Local systems: the infinitesimal perspective April 30, 2009
Posted by David Speyer in Algebraic Geometry, D-modules.12 comments
This is the next in my series of posts on different ways to think about local systems. This time, we will consider an approach where we only build isomorphisms along infinitesimally short paths. If you pursued this line of thought long enough — and thought very hard about finite characteristic issues — you would come to the definition of a crystal. But I don’t plan to go nearly that far; I’ll just give you the intuitions in characteristic zero.
I think this is probably the hardest of the three perspectives I want to explore, but it logically comes second. Things will become a bit easier again when we get to our third perspective, connections.
To try to make this a bit easier, I’ll start with a nonstandard presentation; then reboot and give the actual definitions.
More slides April 27, 2009
Posted by Ben Webster in link homology, Soergel bimodules, talks.1 comment so far
My tendency to write slideshows instead of actual posts continues. If you like to see oodles of subtle variations on the same talk, you can see my slides from speaking at ARTIN in Glasgow (which just happened to be coincidentally scheduled during the breaks of the categorification conference there), which is the 8th time I’ve given that talk this year (I’m giving a talk today which will be my 13th total talk of 2009. You can see why I’ve been spending more time with Beamer than on the blog).
However, if you’re looking for something newer, this time you have a chance to see the slideshow before the people coming to the talk. I’m speaking on my work with Geordie in about 45 minutes, and made a Beamer show to accompany part of the talk.
Notably, this is the first Beamer I’ve made with Tikz. I’m particularly proud of the picture on slide 17, which I’ve posted under the cut:
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Matthew Emerton is smart and helpful April 22, 2009
Posted by David Speyer in Algebraic Geometry, characteristic p.4 comments
In the comments to my previous post, John Mangual raised a number of questions about the relationship between etale and topological cohomology. Matthew Emerton has posted very thorough answers. If you are interested in the topic, and haven’t read his comments yet, you should.
Local systems: the path groupoid approach April 21, 2009
Posted by David Speyer in Algebraic Geometry, D-modules.8 comments
This is the first of the series of posts I promised, on different ways of getting local systems.
In this section, we’ll explain the approach which leads to étale sheaves. I’ll start out by describing the analogous ideas in the topological setting; and then sketch how to make them fully algebraic.
I’ve realized that I need a word for the data which I use to obtain a local system. Because I’m feeling uncreative, I’ll call it the input. Again, 
Tikz April 21, 2009
Posted by Ben Webster in math life, tikz.15 comments
For those of you following the long OT comments discussion on graphics programs, let me second Chris’s endorsement of Tikz. It is by far the best LaTeX graphics package I have ever used. I couldn’t possibly compete with the manual (what other manual starts with 3 fantastic tutorials?), and this page of examples. I have a certain affection for xypic, which has served me well over the years, but Tikz is just lightyears ahead. That is all.
Three ways of looking at a local system: Introduction and connection to cohomology theories April 20, 2009
Posted by David Speyer in Algebraic Geometry, Algebraic Topology.42 comments
Suppose we have a space
Roughly, a local system consists of a vector bundle 
In this series of posts, I want to present several of these ideas in the case of a smooth manifold
I said above that a local system includes the data of isomorphisms between different fibers 
We could work with an arbitrary
We could work with an infinitesimal 
We could work with a 
These give rise to étale cohomology, crystalline cohomology and deRham cohomology. I should point out that, for modern applications, one usually wants to work on stacks, with singularities and in arbitrary characteristic. I won’t be addressing any of those issues; I just want to give the intuition behind each theory.
In this post, I will explain how to build a cohomology theory, given a notion of local system. I will then follow up with three more posts, one for each of the specific approaches above.
As usual, this series comes with a disclaimer: These are tools that are relevant to my work, but their inner workings are not my expertise. If you want to see experts discussing this sort of thing, it looks like that conversation is going on at Urs’ journal club.
Mathematicians beat pornographers! April 16, 2009
Posted by David Speyer in math life.23 comments
For years, whenever I ran a web search for something involving LaTeX, I would throw the word “typesetting” into the search terms in order to screen out the p-o-r-n. I just checked, and this is no longer necessary: Even without safesearch, the first three pages of google hits on “latex” contain only one allusion to the material’s use in fetish wear — and only three references to the rubber material at all.
It is great that google now thinks I am more likely to care about quality typesetting than about rubber clad women. But I wonder whether this is smart behavior on google’s part. It seems to me that a really smart search engine would realize that people searching for “latex” fall into three or four distinct camps — mathematicians, materials scientists, fetishists, and perhaps some group I’m not thinking of — and offer me a few hits focused on each group. And that, in turn, made me wonder how I would design an algorithm to do such clustering. Any ideas?
“I just wanted to make sure you hadn’t gone to the wrong airport” April 14, 2009
Posted by Ben Webster in travel.1 comment so far
A note to those of you who I told I would be Glasgow nowish,and who are wondering what happened: The answer is “mechanical problems.” Thus I’m in air travel limbo in Dublin for the day, instead of categorifying things. Oh well, at least now I get to find out whether Guinness does taste different here.
Go blue! April 13, 2009
Posted by David Speyer in jobs.8 comments
I thought I’d let you all know my plans for the future — I am staying at MIT one more year and then moving to the University of Michigan in Fall 2010, where I will be an associate professor.
I visited Austin, UMass Amherst, Duke, Minnesota and Stony Brook, and I would have been tremendously happy to work at any of these schools. They all made me feel very at home, and excited about the collaborations I could have there. I wish I could split myself in six!
If you are a combinatorialist who is going on the job market next year, particularly one whose interests blend into various flavors of geometry and representation theory, you should be thinking about Austin. Obviously, I have no capacity to speak for Austin, but everyone I talked to was very excited about bringing in someone in that direction. And, with Sean Keel and David Ben-Zvi around, you will never lack for interesting problems to discuss!
Most of my co-bloggers also did fairly well. I know AJ will be at Stony Brook next year, and Noah at Columbia. Any other news?
How to get an algebra from a knot invariant April 13, 2009
Posted by Ben Webster in low-dimensional topology.4 comments
So, a couple of months ago, I gave a talk at the Max Planck Institute on knot homology, and as motivation I tried to explain why anyone studying the HOMFLY polynomial is inexorably led to the Hecke algebra. Nathan Geer, who was in the audience, asked me afterwards if there was anywhere this construction was written down, and lacking a good answer or the ambition to write a paper about it myself, I thought I would try to explain it in a blog post. It’s just applying an old TQFTologist’s trick, but old tricks often still have some new life in them.
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