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 Tran, Student 2021–2023
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 2023, we offered the following seminars: Abstract Algebra; Modular Forms; and Primes and Zeta Functions.
Abstract Algebra (Marjory Baruch, Syracuse University)
In the PROMYS style, we will be exploring abstract algebraic structures as they arise from concrete examples. We will learn about groups in the context of permutations, symmetries, and transformations. We will look at some classical results, and possibly apply some group theory to puzzles and games. Number theory and counting will be used throughout, as will geometry.
Modular Forms (David Rohrlich, Boston University)
Modular forms are certain functions on the complex upper half plane which enter number theory in a startling variety of ways: from representations of integers as sums of squares to combinatorial identities to the proof of Fermat's Last Theorem. The main goal of this course will be to get a feel for what modular forms are and for how they are used in number theory. Since the subject of modular forms is a rather advanced topic, certain statements made in class will have to be taken on faith, with details of proof to be filled in later in your study of mathematics. The goal of the course is not necessarily to be completely self-contained but rather to see why there is any connection between modular forms and number theory at all. One connection which we hope to focus on is the connection with quadratic forms, in particular Jacobi's formula for the number of ways of writing a positive integer as a sum of four squares.
Primes and Zeta Functions (Li-Mei Lim, Boston University)
We know from the PROMYS number theory problem sets that there are infinitely many primes that are 3 mod 4, and that there are also infinitely many primes that are 1 mod 4. Are there more primes that are 1 mod 4 or 3 mod 4? Using methods from calculus and analysis, along with infinite series called zeta functions and L-functions, we will explore this question and other similar ones.
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)
2014 Advanced Seminars
Values of the Zeta Function and p-Adic Analysis
Professor David Geraghty (Boston College)
Algebra
Professor Marjory Baruch (Syracuse University)
Geometry and Symmetry
Professor Steve Rosenberg (Boston University)
2013 Advanced Seminars
Representations of Finite Groups
Professor Robert Pollack (Boston University)
Wavelet Transformations
Professor Marjory Baruch (Syracuse University)
Geometry and Symmetry
Professor Steve Rosenberg (Boston University)
2012 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)
2011 Advanced Seminars
Character Sums
Professor Jay Pottharst (Boston University)
The Mathematics of Computer Graphics
Professor Marjory Baruch (Syracuse University)
Geometry and Symmetry
Professor Steve Rosenberg (Boston University)
2010 Advanced Seminars
Modular Forms
Professor Jon Hanke (University of Georgia)
The Mathematics of Computer Graphics
Professor Marjory Baruch (Syracuse University)
Geometry and Symmetry
Professor Steve Rosenberg (Boston University)
Guest Lectures
Our regular weekly activities are supplemented by diverse lectures by faculty and guests of the program. These lectures introduce participants to related scientific fields and include discussions of the ethics and philosophy of science, the relationship between pure and applied science, and career options.
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 Musings
Joshua 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
Erick 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.
Counting Lattice Points
Mentored by Jeremy Booher (University of Florida), Bryden Cais (University of Arizona), Joe Kramer-Miller (Lehigh University), James Upton (University of California, San Diego)
Heisenberg Continued Fractions
Mentored by Paul Gunnells (University of Massachusetts at Amherst)
Permutations, Tiered Trees, Weights, and their Generalizations
Mentored by Paul Gunnells (University of Massachusetts at Amherst)
Tangles with k Ropes and 2k Hands
Mentored by Minh-Tam Quang Trinh (Massachusetts Institute of Technology)
The Eudoxus Reals
