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 2018, for the further enrichment of our returning students and counselors, PROMYS and the Clay Mathematics Institute will offer advanced seminars in Cryptography, Galois Theory, and Graph Theory.

Past seminars have included: 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 Li-Mei Lim, Boston University

At its heart, public key cryptography relies on *one-way functions*, namely, things that are easy to do, but hard to undo without some secret knowledge. In this course, we will discuss what some of these hard problems are, and how we measure their difficulty. With that in mind, we will look at some computational aspects of number theory. Specific topics may include the discrete logarithm problem, primality testing, knapsack algorithms, and lattice-based cryptography.

Professor David Speyer, University of Michigan

Through a series of problems, I'll aim to take you through the proof of Abel's theorem that there is no universal formula for the solution of quintic equations. 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.

Professor 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.

**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