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\begin{document}
\begin{center}
\bf
Math 553C\hfill Recent Research in Geometric Analysis\hfill
Spring 1994\\
\medskip
SUGGESTED PAPERS
\end{center}
\bigskip




\section{Prescribing scalar and Gaussian curvature}
\begin{itemize}

\item
J. L. Kazdan and F. W. Warner, {\it Curvature functions for compact
2-manifolds}, Ann.\ of Math.\  {\bf 99} (1974) 14--47.  This paper gives
necessary and sufficient conditions on a function $K$ on a compact
2-manifold in order that there exist a Riemannian metric whose Gaussian
curvature is $K$.

\item
M. Gromov and H. B. Lawson, {\it The classification of simply connected
manifolds of positive scalar curvature}, Ann.\ of Math.\ {\bf 111} (1980)
423--434.  The title speaks for itself.

\item
R. Schoen, {\it Conformal deformation of a Riemannian metric to constant
scalar curvature}, J. Differential Geom.\  {\bf 20} (1984) 479--495.  The
solution of the ``Yamabe problem'': every compact Riemannian manifold has a
conformal metric of constant scalar curvature.


\end{itemize}
\section{Uniformization of positively-curved manifolds}
\begin{itemize}

\item
R. Schoen and S.-T. Yau, {\it Complete three-dimensional manifolds with
positive Ricci curvature and scalar curvature}, in ``Seminar on
Differential Geometry'', pp.\ 209--228, Ann.\ of Math.\ Studies {\bf 102},
Princeton University Press, Princeton, 1982.  Using minimal surface theory,
shows that any complete noncompact 3-manifold with positive Ricci curvature
is diffeomorphic to $\R^3$.

\item
R. S. Hamilton, {\it Three-manifolds with positive Ricci curvature}, J.
Differential Geometry {\bf 17} (1982) 255--306.  Uses heat flow techniques
to prove that any compact 3-manifold that admits a metric with positive
Ricci curvature also admits a metric with constant positive curvature, and
hence is covered by the 3-sphere.

\item
R. S. Hamilton, {\it Four-manifolds with positive curvature operator}, J.
Differential Geometry {\bf 24} (1986) 153--179.  Extends Hamilton's theorem
on three-manifolds with positive Ricci curvature to certain 4-manifolds
with positive curvature, while simultaneously simplifying and clarifying
the method of proof used in the 3-manifold paper.



\end{itemize}
\section{Compactness of families of Riemannian metrics}
\begin{itemize}

\item
J. Cheeger, {\it Finiteness theorems for Riemannian manifolds}, Am.\ J.
Math.\  {\bf 92} (1970) 61--74.  Proves that in any family of Riemannian
manifolds with certain bounds on curvature, volume, and diameter, there can
be at most finitely many diffeomorphism types.

\item
M. Gromov, {\it Structures m\'etriques pour les vari\'et\'es
riemanniennes}, Textes Math\'e\-ma\-tiques {\bf 1}, CEDIC, Paris, 1981.
This is the original proof of Gromov's famous ``compactness theorem'': any
sequence of Riemannian manifolds whose curvatures and diameters are
bounded, and whose volumes are bounded below, contains a subsequence that
``converges'' (in a rather weak topology).

\item
S. Peters, {\it Convergence of Riemannian metrics}, Comp.\ Math.\ {\bf 62}
(1987) 3--16.  A more readable proof of Gromov's compactness theorem.

\end{itemize}
\newpage
\section{The inverse spectral problem}
\begin{itemize}

\item
B. Osgood, R. Phillips, and P. Sarnak, {\it Compact isospectral sets of
surfaces}, J.\ Funct.\ Anal.\ {\bf 80} (1988) 212--234.  The first partial
solution to Mark Kac's famous question ``Can one hear the shape of a
drum?''  (In other words, does the spectrum of the Laplacian on a bounded
plane domain determine the domain up to isometry?)  This paper proves that
any family of isospectral domains in the plane (i.e. domains with the same
spectrum) must be compact.

\item
T. Sunada, {\it Riemannian coverings and isospectral manifolds}, Ann.\
Math.\ {\bf 121} (1985) 169--186.  Gave the first systematic construction
of compact isospectral manifolds, introducing ideas that have played a
central role in all subsequent isospectral examples.

