23
Spectral Theory and Global Analysis 14 th - 18 th August 2006 Book of Abstracts Carl von Ossietzky Universität Oldenburg Institut für Mathematik Germany, 26111 Oldenburg
Spectral Theory and Global Analysis
14th − 18th August 2006
Book of Abstracts
Carl von Ossietzky Universität OldenburgInstitut für Mathematik
Germany, 26111 Oldenburg
ii
Organizers:
• Daniel Grieser ([email protected])
• Thomas Krainer ([email protected])
• András Vasy ([email protected])
Website to this conference:http://www.mathematik.uni-oldenburg.de/personen/grieser/stga/
The material in this booklet has been supplied by the authors and has not been corrected by other persons.This document has been typeset with LATEX.
Contents
1 Abstracts 1M. Agranovich: Regularity of Variational Solutions to Linear Problems in Lipschitz Domains . . 1
B. Booss-Bavnbek: Curves of Lagrangian Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1
A. Chorwadwala: Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
B. Colbois: Capacity and Spectral Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
A. Djuraev: Singular-Perturbed Boundary-Value Problems . . . . . . . . . . . . . . . . . . . . . . . 3
G. Francsics: Analysis on Complex Hyperbolic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 5
L. Friedlander: Determinants of Zeroth Order Operators . . . . . . . . . . . . . . . . . . . . . . . . 5
G. Grubb: Basic Zeta Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
E. Hunsicker: Hodge Theorems for Manifolds with Edge Metrics . . . . . . . . . . . . . . . . . . . 6
D. Jakobson: Estimates from below . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
M. Karamehmedovic: Continuation of Solutions to Boundary Problems . . . . . . . . . . . . . . . 7
K. Kirsten: Functional Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Y. Kordyukov: Spectral Gaps for Periodic Schrödinger Operators with Magnetic Wells . . . . . . 8
M. Lesch: Relative Pairing in Cyclic Cohomologyy and Divisor Flows . . . . . . . . . . . . . . . . 9
P. Loya: Zeta Functions on Conic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
X. Ma: Bergman Kernels and Geometric Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 10
P. McDonald: The Eta Invariant for Quantum Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 11
R. B. Melrose: Simplicial Complexes and Pseudodifferential Operators . . . . . . . . . . . . . . . 12
G. Mendoza: b-complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
S. Moroianu: The Index of Dirac Operators on Manifolds with Non-exact Cusp Metrics . . . . . 13
W. Müller: Hodge Theory for Manifolds with Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . 13
B. Mümken: Pseudo Riemannian Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
J. Park: Spectrum of Dirac operator over Cofinite Quotients of PSL(2, R) . . . . . . . . . . . . . . 14
Y. Petridis: Applications of Perturbation Theory to the Spectral Theory . . . . . . . . . . . . . . . 14
P. Piazza: Groups with Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
D. Pliakis: Small Time Heat Expansion for the Laplace-Beltrami . . . . . . . . . . . . . . . . . . . 16
F. Rochon: Eta Invariant and the Determinant Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Y. Safarov: Approximate Spectral Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
A. Savin: On the Homotopy Classification of Elliptic Operators on Manifolds with Corners . . . 17
E. Schrohe: Tangent Groupoid for Manifolds with Boundary . . . . . . . . . . . . . . . . . . . . . 17
B. Schulze: Asymptotics on Manifolds with Corners . . . . . . . . . . . . . . . . . . . . . . . . . . 18
A. Strohmaier: High Energy Limits and Frame Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 18
J. Toth: Maximal Blowup of Modes and Quasimodes . . . . . . . . . . . . . . . . . . . . . . . . . . 19
G. Weingart: Some Remarks about Heat Kernel Coefficients . . . . . . . . . . . . . . . . . . . . . . 19
S. Zelditch: Complex Zeros of Real Ergodic Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . 19
1 Abstracts
Regularity of Variational Solutions to Linear Problems in LipschitzDomains
Mikhail Agranovich
We consider a second order strongly elliptic system in a Lipschitz domain. We discuss the regularity inLebesgue and Besov spaces of variational (weak) solutions to the Dirichlet and Neumann boundary valueproblems as well as the regularity of eigenfunctions of the spectral Dirichlet, Neumann and Poincare–Steklov boundary value problems.
