematics.  Spring 2020.
  Introduces the ideas of rigor and proof through an examination of basic set theory, quantification theory, elementary counting, discrete probability, and additional topics. Prereq., MATH 1300 or APPM 1350.

  • Section 5: Monday, Wednesday and Friday, 2:00 – 2:50 pm, ECCR 131.
  • Course website

• Mathematics 8114.  Topics in Number Theory. Spring 2020.
  (a.k.a. Elliptic Curves)
  May include the theory of automorphic forms, elliptic curves, or any of a variety of advanced topics in analytic and algebraic number theory. Department enforced prerequisite: MATH 6110. Instructor consent required for undergraduates. [Edit: I will cover the arithmetic of elliptic curves.]

  • Monday, Wednesday and Friday, 1:00 – 1:50 pm, MATH 220.
  • Course announcement.
  • Homework listing.

• Mathematics 3110. Introduction to Theory of Numbers. Spring 2019.
  (a.k.a. Number Theory)
  Studies the set of integers, focusing on divisibility, congruencies, arithmetic functions, sums of squares, quadratic residues and reciprocity, and elementary results on distributions of primes. Prerequisite: MATH 2001.

  • Monday, Wednesday and Friday, 2:00 – 2:50 am, ECCR 105.
  • Course Website

• Mathematics 6180. Algebraic Number Theory. Spring 2019.
  Introduces number fields and completions, norms, discriminants and differents, finiteness of the ideal class group, Dirichlet’s unit theorem, decomposition of prime ideals in extension fields, decomposition, and ramification groups. Prereqs., MATH 6110 and 6140. Undergraduates must have approval of the instructor.

  • Monday, Wednesday and Friday, 3:00 – 3:50 am, ECCR 118.
  • Algebraic Number Theory (by Matthew Baker) — these are the notes we base the course on.
  • My own corrections to Baker’s notes — based on my reading last time I taught it.
  • Some Motivational Question in Number Theory (the one-page version without answers is here).

• Mathematics 2001. Introduction to Discrete Mathematics. Spring 2018.
  Introduces the ideas of rigor and proof through an examination of basic set theory, quantification theory, elementary counting, discrete probability, and additional topics. Prereq., MATH 1300 or APPM 1350.

  • Section 1: Monday, Wednesday and Friday, 9:00 – 9:50 pm, ECCR 108.
  • Section 1: Monday, Wednesday and Friday, 10:00 – 10:50 pm, ECCR 108.
  • Course website

• Mathematics 6180. Algebraic Number Theory. Spring 2017.
  Introduces number fields and completions, norms, discriminants and differents, finiteness of the ideal class group, Dirichlet’s unit theorem, decomposition of prime ideals in extension fields, decomposition, and ramification groups. Prereqs., MATH 6110 and 6140. Undergraduates must have approval of the instructor.

  • Monday, Wednesday and Friday, 11:00 – 11:50 am, ECCR 139.
  • Course Announcement.
  • Algebraic Number Theory (by Matthew Baker) — these are the notes we base the course on.
  • My own corrections to Baker’s notes — based on my reading this semester.
  • Some Motivational Question in Number Theory (the one-page version without answers is here).

• Mathematics 4440/5440. Cryptography and Coding Theory. Fall 2016.
  Gives an introduction, with proofs, to the algebra and number theory used in coding and cryptography. Basic problems of coding and cryptography are discussed; prepares students for the more advanced ECEN 5032 and 5682. Prereq., MATH 3130. Recommended prereqs., MATH 3110 and 3140.

  • Monday, Wednesday and Friday, 1:00 – 1:50 pm, ECCR 139.
  • Course website

• Mathematics 2001. Introduction to Discrete Mathematics. Fall 2016.
  Introduces the ideas of rigor and proof through an examination of basic set theory, quantification theory, elementary counting, discrete probability, and additional topics. Prereq., MATH 1300 or APPM 1350.

