Math 497A - Honors MASS Algebra

Integer Partitions

Instructor: George E. Andrews, Professor of Mathematics

MWRF - 10:10-11:00am

Course objectives and Honors Objectives: This course will be an introduction to the theory of paritions. This venerable field (its foundation trace back to Euler in the 17th century) has become of interest to computer scientist and physicists as well as to mathematicians interested in number theory and combinatorics. We shall provide an introduction to both the combinatorial and analytic aspects of the subject.

Mode of Instruction: Lectures and recitation periods with homework, written mid-term examination, oral final examination, and project.

Typical Readings: The central text for the course will by Integer Partitions (by Andrews and Eriksson) which has just been published by Cambridge University Press. In addition original papers in the literature will be assigned.

Work Requirements/Evaluation Criteria: Homework, written mid-term examination, oral final examination, and project.

Math 497B - Honors MASS Analysis

P-adic Analysis in Comparison with Real

Instructor: Svetlana Katok, Professor of Mathematics

MWRF - 11:15-12:05 pm

Suggested Prerequisites: Acceptance into MASS program (see above).

Course Objectives: Both real and p-adic numbers are obtained from the rationals by a procedure called completion, which can be applied to any metric space, by using different distances on the rationals: the usual Euclidean distance for the reals and a new p-adic distance for each prime p, for the p-adics. The p-adic distance satisfies the strong triangle inequality that causes surprising properties of p-adic numbers and leads to interesting deviations from the classical real analysis much like the renunciation of the fifth postulate of Euclid's Elements, the axiom of parallels, leads to non-Euclidean geometry. Similarities, on the other hand, arise when the fact does not depend on the strong triangle inequality, and in these cases the same proof works in the real and p-adic cases. Analysis of the differences and similarities will help the students to better understand the proofs in both contexts.

I included several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire category theorem, surjectiveness of isometries of compact metric spaces). They will enhance the students' understanding of real analysis and intertwine the real and p-adic contexts of the course.

Typical Readings: p-adic analysis in comparison with real by Svetlana Katok, in MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics}, 11-87, AMS, Providence, 2003 (based on a MASS 2000 course)

Mode of Instruction: Lectures, problem solving, projects

Work Requirements/Evaluation Criteria: Homework, written mid-term examination, oral final examination and project.

Math 497C - Honors MASS Geometry

Geometry and billiards

Instructor: Sergei Tabachnikov, MASS Director and Professor of Mathematics

MWRF - 1:25-2:15pm

The course is an introduction to geometry and dynamics of billiards. As a motivation, configuration and phase spaces of mechanical systems with elastic collisions, that can be described as billiards, will be discussed. Topics include: billiard in a rectangle and the dynamics of a circle rotation, optical properties of conics and Poncelet theorem, evolutes and involutes of plane curves and the four-vertex theorem, integral geometry and Hilbert's fourth problem, periodic billiard trajectories and variational principles, billiards in polygons, chaotic billiard dynamics and others.

Prerequisites: acceptance into MASS program

Typical Readings: Billiards by S. Tabachnikov and lecture notes that will be available prior to the lectures

Work Requirements/Evaluation Criteria: weekly homework, written mid-term examination, oral final examination, and research project.

Mode of Instruction: three weekly lectures and one weekly recitation session with a TA

Math 497D - MASS Interdisciplinary seminar

Instructor: Sergei Tabachnikov

T - 10:10-12:05pm

This seminar is designed to focus on selected interdisciplinary topics in algebra, geometry and analysis to coordinate core courses and to prepare students to MASS Colloquium. Seminar sessions may include presentations from student research projects.