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 on Character Sums, The Mathematics of Computer Graphics, and Geometry
and Symmetry.
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, and the Mathematics of Algorithms.
Advanced Seminars for PROMYS 2011
Character Sums (Jay Pottharst, Boston University):
We will work through several beautiful number-theoretic applications of a
family of special constructions called character sums. These include
reciprocity laws, counting solutions to equations modulo primes, and finding
an explicit form for the L-function of a CM elliptic curve. Along the way we
will learn the about finite fields and their Galois theory, groups and their
representations, cyclotomic numbers, and more.
The Mathematics of Computer Graphics (Professor Marjory Baruch, Syracuse University)
The challenge of representing an expansive, moving, 3-D, continuous world on a
finite, pixilated, 2-D screen raises a variety of mathematical problems. We
will look at Linear Spaces and how they can be applied to placing and moving
objects, and changing the viewpoint of the observer. We will explore various
ways to approximate curves given limited information. We will study conic
sections to better represent shadows. We will look at ways to generate normals
in order to work with reflected light. We will look at topological questions
of determining what points are inside a boundary as this relates to coloring
an object. We will look at the use of fractals for producing predictably
irregular results. And we will look at some aspects of projective geometry in
order to draw in perspective. These are just a sampling of topics. Students
with and without programming experience will find various opportunities to put
their new understanding to work, producing graphics.
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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