The Richard E. Phillips Distinguished Lecture Series was established in 1997 with generous endowment from the family of our late distinguished colleague Richard E. Phillips. The purpose of the Phillips Lectures is to advertise the utility and power of mathematics within the university, and to stimulate the interest of graduate students, postdocs and faculty. Interaction with graduate students and postdoctoral fellows is an integral part of the visit. Each series consists of three lectures delivered over a period of 4-5 days. The first lecture is targeted to a broad audience with diverse mathematical background and displays the utility of the subfield of mathematics. The second lecture is at the level of a mathematical colloquium, while the third is more focused and highlights technical aspects of the domain.

**March 20-22, 2017**

Peter Ozsvath

Princeton University

**Lecture 1: An introduction to Heegaard Floer homology**

Monday 3/20/17: Kellogg Center Auditorium 5:30-6:30 pm, reception following in Red Cedar AB.

"Knot theory" is the study of closed, embedded curves in three-dimensional space. Classically, knots can be studied via various computable polynomial invariants, such as the Alexander polynomial. In this first talk, I will recall the basics of knot theory and the Alexander polynomial, and then move on to a more modern knot invariant, "knot Floer homology", a knot invariant with more algebraic structure associated to a knot. I will describe applications of knot Floer homology to traditional questions in knot theory, and sketch its definition. This knot invariant was originally defined in 2003 in joint work with Zoltan Szabo, and independently by Jake Rasmussen. A combinatorial formulation was given in joint work with Ciprian Manolescu and Sucharit Sarkar in 2006.

**Lecture 2: Bordered techniques in Heegaard Floer homology.**

Tuesday 3/21/17: 115 International Center 5:20 – 6:20

Heegaard Floer homology is a closed three-manifold invariant, defined in joint work with Zoltan Szabo, using methods from symplectic geometry (specifically, the theory of pseudo-holomorphic disks). The inspiration for this invariant comes from gauge theory. In joint work with Robert Lipshitz and Dylan Thurston from 2008, the theory was extended to an invariant for three-manifolds with boundary, "bordered Floer homology". I will describe Heegaard Floer homology, motivate its construction, list some of its key properties and applications, and then sketch the algebraic input for the bordered version.

**Lecture 3: Bordered knot invariants**

Wednesday 3/22/17: C304 Wells Hall 4:00 – 5:00pm

I will describe a bordered construction of knot Floer homology, defined as a computable, combinatorial knot invariant. Generators correspond to Kauffman states, and the differentials have an algebraic interpretation in terms of a certain derived tensor product. I will also explain how methods from bordered Floer homology prove that this invariant indeed computes the holomorphically defined knot Floer homology. This is joint work with Zoltan Szabo.

**Bio:**

Peter Ozsvath received his PhD at Princeton University in 1994 under the supervision of John Morgan. He has held professorship positions at Columbia University, University of California at Berkeley, Massachusetts Institute of Technology, and is currently a professor of mathematics at Princeton University. He is a major figure in the modern development of topology. Together with Zoltan Szabo, Peter created Heegaard Floer homology, an extremely influential invariant of a 3-manifold equipped with a spin^c structure. He was an invited speaker at the International Congress of Mathematics, and a recipient of numerous awards, including an Alfred P. Sloan Fellowship, a Guggenheim Fellowship, and the American Mathematical Society’s extremely prestigious triennial Oswald Veblen Prize in Geometry.

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