  • Monday, Wednesday and Friday, 2:00 – 2:50 pm, ECCR 139.
  • Course website

• Mathematics 8114. Topics in Number Theory: The Arithmetic of Kleinian Groups. Spring 2016.
  Please see the Course Announcement.

  • Monday, Wednesday and Friday, 9:00 – 9:50 am, ECCR 118.
  • Course Announcement.
  • Course Notes (ongoing).
  • Moebius Transformations Revealed (video).
  • Hopf fibration (video).
  • Hyperbolic Spaces (notes by John Parker).
  • Indra’s Pearls website (to accompany the book; pictures/code).
  • Computational tools for hyperbolic 3-manifolds and Kleinian groups.
  • Curtis T. McMullen, The evolution of geometric structures on 3-manifolds; also video.
  • A Crash Course on Kleinian Groups (notes by Caroline Series).
  • Hyperbolic Geometry (notes by Caroline Series).

• Mathematics 3170. Introduction to Combinatorics, Fall 2015.
  Covers basic methods and results in combinatorial theory. Includes numeration methods, elementary properties of functions and relations, and graph theory. Emphasizes applications. Prerequisites: Requires prerequisite course of MATH 2001 (minimum grade C-).

  • Monday, Wednesday and Friday, 11:00 – 11:50 am, ECCR 118.
  • Course website
  • Syllabus
  • Worksheet on Inclusion Exclusion
  • Worksheet One on Elementary Enumeration
  • Worksheet Two on Elementary Enumeration
  • Counting Review Problems
  • Generating Functions Worksheet and Solutions
  • Generating Functions Review Sheet
  • Graph Theory Game Worksheet (“col”)
  • Graph Theory Worksheet and Solutions
  • Hamiltonian Hypercube
  • Induction Worksheet
  • Pigeonhole Worksheet
  • Stirling Numbers of the First Kind Worksheet
  • Surreal Numbers
  • FCQ Results: numerical and comments

• Mathematics 2001. Introduction to Discrete Mathematics, Fall 2015.
  Introduces the ideas of rigor and proof through an examination of basic set theory, quantification theory, elementary counting, discrete probability, and additional topics. Prereq., MATH 1300 or APPM 1350.

  • Monday, Wednesday and Friday, 1:00 – 1:50 pm, ECCR 108.
  • Course website
  • Syllabus
  • Proof Grading Sheet
  • Day One: Exploration
  • Worksheet on Counting with Independent Choices
  • Worksheet on Truth Tables
  • Worksheet on Quantifiers and Negation
  • Worksheet on Negating for Contradiction
  • Second Counting Worksheet
  • An Example Combinatorial Proof
  • Worksheet on Combinatorial Proof
  • More Example Combinatorial Proofs
  • Worksheet on the game SET
  • Worksheet on Relations and their Properties
  • Worksheet on Relations and Equivalence Classes
  • Worksheet on Functions
  • Worksheet on Inverse Functions
  • A function is bijective if and only if it has an inverse
  • Take One or Two: Game Theory Worksheet
  • Take One or Two: Solution by Induction
  • Worksheet on Induction Format
  • FCQ Results: numerical and comments

• Mathematics 2001. Introduction to Discrete Mathematics, Spring 2015.
  Introduces the ideas of rigor and proof through an examination of basic set theory, quantification theory, elementary counting, discrete probability, and additional topics. Prereq., MATH 1300 or APPM 1350.

  • Monday, Wednesday and Friday, 9:00 – 9:50 am, ECCR 108.
  • D2L course website
  • Course syllabus
  • FCQ Results

• Mathematics 6130. Algebra 1, Fall 2014.
  Studies group theory and ring theory. Prerequisite, Math 3140. Undergraduates need instructor consent. Prerequisites: Restricted to graduate students only.

