# Mathematics at PROMYS

“

PROMYS taught me to appreciate mathematics in a completely new way. I was taught to investigate results on my own, make bold conjectures, and prove the most basic of theorems.”Richard Chen, Student 2019 and 2020

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 2022, we will offer the following seminars: **Graphs, Matroids, and Polynomial Countability**; **Number-theoretic Cryptography**; and **Linear Algebra, Bernstein Polynomials, and Visualization**.

**Graphs, Matroids, and Polynomial Countability (Professor Melody Chan, Brown University)**

This seminar is centered on topics in graph theory and algebra. Associated to every finite graph *G* there is a polynomial called the *graph polynomial* of *G*, which arises naturally in physics. Next, given a polynomial with integer coefficients, for example *g(x*,*y*,*z)* = *xy* + *yz* + *xz*, one can ask whether there exists a single polynomial *F(t)* such that the number of solutions to *g* over a finite field with q elements is exactly *F(q)*. In this seminar, we shall investigate a conjecture, first publicized in a 1997 lecture of Kontsevich, that all graph polynomials are polynomially countable. A central role in the eventual resolution of this conjecture by Belkale-Brosnan was played by matroids, which are an interesting abstraction of both graphs and the notion of linear independence. We shall define and study matroids, and, time permitting, indicate the role that matroids played in the solution to Kontsevich's conjecture.

**Number-theoretic Cryptography (Professor David Jao, University of Waterloo)**

Cryptography based on elliptic curve arithmetic is the primary means by which secure communication occurs on the internet today. Other areas of number theory and algebraic geometry, including binary quadratic forms, quaternion arithmetic, and elliptic curve isogenies, play a vital role in the development of the next generation of cryptographic standards, which are needed to protect users against future technologies such as quantum computers. We will cover the fundamental concepts of cryptography and how modern researchers use algebraic number theory and arithmetic geometry to design and build secure cryptosystems.

**Linear Algebra, Bernstein Polynomials, and Visualization (Professor Marjory Baruch, Syracuse University)**

How can we create a curve that goes through some particular points? How wiggly or smooth might that curve be? Suppose we have the shape of a curve in mind but we don't know any points on the curve, how might we describe such a curve mathematically and get an equation for it? Or, given a curve or a shape or a solid of some sort, how can we squish it, stretch it, rotate it, and move it along some path? There is so much we can visualize, but how can we describe it and work with it mathematically? And, there is so much we think we can't visualize at all – how can we see 4D or 5D space?

We will look at building some lines, polynomials, Bezier curves and surfaces. We will do some linear algebra – to help us modify and move our constructions. Quaternions, even more imaginary than imaginary numbers, can help us work with rotations without getting stuck. To know what should be "colored in" in the picture in our mind, we need to know which points are inside the curve and which are outside the curve. And, to understand how we are seeing an object and building a picture in our brain, we will look at various ways of projecting and slicing, onto or with a plane or at least a space of lower dimension.

**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)

**Professor David Speyer (University of Michigan)**

**Galois Theory**

**Professor Marjory Baruch (Syracuse University)**

**Graph Theory**

**2017 Advanced Seminars **

**The Analytic Class Number Formula**Professor Jared Weinstein (Boston University)

**Professor Marjory Baruch (Syracuse University)**

**Algebra**

**Professor Steve Rosenberg (Boston University)**

**Geometry and Symmetry**

**2016 Advanced Seminars**

**Modular Forms**Professor David Rohrlich (Boston University)

**Professor Marjory Baruch (Syracuse University)**

**The Mathematics of Computer Graphics**

**Professor Steve Rosenberg (Boston University)**

**Geometry and Symmetry**

**2015 Advanced Seminars**

**Complex Analysis in Number Theory (Dirichlet’s Theorem)**Dr. John Bergdall (Boston University)

**Marjory Baruch (Syracuse University)**

**Galois Theory**

**Professor Steve Rosenberg (Boston University)**

**Geometry and Symmetry**

**2014 Advanced Seminars**

**Values of the Zeta Function and p-Adic Analysis**Professor David Geraghty (Boston College)

**Professor Marjory Baruch (Syracuse University)**

**Algebra**

**Professor Steve Rosenberg (Boston University)**

**Geometry and Symmetry**

**2013 Advanced Seminars**

**Representations of Finite Groups**Professor Robert Pollack (Boston University)

**Professor Marjory Baruch (Syracuse University)**

**Wavelet Transformations**

**Professor Steve Rosenberg (Boston University)**

**Geometry and Symmetry**

**2012 Advanced Seminars**

**The Analytic Class Number Formula**Professor Jared Weinstein (Boston University)

**Professor Marjory Baruch (Syracuse University)**

**Algebra**

**Professor Steve Rosenberg (Boston University)**

**Geometry and Symmetry**

**2011 Advanced Seminars**

**Character Sums**Professor Jay Pottharst (Boston University)

**Professor Marjory Baruch (Syracuse University)**

**The Mathematics of Computer Graphics**

**Professor Steve Rosenberg (Boston University)**

**Geometry and Symmetry**

**2010 Advanced Seminars**

**Modular Forms**Professor Jon Hanke (University of Georgia)

**Professor Marjory Baruch (Syracuse University)**

**The Mathematics of Computer Graphics**

**Professor Steve Rosenberg (Boston University)**

**Geometry and Symmetry**

## 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.

**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.

**Mathematical Card Magic**

Mentored by Matt Baker (Georgia Institute of Technology)**Coexter Nim**

Mentored by Paul Gunnells (University of Massachusetts at Amherst)**Hypergraph Fuss-Catalan Numbers**

Mentored by Paul Gunnells (University of Massachusetts at Amherst)**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)**Heapable Sequences**

Mentored by Mike Mitzenmacher (Harvard University)**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)**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 CurvesMentored 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**

Mentored by Keith Conrad (University of Connecticut)

*F*[*x*]**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

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 *p*

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