# Lectures on Geometry

This new volume from CMI and Oxford University Press contains a collection of papers based on lectures delivered by distinguished mathematicians at CMI events. It contains a collection of papers based on lectures delivered by distinguished mathematicians at Clay Mathematics Institute events over the past few years. It is intended to be the first in an occasional series of volumes of CMI lectures. Although not explicitly linked, the topics in this inaugural volume have a common flavour and a common appeal to all who are interested in recent developments in geometry. They are intended to be accessible to all who work in this general area, regardless of their own particular research interests.

**Two Lectures on the Jones Polynomial and Khovanov Homology
Edward Witten**

Edward Witten works at the Institute for Advanced Study at Princeton. He is one of the leading figures in contemporary theoretical physics. His chapter is based on two lectures he gave at the Clay Research Conference in 2013. It surveys the groundbreaking work of Witten and his collaborators in fitting Khovanov homology into a quantum field theory framework. In the abstract of his contribution, Witten says: ‘I describe a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable *q*. The two main steps are to reinterpret three-dimensional Chern–Simons gauge theory in four-dimensional terms and then to apply electric–magnetic duality. The variable *q* is associated to instanton number in the dual description in four dimensions.’ This hardly does justice to the extraordinary range of ideas and techniques from mathematics and theoretical physics on which his lectures drew in his journey from an elementary starting point in the classical theory of knots. The second lecture was delivered in the Number Theory and Physics workshop at the conference. It takes the story further, describing how Khovanov homology can emerge upon adding a fifth dimension. Along the way, Witten describesmany significant new ideas, such as the Kapustin–Witten equations (important in geometric Langlands) and a new approach to evaluating some Feynman integrals via complexification. Witten’s approach is very natural, and especially attractive to a geometer, using Picard–Lefschetz theory in an essential way.

**Elementary Knot Theory
Marc Lackenby**

Marc Lackenby is a Professor of Mathematics at Oxford, with special interests in geometry and topology in three dimensions. His chapter is partly based on a lecture at the Clay Research Conference in 2012. It focuses on identifying some fundamental problems in knot theory that are easy to state but that remain unsolved. A survey of this very active field is given to place these problems into context. Because the tools that are used in knot theory are so diverse, the chapter highlights connections with many other fields of mathematics, including hyperbolic geometry, the theory of computational complexity, geometric group theory (a large area that connects with Bridson’s chapter) and Khovanov homology (the subject of Witten’s chapter).

**Cube Complexes, Subgroups of Mapping Class Groups and Nilpotent Genus
Martin R. Bridson**

Martin Bridson is the Whitehead Professor of Mathematics at Oxford, well known for his work in geometric group theory. His contribution is based on the lecture he gave as a Clay Senior Scholar at the Park City Mathematics Institute in 2012. This event is organized each summer by the Institute for Advanced Study at Princeton and is supported by the Clay Mathematics Institute through the appointment of Clay Senior Scholars. The PCMI Scholars provide mathematical leadership for the summer programmes and deliver lectures addressed to a wide mathematical audience. Bridson’s chapter focuses on two recent sets of results of his, one on mapping class groups of surfaces and the other on nilpotent genera of groups, both of which illuminate extreme behaviour among finitely presented groups. It provides an extremely useful and readable introduction to an important and lively area.

**Polyfolds and Fredholm Theory
Helmut Hofer**

Helmut Hofer is a member of the Institute for Advanced Study at Princeton. He has played a major part in the development of symplectic topology. The original version of this important and previously unpublished chapter was written following the Clay ResearchConference in 2008, at whichHofer spoke. Since then it has been extended and revised to bring it up to date. The chapter discusses generalized Fredholm theory in polyfolds, an area in whichHofer is a leading figure, with a focus on a particular topic—stable maps—that has a close connection to Gromov–Witten theory. This selection allows Hofer to set his chapter within a broad context.His excellent and full introductionmakes accessible the very detailed exposition that follows.

**Maps, Sheaves and K3 Surfaces
Rahul Pandharipande**

Rahul Pandharipande works at ETH Zürich. He is well known for his work with Okounkov, Nekrasov and Maulik on Gromov–Witten theory and Donaldson– Thomas invariants, for which he received a Clay Research Award from CMI in 2013. Pandharipande’s chapter also arises from a lecture delivered at the Clay Research Conference in 2008, in which he reviewed his work and that of his collaborators on recent progress in understanding curve counting (Gromov–Witten theory and its cousins) in higher dimensions. Gromov–Witten theory is notoriously hard and is only fully understood in dimensions 0 and 1. Pandharipande describes progress in dimensions 2 and 3. The chapter concisely describes a wide variety of important geometric ideas and useful techniques. It ends by bringing the story up to date with a brief account of the successful proofs of some of the principal conjectures covered in the original lecture.