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 2021, for the further enrichment of our returning students and counselors, PROMYS and the Clay Mathematics Institute will offer advanced seminars in Galois Theory; Geometry and Symmetry; and Finding Rational Points on Hyperelliptic Curves.

Past seminars have included: Undecidability and Hilbert’s 10th Problem, Topology, Graph Theory, Cryptography, Values of the Zeta Function and p-Adic Analysis, Modular Forms, Primes and Zeta Functions, 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 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. We'll spend a significant amount of class time proving theorems and doing computations with each other; I hope to do very little lecturing.

Professor Steve Rosenberg, Boston University

Besides the standard high school geometry, there are geometries of finite sets of points and lines, non-Euclidean geometries, and geometries of shortest paths on bumpy surfaces (like the earth's surface). Each geometry has its group of symmetries – the functions from the points of the geometry to the points that preserve the geometric structure. Properties of these symmetry groups explain many deep features of the geometry. We will discuss the classical geometries of Euclidean, spherical, projective and hyperbolic type and develop the group theory techniques needed to understand their symmetry groups.

Professor Jennifer Balakrishnan, Boston University

A hyperelliptic curve (over the rational numbers) can be written as y^2 = f(x) where the degree of the polynomial f(x) is at least 5 and the coefficients are all rational. These curves have finitely many rational points, by a famous theorem of Faltings. But for a given hyperelliptic curve, how do we find its finite set of rational points? This is an area of active research. We will explore the arithmetic and geometry of these curves and investigate some open problems.

**Advanced Seminars in 2020**

Topology with Professor Dev Sinha

Graph Theory with Professor Marjory Baruch

Undecidability and Hilbert’s 10th Problem with Dr. Henry Cohn and Dr. Cameron Freer

**Advanced Seminars in 2019**

Probability, Combinatorics, and Computation with Professor Lionel Levine

Primes and Zeta Functions with Professor Li-Mei Lim

Algebra with Professor Marjory Baruch

**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