Mentored by Matt Baker (Georgia Institute of Technology)
2022 Research Projects
Balancing Centrifuges
Mentored by Matt Baker (Georgia Institute of Technology)
Integer Complexity Generalized to Cyclotomic Fields
Mentored by Joshua Zelinsky (Iowa State University)
Irreducible Polynomials over a Finite Field
Mentored by Jared Weinstein (Boston University)
Konigsberg Pseudoprimes and Fermat’s Little Theorem for Matrices
Mentored by David Lowry-Duda (Institute for Computational and Experimental Research in Mathematics – ICERM)
The Circulation Game
Mentored by Matt Baker (Georgia Institute of Technology)
Tournament of Ten
Mentored by Nathan Kaplan (University of California, Irvine)
2021 Research Projects
Coexter Nim
Mentored by Paul Gunnells (University of Massachusetts at Amherst)
Hypergraph Fuss-Catalan Numbers
Mentored by Paul Gunnells (University of Massachusetts at Amherst)
Mathematical Card Magic
Mentored by Matt Baker (Georgia Institute of Technology)
Prime Sums
Mentored by David Lowry-Duda (Institute for Computational and Experimental Research in Mathematics – ICERM)
2020 Research Projects
Beyond Pick’s Theorem: Ehrhart Polynomials and Mixed Volumes
Mentored by Kiran Kedlaya (University of California, San Diego)
Continuous Functions and Notions of Smallness
Mentored by Cameron Freer (MIT), Rehana Patel (Wesleyan University), and Maya Saran (Ashoka University, Delhi)
Counting Matrices
Mentored by Jayadev Athreya (University of Washington)
Forgetful Fibonacci Sequences
Mentored by Joshua Zelinsky (The Hopkins School, New Haven)
Gerrymandering 1
Mentored by Diana Davis (Swarthmore College)
Gerrymandering 2
Mentored by Diana Davis (Swarthmore College)
Heapable Sequences
Mentored by Mike Mitzenmacher (Harvard University)
Hypergraph Catalan Number
Mentored by Paul Gunnells (University of Massachusetts, Amherst)
Narain Lattices
Mentored by Henry Cohn (Microsoft Research Institute)
Open Problems Related to Voting Power
Mentored by Joshua Zelinsky (The Hopkins School, New Haven)
Some Counting Problems in Finite Fields
Mentored by Wayne Peng and Thomas Tucker (University of Rochester)
2019 Research Projects
Bernoulli Sandpiles on the Infinite Ladder Graph
Mentored by Lionel Levine and Ryan McDermott (Cornell University)
Characterizing D for Integer-Solvable x² − Dy² = −1
Mentored by Erick Knight (University of Toronto)
Exponents of Jacobians of Graphs and Regular Matroids
Mentored by Matthew Baker (Georgia Institute of Technology)
Lower Bounds on a-Numbers of Artin-Schreier Curves
Mentored by Jeremy Booher (University of Arizona)
Metaheuristics for Optimizing Voter Distributions against Partisan Gerrymandering
Mentored by Diana Davis (Swarthmore College)
Representation Theory and Dickson’s Theorem
Mentored by John Bergdall (Bryn Mawr College)
Ulam Sequences
Mentored by Jayadev Athreya (University of Washington)
2018 Research Projects
Averages of Divisors
Mentored by Joshua Zelinsky (Iowa State University)
Characteristics of Hyperfields obtained as Quotients of Finite Fields
Mentored by Matt Baker (Georgia Institute of Technology)
Class Groups of Function Fields
Mentored by Erick Knight (University of Toronto) and Ananth Shankar (MIT)
Expansions of Natural Numbers and of Real Numbers
Mentored by Michael King (Bowdoin College)
Greedy Avoidance of Arithmetic Progressions
Mentored by Lila Greco and Lionel Levine (Cornell University)
Left, Center, Right
Mentored by Nathan Kaplan (University of California, Irvine)
Permutation Statistics
Mentored by Paul Gunnells (University of Massachusetts, Amherst)
2017 Research Projects
2-Torsion in Class Groups
Mentored by Erick Knight (University of Toronto)
Benford's Law and Ulam Sequences
Mentored by Jayadev Athreya (University of Washington)
Benford's Law in Linear Recurrences and Continued Fractions
Mentored by Jayadev Athreya (University of Washington)
Billiard Trajectories on Integrable Polygons and Cutting Sequences in Equilateral Triangles
Mentored by Jared Weinstein (Boston University)
Coxeter Nim
Mentored by Paul Gunnells (University of Massachusetts, Amherst)
On the Gonality Conjecture for Graphs
Mentored by Matt Baker (Georgia Institute of Technology)
Random Game Trees
Mentored by Daniel Jerison and Lionel Levine (Cornell University)
