Mathematics at PROMYS

“PROMYS is a unique mathematical experience—one where you are given a lot of freedom and responsibility to do what you want and still have incredible opportunities to be immersed in, learn, invent, discover, and experience mathematics.” Vincent T.

At PROMYS, the opportunities for mathematical exploration abound!

Number Theory

Each weekday begins with all participants attending Number Theory lecture from 9:00–10:30 a.m. The main activity of first-year participants is their intensive efforts to solve an assortment of challenging problems in Number Theory. Daily problem sets encourage participants to design their own numerical experiments and to employ their own powers of analysis to discover mathematical patterns, formulate and test conjectures, and justify their ideas by devising their own mathematical proofs.

Advanced Seminars

Each summer, returning students and motivated first-year participants take one or more of the Advanced Seminars offered on diverse topics. PROMYS faculty and visiting mathematicians lead the seminars which meet two or three times per week for lecture and also feature engaging problem sets. In 2024, we offered the following seminars: Discrete Dynamical Systems and the Poincaré Recurrence Theorem; Galois Theory; and Graphy Theory.

2024 Advanced Seminars

Discrete Dynamical Systems and the Poincaré Recurrence Theorem (Margaret Beck, Boston University)

A discrete dynamical system is a map f on a space X. Starting with an initial point x_0 in X, one computes the orbit {x_n} of x_0 by iterating the map f: x_0, x_1 = f(x_0), x_2 = f(x_1) = f(f(x_0)), etc. Discrete dynamical systems are interesting in their own right, for example due to the chaotic behavior they can produce and the ways that mathematicians have attempted to characterize that behavior. Curiously, they can also be related to other branches of mathematics, including number theory. We will explore these ideas by introducing a variety of tools, including symbolic dynamics, invariant measures, recurrence, and ergodicity. Along the way we will prove the Poincaré Recurrence Theorem, which states that, under certain conditions, orbits of a dynamical system will necessarily return to their initial state, in fact infinitely often. We will also look at various implications of this theorem.

Galois Theory (David Speyer, University of Michigan)

This course will take you through the proof of Abel's theorem that there is no universal formula for the solution of quintic equations, and the beginnings of Galois theory. Classes will be very active, with a lot of time dedicated to in class problem solving. The background assumed is only the first year PROMYS material and comfort writing proofs. If you've seen linear algebra already, it will help you, but you don't need it, and you may have a different view of it after this course.

Graph Theory (Marjory Baruch, Syracuse University)

Graph theory has many applications, from representing street maps, dictating what shape fair dice can be, to coloring maps and looking for trends on the internet. It is a critical tool for cyber security. We will be studying the underlying classical mathematical graph theory (does it take more colors to color a map on a bagel?). Important numerical relationships will emerge from the study of graphs. In the PROMYS style, there will be opportunity for you to build your own understanding through examples and generating conjectures.

2015-2023 Advanced Seminars

2023 Advanced Seminars

Abstract Algebra
Professor Marjory Baruch (Syracuse University)
Modular Forms
Professor David Rohrlich (Boston University)
Primes and Zeta Functions
Professor Li-Mei Lim (Boston University)

2022 Advanced Seminars

Graphs, Matroids, and Polynomial Countability
Professor Melody Chan (Brown University)
Number-theoretic Cryptography
Professor David Jao (University of Waterloo)
Linear Algebra, Bernstein Polynomials, and Visualization
Professor Marjory Baruch (Syracuse University)

2021 Advanced Seminars

Finding Rational Points on Hyperelliptic Curves
Professor Jennifer Balakrishnan (Boston University)
Galois Theory
Professor David Speyer (University of Michigan)
Geometry and Symmetry
Professor Steve Rosenberg (Boston University)

2020 Advanced Seminars

Undecidability and Hilbert’s 10th Problem
Dr. Henry Cohn (Microsoft Research; MIT) and Dr. Cameron Freer (MIT)
Topology
Professor Dev Sinha (University of Oregon)
Graph Theory
Professor Marjory Baruch (Syracuse University)

