Schedule
Kick-off event TBA
Mid-semester check-in TBA
End-of-semester presentations TBA
Projects
Beginner Project
Mentor: Bob Kuo (he/him)
Mentees: Leila (she/her), Naomi Martineau-Paré (she/her)
Title: "Geometry: Euclid and Beyond"
Abstract: In this project, we will explore geometry from its very beginnings. This means we'll start with the ancient Greek mathematician's study of straightedge and compass constructions, dating back to at least 300BCE. This is a very rich subject on its own, being perhaps the origin of the axiomatic approach to studying mathematics, all recorded within the 13 volumes of Euclid's Elements. Euclid's elements is perhaps the most important textbooks in history, yet there's one point of contention that troubled mathematicians throughout history and even Euclid himself. This is his "parallel postulate", whose controversy was only resolved by Hilbert's Axioms in 1899. We will follow this historical development, studying the axioms Euclid lays out in volumes I-III of Elements, followed by Hilbert's axioms. From then onwards, this project is open to many natural continuations, which I will let you decide on. Possible directions are :
A deeper dive into Euclid's elements.
The connection of straightedge and compass constructions to abstract algebra and Gauss's construction of the regular 17-gon.
Non-Euclidean geometry such as projective, inversive, and hyperbolic geometry.
Our main reference will be Robin Hartshorne's "Geometry: Euclid and Beyond" and various (mainly David E. Joyce's) translations of Euclid's "Elements". Both of these references are available for free online. We may supplement this reading project with other references depending on where we take it.
Prerequisites: None!
Intermediate Project
Mentor: Edouard Heitzmann (he/they)
Mentees: Jacob DeWitt (he/him), Assaf Shaham (he/him)
Title: Graph Theory
Abstract: For this project we would read through a few chapters of Reinhard Diestel’s ‘Graph Theory’ textbook. This is a go-to reference in the field, and it has several nice mid-level results that could make nice topics for a DRP presentation. Among others, these results include Hall’s theorem for perfect matchings, Dirac’s Theorem on Hamiltonian Cycles, and The Five-Colour Theorem.
Prerequisites: Discrete math (introduction to proofs)
Advanced Project
Mentor: Eli Orvis (he/him)
Mentees: Cam Mars (he/him), Caiden Woolery (he/him)
Title: "Quadratic forms, class numbers, and computation"
Abstract: In this project, we will learn the fundamentals of binary quadratic forms. These innocent looking equations can be written as ax^2 + bxy + cy^2, where a, b, and c are integers, and were studied extensively by Gauss. It may come as a surprise that such simple equations lead to such deep mathematics: questions about quadratic forms inspired the creation of algebraic number theory, class field theory, and even modular forms. The project will start with learning the basic theory from David Cox’s book “Primes of the Form x^2 + n y^2.” We’ll get acquainted with the class group, composition, and see how these can answer the question of which primes can be written as p = x^2 + n y^2. From there, we will progress to computational questions, like “how does one actually compute the class group?”, using Cohen’s “A course in computational algebraic number theory.”
Finally, there are several directions that we can go to cap off the project, depending on student interest and time available. One is towards modern algebraic number theory, reformulating the fundamental results about quadratic forms in terms of rings and ideals. Even more algebraic would be to read about the general theory of quadratic forms in n variables over an arbitrary field. If analysis is more enticing, we could read about Jacobi’s theta functions, which encode representation counts for quadratic forms and lead to modular forms and the Riemann zeta function.
Prerequisites: Abstract algebra or Introductory Number Theory. Some familiarity with big-O notation might be helpful but not necessary. No additional prerequisites are necessary, as final topics can be tailored to students’ specific interests and background.