This volume 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.

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Edward Witten is one of the leading figures in contemporary theoretical physics. He has made ground-breaking contributions to Physics and Mathematics. He has been given numerous awards, including the Fields Medal in 2004 and the Fundamental Physics Prize in 2012. He is a Foreign Member of the Royal Society and a Fellow of the American Mathematical Society. Currently he works at the Institute for Advanced Study in Princeton.

Martin Bridson is the Whitehead Professor of Mathematics at Oxford, well known for his work in geometric group theory. He was an invited speaker at the International Congress in 2006 and was made a Fellow of the American Mathematical Society in 2014.

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. He is a member of the National Academy of Sciences, a Fellow of the American Mathematical Society, and a recipient of the Ostrowski (1999) and Heinz Hopf Prizes (2013). He was an invited speaker at the International Congress of Mathematicians in Kyoto.

Mark Lackenby is a Professor of Mathematics at Oxford. He was an invited speaker at the International Congress of Mathematicians in 2010. He was awarded a Whitehead Prize by the LMS in 2003 and a Philip Leverhulme Prize in 2006. He has done important work in geometry and topology in three dimensions.

Rahul Pandharipande is a former student of Joe Harris at Harvard and a frequent collaborator of Andrei Okounkov (Fields Medal 2006). He is well known for his work with Okounkov, Nekrasov and Maulik on Gromov-Witten Theory and Donaldson-Thomas invariants. In 2013, he received a Clay research Award from CMI and the Infosys Prize from the Infosys Foundation.

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