Tau Ideals in Number Fields
Mentored by Joshua Zelinsky (Birmingham-Southern College)
2016 Research Projects
An Interesting Congruence About the Ramanujan Tau Function
Mentored by Kevin Buzzard (Imperial College, London)
Bouncing Superball
Mentored by Jeremy Booher (Stanford University)
Fibonacci Words
Mentored by Lionel Levine (Cornell University)
Integral Eigenvalues of Schreier Coset Graphs of Weyl Groups
Mentored by Paul Gunnells (University of Massachusetts, Amherst)
Monogenic Cubic Fields
Mentored by Ila Varma (Harvard University), Dylan Yott (UC Berkeley), and Jonathan Hanke
On Wythoff’s (a, a) Variation
Mentored by Paul Gunnells (University of Massachusetts, Amherst)
PROMYS Polygon
Mentored by Victor Rotger (Universitat Politècnica de Catalunya, Barcelona)
Ramified Automorphisms of the Field of Laurent Series
Mentored by Laurent Berger and Sandra Rozensztajn (École Normale Supérieure de Lyon)
Representation of of Rational Numbers in Different Numerations Systems
Mentored by Ira Gessel (Brandeis University)
Slopes of Some Newton Polygons
Mentored by John Bergdall (Boston University)
2015 Research Projects
Box-Ball System
Mentored by Lionel Levine (Cornell University)
Degrees and Linear Diophantine Equations in F[x]
Mentored by Keith Conrad (University of Connecticut)
Dynamical Systems and Number Theory
Mentored by Victor Rotger (Universitat Politècnica de Catalunya, Barcelona)
Heisenberg Continued Fractions
Mentored by Paul Gunnells (University of Massachusetts, Amherst)
Improved Bounds for Kissing Numbers in Dimensions 25-31
Mentored by Henry Cohn (Microsoft Research)
Modular Representations of GL2(Fp)
Mentored by Laurent Berger (École Normale Supérieure de Lyon)
Running Sums and Stopping Times
Mentored by Jared Weinstein (Boston University)
Sets of Pairwise Equidistant Points
Mentored by David Rohrlich (Boston University)
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.
How to Write a Proof
How to Give a Talk
Complex Analysis I–IV
Introduction to Category Theory
Jordan-Schoenflies Theorem
Symmetric Functions
Cayley’s Tree Theorem
Irrationality & Transcendence
Knot Theory
Ideals
LaTeX Workshop
Dynamical Systems
Ruler-Compass Constructions
Bezout’s Theorem
Fundamental Groups
Axiom of Choice I–II
Higher Homotopy Groups
Hexaflexagons
The Hasse-Weil Bound
Hilbert’s Third Problem
Galois Theory
Proofs of Quadratic Reciprocity
Voting Theory & Arrow’s Theorem
Hyperbolic Geometry
Dynamic Time-Warping Algorithms
The Banach-Tarski Paradox
The Aperiodic Tiling (T-Shirt Talk)
2022 Counselor Minicourses
Proofwriting Basics
Projective Geometry
Propositions as Types
Knot Theory
Cardinality and The Halting Problem
Determinants and Path Counting
How to Calculate Doomsday
Brouwer Fixed Point Theorem (in 2D)
Linear Algebra
Generating Functions
Spectral Graph Theory
Surreal Numbers
LaTeX Workshop
Martingales and the Optimal Stopping Theorem
Measure Theory
Poisson Distribution
A (Locally) Small Introduction to Category Theory
Art and Fractals
Model Theory and First-Order Logic
Imposternomials
Probabilistic Method
Cantor Set
Kummer’s Theorem
The Tri-Pants Graph
The Irrationality of √2
Banach Tarski Paradox
Combinatorial Species
The Moon is a Planet, and Other Astronomical Hot Takes
Googology
The Weierstraß Elliptic Function
Combinatorial Nullstellensatz
Markov Chains
Isometries
Constructible Polygons (T-shirt Talk)
Arithmetic of Order Types and Ordinals
Ostrowski’s Theorem
2021 Counselor Minicourses
How to Write a Proof
Introduction to Mechanism Design
Ramsey Theory
Knot Theory
Measure Theory
Differential Equations and Recursion
Tropical Arithmetic
LaTeX
Languages and Automata
Axiom of Choice
Proof Assistants
Cutie Pi: Leibniz, Euler, and Warmth from Hugs
Construction of R
Cryptography is Hard
Randomized computations
Continued Fractions
Metric Space and Topology
Riemann Integration
T-shirt Talk: Voronoi Diagram
How to find roots of polynomials mod <em>p</em>
Polynomial Statistics
Erdös Problems
Law of Large Numbers
2020 Counselor Minicourses
Bounds on the Cardinality of Maximal Sidon Sets
Projective Geometry
An Introduction to the Nash Equilibrium
Prime and Shine
Fun with Catalan Numbers and Motivating Generating Functions
Is this a polynomial function?
