## Topics

*Topics in Number Theory*

Instructor: Robert Vaughan, Professor of Mathematics

TA: Victoria Sadovskaya*Geometric Structures, Symmetry and Elements of Lie Groups*

Instructor: Anatole Katok, Raymond N Shibley Professor of Mathematics

TA: Alistair Windsor*Mathematical Methods in Mechanics*

Instructor: Mark Levi, Professor of Mathematics

TA: Roland Gunesch*MASS Seminar*

Instructor: Jeff Raven

## Course Outline

MATH 497A: TOPICS IN NUMBER THEORY (4:3:1)

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

INSTRUCTOR: ROBERT VAUGHAN

TEXT: AN INTRODUCTION TO THE THEORY OF NUMBERS, by I. Niven, H. Zuckerman and H. L. Montgomery

In 1782, Edward Waring wrote Every integer is a cube or the sum of two, three, … nine cubes; every integer is also a biquadrate or the sum of up to nineteen such; and so forth.

By this we presume that he meant that for each `k`>2 there is a number `s(k)` such that every positive integer can be written as the sum of at most `s(k)` `k`^{th} powers, and that if `g(k)` is the smallest possible value of `s(k)`, then we have `g`(3)=9 and `g`(4)=19. We now have essentially a complete solution to Waring's problem, but this has only been through the combined efforts of many mathematicians, including seminal work by Ramanujan, Hardy, Littlewood, Weyl, Vinogradov and Davenport.

Perhaps the most important unsolved problem in mathematics is the Riemann Hypothesis, which states that the complex zeros of the Riemann zeta-function, defined for `Re(s)`>1 by the sum of `n ^{-s}` over all positive integers

`n`, all have real part ½. One deduction from this is that the prime numbers are very evenly distributed, but there are many other consequences. It apparently gives us very profound information about the interplay between the additive structure and the multiplicative structure of the integers. The Riemann zeta-function is important because it is the prototype for numerous functions for which there are corresponding hypotheses and consequences.

The aim of this course is to introduce those topics from number theory algebra and analysis which are relevant to our current understanding of problems such as those above.

MATH 497B: GEOMETRIC STRUCTURES, SYMMETRY AND ELEMENTS OF LIE GROUPS (4:3:1)

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

INSTRUCTOR: ANATOLE KATOK

TEXT: Most likely no single book will be selected as the principal text. Participants will be given handouts and copies of appropriate sections from sources. The following books will be helpful both for the background and as guides for various parts of the course.

1. INTRODUCTION TO GEOMETRY, by H.S.M. Coxeter 1969, John Wiley. A great panorama of geometry written by an outstanding geometer of the twentieth century. Very accessible and contains numerous gems

.

2. MODERN GEOMETRIES, by M.Henle. The analytic approach. Coversmany areas to be treated in the course from a similar perspective but at a more elementary level; great as a background reading.

A unifying approach to most principal classical geometries (Euclidean, spherical, hyperbolic (the original non-Euclidean

), affine, projective, Minkowski geometry of relativistic space-time, as well as many less familiar ones) is the concept of the transformation group which preserves the properties of this geometry. These groups happen to be among the most elementary but also most important examples of Lie groups, which play a central role in modern mathematics, as well as in many fundamental physical theories.

The course will start with an overview of principal geometries mentioned above with developing selected topics in depth. Then we will consider transformation groups associated with these geometries and introduce some basic concepts of Lie group theory based on these examples. Finally we will discuss a more general concept of geometric structure which allows to consider geometries with fewer symmetries and consider some basic examples such as Riemannian metrics on surfaces.

MATH 497C: MATHEMATICAL METHODS IN MECHANICS (4:3:1)

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

INSTRUCTOR: MARK LEVI

TEXT: MATHEMATICAL METHODS OF CLASSICAL MECHANICS, by V. I. Arnold

This course is an introduction to both classical and modern ideas in dynamical systems from the point of view of classical mechanics. The main ingredients of the course will be analysis (ordinary differential equations), geometry and physical interpretations/applications.

The goal of this course is to give a solid foundation in dynamical systems and to paint a clear picture of a fruitful union of analysis, geometry and physics. To emphasize physical relevance of the theory developed, experimental demonstrations will be shown in class. Examples include stabilization of the upside-down pendulum by vibrating up-and-down its suspension point, the motion of rigid bodies, the light passing through lenses and mirrors, and other phenomena.

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

TIME: T 9:30 am - 11:15 am

INSTRUCTOR: JEFF RAVEN

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

## Calendar

- Arrival
- August 21-22
- Orientation and welcome party
- August 23
- First day of classes
- August 24
- Midterms
- October 6-8
- Midterm break
- October 9-12
- Thanksgiving break
- November 25-28
- Last day of regular classes
- December 3
- Study week
- December 4-9
- Final exams
- December 10-16
- Graduation ceremony
- December 17, 10am - 12 noon
- Dorms close
- December 19