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 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.

Advanced Seminars for PROMYS 2021

Galois Theory
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.

Geometry and Symmetry

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.

Finding Rational Points on Hyperelliptic Curves
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