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.

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!

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.

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