Workshop

A brief tour of some key ideas in symplectic geometry and their development over the years, with special reference to programs at MSRI.

From 1987, just 5 years after MSRI's first programs, until now, MSRI has played important roles in the development of commutative algebra. The first major event was the 3-week "MicroProgram" of June 1987. The commutative algebra community, now aware of the possibilities, returned in force for the year-long program in 2002-03, another year-long program in 2012-13, and the Hot Topics workshop of March, 2018. The semester program scheduled for Spring 2024 will continue the sequence. An article in the Mathematics Intelligencer about the MicroProgram, "A mathematician commuting in Berkeley" by Craig Huneke, inspired my title, with the addition justified by subsequent events.

Each of these events featured reports on budding research directions that thus became the property of the whole community, growing into great trends, and inspiring much work and progress. At each of them young mathematicians were introduced to the established figures of the field. Each program strengthened the culture of the field, and passed it on: many of the organizers had been postdocs at the previous programs. 

I will try to give a flavor of this culture, and describe some morsels of mathematics from two of the many areas that were deeply influenced by these programs: the development of methods using Frobenius endomorphisms, and the development of representation theory with the study of syzygies.

In the past 40+ years there have been many revolutions in knot theory and 3-manifold topology. Three of the main successful approaches have been due to the work of Thurston (the Geometrization Conjecture), Vaughan Jones (the Jones polynomial), and Simon Donaldson (Donaldson invariants and Floer homology).  We will summarize some of these topics and the role that MSRI/SLMath has played in their development. Then we will indicate the prospects for discovering connections between these subjects, such as the Volume Conjecture and the L-Space Conjecture. 

For centuries, mathematicians have relied on computers to perform calculations, to suggest conjectures, and as components of mathematical proofs.  In the light of more modern tools such as interactive theorem provers, machine learning algorithms, and generative AI, we are beginning to see machines used in more creative and substantive ways in our work.  In this talk we survey some historical and recent developments, and speculate on the future roles of machine assistance in mathematics.