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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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\end{letter}\end{document}