2019 Advanced Seminars

Probability, Combinatorics, and Computation
Professor Lionel Levine (Cornell University)
Primes and Zeta Functions
Professor Li-Mei Lim (Boston University)
Algebra
Professor Marjory Baruch (Syracuse University)

2018 Advanced Seminars

Cryptography
Professor Li-Mei Lim (Boston University)
Galois Theory
Professor David Speyer (University of Michigan)
Graph Theory
Professor Marjory Baruch (Syracuse University)

2017 Advanced Seminars

The Analytic Class Number Formula
Professor Jared Weinstein (Boston University)
Algebra
Professor Marjory Baruch (Syracuse University)
Geometry and Symmetry
Professor Steve Rosenberg (Boston University)

2016 Advanced Seminars

Modular Forms
Professor David Rohrlich (Boston University)
The Mathematics of Computer Graphics
Professor Marjory Baruch (Syracuse University)
Geometry and Symmetry
Professor Steve Rosenberg (Boston University)

2015 Advanced Seminars

Complex Analysis in Number Theory (Dirichlet’s Theorem)
Dr. John Bergdall (Boston University)
Galois Theory
Marjory Baruch (Syracuse University)
Geometry and Symmetry
Professor Steve Rosenberg (Boston University)

2024 Guest Lectures

The Counting Project
Tim Harford, The Counting Project
The Inaugural Vicky Neale Public Lecture

The King and His Money
Emma Knight (CIBC)

The Rise and Fall of the Local-to-Global Conjecture
Professor Kate Stange (University of Colorado Boulder)

The Cap Set Problem and Hypothesis Generation by Machine
Professor Jordan Ellenberg (University of Wisconsin Madison)
2024 Arnold Ross Lecture sponsored by the American Mathematical Society (AMS)

Proportion Spaces, Spanning Trees, and Chip Firing
Professor Matt Baker (Georgia Tech School of Mathematics)

2015-2023 Guest Lectures

2023 Guest Lectures

Symmetries and Commutativity
Professor Dev Sinha (University of Oregon)
Interpolation Problems for Curves
Professor Isabel Vogt (Brown University)
2023 Arnold Ross Lecture sponsored by the American Mathematical Society (AMS)
Using Unique Factorization to Solve Diophantine Equations
Professor Padma Srinivasan (Boston University)

2022 Guest Lectures

Spherical Packings/Codes in Various Dimensions
Professor Noam Elkies (Harvard University)
2022 Arnold Ross Lecture sponsored by the American Mathematical Society (AMS)
The Riemann-Roch Theorem for Graphs
Professor Matt Baker (Georgia Tech School of Mathematics)

2021 Guest Lectures

A Survey of Diophantine Equations
Professor Edray Goins (Pomona College)
Tilings and Counting
Professor Philip Engel (University of Georgia)
Problems on Rainbow 3-term Arithmetic Progressions
Professor Michael Young (Carnegie Mellon University)
What does a circle know about primes?
Dr. Aditya Karnataki (Beijing International Center for Mathematical Research)

2020 Guest Lectures

Billiards on Regular Polygons
Professor Diana Davis (Swarthmore College)
Ants on Pants: An introduction to manifolds and bordism
Professor Agnes Beaudry (University of Colorado at Boulder)
How Quadratic Reciprocity is Like Dealing Cards
Professor Matthew Baker (Georgia Institute of Technology)
Waring's Problem
Dr. Vicky Neale (Balliol College at the University of Oxford)
Joy & Happiness in Mathematics
Professor Helen Grundman (Bryn Mawr College and Brown University)
The German Tank Problem: Math/Stats at War
Professor Steven J. Miller (Williams College)

2019 Guest Lectures (30th Anniversary of PROMYS)

The Biggest Known Prime
Professor Keith Conrad (University of Connecticut)
Brussels Sprouts and the Euler Characteristic
Professor Jeremy Booher (University of Arizona)
Binary Quadratic Forms and the Conjectures of Gauss
Professor Ila Varma (UC San Diego/University of Toronto)
Designing New Quantum Materials at the Atomic-scale
Professor Julia Mundy (Harvard University)
How to Turn Pure Math into Applied Math: My Quest for the Perfect Application
Professor David Jao (University of Waterloo)
Natural Language Understanding, Deep Learning, and the BERT Revolution
David R.H. Miller (Google)
Perspectives on Law and Mathematics
Professor Alex Lee (Northwestern University)
Towers of Life
Dr. Glen Whitney (Harvard University)
An Introduction to Bernoulli Percolation
Ryan McDermott (Cornell University)