  • Monday, Wednesday and Friday, 9:00 – 9:50 am, KOBL 330.
  • Course website
  • FCQ Results

• Mathematics 4440/5440. Coding and Cryptography, Spring 2014.
  Gives an introduction, with proofs, to the algebra and number theory used in coding and cryptography. Basic problems of coding and cryptography are discussed; prepares students for the more advanced ECEN 5032 and 5682. Prereq., MATH 3130. Recommended prereqs., MATH 3110 and 3140.

  • Monday, Wednesday and Friday, 2:00 – 2:50 pm, FLMG 156 (changed to ECCR 118).
  • Course website
  • FCQ Results
  • Syllabus
  • Magic Box Cheat Sheet
  • RSA Cheat Sheet
  • Midterm Topics
  • Final Topics
  • Legendre Jacobi Cheat Sheet
  • Coding Theory Worksheet Blank
  • Coding Theory Solutions

• Mathematics 3130. Linear Algebra, Fall 2013.
  Examines basic properties of systems of linear equations, vector spaces, linear independence, dimension, linear transformations, matrices, determinants, eigenvalues, and eigenvectors. Prereq., MATH 2300 or APPM 1360. Credit not granted for this course and APPM 3310.

  • Monday, Wednesday and Friday, 1:00 – 1:50 pm, KOBL 220.
  • Course website
  • FCQ Results

• Mathematics 6110. Introduction to Number Theory, Fall 2013
  Examines divisibility properties of integers, congruencies [sic], diophantine equations, arithmetic functions, quadratic residues, distribution of primes, and algebraic number fields. Prereq., MATH 3140. Undergraduates must have approval of the instructor.

  • Monday, Wednesday, and Friday, 3:00 – 3:50 pm, ECST 1B21 (part of the Engineering Center).
  • Course website
  • Course Notes
  • FCQ Results

• Mathematics 2300. Calculus 2, Honours Section, Fall 2012
  Continuation of MATH 1300. Topics include transcendental functions, methods of integration, polar coordinates, conic sections, improper integrals, and infinite series. Prereq., MATH 1300. Credit not granted for this course and MATH 1320 or APPM 1360.

  • Monday, Tuesday, Wednesday, Thursday and Friday, 9:00 – 9:50 am, ECCR 118.
  • Course website
  • FCQ Results

• Mathematics 6110. Introduction to Number Theory, Fall 2012
  Examines divisibility properties of integers, congruencies, diophantine equations, arithmetic functions, quadratic residues, distribution of primes, and algebraic number fields. Prereq., MATH 3140. Undergraduates must have approval of the instructor.

  • Monday, Wednesday and Friday, 3:00 – 3:50 pm, ECST 1B21.
  • Course website
  • FCQ Results

Courses Taught at the University of British Columbia

• Mathematics 317. Vector calculus, Fall 2010       Parametrizations, inverse and implicit functions, integrals with respect to length and area; grad, div, and curl, theorems of Green, Gauss, and Stokes.
  • Course website
  • Virtual office hours (course blog)
  • Midterm I • Midterm II
  • Pencast: An Example Computing Flux (online video)
  • Last-day gameshow questions (answers in the course website)
  • The Big Theorems (handout for end of term)
  • Example homework problems (line integrals in vector fields)
  • Sage plotting commands

Courses Taught at Harvard University

• Mathematics 129. Number Fields, Spring 2009       Algebraic number theory: number fields, unique factorization of ideals,
  finiteness of class group, structure of unit group, Frobenius elements,
  local fields, ramification, weak approximation, adeles, and ideles.

  • Course website was internal to Harvard
  • Course lecture notes (incomplete)

• Mathematics 152. Discrete Mathematics, Fall 2008
  An introduction to finite groups, finite fields, finite geometry,
  discrete probability, and graph theory. A unifying theme of the course
  is the symmetry group of the regular icosahedron, whose elements can be
  realized as permutations, as linear transformations of vector spaces
  over finite fields, as collineations of a finite plane, or as vertices
  of a graph. Taught in a seminar format, and students will gain
  experience in presenting proofs at the blackboard.

