MATH 497A: FINITE GROUPS, SYMMETRY, AND ELEMENTS OF GROUP REPRESENTATIONS (4:3:1)

TIME: MWRF 11:15 am - 12:05 pm

INSTRUCTOR: ADRIAN OCNEANU

TEXT: Representation Theory: A First Course, W. Fulton and J. Harris, Springer-Verlag, Graduate Texts in Mathematics, No. 129, 3rd edition, 1991

Starting with very concrete examples of symmetry groups, the course will develop, in an interactive manner based on examples, the theory of group representations and its relations to different areas of mathematics and physics.

Representation theory studies the ways in which a group acts on vector spaces. The course will emphasize the way in which the set of such representations has itself a group-like structure. There is a deep connection between representation theory and the structure of 3-dimensional space. Structure coefficients coming from representation theory can be used to define topological invariants of knots and 3-dimensional manifolds. These invariants are related to current attempts to model quantum gravity in mathematical physics.

The course will start from the representation theory of the dihedral group, which is the symmetry group of a regular n-gon. It will continue with the study of a continuous group, U(1), for which the representation theory is connected to the Fourier transform.

The main nonabelian example will be SU(2), related to 3-dimensional rotations, and its subgroups, among which are the symmetry groups of the Platonic solids.

It is widely thought that, in the same way in which quantum mechanics used noncommutative mathematics, the long sought unification between gravity and quantum mechanics, called quantum field theory, will require new types of mathematical structures such as the ones described above.

An introduction to this subject was presented in a mini-course in REU 1999.


MATH 497B: PROJECTIVE AND NON-EUCLIDEAN GEOMETRY (4:3:1)

TIME: MWRF 10:10 am - 11:00 am

INSTRUCTOR: VICTOR NISTOR

TEXT: I am planning to use the book Introduction to Geometry by H.S.M. Coxeter, 2nd edition, 1989, John Wiley & Sons, and possibly the notes by Svetlana Katok Arithmetic and Geometry of the Unimodular Group (MASS 1997)

The main topics are: Isometries and similarities in the Euclidean space. Review of coordinates, complex numbers, and matrices. Affine, projective, and hyperbolic geometries. Axioms and geometric transformations in these geometries. The theorems of Desargues and Pappus. Straight lines and circles as geodesics in the euclidean and hyperbolic geometries.

To get a better hold of these concepts, we shall usually study in detail the case of the two dimensional geometries (affine, projective, or hyperbolic plane). The geometric transformations of these planes can be encoded in a two-by-two matrix. Usually, these matrices are elements of the unimodular group SL(2,R). The properties of this group and of its elements will play an important role in this course and will be studied in detail.


MATH 497C: REAL AND P-ADIC ANALYSIS (4:3:1)

TIME: MWRF 1:25 pm - 2:15 pm

INSTRUCTOR: SVETLANA KATOK

TEXT: Recommended books: p-adic Numbers by F.Q. Gouvea, 2nd edition, Springer-Verlag, 1997 and A Course in p-adic Analysis by A. Robert, Springer-Verlag, Graduate Texts in Mathematics, No. 198, 2000

The real numbers are obtained from rationals Q with the usual Euclidean distance by a procedure called completion which can be applied to any metric space. The Euclidean distance, however, is not the only way to measure closeness between rational numbers. Using other (p-adic) distances, we obtain completions of rationals, called p-adic numbers and denoted by Qp. The p-adic distance satisfies the strong triangle inequality |x + y| ≤ max(|x|, |y|) which causes surprising properties of p-adic numbers and leads to interesting deviations from the classical real analysis, for instance, a series ∑an in Qp converges if and only if limn→∞an = 0—a calculus student's dream come true!

In this course we shall develop a theory of p-adic numbers and study p-adic analysis vs. real analysis, in other words, examine what form the basic ideas of calculus take in the p-adic context. Particular topics will include: sequences andseries, elementary functions defined by means of power series, continuity and differentiability, antidirivation and integration.


MATH 497D: MASS SEMINAR (3:3:0)

TIME: T 9:05 am - 11:00 am

INSTRUCTOR: SERGEI TABACHNIKOV

This seminar is designed to focus on selected interdisciplinary topics in algebra, geometry and analysis. These areas will be related to the other MASS courses. Seminar sessions may include presentations from student homework solutions.