Clay Mathematics Institute / PROMYS Advanced Seminars

The Clay Mathematics Institute (CMI) is dedicated to increasing and disseminating mathematical knowledge. Since 1999, the CMI/PROMYS partnership has run the research labs and the advanced seminars.

In 2019, for the further enrichment of our returning students and counselors, PROMYS and the Clay Mathematics Institute will offer advanced seminars in Probability, Combinatorics and Computation; Zeta Functions and Primes; and Algebra.

Past seminars have included: Cryptography, Galois Theory, Values of the Zeta Function and p-Adic Analysis, Modular Forms, Complex Analysis in Number Theory, Combinatorics, Values of the Riemann Zeta Function, Abstract Algebra, Modular Forms, Hyperbolic Geometry, Random Walks on Groups, Dirichlet Series, Graphs and Knots, The Mathematics of Algorithms, and Character Sums.

Advanced Seminars for PROMYS 2019

Probability, Combinatorics, and Computation
Professor Lionel Levine, Cornell University

If a chess knight starts in the corner square of an 8x8 chessboard and makes random legal moves, how many moves on average until it returns to its starting square? Is there a tour of the n-dimensional cube {0,1}^n traversing each of the n*2^{n-1} edges exactly once in each direction? What kinds of functions can you get by composing addition, multiplication and the floor function? And what do these questions have to do with each other? Come to Combinatorics, Probability, and Computation to find out!

Primes and Zeta Functions
Professor 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.

Algebra
Professor 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, possibly including issues of solving a fifth degree equation, tessellations of the plane and higher dimensions, and writing down roots of third degree equations. Number theory and counting will be used throughout, as will geometry. For those interested, there will be opportunities to model some of what we are doing on the computer.

Advanced Seminars in 2018

Cryptography with Professor Li-Mei Lim
Galois Theory with Professor David Speyer
Graph Theory with Professor Marjory Baruch

Advanced Seminars in 2017

The Analytic Class Number Formula with Professor Jared Weinstein
Algebra with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg 

Advanced Seminars in 2016

Modular Forms with Professor David Rohrlich
The Mathematics of Computer Graphics with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg 

Advanced Seminars in 2015

Complex Analysis in Number Theory (Dirichlet’s theorem on arithmetic progressions) with Dr. John Bergdall
Galois Theory with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg 

Advanced Seminars in 2014

Values of the Zeta Function and p-Adic Analysis with Professor David Geraghty
Algebra with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2013

Representations of Finite Groups with Professor Robert Pollack
Wavelet Transformations with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2012

The Analytic Class Number Formula with Professor Jared Weinstein
Algebra with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2011

Character Sums with Professor Jay Pottharst
The Mathematics of Computer Graphics with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2010

Modular Forms with Professor Jon Hanke
The Mathematics of Computer Graphics with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2009

Combinatorics with Dr. Henry Cohn
Topics in Linear Algebra with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2008

Representations of Finite Groups with Professor Robert Pollack
Algebra: Galois Theory with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2007

Modular Forms with Professor David Rohrlich
Abstract Algebra with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2006

Combinatorics with Professor Ira Gessel
Algebra with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2005

Values of Riemann zeta function with Professor David Rohrlich
Algebra with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2004

Graphs & Knots with Professor David Rohrlich
Computer Graphics with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2003 

Combinatorics with Professor Ira Gessel
Algebra with Professor Marjory Baruch
Geometry & Symmetry with Professor Steve Rosenberg

Advanced Seminars in 2002

Modular Forms with Professor David Rohrlich
Algebra with Professor Marjory Baruch
Hyperbolic Geometry with Professor David Fried