\item
C. S. Gordon, D. L. Webb, and S. Wolpert, {\it One cannot hear the shape of
a drum}, Bull.\ Amer.\ Math.\ Soc.\ {\bf 27} (1992) 134--138.  The first
counterexample to Kac's conjecture, showing that non-isometric plane
domains can be isospectral.



\end{itemize}
\section{Spectral asymptotics on Riemannian manifolds}
\begin{itemize}

\item
R. Seeley, {\it Singular integrals and boundary problems}, Amer.\ J.\
Math.\ {\bf 88} (1966) 781--809.  The method of Hadamard is used to get
information on the asymptotic behavior of eigenvalues of the Laplacian for
manifolds with boundary.

\item
J. Chazarain, {\it Formule de Poisson pour les vari\'et\'es riemanniennes},
Invent.\ Math.\ {\bf 1974} 65--82.  Develops an asymptotic formula,
generalizing the Poisson summation formula, relating the eigenvalues of the
Laplacian on a compact Riemannian manifold with the lengths of its closed
geodesics.

\item
S. Marvizi and R. B. Melrose, {\it Spectral invariants of convex planar
regions}, J.\ Differential Geom.\  {\bf 17} (1982) 475--502.  Uses wave
asymptotics to find spectral invariants of convex domains in the plane.



\end{itemize}
\section{Morse theory and calculus of variations}
\begin{itemize}

\item
R. Bott, {\it Nondegenerate critical manifolds}, Ann.\ of Math.\  {\bf 60}
(1954) 248-261.  Extends Morse theory from the case of isolated critical
points to the case of critical submanifolds.

\item
R. S. Palais and S. Smale, {\it A generalized Morse theory}, Bull.\ Amer.\
Math.\ Soc.\ {\bf 70} (1964) 165--172.  Introduced the famous
``Palais-Smale Condition C'', which gives sufficient conditions for
existence of critical points of functionals on infinite-dimensional spaces.

\item
R. S. Palais, {\it The principle of symmetric criticality}, Comm.\ Math.\
Phys.\ {\bf 69} (1979) 19--30.  Given a function space acted on by a
compact Lie group $G$, and a functional on the space that is $G$-invariant,
this paper shows that, under very general conditions, functions that are
critical points among $G$-invariant variations are actually critical among
all variations.  This shows that, in the presence of symmetry, calculus of
variations problems can be solved by restricting attention to $G$-invariant
objects.



\end{itemize}
\newpage
\section{Minimal submanifolds}
\begin{itemize}

\item
H. Federer and W. Fleming, {\it Normal and integral currents}, Ann.\ of
Math.\ {\bf 72} (1960) 458--520.  Introduces the concept of integral
currents (generalized submanifolds), the main tool of geometric measure
theory which is used to study minimal submanifolds.

\item
J. Sacks and K. Uhlenbeck, {\it The existence of minimal immersions of
2-spheres}, Ann.\ of Math.\  {\bf 113} (1981) 1--24.  Introduces the
technique of rescaling to handle the ``bubbling'' phenomena that are
ubiquitous in geometric nonlinear PDE's.

\item
R. Harvey and H. B. Lawson, {\it Calibrated Geometries}, Acta Math.\ {\bf
148} (1982) 47--157.  This paper introduced an important and creative new
point of view, that of ``calibrations'', into the subject of minimal
submanifolds.


\end{itemize}
\section{Harmonic maps}
\begin{itemize}

\item
J. Eells, Jr., and J. H. Sampson, {\it Harmonic mappings of Riemannian
manifolds}, Amer.\ J.\ Math.\  {\bf 86} (1964) 109--160.  Using heat-flow
methods, they prove the existence of a harmonic map in any homotopy class
>From one compact Riemannian manifold to another, provided the target
manifold is negatively curved.

\item
R. Schoen and K. Uhlenbeck, {\it A regularity theory for harmonic maps}, J.
Diff.\ Geom.\ {\bf 17} (1982) 307-336.  The authors prove that the
Hausdorff dimension of the singular set of an energy minimizing harmonic
map from an $n$-manifold is less than or equal to $n-3$.

\item
L. Simon, {\it Asymptotics for a class of non-linear evolution equations,
with applications to geometric problems}, Annals of Math.\ {\bf 118} (1983),
525-572.  First steps in the development of a detailed asymptotic
description of the structure of the singular set for such problems as
harmonic maps or minimal surfaces.

