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\begin{document}
\title{VIGRE Report Part II $\cdot$ Fall 2009}
\author{John Kerl}
%\date{\today}
\date{December 9, 2009}
\maketitle

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\section{Dissertation research}

I presented the progress of my research to the UA mathematical physics seminar
on Wednesday, September 9, 2009.  I put this in the form of a very early,
rough-draft job talk; I received much useful feedback from the audience,
particularly from Bob Sims.

My co-advisor Daniel Ueltschi suggested earlier this fall that I write a draft
paper of my dissertation results before, rather than after, defending my
dissertation.  He suggested this for two purposes:  (1) for him, as a writer of
letters of recommendation on my behalf, to evaluate my ability to work
independently and write up my research, and (2) in order for me to be able to
list a submitted publication on my job-application materials, rather than the
more dubious-sounding ``paper in preparation''.  As of two weeks ago, I have
completed this draft and have received some feedback from Daniel; I am awaiting
additional feedback from my other co-advisor Tom Kennedy before submitting the
paper for publication.  I hope to submit it before year's end.

I have completed the worm-algorithm section of my dissertation.  The
superfluid-fraction, extrapolation, and finite-size-scaling portions are all
complete and contained within my draft paper; I need merely to copy, paste, and
lightly edit these to form missing dissertation chapters.

As described in the paper, I have quantified $\Delta T_c$ for the $r_\ell$
model on the lattice; the sought-after constant described in my VIGRE
application is currently estimated to be $0.665 \pm 0.067$.  The error bar will
only shrink as more compute runs complete on the University of Arizona's
High-Performance Computing cluster, over the next month or so.  The value
itself is of interest: it matches (to within experimental uncertainty) the
analytical result obtained by Ueltschi and Betz for the small-cycle-weight
model with varying point positions.  This suggests that fine details of
interjump interactions and point positions do not affect the constant of
linearity in the shift in critical temperature, although the critical
temperatures themselves do differ noticeably.

Let $\ell_{\textrm{max}}$ be the length of the longest cycle in a permutation;
let $N f_I$ be the number of cycles in macroscopic cycles.  For non-interacting
and $r_2$-interacting spatial permutations, $\langle \ell_{\textrm{max}}\rangle
/ N f_I$ is empirically found to be the same (approximately 63\%) as for
uniformly distributed permutations.  For $r_\ell$ interactions, this is no
longer the case.  In the paper (and thus in my dissertation), I characterize
this phenomenon:  The 63\% fraction in the non-interacting random-cycle model
matches the 1966 result of Shepp and Lloyd for non-spatial permutations;
interactions give rise to a weak linear increase in the fraction.

Off-lattice computations will not be pursued for my dissertation work; these
are a possible postdoctoral project.  For my dissertation, points are held
fixed on the lattice, with cycle-length-weighed interactions between
permutation jumps.  This is an artificial model, of its own probabilistic
interest; furthermore, it sheds light on which properties of the random-cycle
model do and do not depend on the fine details of point positions or
interactions.  For the Bose-gas model, one should pursue off-lattice point
positions as well as permutation-jump interactions which are derived from the
Bose-gas model.

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\section{Items other than dissertation research}

I am currently completing my final course:  Probability and Random Processes in
Engineering, in the Electrical Engineering department.  This satisfies an
out-of-department requirement.

The percolation project alluded to in my application, which is an
analysis and write-up of work with Janek Wehr in spring 2008, has been on hold
this semester in favor of job applications and dissertation work.  Nonetheless,
I will bring this to a close by the end of the spring semester.  The remaining
work for the percolation project consists of (1) finite-size-scaling analysis,
which I now have a good handle on, and (2) writing up the paper, which will not
be difficult.

I have been meeting weekly with Tom Kennedy's bridge group.  My contributions
have been twofold:  (1) mentoring an undergraduate, Howard Cheng, to produce
computational results, and (2) analyzing those results.  Howard already had
facility with the Java programming language; Tom's bridge libraries are in C++,
so Howard had the opportunity to learn some C++ programming.  We are examining
empirical distributions of heights of irreducible bridges for upper-half-plane
self-avoiding walks.  In particular, it is believed that UPSAWs are
decomposable as a concatenation of IID irreducible bridges, and their
concatenated heights are expected (following recent work of Alberts) to adhere
to a stable distribution.  My analysis at present indicates that (a) the
distributions do indeed appear to be stable; (b) the stable-distribution
parameters are estimatable; (c) I will need to do further work to confirm or
deny that the stable-distribution parameters are, to within experimental
uncertainty, the ones predicted by Alberts.  It has been a pleasure working
with Howard; he and I both plan to continue working on this project in the
spring semester.

The Joint Meetings of the AMS/MAA will be Jan. 13-16, 2010, in San Francisco.
I have a 15-minute talk scheduled for 8 a.m. Wed. Jan. 13, and will be spending
much of the rest of my time in the Job Center.

David Landau's Center for Simulational Physics in Athens, Georgia, will be
holding a weeklong conference Mon.-Fri. Feb. 22-26 2010.  I have applied to
participate, and have been accepted.  I will focus on my talk on my worm
algorithm:  this offers an added feature over the algorithm used for the
computational results in my dissertation, but it has a stopping-time problem.
At the conference I hope to get some ideas about that problem.  I also hope to
make many professional contacts.

I completed my CV, teaching statement, and research statement in October; since
then, I have continued to search for and apply for jobs.  Due to the breadth
and depth of my experience, I have the opportunity to apply for various types
of positions; given the sad state of the current job market, I moreover have
the necessity of doing so.  I have approximately 40 applications out, several
of them with a Dec. 1 application deadline; at least a dozen more will go out
with Jan. 1st deadlines.  I am considering postdoc, visiting, and tenure-track
academic positions, as well as industrial and  government-laboratory work.  To
date I have had three phone interviews, one on-site interview, and two letters
of rejection.  This is not bad; yet, I will be happy to receive a job offer,
and happier yet to receive a marginal offer and be able to turn it down in
favor of a better one.

\end{document}