Curves of Lagrangian Subspaces in Symplectic Hilbert Space
Bernhelm Booss-BavnbekThis is about a joint work with Chaofeng Zhu
Generalizing approaches due to M. Morse and A. Floer, there have been various formulas expressingthe spectral flow of a curve of self-adjoint elliptic operators on a partitioned manifold by the Maslov indexof the induced curve of Cauchy data spaces along the separating hypersurface. In this talk we give anovel approach based on the classic concept of symplectic reduction which points out the necessary andsufficient conditions for such spectral flow formulas for closed symmetric Fredholm relations.
Extremum of the Energy Functional over Family of Domains inSpace Forms
Anisa M.H. ChorwadwalaThis is about a joint work with A.R. AithalAMS subject classification: 35J25, 35P15, 53C21, 58J32, 58J50
Let (M, g) be a Riemannian manifold, Ω ⊂ M an open set such that Ω is a smooth compact submanifoldof M. For a smooth function f : M −→ R, let ∇ f denote the gradient of f . The (positive) Laplacian ∆ isdefined by ∆ f = −div(∇ f ). Let y := y(Ω) ∈ H1
0 (Ω) denote the weak solution of the Dirichlet BoundaryValue Problem:
∆u = 1 on Ω,
u = 0 on ∂Ω
)(I)
2
Let Sn denote the unit sphere in Rn+1 centered at the origin with induced Riemannian metric 〈 , 〉 fromthe Euclidean space Rn+1. Consider the hyperbolic space
Hn :=
((x1, x2, . . . , xn+1) ∈ Rn+1
nX
i=1
x2i − x2
n+1 = −1 ∧ xn+1 > 0
)
with the Riemannian metric induced from the quadratic form
(x, y) :=nX
i=1
xi yi − xn+1 yn+1
where x = (x1, x2, . . . , xn+1), y = (y1, y2, . . . , yn+1).Let B1 be a ball of radius r1 in Sn(Hn), and let B0 be a smaller ball of radius r0 such that B0 ⊂ B1. For
Sn we consider r1 < π. Let Ω = B1 \ B0. Let F denote the family of such domains Ω. Consider the EnergyFunctional
E(Ω) :=Z
Ω||∇y(Ω)||2 dVol =
ZΩ
y dVol
on F , associated to problem (I).Following [3], using the results of Shape calculus (developed for Riemannian Geometry from [5]) and
maximum principles for elliptic PDEs [4] we prove our following main result :
Put Ω0 = B(p, r1) \ B(p, r0) for any fixed p ∈ Sn (Hn).
1.3.1 Theorem
The Energy functional E(Ω) on F assumes minimum at Ω if and only if Ω = Ω0, i.e., when theballs are concentric.
References(1) Aubin, T., Nonlinear Analysis on Manifolds - Monge-Ampere Equations, Springer-Verlag, 1982.
(2) Folland, G., Introduction to Partial Differential Equations, Prentice-Hall of India Private Limited, New Delhi,2001.
(3) Kesavan, S., On Two Functionals Connected to the Laplacian in a Class of Doubly Connected Domains,Proceedings of the Royal Society of Edinburgh 133A (2003) 617–624.
(4) Protter, M. and Weinberger H., Maximum Principles in Differential Equations, Prentice-Hall Inc., New Jersey,1967.
(5) Sokolowski, J., Zolesio, J-P., Introduction to Shape Optimization - Shape Sensitivity Analysis, Springer-Verlag,1992.
(6) Todhunter, I., Spherical Trigonometry, Macmillan and Co. Ltd., London, 1949.
3
Capacity and Spectral Stability
Bruno ColboisThis is about a joint work with Jérôme BertrandAMS subject classification: 31C15, 58G25, 35P15
By the Faber-Krahn inequality, the first eigenvalue of the Dirichlet problem for a domain of constantvolume in Rn is minimum for the ball. If a domain has its first Dirichlet eigenvalue close to the firstDirichlet eigenvalue of the ball, I will show that the same is true (non uniformly) for all the correspondingeigenvalues. The proof uses a notion of capacity adapted to the Dirichlet problem.
