403/503 – HW6

February 24, 2015

This exercise is due Tuesday, March 3, at the beginning of lecture.

Recall that the nth Jordan block for \lambda, J(\lambda,n), is the n\times n matrix whose entries along the main diagonal are \lambda, along the diagonal immediately below the main one are 1, and all other entries are 0. For example, J(5,4) is the matrix

\displaystyle \left(\begin{array}{cccc}5&0&0&0\\ 1&5&0&0\\ 0&1&5&0\\ 0&0&1&5\end{array}\right).

Find a general formula for the powers of Jordan blocks, i.e., compute J(\lambda,n)^k.

Baby shower

February 12, 2015

My wife had a surprise baby shower at work.


Just a bit more than a month to go.

“A survey of the density topology on the reals”

February 12, 2015

Stuart Nygard, one of my current Master’s students, just returned from attending this year’s Winter School in Abstract Analysis. He gave a survey talk on the general topic of his thesis, the density topology. His slides can be found here.

Slides of most of the talks can also be downloaded at the conference website, here. Sadly, there do not seem to be slides for the tutorials on the analysis section.

311 – Another extra credit exercise

February 11, 2015

This optional set is due Tuesday, February 24, at the beginning of lecture. Some of these problems are harder than others, you do not need to address all of them. Contact me if you would like additional time to keep working on some of the more difficult ones.

Recall the theories T_n introduced in homework 2:

  • For any two distinct points passes a unique line.
  • Any line has at least n points.
  • There are three non-collinear points.

Recall also the axioms of projective geometry:

  1. For any two distinct points passes a unique line.
  2. Any two distinct lines meet at a point.
  3. There are at least four points, no three of them collinear.

Theorem (T_{n+1}). There are at least n^2+n+1 points.

Proof. There is a line \ell with at least n+1 points a_1,\dots,a_{n+1}, and there is at least one point a not in \ell. The lines \overleftrightarrow{aa_1},\dots,\overleftrightarrow{aa_{n+1}} are all different, and each of them has at least n-1 points other that a and the a_i. It is not hard to see that all these points are different, giving us at least: n+1 points in \ell, n-1 points in each of the lines \overleftrightarrow{aa_i}, and a, that is, (n+1)+(n-1)(n+1)+1=n^2+n+1 points. \Box

Recall that if P is a finite model of the projective axioms, then there is a natural number n (the order of P) such that P has exactly n^2+n+1 points and n^2+n+1 lines, each point belongs to exactly n+1 lines, and each line goes through exactly n+1 points. (See here for a reference with proofs of these facts, from Perspectives on projective geometry, by Jürgen Richter-Gebert.)

  • Show that the only model of T_3 with 7 points is the Fano plane.
  • Show that all finite projective planes of order three are isomorphic (an example is described here).
  • Is any model of T_4 with 13 points the finite projective plane of order three?
  • Is any model of T_5 with 21 points the finite projective plane of order four? More generally, if n>1, is any model of T_{n+1} with n^2+n+1 points a finite projective plane of order n?

311 – Extra credit exercise

February 10, 2015

This optional exercise is due Thursday, February 12, at the beginning of lecture.

Show that the projective plane \mathbb Z_2P^3 obtained through the vector space construction applied to the field \mathbb Z_2 of two elements is (isomorphic to) the Fano plane.

I review the construction: To form KP^2 we start with a field K and consider K^3.

A point in the projective plane is just a line in K^3 through the origin. Note that any such line is obtained by fixing a non-zero element (x,y,z) of K^3 and considering all its scalar multiples, all the vectors (\alpha x,\alpha y,\alpha z) with \alpha\in K.

(When K=\mathbb Z_2, the only values of \alpha are 0 and 1, so a line consists of only two points, one of which is the origin.)

A line in the projective plane is actually the collection of lines contained in a plane in K^3 through the origin. In other words, a set L of points (that is, of lines through the origin) is a line iff there is a plane in K^3 that contains the lines that are the elements of L and no others. Note that any such plane can be described by fixing coefficients \alpha,\beta,\gamma in K, not all equal to zero, and considering the set of vectors (x,y,z)\in K^3 such that \alpha x+\beta y+\gamma z=0.

To show that when K=\mathbb Z_2 this construction produces the Fano plane it is not enough to indicate that the resulting space consists of 7 points and 7 lines, you also need to verify that these can be identified with the points and lines of the Fano plane in such a way that all incidence relations in the Fano plane are verified in this space and no, others.

403/503 – Extra credit homework

February 5, 2015

This set is optional, and due February 12 at the beginning of lecture.

From Chapter 3 of Axler’s book, solve exercises 1,2,3,9,10,16, and 25.

(The numbering is as in the second edition. If you own the third edition, let me know and I’ll check in case the statements have changed.)

Cryptic marks

February 5, 2015

New scientist recently ran a series on articles on “How to think about…” One of them, by Richard Webb and published December 13, 2014, was about infinity. It contains this quote:

Woodin’s notepads consist mainly of cryptic marks he uses to focus his attention, to the occasional consternation of fellow plane passengers. “If they don’t try to change seats they ask me if I’m an artist,” he says.

David Roberts wondered on Google+ what these cryptic marks look like. This reminded me of some pictures I took of them at the Conference on inner model theory at UC Berkeley last year.

2014-06-10 17.59.20

2014-06-10 17.37.00

2014-06-10 17.36.08