2018 Guest Lectures

Generalized Catalan Numbers
Professor Paul Gunnells (University of Massachusetts at Amherst)
Tilings and Counting
Dr. Philip Engel (Harvard University)
Paradoxes in Probability
Lila Greco (Cornell University)
Checker Stacks and Related Musing
sJoshua Greene (Weiss Asset Management)

2017 Guest Lectures

How many times to shuffle a deck of cards?
Daniel Jerison (Cornell University)
The Emperor and His Money
Emma Knight (University of Toronto)
What's so special about eight dimensions anyway?
Jonathan Hanke (Goldman Sachs)

2016 Guest Lectures

Impossibility by Modular Arithmetic
Professor Keith Conrad (University of Connecticut)
Applying Physics to Mathematics
Professor Tadashi Tokieda (Stanford University/University of Cambridge)
Summing It Up: Amazon Rank 20,948
Professor Rob Gross (Boston College)
Hasse’s Local-Global Principle
Dr. Ila Varma (Harvard University/Columbia University)
The Future of Prediction
Professor Lionel Levine (Cornell University)

2015 Guest Lectures

Continued Fractions
Professor Keith Conrad (University of Connecticut)
Can you hear the shape of a drum?
Professor Moon Duchin (Tufts University)
Triangles
Professor Melody Chan (Brown University)

Research Projects

All students have the opportunity to participate in the process of scientific research — PROMYS-designed exploration labs for first-year students and research projects mentored by professional mathematicians for returning students. Every summer, research mathematicians propose original problem statements for the PROMYS program. Each returning student selects a problem, then teams of four engage in open-ended exploration under the mentor's guidance. At the end of the summer, students write up and present their research to the entire PROMYS community. Some of these papers have been published or presented at conferences like the Joint Mathematics Meetings.

2024 Research Projects

Letter Counting in Groups
Proposed by Professor Dev Sinha (University of Oregon)

Necklaces and Permutations
Proposed by Professor Matt Baker (Georgia Institute of Technology)

Point-Hyperplane Incidence Matrix
Proposed by Professor Alan Chang (Washington University in St. Louis)

Snake on General Graphs
Proposed by Harry Altman

Summing Primes
Proposed by David Lowry-Duda (ICERM)

Tiered Hypertrees
Proposed by Professor Paul Gunnells (UMass Amherst)

Exploration Labs

First-year students may also choose to participate in open-ended projects called Exploration Labs. They work in small groups, guided by a counselor and faculty member. At the end of the summer, the students write up their findings and make a presentation of their research to the assembled PROMYS community.

Counselor Minicourses

Counselors contribute to the mathematically rich environment at PROMYS by doing their own research and by designing and presenting a wide range of lectures on topics of special interest. They organize minicourses open to all participants and counselor seminars intended for their peers.

2024 Counselor Minicourses

Proofwriting Basics
Linear Algebra
Sphere Packing and Sphere Kissing
Why is Pi Irrational
So, uh, what exactly happened to Silicon Valley Bank?
Group Theory I–II
The Curry-Howard Correspondence
Projective Geometry
Metric Spaces
How to LaTeX
Mathematical Music Theory
Axiomatic Theories and First-Order Languages
The Fundamental Group
Cryptograph
Introduction to Homology
Matrix Tree Theorem
Countability and Measurability
Cyclotomic Polynomials
The Riemann Hypothesis for Zp[x]
Elliptic Curves
3264 Conics
Constructing the Reals
Behrend’s Construction
Hyperbolic Geometry
Proof Assistants 101
Introduction to Abstract Topology
The Geometry of Continued Fractions (T-Shirt Talk)