References(1) Bertrand, J.; Colbois, B. Capacité et inégalité de Faber-Krahn dans Rn . (French) [Capacity and Faber-Krahn
inequality in Rn] J. Funct. Anal. 232 (2006), no. 1, 1–28
Singular-Perturbed Boundary-Value Problems with a MultiplePure Imaginary Spectrum
Abubakir Djuraev
Using results received in [1] - [5], we shall develop of the method asymptotical integration of a boundary-value problem with Jordan many crate matrix. The equation(1.5.0.1)
εy′(t, ε)− A(t)y(t, ε) = h(t),
with a boundary-value condition(1.5.0.2)
Gy ≡n
y1(0, ε), . . . , yn0 (0, ε), yn0+1(1, ε), . . . yn(1, ε)o
= y0.
The following conditions let are executed:
10. A(t) ∈ C∞([0, 1], Cn×n), h(t) ∈ C∞([0, 1], Cn)
20. The spectrum λij(t) with i = 1, . . . , r, j = 1, . . . , ni and n1 + · · ·+ nr = n of a matrix A(t) meetsthe following requirements:
(a) λij(t) = λi(t) for i = 1, . . . , r and j = 1, . . . , ni
(b) Re λi(t) ≡ 0 for i = 1, . . . , r
(c) λi(t) 6= 0 for i = 1, . . . , r
30. The matrix A(t) has of a Jordan chain of vectors r of length ni with i = 1, . . . , r, i.e. ∀t ∈ [0, 1] :A(t)ϕi1(t) = λi(t)ϕi1(t), A(t)ϕij(t) = λi(t)ϕij(t) + ϕi,j−1(t) for i = 1, . . . , r and j = 2 . . . , ni
40. The initial structure of a matrix A(t) does not vary on a piece [0, 1].Derivative from eigen- an ad joint vectors we shall spread out on basis:
ϕij(t), i = 1, . . . , r , j = 1, . . . , ni , ϕ′ij(t) =rX
k=1
nkXs=1
Cksij (t)ϕks(t).
4
50. Shall require that for Cinii1 was satisfied condition
∀t ∈ [0, 1] : Re ni
r−Cini
i1 (t)ξki < 0 i = 1, . . . , r , k = 1, . . . , n0
i
∀t ∈ [0, 1] : Re ni
r−Cini
i1 (t)ξki > 0 i = 1, . . . , r , k = n0
i + 1, . . . , ni
with n01 + n0
2 + · · · n0r = n0,
where ξi –primitive roots ni – of a degree from unit.
For asymptotical integration of a boundary value problem 1.5.0.1, 1.5.0.2 at performance of conditions10.–40. we shall enter additional independet variable under the formulas:
τik =1ε
Z t
0
hλi(x) + ni
√εg1,i,k(x) + · · ·+ ni
√ε
ni−1gni−1,i,k(x)i
dx ≡ Ψik(t, ε) i = 1, . . . , r , k = 1, . . . , n0i
τik =1ε
Z t
1
hλi(x) + ni
√εg1,i,k(x) + · · ·+ ni
√ε
ni−1gni−1,i,k(x)i
dx ≡ Ψik(t, ε) i = 1, . . . , r , k = n0i + 1, . . . , ni
where n0i =
h ni2
ifor i = 1, . . . , r and instead of the required decision y(t, ε) tasks 1.5.0.1, 1.5.0.2 we shall
study the »extended« function u(t, τ, ε) such that u(t, τ, ε)|τ=Ψ(t,ε) ≡ y(t, ε) where τ = (τ11, . . . , τrnr ),Ψ(t, ε) = (Ψ11(t, ε), . . . , Ψrnr (t, ε)).For definition of function u(t, τ, ε) we shall put the following problem(1.5.0.3)
ε∂u∂t
+rX
i=1
niXk=1
λi(t) + ni
√εg1,i,k(t) + · · ·+ ni
√ε
ni−1gni−1,i,k(t) ∂u
∂τik− A(t)u = h(t)
(1.5.0.4)G0u(M0, ε) + G1u(M1, ε) = y0
at ε → 0. As the problem 1.5.0.3,1.5.0.4 is regular on ε at ε → 0, naturally its decision we shall define asseries(1.5.0.5)
u(t, τ, ε) =∞X
s=−s02
s0√
εsus(t, τ)
where s02 =
hs0−1
2
i, s0 = LCMn1, . . . , nr. With factors from without resonance space of the decisions
U =
8<:u(t, τ) : u =rX
i,k=1
niXj=1
nkXs=1
ui,j,k,s(t)ϕij(t)eτks +rX
i=1
niXj=1
ui,j(t)ϕij(t), ui,j,k,s(t), ui,j(t) ∈ C∞([0, 1], C)
9=;1.5.1 Theorem
Let for a boundary-value problem 1.5.0.1, 1.5.0.2, the conditions 10.–50. and decision of a problem1.5.0.3 are executed, 1.5.0.4 is determined as a sedate series 1.5.0.5 of space U.Then the narrowing of a series 1.5.0.5 at τ = Ψ(t, ε) and ε → 0 is asymptotical series for thedecision of a problem 1.5.0.1, 1.5.0.2.