\end{itemize}
\section{Gauge theory and 4-manifolds}
\begin{itemize}

\item
M. Atiyah, N. Hitchin, and I. Singer, {\it Self-duality in four-dimensional
Riemannian geometry}, Proc.\ Roy.\ Soc.\ London A {\bf 362} (1978)
425--461.  This paper uses twistor methods to construct instantons
(Yang-Mills connections) on $S^4$.

\item
C. Taubes, {\it The existence of self-dual connections on non self-dual
4-man\-i\-folds.}, J. Differential Geom.\ {\bf 17} (1982) 139--170.  Proof
of the existence of solutions to the Yang-Mills equation on a large class
of 4-man\-i\-folds.

\item
S. K. Donaldson, {\it Self-dual connections and the topology of smooth
4-manifolds}, J. Differential Geom.\ {\bf 18} (1983) 279--315.  Donaldson's
spectacular result on the intersection form of simply-connected
4-manifolds, proved by analyzing the set of solutions to the Yang-Mills
equation.  He showed that most topological 4-manifolds don't have smooth
structures.



\end{itemize}
\newpage
\section{Conformal invariant theory}
\begin{itemize}

\item
C. Fefferman, {\it Parabolic invariant theory in complex analysis}, Adv.\
Math.\ {\bf 31} (1979) 131--262.  Fefferman initiated the task of
generalizing Weyl's invariant theory for Riemannian manifolds.  In this
paper, he concentrates on CR geometry, which is formally very similar to
the conformal case.  He described his approach to the problem, and, with a
very long and difficult argument, proved that it works for some invariants
in some dimensions.

\item
C. Fefferman and C. R. Graham, 
{\it Conformal invariants},
in ``\'Elie Cartan et les Math\'e\-ma\-tiques d'Aujourd'hui'', Ast\'erisque,
1985, pp.\ 95--116.  Here Fefferman's framework for constructing invariants
is extended to the conformal case.

\item
T. N. Bailey, M. G. Eastwood, and C. R. Graham, {\it Invariant theory for
conformal and CR geometry}, Ann.\ Math.\ (to appear).  The problem of
describing scalar conformal invariants is completely solved in odd
dimensions.



\end{itemize}
\section{Complex structures and their deformations}
\begin{itemize}

\item
A. Newlander and L. Nirenberg, {\it Complex coordinates in almost-complex
manifolds}, Ann.\ of Math.\  {\bf 65} (1957) 391--404.
The proof that every integrable almost-complex manifold is actually a
complex manifold.

\item
K.  Kodaira and D. C.  Spencer, {\it On deformations of complex analytic
structures I, II}, Ann.\ of Math.\  {\bf 67} (1958) 328--401, 403--466.
The first general result on the local structure of the space of
complex structures on a complex manifold.

\item
M. Kuranishi, {\it On the locally complete families of complex analytic
structures}, Ann.\ Math.\ {\bf 75} (1962) 536--577.  Here Kuranishi
developed a technique to completely describe the local structure of the
space of inequivalent complex structures on a given complex manifold.  The
proof has since been simplified somewhat by Kuranishi and T. Akahori.



\end{itemize}
\section{Uniformization of K\"ahler manifolds}
\begin{itemize}

\item
Y.-T. Siu and S.-T. Yau, {\it Complete K\"ahler manifolds with nonpositive
curvature of faster than quadratic decay}, Ann.\ Math.\ {\bf 105} (1977)
225--264.  A first step in a ``uniformization theory'' for complete
K\"ahler manifolds in higher dimensions, showing that any negatively-curved
complete K\"ahler manifold whose curvature decays sufficiently fast at
infinity must be biholomorphic to $\C^n$.

\item
N.-M.\ Mok, Y.-T. Siu and S.-T. Yau, {\it The Poincar\'e-Lelong equation on
complete K\"ahler manifolds}, Comp.\ Math.\ {\bf 44} (1981) 183--218.
Shows, under the hypotheses of the Siu-Yau theorem above, that the manifold
must in fact be isometric to $\C^n$.

\item
Y.-T. Siu and S.-T. Yau, {\it Compact K\"ahler manifolds of positive
bisectional curvature}, Invent.\ Math.\ {\bf 59} (1980) 189--204.  Proof of
the Frankel conjecture: every compact K\"ahler manifold with positive
bisectional curvature is analytically isomorphic to complex projective
space.  Uses harmonic map techniques.


