**Math 497A - Honors MASS Algebra**

**Elliptic functions and elliptic curves**

*Instructor:* Yuriy Zarkhin, Professor of Mathematics

*Teaching Assistant:*

113 McAllister Building, MWRF 11:15 am - 12:05 pm

*Description:*
In this course we will study analytic, geometric and arithmetic properties of elliptic curves. From the analytic point of view elliptic curve is the special doubly-periodic (aka elliptic) meromorphic functions in one complex variable (with given lattice of periods) that may be viewed as counterparts of (periodic) trigonometric functions (like cotangent). From the algebraic point of view an elliptic curve is a smooth degree 3 curve on the projective plane. The arithmetic approach deals with the existence and construction of rational points on such curves and/or with solutions of congruences of degree 3. An interaction of different approaches makes the very subject of elliptic curves extremely deep and beautiful and opens the way for unexpected applications.
Elliptic functions and elliptic curves play an important role in complex analysis, mathematical physics, algebraic geometry, number theory, mathematical logic and mathematical cryptography. A delicate analysis of elliptic curves played a crucial role twenty years ago in the proof of Fermat Last Theorem by Andrew Wiles.

*Reading:*
L.C. Washington ``Elliptic curves: Number Theory and Cryptography", 2nd edition. Chapman & Hall, CRC Press, 2008.

**Math 497B - Honors MASS Analysis**

**Geometry of in nite dimensional spaces
(functional analysis and its applications)**

*Instructor:* Moisey Guysinsky, Professor of Mathematics

*Teaching Assistant:*

113 McAllister Building, 1:25 pm - 2:15 pm

*Description:*
In this course, we study how geometric methods could be used to understand properties of functions.
Spaces of functions could be thought as infinite dimensional linear spaces. Geometry of those spaces will be studied using both analogies and insights coming from the familiar finite-dimensional situations (such as convexity) and new features associated with infinite dimension, e.g. reflexivity or its absence. We also see the applications to problems from analysis and differential equations.

**Math 497C - Honors MASS Geometry**

**Knot Theory**

*Instructor:* Sergei Tabachnikov, Professor of Mathematics and MASS Director

*Teaching Assistant:*

113 McAllister Building, 10:10 am - 11:00 am

*Description:*
In this course, we shall study geometry, topology, and combinatorics of knots and links. We shall start with the classical period (late 19th - first half of 20th century), and discuss tabulating of knots, unknotting numbers, linking numbers, knots and surfaces, operations on knots. Then we shall study quantum knot invariants, including knot polynomials and their applications, and relations with models of statistical mechanics. We shall also study knot invariants of finite type and their relation with Lie algebras. We shall present some applications of knot theory in biology and chemistry, including the structure of DNA.

*Reading:*
C. Adams "The Knot Book", Amer. Math. Soc., 2004.