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.
This year, for the further enrichment of our returning students and counselors, PROMYS and the Clay Mathematics Institute are offering advanced seminars in
Combinatorics; Topics in Linear Algebra; and Geometry and Symmetry.
Past seminars have also included: Combinatorics; Values of the Riemann zeta function; Hyperbolic Geometry; Random Walks on Groups; Dirichlet Series; Mathematics of Computer Graphics; Graphs and Knots; and the Mathematics of Algorithms.
Advanced Seminars for PROMYS 2009
Combinatorics (Henry Cohn, Microsoft Research New England):
Combinatorics is the study of discrete, as opposed to continuous, mathematical structures. Such structures occur everywhere in mathematics and have become increasingly important over the last fifty years, both for pure mathematics and for applications in computer science. In this course, we'll sample topics from a wide range of areas. We'll start with enumerative combinatorics, the art of counting how many objects satisfy given constraints, and we'll branch out to other subfields from there. Part of the beauty of combinatorics is the breadth of examples, and we'll examine some of my favorites, including voting paradoxes, a discrete analogue of calculus, error-correcting codes for communicating over a noisy communications channel, and how to convince someone something is true without revealing any information whatsoever about why.
Topics in Linear Algebra: (Professor Marjory Baruch, Syracuse University):
We will be looking at the beautiful mathematics that results from trying to solve the simplest equations. Linear algebra is critical to the study of algebra, geometry, calculus, - every field of mathematics. Linear algebra controls the way we display objects on a computer screen, allows us to accurately encode photos from the space, provides tools for processing cell phone signals, and determines which website should appear first in a google search. While we cover the classical mathematics of linear algebra, we will also look at some clever new strategies that arise from the need to do efficient calculations. If you are so inclined you will have the opportunity to do computer experimentation and visualization.
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.
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