- 4-manifold Abel Lecture algebraic K-theory axiomization calibrated geometry Cerf theory Clark Barwick cobordism hypothesis comparison problem Costello emily riehl Frobenius algebra generators and relations higher category theory homotopy category homotopy theory Mike Hopkins Morse 2-function ninja open-closed quantum field theory Stolz-Teichner symmetric monoidal bicategories topological conformal field theory topological field theories unicity Yasha Savelyev
The history of higher category theory has lead to a wealth of definitions, each built on differing ideas and principles. Until recently there has been very little in the way of machinery to compare them. My ongoing work, joint with Clark Barwick, is changing this.
The semester here in Bonn just ended, which means that the topics course I have been teaching has come to an end. It ended up being about two things. The first half was about approximations of functions. We proved the smooth approximation theorems and Jet transversality theorems. With this we were abel to prove the Whitney embedding and immersion theorems as well as the Pontryagin-Thom construction.
In the second half of the course we covered the h-principle including the proof that the sheaf of solutions to any open partial differential relation satisfies the h-principle for any open manifold. This includes sheaves of immersions, submersions, Morse functions, and many others.
The course notes can be found here.
This paper by Martin Brandenburg and Alexandru Chirvasitu looks interesting. There is a connection to a categorified algebraic geometry. Continue reading
There are a couple interesting courses happening this Spring (2012) in the Boston area.
I almost always love Michael Hopkins’ lectures and his 2011 abel lecture is no exception. In the video at around the 13 and a half minute mark you will find this choice quote:
That may not seem like much, but you have to think of a great mathematician as like a ninja. As long as he can just touch his opponent with his toe and get any relationship at all, he can defeat the opponent.
The video is worth checking out too. I hope you enjoy.
A Morse function is a generic map to the real line. A Morse 2-function is generic map to the plane, or other 2-manifold. These functions featured prominently in my dissertation where I used them to give a classification of 2D topological field theories in terms of generators and relations. Now David Gay and Robion Kirby are using them to study 4-manifolds.
This paper by Yasha Savelyev caught my eye. It appears to blend together ideas from the Segal-Stolz-Teichner approach to elliptic cohomology and Costello’s notion of topological conformal field theories.
Ph.D. Dissertation (UC Berkeley) 2009
This work proves the cobordism hypothesis in dimension two. Specifically, this work classifies 2-dimensional extended topological field theories in terms of generators and relations. In this context, a topological field theory is a functor from a bordism category to some target category and an extended field theory is a higher categorical version of this. Why use higher categories? They allow you to mathematically encode the locality of the theory, something which is predicted from the physical point of view and which is very helpful mathematically.
The methods combined algebraic results on symmetric monoidal bicategories with a generalization of Cerf theory. This provides and alternate approach to the cobordism hypothesis to the one given by Hopkins and Lurie, and to date (Feb 2012) remains the only complete account.
In this work, which lead to my undergraduate thesis and written jointly with my undergraduate advisor, I studied Cayley 4-submanifolds of 8-dimensional Euclidean space. These are certain calibrated submanifolds, and play a role in string theory and its generalizations. The main technique was to use 3-dimensional subgroup of Spin(7) to simplify and solve the PDEs defining these Cayley 4-manifolds.