\end{itemize}
\newpage
\section{Riemannian manifolds with special holonomy groups}
\begin{itemize}

\item
M. Berger, {\it Sur les groupes d'holonomie des vari\'et\'es \`a connexion
affine et des vari\'et\'es Riemanniennes}, Bull.\ Soc.\ Math.\ France {\bf
83} (1955) 279--330.  Shows that the only Lie groups that can occur as
holonomy groups of Riemannian manifolds are $SO(n)$, $U(m)$, $SU(m)$,
$Sp(k)$, $Sp(k)Sp(1)$, $G_2$, and ${\it Spin}(7)$.

\item
S. M. Salamon, {\it Differential geometry of quaternionic manifolds}, Ann.\
Scient.\ Ec.\ Norm.\ Sup.\ {\bf 19} (1986) 31--55.  One of the most
interesting classes of Riemannian manifolds are those with holonomy group
$Sp(k)Sp(1)$, called quaternionic-K\"ahler manifolds.  This is supposed to
be a very good introduction to the subject.


\end{itemize}
\section{K\"ahler-Einstein metrics}
\begin{itemize}

\item
S.-T. Yau, {\it On the Ricci curvature of a compact K\"ahler manifold and
the complex Monge-Amp\`ere equation I}, Comm.\ Pure Appl.\ Math.\ {\bf 31}
(1978) 339--411.  Solves the ``Calabi conjecture'' for compact K\"ahler
manifolds with non-positive first Chern class: on any such manifold, there
exists a K\"ahler-Einstein metric in each admissible K\"ahler class.

\item
A. Futaki, {\it An obstruction to the existence of Einstein K\"ahler
metrics}, Invent.\ Math.\  {\bf 73} (1983) 437--443.  The first known
obstruction to the existence of K\"ahler-Einstein metrics on compact
K\"ahler manifolds with positive first Chern class.

\item
S.-Y.\ Cheng and S.-T.\ Yau, {\it On the existence of a complete
K\"ahler-Einstein metric on noncompact complex manifolds and the regularity
of Fefferman's equation}, Comm.\ Pure Appl.\ Math.\ {\bf 33} (1980)
507--544.  Proves a non-compact version of Yau's theorem on the Calabi
conjecture: any strictly pseudoconvex domain in $\C^n$ admits a complete
K\"ahler-Einstein metric.


\end{itemize}
\section{Geometry of strictly pseudoconvex domains in $\C^n$}
\begin{itemize}

\item
C. Fefferman, {\it The Bergman kernel and biholomorphic mappings of
pseudoconvex domains}, Invent.\ Math.\ {\bf 26} (1974) 1--65.  Fefferman's
celebrated extension theorem, which shows that biholomorphic mappings
between strictly pseudoconvex domains extend smoothly to the boundary.  His
complicated proof has since been considerably simplified and generalized by
Bell, Ligocka, Forsterni\v{c}, and others.

\item
C. Fefferman, {\it Monge-Amp\`ere equations, the Bergman kernel, and
geometry of pseudoconvex domains}, Ann.\ of Math.\ {\bf 103} (1976)
395--416.  Generalizes the Poincar\'e metric to arbitrary strictly
pseudoconvex domains in $\mathbf C^n$, and uses it to give a new definition
of the biholomorphically invariant curves called ``chains'' in their
boundaries.

\item
L. Lempert, {\it Solving the degenerate complex Monge-Amp\`ere equation
with one concentrated singularity}, Bull.\ Soc.\ Math.\ France {\bf 109}
(1981) 427--454.  Proved that, for any strictly convex domain in $\C^n$,
the exponential map from an interior point is a diffemorphism onto the
domain, thus providing a ``canonical'' parametrization for such domains.
This has been applied recently by John Bland and Tom Duchamp (UW) to
construct ``normal forms'' for convex domains and for the CR manifolds that
bound them.


\end{itemize}
\newpage
\section{CR manifolds}
\begin{itemize}

\item
S. S. Chern and J. K. Moser, {\it Real Hypersurfaces in complex manifolds},
Acta Math.\ {\bf 133} (1974) 219--271.  Solved of the ``equivalence
problem'' for CR manifolds (abstract models of real hypersurfaces in
$\C^n$), based on the construction of CR invariants in two different ways:
by putting the power series of a real hypersurface in a ``normal form''
(Moser); and by constructing a canonical connection on a certain fiber
bundle over the CR manifold (Chern).

