More Unicity

I am on youtube talking about the Unicity theorem and my work with Clark Barwick.

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The Comparison Problem in Higher Category Theory

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.

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Summer Time

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.

Tensor functors between categories of quasi-coherent sheaves – arXiv:1202.5147

This paper by Martin Brandenburg and Alexandru Chirvasitu looks interesting. There is a connection to a categorified algebraic geometry.  Continue reading

Courses in Cambridge (MA), Spring 2012

There are a couple interesting courses happening this Spring (2012) in the Boston area.

Quote

“A great mathematician is like a Ninja”

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.

Abel Lectures 2011: Michael Hopkins, “Bernoulli numbers, homotopy groups, and Milnor”

Reconstructing 4-manifolds from Morse 2-functions – arXiv:1202.3487

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.
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TCFT Elliptic Objects – arXiv:1202.4118

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. 

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The Classification of Two-Dimensional Extended Topological 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.
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Examples of Cayley 4-manifolds

Houston Journal of Mathematics 30 (2004), no. 1, 55–87. Joint with Weiqing Gu.

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.