  • Course website was internal to Harvard
  • Course weblog (by the students and myself)
  • Course syllabus (the course has a very unusual structure driven by student seminars)
  • Tips for proofs and presenting (my advice to students)
  • Proofs (example outline I wrote for student preparation of presentations)
  • Change ringing field trip (student account of the trip)
  • RSA encryption (exploratory homework project I designed)
  • Homeworks, quizzes, outlines available upon request.

Courses Taught at Brown University

• Mathematics 52. Linear Algebra, Spring 2006       Vector spaces, linear transformations, matrices, systems of linear equations, bases, projections, rotations, determinants, and inner products. Applications may include differential equations, difference equations, least squares approximations, and models in economics and in biological and physical sciences.
  • Course website
  • The Great Hamster Caper: Dr. Echelon’s Adventures in Basis Change (handout)
  • Advice on doing your math homework
  • Advice on reading your math textbook
  • Course goals and grading
• Mathematics 18. Multivariable Calculus, Fall 2004       Three-dimensional analytic geometry. Differential and integral calculus of functions of two or three variables: partial derivatives, multiple integrals, Green’s Theorem.
  • Course website
  • Green’s Theorem lecture (video taped as part of Sheridan Centre Teaching Certificate program)
  • Worksheet on surfaces (for in-class use)
  • Lecture notes on Lagrange multipliers (for in-class use)
• Mathematics 9. Introductory Calculus, Fall 2003       An intensive course in the calculus of one variable including limits; differentiation; maxima and minima, and the chain rule for polynomials, rational functions, trigonometric functions, and exponential functions. Introduction of integration with applications to area and volumes of revolution.
  • Course webpage
  • Study suggestions
• Teaching Assistant, Mathematics 9 (Introductory Calculus) in Fall 2002, Spring 2003       An intensive course in the calculus of one variable including limits; differentiation; maxima and minima, and the chain rule for polynomials, rational functions, trigonometric functions, and exponential functions. Introduction of integration with applications to area and volumes of revolution.
  • Example quiz with solutions

Other Teaching

• A Taste of Pi: Clocks, Set, Elliptic Curves and the Secret Math of Spies, Fall 2010       A presentation and workshop for interested high school students, as part of A Taste of Pi. I recommend to interested high school students Joe Silverman’s book A Friendly Introduction to Number Theory and also the summer program, PROMYS.
  • Lecture component
  • Worksheet
  • News brief (sidebar page 7)

Course Evaluations

• The University of Colorado conducts the Faculty Course Questionnaire. Numerical Summary accessible via CU’s FCQ website: [ web ]
  See individual courses above for full comments
• The University of British Columbia Mathematics Department conducts departmental evaluations by email questionnaire. Report for Math 317 – Fall 2010 [ pdf ]
• The Harvard University Mathematics Department conducts departmental evaluations by email questionnaire. Math 129 had too few students to qualify for evaluation.Numerical Breakdown for Math 152 – Fall 2008 [ pdf ]
  Complete Record of Reviews for Math 152 – Fall 2008 [ pdf ]
• The Harvard Q is a student organisation that conducts evaluations by email questionnaire.Harvard Q Review for Math 152 – Fall 2008 [ numerical | comments ]
• The Brown University Mathematics Department conducts reviews by paper questionnaire. Although I have a record of the whole stack of reviews, I’ve only entered the breakdown and a selection of comments into the computer. Full reviews are available upon request.Selected Quotes and Student Reviews (Math 52) [ html ]Numerical Breakdown of Departmental Evaluations (Math 9, 18, 52) [ html ]
• The Brown University Critical Review is a student-run publication which
  independently evaluates courses and instructors.  Evaluations
  include a numerical breakdown and a summary of student comments.
  Critical Review for Math 9 – Introductory Calculus – Fall 2003 [ pdf ]
  Critical Review for Math 18 – Multivariable Calculus – Fall 2004 [ pending ]
  Critical Review for Math 52 – Linear Algebra – Spring 2006 [ pdf ]