In the Spring 2020 semester, the CMSA will be hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars. Talks will take place throughout the semester. All talks will take place virtually.
Written articles will accompany each lecture in this series and be available as part of the publication “The Literature and History of Mathematical Science“
The schedule below will be updated as talks are confirmed.
|Camillo De Lellis (IAS)|
|William Casselman (University of British Columbia)|
Title: The origins of Langlands’ conjectures
|Michael Freedman (Microsoft – Station Q)|
|Harry Shum |
|Vaughan Jones (Vanderbilt University)|
|Ralph Cohen (Stanford University)|
|Claire Voisin (Collège de France)|
|Vyacheslav Shokurov (Johns Hopkins University)|
|Shing-Tung Yau (Harvard) |
Title: Shiing-Shen Chern as a Great Geometer of 20th Century
Video | Slides | Article
| Donald Rubin (Harvard)|
Title: Why do some universities have separate departments of statistics? And are they all anachronisms, destined to follow the path of other dinosaurs?
Video | Slides
| Joe Harris (Harvard)|
Title: Rationality questions in algebraic geometry
Abstract: Over the course of the history of algebraic geometry, rationality questions — motivated by both geometric and arithmetic problems — have often driven the subject forward. The rationality or irrationality of cubic hypersurfaces in particular have led to the development of abelian integrals (dimension one), birational geometry (dimension two) and Hodge theory (dimension 3). But there remained much we didn’t understand about the condition of rationality, such as how it behaves in families. However, there has been recent progress: work of Hassett, Tschinkel, Pirutka and others, working with examples in dimension 4, showed that it is in general neither an open condition nor a closed one, but does behave well with respect to specialization.
In this talk I’ll try to give an overview of the history of rationality and the current state of our knowledge.
|Simon Donaldson (Stony Brook)|
Title: The ADHM construction of Yang-Mills instantons
Abstract: In 1978 (Physics Letters 65A) Atiyah, Hitchin, Drinfeld and Manin (ADHM) described a construction of the general solution of the Yang-Mills instanton equations over the 4-sphere using linear algebra. This was a major landmark in the modern interaction between geometry and physics, and the construction has been the scene for much research activity up to the present day. In this lecture we will review the background and the original ADHM proof, using Penrose’s twistor theory and results on algebraic vector bundles over projective 3-space. As time permits, we will also discuss some further developments, for example the work of Nahm on monopoles and connections to Mukai duality for bundles over complex tori.
Video | Slides
|Lydia Bieri (University of Michigan)|
Title: Black Hole Formation
Abstract: Can black holes form through the focusing of gravitational waves?
This was an outstanding question since the early days of general relativity. In his breakthrough result of 2008, Demetrios Chrstodoulou answered this question with “Yes!”
In order to investigate this result, we will delve deeper into the dynamical mathematical structures of the Einstein equations. Black holes are related to the presence of trapped surfaces in the spacetime manifold.
Christodoulou proved that in the regime of pure general relativity and for arbitrarily dispersed initial data, trapped surfaces form through the focusing of gravitational waves provided the incoming energy is large enough in a precisely defined way. The proof combines new ideas from geometric analysis and nonlinear partial differential equations as well as it introduces new methods to solve large data problems. These methods have many applications beyond general relativity. D. Christodoulou’s result was generalized in various directions by many authors. It launched mathematical activities going into multiple fields in mathematics and physics. In this talk, we will discuss the mathematical framework of the above question. Then we will outline the main ideas of Christodoulou’s result and its generalizations, show relations to other questions and give an overview of implications in other fields.
|Pavel Etingof (MIT)|
Title: Quantum Groups
Abstract: The theory of quantum groups developed in mid 1980s from attempts to construct and understand solutions of the quantum Yang-Baxter equation, an important equation arising in quantum field theory and statistical mechanics. Since then, it has grown into a vast subject with profound connections to many areas of mathematics, such as representation theory, the Langlands program, low-dimensional topology, category theory, enumerative geometry, quantum computation, algebraic combinatorics, conformal field theory, integrable systems, integrable probability, and others. I will review some of the main ideas and examples of quantum groups and try to briefly describe some of the applications.
Video | Slides
|Robert Griess (University of Michigan)|
Title: My life and times with the sporadic simple groups
Abstract: Five sporadic simple groups were proposed in 19th century and 21 additional ones arose during the period 1965-1975. There were many discussions about the nature of finite simple groups and how sporadic groups are placed in mathematics. While in mathematics grad school at University of Chicago, I became fascinated with the unfolding story of sporadic simple groups. It involved theory, detective work and experiments. During this lecture, I will describe some of the people, important ideas and evolution of thinking about sporadic simple groups. Most should be accessible to a general mathematical audience.
Video | Slides
|Ciprian Manolescu (Stanford)|
Title: Four-dimensional topology
Abstract: I will outline the history of four-dimensional topology. Some major events were the work of Donaldson and Freedman from 1982, and the introduction of the Seiberg-Witten equations in 1994. I will discuss these, and then move on to what has been done in the last 20 years, when the focus shifted to four-manifolds with boundary and cobordisms. Floer homology has led to numerous applications, and recently there have also been a few novel results (and proofs of old results) using Khovanov homology. The talk will be accessible to a general mathematical audience.
|Bong Lian (Brandeis)|
Title: From string theory and Moonshine to vertex algebras
Abstract: This is a brief survey of the early historical development of vertex algebras, beginning in the seventies from Physics and Representation Theory. We shall also discuss some of the ideas that led to various early formulations of the theory’s foundation, and their relationships, as well as some of the subsequent and recent developments. The lecture is aimed for a general audience.
Slides | Video