The First National Forum of Young Topologists October 5, 2009
Posted by Ben Webster in conferences, topology.8 comments
So, I’ve added to the sidebar a new conference: The National Forum of Young Topologists in new Orleans next month. This actually sounds like a somewhat intriguing conference, concentrating on a mix of math and what I would call “professional development,” for lack of better word. Rather than a laser-like focus on the most recent stuff, it sounds like the math talks are supposed to be more historical in nature, and apparently the lectures by junior people are supposed to also “have the effect of showing examples of success stories starting out as graduate students through obtaining a tenure track position.”
While this all sounds very nice (I’ll admit that as a graduate student I never felt “isolated with [my] entire mathematical universe being [my] Ph.D. adviser,” and thus never felt the need for a conference to correct this, I know some people who’ve had that experience), I just can’t help but mock the name. I mean, I cannot imagine what on Earth the organizers were thinking when they came up with that one (and this comes from a guy who freely admits that the name of this blog was arrived at under the influence of spirituous beverages). I mean, did they put that in the grant application? It sounds like the world’s oddest Stalinist youth organization. The website could really use a couple of posters of square-jawed youths with sickles (or maybe just this guy) with a diagram of the Kirby calculus on the flag. I’m glad they want to have more such conferences (or at least, so they claim on the website), but I hope they don’t feel obligated to stick with the name.
Geometry and triply graded knot homology April 9, 2009
Posted by Ben Webster in Algebraic Geometry, D-modules, topology.24 comments
So, I thought I would actually get back to blogging my research (by which I mean, shameless self-promotion) a bit. Probably the problem that I’ve focused the most on in the past few years is how to understand knot homology geometrically. While this still has a lot of mysteries, Geordie Williamson and I are finishing up a pair of papers that I think are a big step forward in this area.
The one I’d like to talk about in this post is “A geometric construction of colored HOMFLYPT homology.” (It’s not on the arXiv yet, but we’re close. Comments would be helpful, hint, hint). I’m mostly just going to talk about the consequences of this paper for the triply graded homology of Khovanov-Rozansky, though I think one of its most exciting features is how easily it generalizes to the colored situation.
In my last post, I described a complex of sheaves 
And that’s exactly what we do; (more…)
Math + Legos December 16, 2008
Posted by Chris Schommer-Pries in Legos, elementary topology, topology.2 comments
This link was posted on the topology list a while ago and I thought it was too cute not to pass along.
http://www.andrewlipson.com/mathlego.htm
My favorite is Boy’s surface!
Embedded TQFT? January 7, 2008
Posted by Ben Webster in Algebraic Topology, QFT, combinatorics, low-dimensional topology, topology, tqft.10 comments
So, a subject rather near and dear to the hearts of many of my fellow co-bloggers is that of 1+1-dimensional TQFT: that is, of monoidal functors from the category of 1-manifolds with morphisms given by smooth cobordisms to the category of vector spaces over your favorite field 
There’s a rather remarkable theorem about such functors, which really deserves a post of its own for proper explanation, but I’ll spoil the surprise here.
Any such functor associates a vector space 
Furthermore, the cap, thought of as a cobordism from the empty set to a circle gives a map 
Theorem. A commutative algebra with counit 
Kronheimer on “Knot Groups and Lie Groups” November 11, 2007
Posted by Ben Webster in link homology, low-dimensional topology, representation theory, talks, topology.12 comments
So, I’m in lovely Edinburgh, Scotland (everyone I’ve told about this said “Scotland? In November?” but it’s not actually worse than New Jersey) in advance of the Maxwell Colloquium on Knot Homology.
By sheer luck, my trip here happened to overlap with the University of Edinburgh’s Whittaker Lecture which is a bit like the Bowen Lectures at Berkeley, except that there’s only one of them. By even more luck, the speaker with Prof. Peter Kronheimer (from Harvard) and the topic was “Knot Groups and Lie Groups.”