\item
S. M. Webster, {\it Pseudohermitian structures on a real hypersurface}, J.
Differential Geometry {\bf 13} (1978) 25--41.  Carries out the analogue of
the Chern connection for a CR manifold endowed with a specific choice of
Levi form (a ``pseudohermitian structure'').  

\item
N. Tanaka, ``A Differential-Geometric Study on Strongly Pseudoconvex
Manifolds'', Kinokuniya Company Ltd., Tokyo, 1975.  An alternative
construction of the canonical connection associated with a pseudohermitian
structure, carried out more or less simultaneously with and independently
of Webster's.



\end{itemize}
\section{Dynamical systems and symmetry}
\begin{itemize}

\item
J. E. Marsden and A. Weinstein, {\it Reduction of symplectic manifolds with
symmetry}, Reports on Math.\ Phys.\ {\bf 5} (1974) 121--130.  Extends
techniques used classically for dynamical systems (systems of ODE's) with
abelian symmetry groups to the nonabelian case; these ideas have become
useful in classical and quantum mechanics and in representation theory.

\item
J. M. Arms and D. C. Wilbour, {\it Reduction procedures for Poisson
manifolds}, in ``Symplectic Geometry and Mathematical Phyics'', pp.\
462--475, Birkhauser, Boston, 1991.  Extends the notion of reduction of
dynamical systems with symmetry to the case in which the orbit space has
singularities. 

\item
J. J.  Duistermaat, {\it Global action angle coordinates}, Comm.\ Pure
Appl.\ Math.\ {\bf 33} (1980) 687-706.  Action-angle variables are
particularly nice coordinates that can be used locally for dynamical
systems with enough symmetry (i.e.\ an abelian symmetry group of the same
dimension as the configuration space).  This article shows that there is an
obstruction to extending these coordinates globally; as an example, the
obstruction is shown to be nontrivial in the case of the spherical
pendulum.


\end{itemize}
\newpage
\section{Symplectic geometry and pseudo-holomorphic curves}
\begin{itemize}

\item
M. Gromov, {\it Pseudo holomorphic curves in symplectic manifolds},
Invent.\ Math.\ {\bf 82} (1985) 307--347.  Here Gromov introduced a new and
fruitful approach to studying global invariants of symplectic manifolds:
``pseudo-holomorphic curves'' are maps from a Riemann surface into a
symplectic manifold that are ``holomorphic'' with respect to some (not
necessarily integrable) almost-complex structure on the symplectic
manifold.  As an application, he proves that no symplectic map can squash a
unit ball into a ``thin'' slice of Euclidean space.

\item
T. H. Parker and J. G. Wolfson, {\it Pseudo-holomorphic maps and bubble
trees}, J. Geom.\ Anal.\ {\bf 3} (1993) 63--98.  Using Sacks \& Uhlenbeck's
technique of ``bubbling'', this paper simplifies and considerably refines
Gromov's proof of compactness of sequences of pseudo-holomorphic curves,
showing that the limit can be viewed as a ``bubble tree'', i.e.\ a map from
a family of Riemann surfaces glued together at points into the symplectic
manifold.




\end{itemize}
\section{Geometric quantization}
\begin{itemize}

\item
B. Kostant, {\it Quantization and unitary representations, part I:
prequantization}, in ``Lectures in Modern Analysis and Applications, Volume
III'', pp.\ 515--538, Lecture Notes in Mathematics {\bf 170},
Springer-Verlag, New York, 1970.  Introduced the basic ideas of the
procedure for turning classical dynamical systems into quantum systems, now
called ``geometric quantization''.

\item
B. Kostant, {\it Graded manifolds, graded lie theory, and prequantization},
in ``Differential Geometrical methods in Mathematical Physics'', Springer
Lecture Notes in Math.\ \#570, 1977, pp.\ 177--306.  Introduced
``supermanifolds,'' now studied by both mathematicians and physicists.



\end{itemize}
\section{General Relativity}
\begin{itemize}

\item
S. W. Hawking and R. Penrose, {\it The singularities of gravitational
collapse and cosmology}, Proc.\ Roy.\ Soc.\ Lond.\ A {\bf 300} (1970)
529--548.  Proves that, under fairly general assumptions, any ``physically
reasonable'' spacetime (4-manifold with a Lorentz metric) must have
singularities (black holes).

