### MATH 5030: Topics in Geometry and Topology I

**Description:** Advanced topics in geometry and topology. With a change of content this course is repeatable to a maximum of twelve credits.

**Credits:** 3

### MATH 5030 - Section 1: Topics on Ricci Flow (Munteanu)

**Description:** The Ricci flow is a heat-type equation for Riemannian metrics that was introduced by Hamilton in the early '80's to study the structure of three dimensional manifolds. The first major success of this flow was to prove that three dimensional manifolds that have positive Ricci curvature must be diffeomorphic to quotients of the sphere. Shortly after, Hamilton devised a program to prove the famous Poincare conjecture in topology (a millennium prize problem); this program was finalized about 15 years ago by Perelman. Other major successes in geometry followed soon after through the work of Brendle, Schoen, Bohm, Willking, Marques and others. The course will present the basics of this theory. We will start with a brief survey of the differential geometry and PDE's needed here, then proceed to introducing the basic properties of the flow: short time existence and uniqueness, evolution equations of various geometric quantities. We will describe in some detail Hamilton's '82 result mentioned above, and the subsequent development such as compactness theorems and the Harnack estimate. The course will then go through some of Perelman's major accomplishments about the structure of singularities for the flow.

**Offered:** Fall

**Credits:** 3

**Sections: **Fall 2017 on Storrs Campus

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

09424 | 5030 | 001 | Lecture | MWF 12:20:00 PM-01:10:00 PM | MONT421 | Munteanu, Ovidiu |