Math 497A - Honors MASS Algebra

Elliptic Curves and Applications to Cryptography

Instructor: Kirsten Eisentraeger, Assistant Professor of Mathematics

MWRF - 1:25-2:15pm

**Description**: The study of diophantine equations is an area of number theory that deals with finding solutions to polynomial equations. Looking for solutions of equations in integers or rationalnumbers has a long history that goes back to ancient Greece. In this class we will focus on elliptic curves, a special class of diophantine equations given by certain cubic equations in two variables. We will study these equations through a combination of techniques from number theory and algebraic geometry.

We will cover the group law for elliptic curves, both in terms of the geometry of the curve and in terms of explicit equations. Then we will discuss points of finite order and isogenies. We will also study elliptic curves over the rationals and prove the Mordell-Weil theorem, which says that the group of rational points on an elliptic curve is finitely generated.

After that, we will talk about elliptic curves defined over finite fields. Elliptic curves over finite fields have many applications to cryptography, and we will discuss their use in discrete-log based cryptosystems and in applications of the Weil and Tate pairings.

**Readings**: Silverman-Tate, *Rational points on elliptic curves* or Silverman, *The Arithmetic of elliptic curves*

Math 497B - Honors MASS Analysis

Elements of Fractal Geometry and Dynamics

Instructor: Yakov Pesin, Distinguished Professor of Mathematics

MWRF – 10:10-11:00am

**Description**: Fractals are strange but beautiful objects that appear in nature and arts as results of self-organization and self-similarity. In dynamics they are responsible

for the presence of highly-irregular, chaotic motions. The course is an introduction to a circle of topics in fractal geometry and chaotic dynamics.

**Readings**: K. Falconer, *Fractal Geometry, Mathematical Foundations and Applications*, John Wiley & Sons, 1990

M. Schroeder, *Fractals, Chaos, Power Laws*, W. H. Freeman & Company, 1991

Math 497C - Honors MASS Geometry

Introduction to Symplectic Geometry

Instructor: Kris Wysocki

MWRF – 11:15-12:05pm

**Description**: This course is intended to give an introduction to the field of symplectic topology. Symplectic topology has its roots in the study of classical mechanics and these days plays an important role in many areas of modern mathematics. The course will begin with linear symplectic geometry and Hamiltonian flows. After that we will define symplectic invariants and discuss some examples of symplectic invariants in details, then will proceed to study the existence of periodic orbits of Hamiltonian vector fields.

**Readings**: Chapters from the book: *Hamiltonian Dynamics and Symplectic Invariants* by H. Hofer and E. Zehnder

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.

Typical Readings: N/A

Math 497E - MASS Colloquim

Instructor: Multiple visiting speakers

R 2:30-3:30 pm

Covers selected topics in mathematics.

Typical Readings: N/A