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 Yang-Mills 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
Bernstein-Markov 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 non-degeneracy 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.

• Overcrowding and hole probabilities for random zeros on complex manifolds, with  Bernard Shiffman and  Scott Zrebiec  (Indiana Univ. Math. J. 57 (2008), 1977-1997 ).

We give large deviations estimates on probabilities of having no zeros or too many zeros in open domains of a \Kahler manifold for random polynomials or holomorphic sections, as the degree tends to infinity. These are higher dimensional generalizations of results of M. Sodin and others, which concerned the behavior as the radius tends to infinity of zeros of random holomorphic functions of one complex variable. (It would be interesting in the future to relate large degree and large radius asymptotics).

•  On the Spectrum of geometric operators on Kähler manifolds, with Dmitry Jakobson and Alexander Strohmaier (J. Mod. Dyn. 2 (2008), no. 4, 701--718. ) :

It is classical to use the Lefschetz operator and its adjoint to study the Lefschetz decomposition for harmonic forms. We do this for eigenforms of the Laplacian with eigenvalue tending to infinity. The Lefschetz decomposition of eigenspaces has an asymptotic statistical pattern. Moreover, there is a larger super-symmetry algebra acting on eigenspaces, which appears to originate in work of Frohlich et al, but which owes a lot to the analysis of my co-author A. Strohmaier. By studying this completely quantum symmetry, one finds that even if the unitary frame flow is ergodic, the Laplacian or Dirac operator fail to be quantum ergodic.

• Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties  (with Yanir A. Rubinstein).

In earlier work with Jian Song, we proved that the Monge-Ampere geodesic segments of the infinite dimensional symmetric space $\hcal$ of Kahler metrics in a fixed class could be $C^2$ approximated by the usual ODE geodesics of the spaces of Bergman metrics. Here we prove the same kind of result for harmonic maps of any Riemannian manifold with boundary into $\hcal$. A very special case is the Wess-Zumino-Witten model, where the source is a Riemann surface with boundary.

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 Phong-Sturm 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 Monge-Ampere equation on an A x M where A is an annulus.  Song and I analyze the Phong-Sturm 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 (Nazarov-Sodin) 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 two-part (or more part) series on the problem posed by Arezzo-Tian and Phong-Sturm of approximating geodesics in the infinite dimensional symmetric space of Kahler metrics in a fixed class by one-parameter subgroup geodesics in spaces of Bergman metrics. Phong-Sturm 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, 2209--2237, 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. Vice-versa, the classical Bernstein polynomials are intimately related to Bergman kernels for the Fubini-Study 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 Yau-Tian-Donaldson 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 Euler-MacLaurin (Dedekind-Riemann) sums over lattice points in a polytope of a kind studied by Guillemin-Sternberg and others. At least, it's cheap  for those who study Bergman kernels. It is not obvious that our asymptotics agree with those of Guillemin-Sternberg; 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 self-averaging 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 Sodin-Tsirelson 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) Patterson-Sullivan distributions $PS_{ir_j}$, which are
the residues of the dynamical zeta-functions $\lcal(s; a): = \sum_\gamma \frac{e^{-sL_\gamma}}{1-e^{-L_\gamma}} \int_{\gamma_0} a$ (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 Calabi-Yau 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 Calabi-Yau 3-fold  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 Calabi-Yau 3-fold 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 Yang-Mills theory, Gross-Matytsin and Kazakov-Wynter used a conjectured analytic continuation of Matytsin's asymptotics of   Itzykson-Zuber 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. Guionnet-O. 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, 611--626) 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 Yang-Mills 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 Gross-Matytsin and Kazakov-Wynter.  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 vary--how 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 Calabi-Yau 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 N-dependent  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 log-convexity 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 Balian-Bloch 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 semi-classical 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 Forges-Les-Eaux 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 Seminaire1997-8 at the Ecole Polytechnique. It is published as Expos\'e XXII of their seminar proceedings.

This is a talk given at Saint-Jean-de-Monts  and is published as expose XV in the 1998 Journees Equations aux Derivees Partielles Saint-Jean-des-Monts.

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 14-18 2008.

For a talk in the Foundations of Computational Mathematics, Hong Kong 16-26 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 Gross-Matytsin and Kazakov-Wynter 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 non-convex 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 Bernstein-Kouchnirenko 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.