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
Modular Forms; The Mathematics of Computer Graphics; and Geometry and Symmetry.
Past seminars have also included: Combinatorics; Values of the Riemann zeta function; Hyperbolic Geometry; Random Walks on Groups; Dirichlet Series; Graphs and Knots; and the Mathematics of Algorithms.
Advanced Seminars for PROMYS 2010
Modular Forms (Professor Jonathan Hanke, University of Georgia):
Modular forms are certain functions on the complex upper half plane which
enter number theory in a startling variety of ways: from representations of
integers as sums of squares to combinatorial identities to the proof of
Fermat's Last Theorem. The main goal of this course will be to get a feel for
what modular forms are and for how they are used in number theory. Since the
subject of modular forms is a rather advanced topic, certain statements made
in class will have to be taken on faith, with details of proof to be filled in
later in your study of mathematics. The goal of the course is not necessarily
to be completely self-contained but rather to see why there is any connection
between modular forms and number theory at all. One connection which we will
certainly make is between certain modular forms called Eisenstein series and
the values of the Riemann zeta function at positive even integers.
The Mathematics of Computer Graphics : (Professor Marjory Baruch, Syracuse University):
Much of mathematics has an interpretation in the world of geometry.
Matrices in 4-dimensional space can be used to represent a variety of
transformations in 3-d space. Solving systems of equations helps us find
normals to surfaces, which is critical for understanding how light reflects
off surfaces. How much information do we need to represent a desired curve?
And what do we want to preserve when we splice curves together? When we want
to fill in a curve with color we have to figure out what is inside and what is
outside. What algorithms can we use to descibe the intersections of deformed
spheres? If we slice the equation of a "circle" in complex 2-space by real
planes, what do we get? Can we build an elliptic curve "adding machine"?
In this course we will be exploring the mathematics of computer graphics.
This involves linear algebra, geometry, topology, finite mathematics, and
more. This is a mathematics course, not a course in programming. While we
will work with and modify programs, no previous programming is required.
Students will spend some time in lab experimenting and connecting the
mathematics with the computer graphics. I anticipate the productions of some
wonderful pictures, illustrating some beautiful mathematics. Who knows? Maybe
we will even end up with a T-shirt design!
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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