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 2014, for the further enrichment of our returning students and counselors, PROMYS and the Clay Mathematics Institute offered advanced seminars in Values of the Zeta Function and p-Adic Analysis, Algebra, Geometry and Symmetry.

Past seminars have included: Galois 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 2014

Values of the Zeta Function and p-Adic Analysis 
Professor David Geraghty, Boston College 

This course is concerned with the values of the Riemann zeta function at positive even integers.  These values can be expressed as powers of pi times certain rational numbers called Bernoulli numbers, and much of the course will be devoted to some remarkable properties of these numbers.  We will also be interested in a closely related family of polynomials known as the Bernoulli polynomials. The ultimate goal of the course is to introduce the "p-adic numbers" (p denotes a prime here) and to show that the Bernoulli numbers can be interpolated to give a "p-adic zeta function."

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.

Geometry and Symmetry 
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 maps from the points of the geometry to itself 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 and the nonexistence of paradoxical decompositions of the circle.

 

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