\item
R. Schoen and S.-T. Yau, {\it On the proof of the positive mass conjecture
in general relativity}, Comm.\ Math.\ Phys.\  {\bf 65} (1979) 45--76.  Here
is the original proof of the famous positive mass conjecture from physics:
roughly speaking, any asymptotically flat 4-dimensional space-time with
positive local mass density looks from infinity like a black hole with
positive mass.  Uses minimal surface theory.



\end{itemize}
\newpage
\section{The Nash-Moser inverse function theorem}
\begin{itemize}

\item
J. Nash, {\it The embedding problem for Riemannian manifolds}, Ann.\ of
Math.\  {\bf 63} (1956) 20--63.  Proved that every Riemannian manifold can
be isometrically embedded in Euclidean space, and introduced the iterative
technique that was later formalized as the Nash-Moser implicit function
theorem.

\item
J. Moser, {\it A rapidly convergent iteration method and nonlinear PDE's I,
II}, Ann.\ Scuola Norm.\ Sup.\ Pisa {\bf 20} (1966) 265--315, 499--535.
Moser's version of the Nash-Moser inverse function theorem, now called
Moser iteration.

\item
R. S. Hamilton, {\it The inverse function theorem of Nash and Moser},
Bull.\ Amer.\ Math.\ Soc.\ {\bf 7} (1982) 65--222.  A very general
treatment of the celebrated Nash-Moser theorem, which extends the inverse
function theorem to spaces of $C^\infty$ functions.



\end{itemize}
\section{Elliptic PDE's and pseudodifferential operators}
\begin{itemize}

\item
L. Nirenberg, {\it The Weyl and Minkowski problems in differential geometry
in the large}, Comm.\ Pure Appl.\ Math.\ {\bf 6} (1953) 337--394.  This is
essentially Nirenberg's thesis, in which he solved the Weyl problem (given
a positively-curved Riemannian metric on the 2-sphere, find an isometric
embedding into $\mathbf R^3$) and the closely-related Minkowski problem.  He
derived some fundamental results on variable-coefficient elliptic PDE's,
introducing techniques that were later formalized into the theory of
pseudo-differential operators.

\item
J. J. Kohn and L. Nirenberg, {\it An algebra of pseudo-differential
operators}, Invent.\ Math.\ {\bf 24} (1974) 65--82.  Here the concept of
pseudo-differential operators was formally introduced.  These operators are
now the main tool in studying elliptic and subelliptic PDE's.

\item
R. B. Melrose, {\it Transformation of Boundary Problems}, Acta Math.\  {\bf
147} (1981) 149--236.  Introduces a new pseudo-differential operator
calculus, that of ``totally characteristic'' operators, to handle boundary
problems for differential operators that degenerate along the boundary.


\end{itemize}
\section{Fourier integral operators}
\begin{itemize}

\item
P. D. Lax, {\it Asymptotic solutions of oscillatory initial value
problems}, Duke Math.\ J.\ {\bf 24} (1957) 627--646.  Here the method of
geometric optics is formalized and used to solve hyperbolic PDE's.

\item
L. H\"ormander, {\it The spectral function of an elliptic operator}, Acta
Math.\ {\bf 121} (1968) 193--218.  Geometric optics is generalized and
developed into a precursor to Fourier integral operators, with applications
to spectral theory.

\item
L. H\"ormander, {\it Fourier integral operators I}, Acta Math.\ {\bf 127}
(1971) 79--183.  This paper and the next formally introduced the concept of
Fourier integral operators, now widely used to study hyperbolic PDE's.

\item
J. J. Duistermaat and L. H\"ormander, {\it Fourier integral operators II},
Acta Math.\ {\bf 128} (1972) 183--269.



\end{itemize}
\newpage
\section{Local solvability of PDE's}
\begin{itemize}

\item
H. Lewy, {\it An example of a smooth linear partial differential equation
without solution}, Ann.\ of Math.\  {\bf 66} (1957) 155--158.
A simple first-order linear partial differential equation that has no
solution in any open set.  This started the whole subject of local
solvability.

