RESEARCH
: PREPRINTS AND TALKS
My research centers around applications of microlocal
analysis to problems concerning

asymptotics of eigenfunctions/eigenvalues
on Riemannian manifolds,

statistical algebraic geometry,

problems of mathematical physics
ranging from quantum chaos to 2D YangMills to string/M theory.

asymptotics in Kahler geometry (Bergman kernel
approximations)
I post all of my recent articles on the arXiv.
Here I am posting expository articles, lectures, and new articles
in reverse chronological order.
NEW
PREPRINTS
We prove a large deviation principle for empirical measures $$
Z_s: = \frac{1}{N} \sum_{\zeta: s(\zeta) = 0}
\delta_{\zeta}, \;\;\; (N: = \# \{\zeta: s(\zeta) = 0)\}$$
of zeros of random polynomials in one variable. By random polynomial, we mean a
Gaussian measure
on the space $\pcal_N = H^0(\CP^1, \ocal(N))$ determined by inner products $G_N(h,
\nu)$ induced
by any smooth Hermitian metric $h$ on $\ocal(1) \to \CP^1$ and any probability
measure $d\nu$ on $\CP^1$ satisfying the weighted
BernsteinMarkov inequality. The speed of the LDP is $N^2$ and the rate function
is closely related to the weighted energy of probability measures on $\CP^1$,
and in particular its unique minimizer is the weighted equilibrium measure.
In a sequel, I prove an LDP for higher genus Riemann surfaces.
The higher dimensional case is out of sight at this time.
We prove that bounded analytic domains $\Omega \subset
\R^n$ with $\pm $ mirror symmetries across all coordinate axes, and with one
axis height fixed (and also
satisfying some generic nondegeneracy conditions) are spectrally determined
among other such domains. That is, you can hear the shape of a real
analytic drum in any number of dimensions if you know in advance that the
mystery drums have the symmetries of an ellipsoid. To our knowledge, it is the
first positive higher dimensional inverse spectral result for Euclidean domains
which is not restricted to balls.
This article compares the distribution of
real and complex zeros for Gaussian random combinations of eigenfunctions
of the Laplacian with frequencies taken from short intervals. The
distribution of real zeros is a rather straightforward application of the
formalism developed with P. Bleher and B. Shiffman for Gaussian random functions
in geometric settings. We also determine the distribution of complex zeros of
the analytic continuations of the Riemannian random waves to a Grauert
tube.
This is only an initial study of the subject. There are
many obvious directions to explore: (i) the variance of the random zeros; (ii)
more detailed local Weyl laws for analytic continuations of eigenfunctions;
(iii) distribution of critical points.
This article is about geodesic rays in the space of Kahler
metrics on a Kahler manifold (M, omega) which were defined by PhongSturm from a
test configuration in the sense of Donaldson. It is one of the few, if not the
only, known way to define an infinite geodesic ray. Such a geodesic is a
solution of the homogeneous complex MongeAmpere equation on an A x M where A is
an annulus. Song and I analyze the PhongSturm ray in detail in the case
of a toric test configuration on a toric variety. We give explict formulae for
the ray and for the approximating Bergman geodesic rays. We use large deviations
and Varadhan's Lemmas to prove C^1 convergence of the approximating rays to the
limit ray.
Suppose that $\Omega \subset \R^2$ is a piecewise real
analytic plane domain.
Then the number $n(\lambda_j) = \# \ncal_{\phi_j} \cap \partial
\Omega$ of boundary nodal points of the $j$th Neumann eigenfunction satisfies
$n(\lambda_j) \leq C
\lambda_j$, where $C$ is a constant depending only on $\Omega$ and $\Delta \phi_j
= \lambda_j^2 \phi_j$. The number of critical points of a Dirichlet or
Neumann eigenfunction satisfies the same bound. It follows that the number of
nodal components (`nodal lines') which intersect the boundary is of this order.
It is known (NazarovSodin) that the average number of nodal components of a
random spherical harmonic is of order $\lambda_j^2$, so our result suggests that
nodal loops have one higher order in $\lambda_j$ than nodal lines which touch
the boundary. I. Poletorvich has used this result to prove an old conjecture of
Pleijel on the number of nodal domains of a Neumann eigenfunction.
Joint work with Jian Song, proving $C^2$ convergence of Bergman
geodesics to Monge Amp\`ere geodesics on general toric Kaehler manifolds.
