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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 2015, for the further enrichment of our returning students and counselors, PROMYS and the Clay Mathematics Institute will offer advanced seminars in Complex Analysis in Number Theory, Galois Theory, 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.

Dr. John Bergdall, Boston University

Euclid famously explained that there were infinitely many prime numbers. Two thousand years later, Lagrange had conjectured a stronger statement: for a fixed modulus m and chosen integer x coprime to m, there are infinitely many primes among the list x, x + m, x + 2m, x + 3m, … It was in 1837 that Dirichlet definitely proved this beautiful statement, and the result and its techniques are a landmark in modern number theory.

Gauss had famously used the *algebra* of complex numbers to prove statements about ordinary integers, like his criterion for when a prime is a sum of two squares. Working just after Gauss, Dirichlet’s proof proceeds by mixing the *calculus* of a complex variable with ideas coming from number theory. The use of complex calculus has continued ever since, with 21st century mathematicians still grappling with the 19th century investigations of Riemann, Hadamard and de la Vallee-Poussin on the distribution of prime numbers in the large (I'm referring to the infamous Riemann hypothesis).

This course will consist of three parts. First, we will scrutinize Dirichlet’s theorem in elementary cases. Second we will develop the analytic techniques necessary to give a complete proof of the theorem. This will serve as an introduction to the beautiful world of complex analysis. Finally we will prove Dirichlet’s theorem and discuss further techniques: questions about densities of natural numbers, the role of so-called zeta functions (generalizing Riemann's famous zeta function).

The prerequisites for the course will consist only of elementary number theory and ideas from calculus on the real line, such as integration and Taylor series.

Professor Marjory Baruch, Syracuse University

The tools that mathematicians use to attack a problem change with time, technology, and the nature of a solution we are seeking. We will use the tools of abstract algebra to address classical problems like trisecting an angle and solving the general quintic polynomial. Our focus will be on using groups to "transform" a variety of structures, algebraic and geometric, with an emphasis on fields, as was suggested by Galois in the 19th century. While we have new tools, computers, to "solve" these problems to as many digits as we like, we require the insights of algebra for many of our algorithms, like the Fast Fourier Transform. We will be studying groups and fields, their interconnections, and how they can be applied in a variety of settings. There will be numerical and theoretical problems, and for those interested, the opportunity to model some of our understanding on computers.

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 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