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 2020, for the further enrichment of our returning students and counselors, PROMYS and the Clay Mathematics Institute will offer advanced seminars in Topology, Graph Theory, and Mathematical Logic.

Past seminars have included: Cryptography, Galois Theory, 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 Dev Sinha, University of Oregon

The cliché is that a topologist cannot tell the difference between a doughnut and a coffee cup. We prefer to highlight a different aspect of breakfast and topology: when one stirs a cup of coffee there is always a point (molecule) of the liquid which may have moved but ends up where it starts—at all times, regardless of how one stirs! That last proviso “regardless of how one stirs” gets to the essence of topology, a subject which explores what one can say allowing for a large amount of deformation. It is also known as a study of “shape”, which is reflected in this example: this stirring fact wouldn’t be true if one were (for some unknown reason) drinking from a bundt cake pan—one can stir “around” and all the liquid is in a new place.

For proofs, we will use the “digital” model of topology, using polyhedra. The data to describe a shape in this way is combinatorial. But our intuition will draw from the “analog” model of topology, the study of manifolds, which historically first arose in studying the shape of solutions of polynomials. It is through the latter where we will visualize the argument for our stirring theorem and most others. We will focus more on low dimensions: graphs, surfaces and the like, where visualization can aid understanding. We will touch on deep analogies between surfaces and number systems such as the Gaussian integers. Our last topic will be persistent homology, a piece of math which has been applied to science including the study of nanoporous materials. We will become “chemical topologists” ourselves, and thus will break new ground not only by being an advanced seminar on topology but one which touches on current applied mathematics.

Professor Marjory Baruch, Syracuse University

Graph theory has many applications, from representing street maps, dictating what shape fair dice can be, to coloring maps and looking for trends on the internet. It is a critical tool for cyber security. We will be studying the underlying classical mathematical graph theory (does it take more colors to color a map on a bagel?). Important numerical relationships will emerge from the study of graphs. In the PROMYS style there will be opportunity for you to build your own understanding through examples and generating conjectures.

Dr. Henry Cohn, Microsoft Research and MIT and Dr. Cameron Freer, MIT

This will be a problem-solving seminar in mathematical logic, with strong connections to the PROMYS number theory class. The basic question is which sorts of problems can be solved algorithmically. For example, is there an algorithm that will determine whether a given Diophantine equation has a solution? Hilbert proposed this problem as the tenth on his famous list of problems from the beginning of the 20th century. Hilbert expected a positive answer, but it turns out that no such algorithm exists. The proof by Matiyasevich, Davis, Robinson, and Putnam (MDRP) of the unsolvability of Hilbert’s 10th problem is one of the great achievements of 20th century mathematics. Our seminar will focus on the theme of undecidability in mathematics, with the MDRP theorem and its consequences as the main results. The seminar will be based on problem sets that guide students through an exploration of these topics, by building on and extending some of the key results of the first-year number theory course and combining them with the theory of computability. The seminar will be run around discussions and Q&A sessions, with few formal lectures. Instead, all the content will be developed through the problem sets, with support from faculty and counselors.

**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