This paper generalizes the $\CP^1$ paper to any toric \Kahler manifold. The
proof of $C^2$ convergence uses somewhat different tools than the previous
paper.
Joint work with Jian Song. This is the first of a twopart (or
more part) series on the problem posed by ArezzoTian and PhongSturm of
approximating geodesics in the infinite dimensional symmetric space of Kahler
metrics in a fixed class by oneparameter subgroup geodesics in spaces of
Bergman metrics. PhongSturm showed that the finite dimensional geodesics
approach the infinite dimensional ones in C^0 and we prove it occurs in C^2 in
the case of S^1 invariant metrics on CP^1. In the sequel we will prove the
same for all toric varieties. In a sense we are solving the complex
homogeneous Monge Ampere equation by means of polynomial approximations. The
method uses the fact that on toric varieties, one can linearize the Monge Ampere
equation and use real convex analysis to obtain results on complex
plurisubharmonic functions. In a third paper in the series, we study the
geodesic rays associated to Donaldson's toric degeneration test configurations
by the same methods. Appeared in Ann. Inst. Fourier (Grenoble) 57 (2007),
no. 6, 22092237, volume in honor of Y. Colin de Verdiere's 60 birthday.
Bernstein polynomials $B_N f(x)$ give explicit degree N
polynomial approximations to continuous functions $f$ on $[0,1]$ using the
values of $f$ at the points $j/N$. While working on approximations of
transcendental metrics by Bergman metrics on toric varieties with J. Song, I
noticed that our Bergman approximations were very similar to Bernstein
polynomials. Viceversa, the classical Bernstein polynomials are intimately
related to Bergman kernels for the FubiniStudy metric on $\CP^m$. Has anyone
noticed this before? The relation leads one to define Bernstein polynomials for
any toric Kahler variety, and indeed on toric varieties the YauTianDonaldson
program is in some sense about Bernstein polynomial approximations. Moreover if
one integrates these polynomials over the polytope of the toric variety, one
obtains a rather cheap way to get asymptotics for EulerMacLaurin (DedekindRiemann)
sums over lattice points in a polytope of a kind studied by GuilleminSternberg
and others. At least, it's cheap for those who study Bergman kernels. It
is not obvious that our asymptotics agree with those of GuilleminSternberg; the
fact that they do amounts to a sequence of integration by parts identities on
the polytope.
Final and early version of joint work with B. Shiffman .
The article continues our series in statistical algebraic geometry.
In earlier articles, we showed that the simultaneous zeros of a system of
polynomials, or holomorphic sections of any positive line bundle over any Kahler
manifold, are on average uniformly distributed with respect to the
curvature volume form. In this article we study the random variable $N_U$
counting the number of simultaneous zeros in an open set $U$. This
measures the extent to which the zeros of a typical system conform to the
expected value. As one varies the system, the zeros can cross the boundary of
$U$ in either direction and this makes the fluctuations much larger than for
smooth statistics, where one sums a smooth function over the zeros.
However, $N_U$ is proved to be selfaveraging in a precise sense, so in a
computer simulation, one should see the zeros of an individual system quite
close to the expected distribution.
The newer
version concentrates completely on the point case, i.e. where the simultaenous
zeros form a point process; it will appear in GAFA. The older contains results
which won't appear in the GAFA article (e.g. on smooth statistics; our earlier
arxiv posting also extended a result of SodinTsirelson on asymptotic normality
of smooth statistics in various ways). The GAFA version also contains an
entirely new treatment of various technical issues regarding smoothing of
currents (due entirely to B. Shiffman); this has an independent interest.
We relate the two types of
phase space distributions associated to eigenfunctions
$\phi_{ir_j}$ of the Laplacian on a compact hyperbolic surface
$X_{\Gamma}$:
(a) Wigner distributions $\int_{S^*\X} a \;dW_{ir_j}=\langle
Op(a)\phi_{ir_j}, \phi_{ir_j}\rangle_{L^2(\X)}$, which arise in
quantum chaos. They are invariant under the wave group.
(b) PattersonSullivan distributions $PS_{ir_j}$, which are
the residues of the dynamical zetafunctions $\lcal(s; a): = \sum_\gamma
\frac{e^{sL_\gamma}}{1e^{L_\gamma}}
\int_{\gamma_0} a$ (where the sum runs over closed geodesics) at the poles $s =
\frac{1}{2} + ir_j$. They are invariant
under the geodesic flow.