Quantum Topology and Classifying Manifolds October 11, 2007
Posted by Noah Snyder in low-dimensional topology, quantum algebra, topology, tqft.25 comments
Grepping the Knot Atlas August 20, 2007
Posted by Scott Morrison in knot atlas, topology, wiki.3 comments
I want to make a brief plug for the Knot Atlas, and in particular a new way you can get hold of all the knot theory data hidden away inside it.
First of all, what is the Knot Atlas? Officially, it’s aiming to be “a complete user-editable knot atlas, in the wiki spirit of Wikipedia“. In more mundane technicalities, it’s a machine sitting underneath Dror Bar-Natan’s desk, running Mediawiki (and other) software maintained by Dror and myself, hosting a wiki with lots of information about knots. You might like to take a self guided tour, or just read the About page.
Next I’ll explain how the Knot Atlas stores and displays its data, and then introduce a new and simple mechanism we’ve devised for using this data — the classic command line tool “grep“, secretly operating on an RDF data dump.
Homotopy theory and higher categories in Barcelona July 26, 2007
Posted by Ben Webster in Algebraic Topology, Category Theory, conferences, topology.1 comment so far
Well, this looks like it should be an exciting year-long program. Not to mention that Barcelona is a fair approximation of heaven on earth (if heaven included a few too many obnoxious tourists).
Too bad I’m already going to an even more exciting program (for me, no offense to you higher categories types), admittedly in a somewhat less appealing locale.
I’m also pretty interested in MSRI’s programs this year. This whole “only being in one place at a time” thing is getting to be a real drag. Shouldn’t the internet have fixed that by now?
A Bicategory of Groupoids July 23, 2007
Posted by Chris Schommer-Pries in Category Theory, groupoids, topology.16 comments
I want to talk about an interesting 2-category of topological groupoids that I’ve been thinking about recently. Let’s start with the basics; what is a groupoid? Well a group can be thought of as a category with one object and with every morphism invertible. A groupoid is the same thing, except that there can be multiple objects. Why are these interesting? well groupoids generalize three different notions at once: sets, equivalence relations, and groups. We’ve seen how groups enter. Equivalence relations on sets can also be viewed as categories where there are exactly one or zero morphisms between two objects: one if the objects are equivalent, zero otherwise.
Obvious Fact: Since groupoids are categories we can talk about functors between groupoids and natural transformations between these. Hence groupoids form a 2-category.
What happens if we add topology? What if we want a space of objects and a space of morphisms? (source, target and identiy maps are countinuous, of course). Well things start to break down.
Spheres in Higher Dimensions July 21, 2007
Posted by Chris Schommer-Pries in Euclidean geometry, fun problems, topology.10 comments
I just reread this problem on the Harvey Mudd Math Fun Facts website. It’s about your basic intuition in higher dimensions. I remember when I first heard it I was so shocked.
Suppose you have a square in the plane. Cut it into four quadrants, then inscribe a circle in each. Finally add a fifth circle which is just tangent to the other circles.
As you can see, it is much smaller then the other circles and the square.
You can repeat this in higher dimensions. For example, you can cut the cube into octants and inscribe eight spheres in these, then add a last final sphere, tangent to the eight inscribed spheres.
So how does the volume of this final sphere change with respect to the volume of the cube when we change dimensions? How does its size compare? It’s always smaller right? WRONG!
As you increase dimension, the final sphere gets bigger and bigger. In dimension 9 it is tangent to the original n-cube and in dimensions bigger than 9 it pushes outside the original cube. Eventually as the dimension increases, the volume of the final sphere exceeds that of the n-cube!
The details are pretty easy and can be found here:
Su, Francis E., et al. “High-Dimensional Spheres in Cubes.” Mudd Math Fun Facts. http://www.math.hmc.edu/funfacts.
There are all sorts of other cool math fun facts on this webpage. You should check it out.
