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 2016, for the further enrichment of our returning students and counselors, PROMYS and the Clay Mathematics Institute will offer advanced seminars in Modular Forms, The Mathematics of Computer Graphics, and Geometry and Symmetry.

Past seminars have included: 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 Rohrlich, Boston University

Modular forms are certain functions on the complex upper half plane which enter number theory in a startling variety of ways: from representations of integers as sums of squares to combinatorial identities to the proof of Fermat's Last Theorem. The main goal of this course will be to get a feel for what modular forms are and for how they are used in number theory. Since the subject of modular forms is a rather advanced topic certain statements made in class will have to be taken on faith, with details of proof to be filled in later in your study of mathematics. The goal of the course is not necessarily to be completely self-contained but rather to see why there is any connection between modular forms and number theory at all. One connection which we will certainly make is between certain modular forms called Eisenstein series and the values of the Riemann zeta function at positive even integers.

Professor Marjory Baruch, Syracuse University

When Merida races around in the movie Brave, her thick, red, corkscrew hair bounces and flows with her. No, each hair is not drawn individually by hand on each new frame. There are helixes and fractals, slopes and bounds controlling that head of hair. Whether it is flapping birds racing through pipes, rotating worlds in video games, or pasting together panoramas on a smartphone camera, there is mathematics working behind the scenes. This will not be a course in video games or a programming course. We will be studying mathematics. We will look at motion in a 3-D world, jumping to 4-D for help, and investigate different ways to represent this all in 2-D.

Circles and ellipses are nice, but most of the curves we see have bumps and valleys, wiggles and waves. Is it reasonable to work with polynomials of degree 100 to represent a complicated curve or surface, or are there other techniques for describing complicated curves? When I use a drawing program and draw with splines, what is going on? This course will combine concepts from linear algebra, geometry, topology, and calculus to explore some mathematical questions arising from computer graphics. When we need a topic we will develop it from the beginning, so previous knowledge is not required.

Professor Steven 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 this group of symmetries 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. We will also relate area and volume to matrix groups and linear algebra. Finally, we will use properties of the symmetry groups of Euclidean space to study paradoxical decomposition of spheres.

**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