Rolla Spring Topology Conference: Day Two.

Workout notes I started with 3100 yards in the pool, including 5 x 200 on the 3:30 (3:16, 14, 14, 15, 13). The water was somewhat warmer than I was used to. Then I did some yoga; it all felt good.

Afterward, some women asked me about our Prius (hybrid car); hmmm…the muscle car to pick up chicks with is a hybrid? 🙂

When I got up, the outter side of my left leg was stiff and tingly up and down the side; kind of along the IT band. My guess is that trauma from the bike crash is on the mend, but I let things get too stiff yesterday. Daily yoga is essential for me at this time.

Update Over lunch I went on a 3 mile plus walk (45 minutes) which included the sidewalk around the golf course and the 400 meter track on campus. I managed a pathetic 2:45 400 meter segment (11:00/mile) and that was picking up the pace! Afterward I had a few (expected) aches and tingles on the outer side of the left leg from the middle of the hip to the knee; the butt/piriformis area feels ok. It was a pretty day for it.

Mat took in a 5 mile run at that time.

Math conference notes

Talks ran late into the evening (typically, talks are over by 5 pm; this time they ran until 7). But the latter talks were worth it; Deneise Halverson (from BYU) gave a nice talk on the problem on the following kind of problem: given a finite set of n points, find an n+1’st point so that the sums of the distances from the n points to the n+1’s point is minimal. This had been done for the plane; she worked on this problem for surfaces of constant Gaussian curvature (reason: in terms of measuring distance ON THE SURFACE, one should be on a surface where the metric doesn’t vary much from location to location; example: if one tried to do this problem on the typical embedded torus in 3 space, the problem would be very difficult if, say, the points were near the saddle point or away from it. She solved this issue by using the flat torus (which embedds in 4-space).

Then Ales Vavpetic gave a talk on grope groups; that is, groups which are the fundamental groups of gropes. A “grope” is, very roughly speaking, a type of infinite construction using 2-surfaces. One might start with, say, a 3 holed torus. Then to the loops generating the first homology, attach another 3 punctured torus and do the same to the next round of 3 tori and so on.

Gropes were used in the work that Freedman did to win his Field’s medal (he used them to get rid of interesection homology of 4 manifolds to prove the topological 4-d Shoenfies conjecture, which says that every PL 3 sphere in 4 space bounds a topological 4-ball. It is unknown if those spheres bound a PL 4 ball.

Update I went to two talks in the morning. One is on continua (compact, connected sets). Some strange examples were discussed. On of the fun ones was this: consider a basic “middle thirds” Cantor set. draw horizontal line segments on the deleted “thirds” points, connect the segments by a segment in a way that alternates “up”, “down”.

Now shrink the joining segments to points; one now gets an infinite collection of arrowheads which alternate direction.

Now take these arrowheads and replace by pseudo arcs, and connect the ends of the original [0,1] segment to make a cirlce; one gets a “circle of pseudoarcs”.

Next, I went to a talk which was about metrization theory of “manifolds” (here, “manifold” means a Hausdorff space which is locally homeomorpic to R^n for some “n”; e. g. the long line would be one such).

There was an example of the product of a long line with the circle, and then of a wierd fiber buncle of the “long tube” over the long ray by some twisting map.

The idea was to show that being T6 (every discrete collection of closed sets can be expanded to a discrete collection of disjoint open sets) and a manifold doesn’t imply that the space is metric.

In this talk, different set theory axioms (continuum hypothesis, diamond) were used; assuming one axiom instead of another lead to different results.

On another note: I should be able to link to a video of some of the talks; if I can download it I’ll put it on my youtube account.

Update After lunch was brutal, given that I ate a small cheese pizza. Still, the talk about the application of dynamical systems to macroeconomics was interesting. They are interested in a utility function (how “good” some state is) that one gets from an inverse limit space (let xt = f(x(t+1) where f is not an invertible function; that is, obtain one point from a point in a future state).

It turns out that calculating the Riemann integral over such a space that oscillates so much is very hard, but using a measure and using Lesbegue integration makes the problem simplier!

Who would have thought? 🙂


March 30, 2007 Posted by | injury, mathematics, swimming, yoga | 1 Comment