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\location{Professor X\\ \smallskip
Courant Institute of Mathematical Sciences\\
251 Mercer Street\\New York, NY 10012--1185\\
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{\setbox9=\hbox{Tel. }\setbox8=\hbox{Fax}\dimen0=\wd9\advance\dimen0by -\wd8
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Tel. (212) 998--3192 \\
Fax\hskip \dimen0(212) 995--4121 \\
Telex 235128 NYU UR \\
Internet: profx@cims.nyu.edu}}
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\signature{Professor X}
\date{\vskip 4cm September 9, 1996}
\begin{document}
\begin{letter}{Professor H. M.\\
Department of Mathematics\\
The University of Arizona\\
Building \#89\\
617 N.~Santa Rita\\
P.O.~Box 210089\\
Tucson, AZ~~85721}
\opening{Dear Professor M:}
\h I'm answering quickly since I'll be away for 2 weeks.
\h Of course you should promote J. Q. to Associate Professor with tenure.
Since his first rate Ph.D.\ thesis, which appeared as 4 papers, he's been
going like a house afire. He is a truly talented mathematician: great
ideas and terrific technical power.
\h His Ph.D.\ thesis, here, was exceptional. It studied travelling front
solutions in inhomogeneous media. While he was working on it, B.
and I were also working on a problem involving travelling waves in combustion
theory. So I knew him well then---he struck me as already a mature mathematician.
He studied travelling waves for
$$
u_t = \partial_{x_i} (a_{ij}(x) \, u_{x_j}) + b_i(x) \,u_{x_i} + f(u) \quad
\hbox{ in }\;\RR^n\ ;
$$
the coefficients are periodic in each variable and $\{a_{ij}(x)\}$ a matrix
close to the identity matrix. B. and I considered only the identity
matrix but we treated a problem in a cylinder. His thesis has many beautiful
arguments. In a later paper he showed that if $a_{ij}(x)$ is positive
definite but not close to $\Id$ then travelling wave solutions
need not exist. In several other papers he studied the stability of travelling
waves for certain classes of functions $f$ if the initial values of $u$, at
time $t=0$, are close to the travelling wave one. Some problems give rise to
certain degenerate equations, and the degeneracy is handled very cleverly---as
in papers 7 and 11. In paper 11, for other $f$, Jack involves the Lax--Oleinik
entropy condition from shock wave theory---a striking paper. All his papers
involve hard technical estimates---he is a master. All these papers are
fundamental. You ask about intellectual independence and leadership, Jack
is simply at the forefront of the field. He is well-known internationally and
has been invited to speak at many conferences.
\h It's very interesting that Jack's work has become increasingly applied. He
can talk with engineers, construct \emph{good} model equations for their
problems and then use \emph{powerful} mathematics to analyse them. Few people
have these combination of talents.
\h Paper 15 studies a thermal diffusion combustion system. Under conditions on
the initial data, as $t\longrightarrow \infty$, the solution converges to a self
similar one for a reduced system. Hard estimates are derived, which play an
essential role. Paper 16 takes up a problem for lasers in an optical ring cavity.
The system of equations, due to Maxwell and Bloch, is a very interesting one:
hyperbolicity and dispersivity both occur---quite new to me. Again there are
lots of hard estimates.
\h Paper 17 goes again in a quite new direction. It studies viscous shocks for the
Burgers equation under random perturbations. It's an extremely interesting paper,
and I believe the ideas will play an important role in future work. 18 is a
terrific paper. Paper 19 is also a striking piece of work. A number of people
have studied this problem and 19 goes far beyond earlier results. The estimate
$O$ (log log $t$) for the $L^\infty$ norm of one component in the system is most
interesting.
\h Jack's work is simply first class and covers more and more directions. He is
also a very clear speaker, a most conscientious teacher and responsible member
of the faculty. He has a very warm outgoing personality and interacts wonderfully
with others---as is evident from his extensive list of collaborators. I would
be very happy to have him as a tenured colleague at Courant.
\h I recommend his promotion and tenure in the strongest way.
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\closing{Sincerely yours,}
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