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
Geometry and Symmetry; Algebra: Galois Theory and Representation of Finite Groups.
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 2008
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 which 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.
Algebra: Galois Theory: (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 an 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.
Representations of Finite Groups (Professor Robert Pollack, Boston University)
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.
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