### 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:** This course will be an introduction to elliptic curves, which roughly speaking are smooth cubic curves in the projective plane (turns out they have a simple model of the form y^2=x^3+ax+b). The surprising feature of elliptic curves is that their points can be made into an abelian group, and this group is finitely generated when we focus on points with coordinates in the rational numbers lying on an elliptic curve with rational coefficients. Elliptic curves are central in modern number theory, e.g., they were essential in the proof of Fermat's Last Theorem. The goal of the course will be to understand and calculate the group of all rational points on an elliptic curve (i.e. calculate its torsion and rank), and a number of more refined invariants (such as the order of the Shafarevich-Tate group).

The prerequisites for this course are the Abstract Algebra sequence (Math 5210 and 5211) and a basic understanding of algebraic number theory and algebraic geometry, although I will adjust the material to the audience background as much as I can. Our textbook will be "The Arithmetic of Elliptic Curves," by Silverman, which is the standard graduate-level textbook for the subject.

**Prerequisites:** MATH 5210 and 5211.

**Offered:** Fall

**Credits:** 3

### MATH 5020 - Section 2: Commutative Algebra

**Description:** Commutative Algebra studies properties of commutative rings. It is essential in many areas of algebra, including algebraic geometry (polynomial functions form a commutative ring) and number theory (algebraic integers are a commutative ring).

Often in commutative algebra the ideas are motivated by geometry. For example, the behavior of functions defined in the neighborhood of a point reflect the geometry near the point and motivate the definition of a local ring. We will study regular local rings, which correspond to nonsingular points (around which a space looks nice and smooth, in a suitable sense). After this we will study classes of local rings that correspond to the more subtle properties of singular points.

Topics covered in the course will include ideals, Noetherian rings and modules, the Hilbert Basis Theorem, the Nullstellensatz, primary decomposition, the Krull dimension of a ring, Hilbert functions, regular sequences and depth of an ideal, regular local rings and Auslander-Buchsbaum-Serre theorem, Cohen-Macaulay rings, and Gorenstein rings.

Commutative Algebra studies properties of commutative rings. It is essential in many areas of algebra, including algebraic geometry (polynomial functions form a commutative ring) and number theory (algebraic integers are a commutative ring). Topics covered in the course will include ideals, Noetherian rings and modules, the Hilbert Basis Theorem, the Nullstellensatz, primary decomposition, the Krull dimension of a ring, Hilbert functions, regular sequences and depth of an ideal, regular local rings and Auslander-Buchsbaum-Serre theorem, Cohen-Macaulay rings, and Gorenstein rings.

The textbook for the course will be Atiyah and MacDonald's "Commutative Algebra."

**Prerequisites:** MATH 5210 and Math 5211.

**Credits:** 3

**Sections: **Fall 2015 on Storrs Campus

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

05190 | 5020 | 001 | Lecture | TuTh 02:00:00 PM-03:15:00 PM | MSB315 | Lozano-Robledo, Alvaro |

08311 | 5020 | 002 | Lecture | MWF 01:25:00 PM-02:15:00 PM | MSB415 | Weyman, Jerzy |