## Topics

*Number Theory: From Fermat's Little Theorem to his Last Theorem*

Instructor: Ken Ono, Assistant Professor of Mathematics

TA: Jeff Raven*The Exponential Universe*

Instructor: John Roe, Professor of Mathematics

TA: Heath Emerson*Functions and Dynamics in One Complex Variable*

Instructor: Greg Swiatek, Professor of Mathematics

TA: Boris Kalinin*MASS Seminar*

Instructors: Anatole Katok and Alexander Mezhirov

## Course Outline

MATH 497A: NUMBER THEORY: FROM FERMAT'S LITTLE THEOREM TO HIS LAST THEOREM (4:3:1)

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

INSTRUCTOR: KEN ONO

TEXTS: AN INTRODUCTION TO THE THEORY OF NUMBERS, by I. Niven, H. Zuckerman and H. Montgomery, published by Wiley, New York, 1991;

RATIONAL POINTS ON ELLIPTIC CURVES, by J. Silverman and J. Tate, published by Springer Verlag, New York, 1992

* Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hance marginis exiguitas non caperet.* —Pierre de Fermat

This is Femat's famous quote asserting that there are no non-zero integers `a`, `b`, and `c` for which `a ^{n}` +

`b`=

^{n}`c`when

^{n}`n`> 2. He claims

[…] to have found a truly wonderful proof, but the margin of his book is not large enough to contain it.Thousands of amateur and professional mathematicians have failed to find this proof, or any proof for that matter.

Then on June 23, 1993 Andrew Wiles, a number theorist at Princeton University, announced that he had indeed proved Fermat's Last Theorem

. The final manuscript has been accepted by the mathematical community and appeared in the *Annals of Mathematics*, (141), 1995.

In this course we will discuss the theory of elliptic curves and introduce enough mathematics so we can appreciate the overall picture of the strategy that ultimately lead to the proof.

**Topics covered in the course** (from MASS-98 supplement): congruences; parameterizing rational points on conics; Pell's equations; elementary Diophantine equations; theory of multiplicative functions; Fermat's last theorem is true for almost all exponents; group theory; geometry of points on cubic curves; Mordell's theorem that the group of points of an elliptic curve is finitely generated; torsion structures in Mordell-Weil groups; two isogeny and two descents; Gauss and Jacobi sums; examples of modular elliptic curves; Jacobi triple product and elementary q-series.

MATH 497B: THE EXPONENTIAL UNIVERSE (4:3:1)

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

INSTRUCTOR: JOHN ROE

TEXT: GEOMETRY OF SURFACES, by John Stillwell, published by Springer Verlag, New York, 1992

Over two thousand years ago, the Greek mathematician Euclid attempted to set geometry on a systematic foundation. Among his foundational statements was the Fifth Postulate

, which reads as follows: * That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles*.

Many mathematicians attempted to *prove* the Fifth Postulate from other, simpler assumptions, but their efforts led in an unexpected direction. In the nineteenth century, it was discovered that a perfectly coherent alternative geometry can be set up in which the Fifth Postulate is false. This alternative geometry, now called *hyperbolic* geometry, is the subject of this course.

The title, The Exponential Universe

, comes from one of the characteristic properties of hyperbolic space: it grows

at an exponential rate (whereas the ordinary Euclidean plane grows

at a quadratic rate). Hyperbolic space is so much roomier than Euclidean space that all kinds of strange mathematical objects can exist within its borders: in the course, we will encounter (among other things) formally successful Ponzi schemes, triangles with three angles all of which are zero, and a geometric theory of two-ring doughnuts.

**Topics covered in the course** (from MASS-98 supplement): the Euclidean plane and its isometry group; actions of groups on spaces; orbits, stabilizers, orbit space; actions by isometries; free actions, discontinous actions; complete locally Euclidean surfaces and their expression as quotients by group actions on the plane; classification: cylinder, twisted cylinder, torus, Klein bottle; groups as metric spaces: generating sets, word length metrics, coarse equivalence; Milnor-Wolf theorem; Riemannian metrics; the hyperbolic plane: metric, models, trigonometry, isometry group, the angle of parallelism; classification of isometries of the hyperbolic plane; areas, defect relation; discontinuous groups of hyperbolic isometries; cusps; the triply punctured sphere, surfaces of genus > 1, the Euler characteristic and curvature; free groups; amenability as a metric property: uniformly finite homology; invariant means on groups; examples of amenable and non-amenable groups; the axiom of choice.

MATH 497C: FUNCTIONS AND DYNAMICS IN ONE COMPLEX VARIABLE (4:3:1)

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

INSTRUCTOR: GREG SWIATEK

TEXT:

This course will provide an introduction to the function theory in one complex variable and will illustrate how dynamical ideas play a role in this theory. Hence, we will first discuss basic ideas of complex analysis. Then, dynamics of groups of automorphisms of the Riemann sphere will be covered. Finally, we will get to automorphic functions, the theory of which is connected with the dynamics of groups.

Specifically, I am planning to cover the following topics: Complex numbers and complex functions; Analytic functions (examples), entire functions; Line integrals and Cauchy's formula; Power series expansions; Properties of analytic functions: maximum principle, openness, Schwarz' Lemma; Analytic continuation; Linear-fractional maps of the Riemanns sphere; Dynamics and geometry of groups of linear-fractional maps; Fuchsian groups; The concept of an automorphic function.

**Topics covered in the course** (from MASS-98 supplement): complex numbers and complex functions; analytic functions, entire functions; line integrals and Cauchy's formula; power series expansions; properties of analytic functions: maximum principle, openness, Schwarz' Lemma; isolated singularities; the residue theorem; the argument principle and its consequences; linear-fractional maps of the Riemann sphere; Kleinian groups and their limit sets.

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

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

INSTRUCTOR: ANATOLE KATOK

TEXT: Instructor's Handouts

## Calendar

- Arrival day / Dorms open
- August 22
- Orientation and welcome party
- August 26, 11 am
- First day of classes
- August 27
- Midterms
- October 12-14
- Midterm break
- October 15-18
- Thanksgiving break
- November 26-29
- Final exams
- December 11-17
- Graduation ceremony
- December 18, 10am - 12 noon
- Dorms close
- December 20