We prove that these distributions (when suitably normalized) are
asymptotically equal as $r_j \to \infty$. We also give exact
relations between them. This correspondence gives a new relation
between classical and quantum dynamics on a hyperbolic surface,
and consequently a formulation of quantum ergodicity in terms of
classical ergodic theory.
Joint work with M. R. Douglas and B. Shiffman. Pdf file of
our third article in a series. It is concerned with a basic (and rather
controversial) problem in string/M theory: count the number of vacua. Each
vacuum is a candidate for the vacuum state of our universe. We are counting
vacua in type IIb string/M theory compactified on a CalabiYau manifold with
flux. It was originally hoped in string/M theory that a unique vacuum would
emerge as the `right' one to model our universe. But this time, no
selection principle is known, and there seem to exist a lot of candidates and no
selection principle. For each CalabiYau 3fold X one has a
different counting problem, and one should also sum up the solutions over the
possible topological types. In this article we fix X and count the vacua in the
associated model.
Vacua are defined mathematically as critical points of flux
superpotentials, which are holomorphic sections of a line bundle over the moduli
space of complex structures on a CalabiYau 3fold X times the moduli space of
elliptic curves. More precisely, they are restrictions to a fundamental domain
of sections over the Teichmuller space.The line bundle is dual to the H^{3,0}
form bundle over moduli space and the flux superpotentials W_G correspond
to fluxes G in H^3(X, \Z + i Z). The fluxes thus form a lattice in C^{b_3} where
b_3 = the third betti number of X. Each flux corresponds to a section and each
section has a number of critical points on moduli space. The aim is to
count the total number of critical points when the flux satisfies a hyperbolic
constraint known as the `tadpole constraint'.
The counting argument involves two ingredients. First, we need
to approximate the discrete lattice point ensemble by a continuous Gaussian one.
This is a purely lattice point problem: if one projects lattice points in a
region onto a hypersurface, how fast do they become equidstributed as one
dilates the region? Second, one needs to find and analyze the formula for the
expected density of critical points in the Gaussian ensemble. The analysis is
very complicated due to the high dimensionality of the problem. Counting
critical points (i.e. vacua) in this model bears some resemblence to counting
metastable states in a glass and gives rise to a similar integral formula.
The distribution of nodal hypersurfaces of eigenfunctions
is an all but impossible problem. But it turns out to simplify on real
analytic manifolds (M, g) if we holomorphically extend eigenfunctions to the
complexification of M and consider complex nodal hypersurfaces.
It is similar to the simplifying effect of complex over real algebraic
geometry. When the geodesic flow is ergodic,
we obtain an equidistribution law for complex nodal hypersurfaces.
Pdf file of a new article with T. Tate in which we give a
counterexample to the asymptotics of character values chi_R(U) of SU(N)
characters. To evaluate partition functions in 2d YangMills theory, GrossMatytsin
and KazakovWynter used a conjectured analytic continuation of Matytsin's
asymptotics of ItzyksonZuber integrals to obtain asymptotics of
SU(N) characters. Matytsin's asymptotics of chi_R(e^A) have recently been
proved to be correct by A. GuionnetO. Zeitouni, where A is a Hermitian matrix.
The asymptotics of chi_R(e^{i A}) are shown to be different from the
predictions in this article. The counterexample is closely related to the one in
the next article, although the calculations are quite different.
Pdf file of a new article (now in Comm.
Math. Phys. 245 (2004), no.
3, 611626) which
gives a counterexample to asymptotics of (central) heat kernel values for SU(N).
The central heat kernel was shown by Migdal to be the partition function of 2D
YangMills theory on the cylinder. The Macdonald identities in the title allow
for explicit evaluation of the large N asymptotics Z_N(A, a_N, V_N) of the
partition function (central heat kernel) when one argument is evaluated at the
conjugacy class of the Coxeter element of SU(N). The other argument can be any
sequence. The asymptotics are not of the type exp(N^2 F) as expected but
exp(N^3 F). This does not appear consistent with some ideas on how the large N
limit of gauge theory should be a string theory. A possible explanation is that
the large N limit should not be taken in this kind of pointwise sense, though
that is how it was taken in the articles of GrossMatytsin and KazakovWynter.
M. Douglas' ideas on the large N limit being a conformal field theory offer an
alternative approach to large N limits which might cure the disease. Comparison
with the article above suggests that the original conjectured asymptotics could
be valid if the partition function is analytically continued and evaluated on
positive matrices rather than unitary matrices. What does that mean to string
theory? (Recently, A. Guionnet and M. Maida have proved this conjecture).