Fractal Dimensions
Geometry on Surfaces
Complexity, or why solving one problem solves them all
Combinatorical Games
Fractal Design
Averaging and inequality
Tour of Philosophy of Science
Factoring Large Numbers
Algebraic Dynamic Programming
PROMYS 2020 T-Shirt Talk
2019 Counselor Minicourses
Curves of Constant Width
Ramsey Numbers
Applying Information Theory to Combinatorics
Voting Theory
Unnecessary Proofs of the Infinitude of Primes
Frisbee Minicourses
Tropical Geometry
Triternions
Influence in Referendum Elections
Elliptic Curves
Markov Chains and Hidden Markov Model
Partition Functions
Rings and Ideals
Soddy-Gosset Theorem
How many proofs of AM-GM is too many?
Homogeneous Polynomials over Finite Fields
T-shirt Talk: Geometry and Splitting of Primes
2018 Counselor Minicourses
Dividing Squares Into Triangles of Equal Area
Linear Algebra and Face Recognition
Metropolis-Hastings Algorithm
The Inscribed Rectangle Problem
The Dehn Invariant
Introduction to Smooth Manifolds
Crossing Numbers of Alternating Knots
The 5-Color Theorem
Random Walks
Burnside’s Lemma
LaTeX Minicourse
The AKS Primality Test
Frisbee Rules and Strategy
Visualizing Polytopes in 4D
Period Three Implies Chaos
ABC Conjecture and its Consequences
Doodles
Introduction to Special Relativity and Group Theory
The Skolem-Mahler-Lech Theorem
T-Shirt Minicourse: Families
2017 Counselor Minicourses
Shnirelman’s Theorem
Infinite Dimensional Topologies
Fermat’s Last Theorem for Polynomials
Fundamental Groups and Covering Spaces
What is R?
Hyperplane Separation Theorem and Intro to Convex Analysis
Let’s Talk About Sets, Baby!
Hyperplane Arrangement
LaTeX Minicourses
Frisbee Minicourses
Extending Platonic Solids
Quantum Computing
General Relativity
Special Relativity
Sharkovsky’s Theorem
Sylow Theorem and Unique Groups of Order n
Linear Algebra and Face Recognition
Introduction to Measure Theory
Goodstein’s Theorem
An Introduction to Generating Functions
T-Shirt Talk: The j –Invariant
2016 Counselor Minicourses
Combinatorial Game Theory
An Interesting Hat Problem
Ergodic Theorem for Markov Chains
Hex and the Brouwer Fixed Point Theorem
What are Elliptic Curves and Why Do We Care?
Gödel & the Halting Problem
Löb’s Theorem
How to Play Ultimate
Covering Spaces
Unique Prime Factorization of Knots
Cubic Curves and Bezout’s Theorem
Lucas’s Theorem and its Applications
Kirchhoff ’s Matrix-Tree Theorem
The Fundamental Group
ABC Conjecture and its Consequences
Brownian Motion and Liouville’s Theorem
A Dip into Galois Theory
PROMYS T-shirt Talk: Dedekind sums
2015 Counselor Minicourses
Quotient Maps
Probabilistic Method
Exact Sequences
Lebesgue Integrals
Cayley Graphs
Linear Algebraic Methods in Combinatorics
Mandelbrot Set
Infinite Graphs
Knot Theory
An Introduction to Topology
Topology and the Intermediate Value Theorem
Planar Graphs and the Three Cottages Puzzle
Introduction to Special Relativity
Group Actions
Game Theory
Schur Polynomials and Quadratic Reciprocity
Arrow’s Impossibility Theorem
Matrix Methods in Population Dynamics
Introduction to Representation Theory
Origami Constructions
T-shirt Talk: The Arithmetic of Pell Conics