### MATH 5020: Topics in Algebra

**Description:** Advanced topics chosen from group theory, ring theory, number theory, Lie theory, combinatorics, commutative algebra, algebraic geometry, homological algebra, and representation theory. With change of content, this course may be repeated to a maximum of twelve credits.

**Prerequisites:** MATH 5211.

**Credits:** 3

### MATH 5020 - Section 1: The Arithmetic of Elliptic Curves

**Description:** The arithmetic of elliptic curves. This course will be an introduction to the theory of elliptic curves. An elliptic curve is a smooth projective curve of genus 1 (with a rational point). Elliptic curves originally arose in complex analysis as the simplest nontrivial examples of Riemann surfaces. This course will focus on elliptic curves over the rational numbers and finite fields (and p-adic fields), which have a central place in modern number theory. Our textbook will be "The Arithmetic of Elliptic Curves", by J. Silverman, which is the standard reference on the subject. The goal of the course will be to understand and calculate the set of all rational points on a given elliptic curve (i.e. calculate its torsion and rank), and a number of refined invariants such as the order of the Tate-Shafarevich group. The prerequisites for this course are the Abstract Algebra sequence (5210 and 5211) and a basic understanding of algebraic number theory and algebraic geometry, although I will adjust the material to the audience background (including some review of background material) as much as I can.

**Prerequisites:** MATH 5211.

**Offered:** Spring

**Credits:** 3

### MATH 5020 - Section 2: Enumerative and Algebraic Combinatorics

**Description:** The course will give an introduction to enumerative and algebraic combinatorics at the graduate level. Topics will include: basic enumeration, generating functions, bijective proofs, sieve methods, theory of partially ordered sets, Moebius functions on posets, rational generating functions.

**Offered:** Spring

**Credits:** 3

**Sections: **Spring 2009 on Storrs Campus

Course | Sec | Comp | Time | Room | Instructor |
---|---|---|---|---|---|

5020 | 1 | Lecture | TuTh 11:00-12:15 | MSB311 | Lozano-Robledo, Alvaro |

5020 | 2 | Lecture | TuTh 9:30-10:45 | MSB118 | Roby, Thomas |