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Starting Point

Sunday, 31 May 2009 10:55 | Author: Anton

This summer I'll be working on illustrations for John Parker's upcoming book "Complex Hyperbolic Kleinian Groups". Last spring with Ryan Hoban and I worked out a general approach for the illustrations and put together most of the following notebook, which will be the starting point for this summer's work:

ComplexHyperbolic0.1.nb

The notebook so far can draw Cygan spheres, chains (C-circles), R-circles, spinal spheres, and HeisenMan, as well as left Heisenberg translations of these objects. In particular, Ryan and I replicated most of the objects in "A Tour of Heisenberg Space".

I'm also planning on running a small seminar on complex hyperbolic geometry (with Jeremy Tyson and Noel DeJarnette). We'll go through at least portions of the book, and I'll be making illustrations for the book and using them in the seminar. I'll post notes from the seminar and the corresponding pictures and animations in this section of the website.

Last Updated ( Monday, 01 June 2009 23:00 )

UIUC Undergrad Admissions

Et Cetera Res - Ridiculous

Friday, 29 May 2009 14:36 | Author: Anton

There's a quick way to get into your favorite public university: just email a state representative. They forward your name to the school, and you end up on the magic preferred admission list.

The Chicago Tribune recently ran a story on this process at UIUC. In short,

There's a separate "Category 1" for people with political support.
Category 1 has higher admission rates than average.
Category 1 has lower credentials than the average student.
Obviously bad students are admitted without notifying the high school to avoid attention.
Often, the politicians don't know much or anything about the students supported.

And what does the UIUC chancelor have to say about this? He sent out an email to the whole campus with the following letter which ignores the issue entirely:

Last Updated ( Friday, 29 May 2009 14:43 )

Riemannian Approximation - Lie groups and metric spheres

Math - Heisenberg09

Friday, 22 May 2009 11:38 | Author: Anton

Riemannian Approximations as Lie groups

The Heisenberg group is given by the group law

$(x_1,y_1,t_1)+(x_2,y_2,t_2) = \left(x_1+x_2, y_1+y_2, t_1+t_2+\frac{1}{2}( x_1 y_2 - y_1 x_2) \right)$

The left action leaves the sub-Riemannian structure, and thus the metric, invariant.

The frame fields for the Riemannian approximations is also invariant under a group law (one for each s):

$(x_1,y_1,t_1)+(x_2,y_2,t_2) = \left(x_1+x_2, y_1+y_2, t_1+t_2-\frac{s}{2}( x_1 y_2 - y_1 x_2) \right)$

Note that the negation in the t coordinate. This just means I flipped the t coordinate when writing down the approximations, and I'll go through and fix that.

The approximationg spaces $\mathbb{H}_s$ are then Lie groups in their own right, endowed with a Riemannian structure. These groups are again two-step nilpotent (except for s=0), so in particular not semi-simple. So I guess what we have is an approximation of the Heisenberg group by other Carnot groups very similar to $\mathbb{H}$ and degenerating metrics.

Last Updated ( Monday, 25 May 2009 17:51 )

Naan

Food - Breads

Tuesday, 19 May 2009 09:30 | Author: Anton

Indian flat bread that puffs up in oven. From "1000 Indian Recipes".

Ingredients:

2 tsp yeast
1 tsp sugar
1/4 cup warm water
1/2 cup yogurt
2 Tbsp oil
2 cups flour
1/4 tsp salt
1/4 cup melted butter

Instructions:

1. Dissolve yeast and sugar in water, after 5 minutes mix in yogurt and oil.
2. Mix flour and salt, slowly mix in (1), and form into a ball.
3. Let rise until doubled, 3-4 hours.
4. Split dough into 12 balls.
5. Take each ball, coat with flour and form into 7-8 inch triangle (equilateral?).
6. Place 3-4 triangles on baking tray, lightly baste with water to prevent drying out.
7. Broil 4-5 inches below heating element: about 1 minute until brown spots start to appear, then about 30 seconds turned over until golden.

Hummus

Food - Hors d'Oeuvres and Such

Sunday, 17 May 2009 15:25 | Author: Anton

Blend:

4 minced garlic cloves
2 15-oz cans of chickpeas, drained and rinsed
2/3 cup tahini (sesame seed paste)
1/3 cup lemon juice
1/2 cup water
1/4 cup olive oil
1/2 tsp salt

Last Updated ( Sunday, 17 May 2009 15:34 )