\item
L. Nirenberg and F. Tr\`eves, {\it On local solvability of partial
differential equations, Part I: necessary conditions}, Comm.\ Pure Appl.\
Math.\ {\bf 23} (1970) 1--38; {\it Part II: sufficient conditions}, Comm.\
Pure Appl.\ Math.\ {\bf 23} (1970) 459--510.  Characterized locally
solvable partial differential operators.



\end{itemize}
\section{The $\overline\partial$ problem}
\begin{itemize}


\item
J. J. Kohn, {\it Harmonic integrals on strongly pseudoconvex manifolds, I},
Ann.\ Math.\ {\bf 78} (1963) 112--148; {\it II}, {\bf 79} (1964) 450--472.
Solved the $\overline\partial$-Neumann problem: on a strictly pseudoconvex
domain $\Omega\subset\C^n$, given a smooth $\overline\partial$-closed
differential form $\eta$ on $\overline\Omega$, find a smooth form $\omega$
on $\overline\Omega$ such that $\overline\partial \omega=\eta$.  This has
many applications to several complex variables, including solving the Levi
problem of characterizing domains of holomorphy, and solving the Cousin
problems of extending holomorphic functions.

\item
J. J. Kohn and L. Nirenberg, {\it Non-coercive boundary problems}, Comm.\
Pure Appl.\ Math.\ {\bf 18} (1965) 443--492.  Extends Kohn's work on the
$\overline\partial$-Neumann problem to a very general class of elliptic
boundary problems that are not uniformly elliptic up to the boundary.

\item
L. H\"ormander, {\it $L^2$ estimates and existence theorems for the
$\overline\partial$ operator}, Acta Math.\ {\bf 113} (1965) 89--152.
Presents an alternative solution to the $\overline\partial$ problem.



\end{itemize}
\section{Subelliptic PDE's}
\begin{itemize}

\item
J. J. Kohn, {\it Boundaries of complex manifolds}, in ``Proc.\ Conf.\ on
Complex Analysis'', Minneapolis 1964 Springer-Verlag, New York, 1965, pp.\
81--94.  Proves a ``Hodge theorem'' for the boundary Cauchy-Riemann
operator $\overline\partial_{b}$ on a CR manifold, analogous to Kohn's earlier
results for $\overline\partial$ on strictly pseudoconvex domains.  One of
the first systematic treatments of ``subelliptic'' differential operators.

\item
L. H\"ormander, {\it Hypoelliptic second-order differential equations},
Acta Math.\ {\bf 119} (1967) 147--171.  Shows that a real second-order
partial differential operator that can be written as a sum of squares of
vector fields is hypoelliptic provided the vector fields and their brackets
span the tangent space at each point.  This is the prototype of subelliptic
behavior which is now extremely important in several complex variables.

\item
G. B. Folland and E. M. Stein, {\it Estimates for the $\overline\partial_b$
complex and analysis on the Heisenberg group}, Comm.\ Pure Appl.\ Math.\
{\bf 28} (1974) 429--522.  Greatly extends and generalizes the results of
Kohn on subellipticity of the $\overline\partial_{\text{b}}$ complex.



\end{itemize}
\newpage
\section{Index theorems for elliptic operators}
\begin{itemize}

\item
M. Atiyah and I. M. Singer,
{\it The index of elliptic operators I}, Ann.\ Math.\ {\bf 87} (1968)
484--530.  This paper and the next four gave the initial proof of the
celebrated Atiyah-Singer index theorem, which relates the index of an
elliptic operator on a manifold (i.e.\ the dimension of the kernel minus
the codimension of the image) to topological invariants.

\item
M. Atiyah and  G. B. Segal, 
{\it The index of elliptic operators II}, Ann.\ Math.\ {\bf 87} (1968)
531--545.

\item
M. Atiyah and I. M. Singer,
{\it The index of elliptic operators III}, Ann.\ Math.\ {\bf 87} (1968)
546--604; {\it IV}, {\bf 93} (1971) 119--138; {\it V}, {\bf 93} (1971)
139--149. 

\item
M. Atiyah, R. Bott, and V. K. Patodi, {\it On the heat equation and the
index theorem}, Invent.\ Math.\ {\bf 19} (1973) 279--330; {\it Errata},
{\bf 28} (1975) 277--280.  Uses classical invariant theory to give a
greatly simplified proof of a key step in the Atiyah-Singer index theorem.
Good overview of the history of the index theorem and its proof.


\end{itemize}





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