Minutes for 1/27/04
Notetaker today: Paul
Snacks today: Jim
Notetaker for 1/29/04: Hal
Snacks 1/29/04: Sam
(Meeting commences)
Jim: Brendan is our new person. Since he's not here, we wont go around with
names.
(We go around with what we did over break)
Jim: I went to Boston, Connecticut, and gave talks about Roter-rooters. I
also am looking for someone to code some Java for me. It would be a paid
position at 10.00/hour.
Sam: Had a hectic break. Looked at everyone's websites.
Paul: Visited my grandparents.
Carl: Not hectic, but full break. Didn't do a whole log of SSL stuff. But
did write a creative summary of my work last semester.
Emilie: Worked on Markoff graphs. Worked with the "grid displacement/vertex
preserving" operations. Found out which way you can move the grid and still
create a triple.
(Enter Brendan. Brendan walks gallantly from the door to an empty seat
between John and Hal. Brendan sits.)
John: Had the flu. Long recovery.
Brendan: Made 7'x2'x4' paper mache bucky and placed it on Bascom hill.
Jim: I had some regrets about not pranking at MIT. Wanted to put a function
on a grid in an elevator that read L(eta).
Hal: no SSL stuff. Did some programming. Visited family.
Martin: Wrote up a document about run times of different ways to do the
Fibonacci numbers. Wants to calculate automorphism groups. Wants Magma.
Jim: Nigel Boston has Magma.
Jim will ask Nigel about MAGMA.
Brendan will ask Isaacs about what he uses to compute automorphism groups.
(Group goes around with math related courses for the semester)
Jim: teaching Math 475.
Sam: Number theory, second semester algebra.
Paul: Topology
Carl: Cryptography
Emilie: Number theory, CS 367
John: not teaching, but researching.
Brendan: 2nd sem Topology, 2nd sem Algebra(742)
Hal: no classes
Martin: theoretical CS, Grading for 475.
(Group goes around with projects they are interested in)
Jim - Not doing SSL research. Wants to write up some stuff done in REACH.
Sam - Newton's method.
Carl - Lifting hypothesis. Tilings of the plane. All attempts to prove the
formula for coeffs of Newtons method inductively have failed.
(Discussion ensues between Sam and Carl about the trivialness of the induction
process as it relates to Newton's method)
Jim mentions the generalization of Newton's method with q numbers.
Paul: Newton's method.
Emilie: Continue with work on tilings of Markoff and his brothers. Main
question: How to get all triples from the lattices?
John: Nothing specifically. Interested in Newton's method and crossover
between dynamical systems and combinatorics.
(Jim explains a connection in billiards. To figure out if a billiards ball,
allowed to bounce for a large amount of time will "fill" the table, that is,
the ball's position will be dense, we can extend the table into imaginary
tables and look at the cutting sequence. This is reminiscent of the Markoff
grid.)
Martin, Hal and Jim discuss the motivation and implications of finding if
the Markoff graph is an expander graph. Jim explains how one can figure
it out by looking at the eigenvalues of the adjacency matrix of a particular
graph.
Say you have eigenvectors v1, v2, v3,... to the adjacency matrix of a graphs
with eigenvalues n, a, b, c,... respectively, in descending magnitude. Then
any vector v will be a linear combination of those eigenvectors,
so v = Av1+Bv2+...
So Mv = Anv1+Bav2+...
so (nI-M)v = B(n-a)v2+C(n-b)v3+...
and (nI-M)^2 v = B (n-a)^2 v2 + C (n-b)^2 v3 + ...
We can keep multiplying like this to approximate the value of a (which is the
second largest eigenvalue, and therefore the spectral gap. The only condition
is that B not be zero.
Three more things
1. Read each others stuff and give feedback.
2. Photos
3. Gaskets.