Joint work with M. Douglas and B. Shiffman. A continuation of
the article below. In this one, we study a new metric invariant of a
positive Hermitian holomorphic line bundle L > M over a Kahler
manifold: the average number of critical points of a Gaussian random holomorphic
section. The critical point equation is that D s (z) = 0 and it depends on the
connection D associated to the metric. Although it is clear that the set and the
number of critical points of a holomorphic section will vary as the metric (or
connection) varies, it is not clear that the average number should varyhow
does one know the average number isn't a topological invariant? To
investigate this, we consider powers L^k of L. It turns out that the
average number N( L^k) of critical points of random holomorphic sections of L^k
has a complete asymptotic expansion in k, which is topological to two orders in
k. So the metric dependence is rather subtle. We find that the metric dependence
is in the Calabi functional, the integral of the square of the scalar curvature
of the curvature (1,1) form of (L, h). We prove that Calabi extremal
metrics are unique asymptotic minimizers of N(L^k) in dimensions < 4. We
conjecture that the same is true in higher dimensions, but the proof requires a
computer evaluation of a difficult integral which we could only complete in low
dimensions.
Joint work with M. Douglas and B. Shiffman. We study
the statistics of critical points of holomorphic sections of Hermitian line
bundles over complex manifolds. Critical points depend on the choice of
metric or connection, and that gives them quite a different flavor from the
study of statistics of zeros of holomorphic sections. Physically, the
superpotential of a string/M theory is a holomorphic section of a line bundle
over the configuration space (the moduli space of complex structures on a
CalabiYau manifold). Our universe is a vacuum of the superpotential, loosely
speaking a critical point of the superpotential (precisely, a critical point is
a supersymmetric vacuum). Physicists don't know which vacuum of which
superpotential, so M. R. Douglas has proposed to study the string/M theory
statistically. B. Shiffman and I have been studying statistical algebraic
geometry and our methods apply to this problem, so we have joined forces. There
are quite a lot of open problems to explore here.
Joint work with T. Tate, to appear in Jour. Funct. Anal.
We give a detailed asymptotic analysis of the multiplicity of an
Ndependent weight nu_N in a highe tensor power V_{\lambda}^N of an
irreducible representation of a compact Lie group G. It turns out that such
multiplicities are of exponential growth depending on where nu_N lies in the Nth
dilate of the Weyl orbit of N lambda. The asymptotics are equivalent to
those for lattice paths with steps coming from lattice points in a convex
polytope. They are similar to multinomial coefficients. Relations to Okounkov's
logconvexity of multiplicities conjecture are discussed.
Second paper in a series on the inverse spectral problem for
analytic plane domains. In the first paper in the series, I give rigorous
version of the BalianBloch trace asymptotics for the resolvent. It gives a new
algorithm for calculating wave invariants associated to periodic reflecting rays
of bounded domains. In this paper, I calculate the wave invariants for a
bouncing ball orbit (among others), and use them to determine the domain when
the domain is analytic and possesses one symmetry. The symmetry is assumed
to interchange the two endpoints of the bouncing ball orbit. The calculation
uses some new methods and I have spent a fair amount of effort verifying the
different steps before submitting the article. Any comments would be much
appreciated.
EXPOSITORY ARTICLES
This is a survey of results on eigenfunctions. The idea of the
survey is to include both local results, e.g. doubling or vanishing order
estimates, Bernstein inequalites and global results using microlocal or
semiclassical methods. It is still in progress and has numerous gaps,
typos, inconsistencies and (probably) errors. There is no original material
here. It is taken from original sources almost verbatim. I am posting it now to make
it easier for people to help me correct it. It is intended for the Handbook of
Differential Geometry
An article expanding on my AMS address in Atlanta, January 2005. It describes
joint work with M. R. Douglas and B. Shiffman.
My address at QMath9 on counting vaca.
An article for the Elsevier Encyclopedia of Mathematical Physics.
Pdf file of a long survey to appear in the Journal
of Differential Geometry Surveys.
Pdf file of a talk at ForgesLesEaux 2003 on joint work with
A. Hassell and C. Sogge on boundary traces (Cauchy data) of eigenfunctions of
bounded domains.
Pdf file of my ICM talk. It describes extremal results on eigenfunctions I
proved with C. Sogge and J. A. Toth, and statistical results on zeros, norms and
other features of polynomials and holomorphic sections which I proved with P.A.
Bleher and B. Shiffman.
