In 2013, for the further enrichment of our returning students and counselors, PROMYS and the Clay Mathematics Institute offered advanced seminars in Geometry and Symmetry, Wavelet Transformations, and Representations of Finite Groups.
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.
This summer we will look at discrete wavelet transformations. We will study some classical pure mathematics - topics in linear algebra, analysis, function spaces, and topology - and see how they can be applied in the very contemporary settings of image processing and data compression. The development of computers has forced us to look at some old mathematics on a much grander scale - for instance working with 3000 x 4000 matrices instead of just 3x3 matrices. At the same time, computers allow us to do massive calculations quickly, though requiring some cleverness to come up with new efficient methods to perform old world tasks. Our focus will be on the mathematics, but we will be guided by challenges in digital photography, and we will see this mathematics in action. Those interested will have the opportunity to do computer experimentation on some examples. Knowledge of programming is not required. There will also be plenty of opportunities for those who know programming to put these skills to work should they choose.
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.
Representation theory is a subject that lies at the crossroads of group theory and linear algebra. The basic objects of study are the actions of groups on vector spaces. On the one hand, this study yields beautiful theorems -- every representation decomposes uniquely into a sum of irreducible representations (a theorem very reminiscent of the unique factorization of numbers into primes!). On the other hand, representation theory is extremely concrete. For a finite group, a representation is nothing more than a finite list of matrices whose products mimic the multiplication law of the group. In addition to proving some of the key theorems of the subject, a main goal of this course will be to obtain as concrete as possible an understanding of these theorems through numerous examples.