Pdf file of my ICMP2000 talk in London. It describes results
on random waves and their applications and motivations in quantum chaos, based
on joint work with P.A. Bleher and B. Shiffman. Since then, M. Douglas has
proposed new types of applications in String/M theory, which were the subject of
his ICMP2003 talk in Lisbon.
This is a seminar report given at the EDP Seminaire19978 at the
Ecole Polytechnique. It is published as Expos\'e XXII of their seminar proceedings.
This is a talk given at SaintJeandeMonts and is published as
expose XV in the 1998 Journees Equations aux Derivees Partielles SaintJeandesMonts.
This is a lecture given at the 1997 International Congress of Math Phys
(Brisbane), and appeared in the Proc. 1997 ICMP.
This is a collection of lectures given in 1996 at the Centre E. Borel
(Institut H. Poincare) during the semester on Disordered systems and quantum chaos.
It is an unpublished survey of some rigorous mathematical results in the area of quantum chaos.
LECTURES
For a talk in the 7th World Congress in Probability and Statistics,
Singapore, July 1418 2008.
For a talk in the
Foundations of Computational Mathematics,
Hong Kong 1626 June 2008.
An introduction to Gaussian random polynomials, sections of line bundles,
analytic functions, mostly of one variable. Emphasis on geometric and asymptotic
aspects.
An invited address for a general audience: Starting with random polynomials
in one complex variable, it introduces the distribution of zeros and critical
points of random holomorphic sections of line bundles. It then describes the
relevance of critical points of random holomorphic section to the vacua
statistics problem (M. R. Douglas). Some sample results from joint work of
the author with B. Shiffman and M.R. Douglas are given. Finally, some purely
geometric results relating distribution of critical points to Calabi extremal
metrics are presented.
Special session talk on distribution of zeros of complexifications of real
eigenfunctions on analytic Riemannian manifolds with ergodic geodesic flow.
Special session talk on the inverse spectral problem for analytic plane
domains, emphasizing the use of Feynman diagrams to calculate and enumerate wave
trace invariants. Also describes use of Dedekind sums to determine the domain
from the wave invariants.
Talk on joint work with Mike Douglas and Bernie Shiffman on statistics
of vacua in string theory. It describes our recent rigorous results on counting
vacua.
Talk on joint work with Tatsuya Tate on multiplicities of weights and irreps
in high tensor powers of an irrep of a compact Lie group. It goes over
asymptotic formulae and connections to lattice paths.
Talk on joint work with M. R. Douglas and B. Shiffman on the complex geometry
of the distribution of supersymmetric vacua in string/M theory.
Recalls some conjectures of GrossMatytsin and KazakovWynter on the
asymptotics of the partition function and then uses Macdonald's identity to give
a counterexample. Also includes a counterexample to their conjectured
asymptotics of characters of $SU(N)$ in the large $N$ limit. Discusses possible
corrections to the conjectures.
Lecture giving a survey of results on the inverse spectral
problem and its relations with dynamics.
October 23, 2003 Lecture at AIM on the zeros of systems of random
polynomials with fixed Newton polytope, and of systems of random fewnomials with
a fixed number of monomials. In recent work, B. Shiffman and I showed that by
fixing the Newton polytopes of a random system of polynomials, one induces a
`classically allowed' region for zeros and a `classically forbidden
region'. This has implications for tentacles of random amoebas. Moreover,
we found that our methods work if we constrain the spectra of the polynomials
not just to convex lattice polytopes but also nonconvex ones, possibly with
empty interiors. This allows us to study random fewnomial systems, where the
number of monomials is fixed as the degree tends to infinity. Khovanskii long
ago showed that random real fewnomial systems of large degree have a number of
real zeros depending only on the number of monomials. Our eventual goal is to
find the expected number. This lecture describes results on complex zeros
and results in progress on real zeros.
This May 2003 talk describes some representative results of my joint work
with P. Bleher and B. Shiffman on the distribution of and correlations between
zeros of polynomials of high degree in several variables. We prove that the
discrete zeros of systems of m polynomials in m variables change from repulsion
to neutrality to attraction as the dimension m increases from 1 to 2 to 3 and
higher. This (to us unexpected result) might be explained in terms of the
geometry of discriminant varieties in these different dimensions. We also
discuss how the Newton polytope impacts on the spatial distribution of zeros,
giving a new perspective on the BernsteinKouchnirenko theorem.
Spring 2002 Bergman lectures on statistical algebraic geometry. More detail
(though at an earlier phase) on our joint work with P. Bleher and B. Shiffman.