The received results can be shown with the help of the follwing circuit:
5
From the circuit it is visible, that with the help of the asymptotical integrate method developed thesingular-perturbed equation 1.5.0.1 with a matrix A(t), when A(t) consists of any quantity of Jordancrates and has any quantity of zero.
References(1) Djuraev A.M. Asymptotical integration of a boundary-value problem with a multiple only imaginary
spectrum // The collection of the proceedings.- M., 1987. No. 141. – P. 30-34.
(2) Djuraev A.M. A boundary-value problem with an astable spectrum // Researches on the integro-differentialequations. - Bishkek: Ilim, 1992. – 24. - P. 193-197.
(3) Djuraev A.M. A boundary-value problem in case of spectral features of the limiting operator // Researcheson the integro-differential equations. – Bishkek: Ilim, 1998. – 27. - P.190-194.
(4) Djuraev A.M. A problem with a multiple pure imaginary spectrum // Ist Turkish world mathematicssymposium. - Elazig: Firat University, 1999. – P.124-127.
(5) Djuraev A.M. Asymptotical integration of the singular-perturbed problems with a multiple spectrum. -Osh, 1999. – 108 p.
Analysis on Complex Hyperbolic Spaces
Gabor FrancsicsThis is about a joint work with Peter LaxThanks to the Mathematical Institute of the Oxford University and the Schödinger Institute of Vienna for theirhospitality. The research was supported in part by a Michigan State University Intramural Research Grant
Our main interest is the spectral and scattering analysis of the automorphic Laplace-Beltrami operatorfor a class of discrete quotients of the complex hyperbolic space with finite volume. This class containsthe Picard modular group with Gaussian integers.
Determinants of Zeroth Order Operators
Leonid FriedlanderThis is about a joint work with Victor Guillemin
Let Q be a positive first order elliptic pseudodifferential operator on a closed manifold X. Assume thatall trajectories of the bicharacteristic flow on T∗X \ X generated by the principal symbol of Q are periodic,with the same period 2π. One can assume that the spectrum of Q consists of positive integers. By Pn wedenote the orthogonal projection on the space generated by eigenfunctions of Q that correspond to theeigenvalues that do not exceed n. Let B be a zeroth order pseudodifferential operator that is close enoughto the identity. A theorem of Guillemin and Okikiolu sais that
log det PnBPn ∼ b +−∞X
k=d,k 6=0
bknk + b0 log n
6
The constant term, b, in this expansion can be called the logarithm of the Szego regularized determinantof B.
On the other hand, trace log BQz is a meromorphic function in the complex plane; 0 is either its regularpoint or a simple pole. The finite part of this function at z = 0, wQ(B), is the logarithm of the zeta-regularized determinant of B. Both the number b and the number wQ(B) are not local invariants. However,and this is our main result, the difference between them is an explicitly computable quantity.
The Local and Global Ingredients in the Basic Zeta Coefficient forBoundary Problems
Gerd Grubb
On a compact manifold X with boundary ∂X = X′, let B = P+ + G be an operator in the Boutet deMonvel calculus, and let P1 be a second-order elliptic differential operator on X and T a differential traceoperator from X to X′, defining an elliptic realization P1,T having R− as a spectral cut. The basic zetacoefficient is the regular value C0(B, P1,T) at s = 0 of the zeta function
ζ(B, P1,T , s) = Tr(B(P1,T)−s)
(meromorphic extension), corrected for a nullspace contribution.The aim of the talk is to show how C0(B, P1,T) can in general be written in local coordinates as the
sum of a finite-part integral of the symbol of B (as in Kontsevich and Vishik’s canonical trace) and anoncommutative-residue-like contribution from B log P1,T . The first ingredient depends on the full struc-ture (is global) and the second ingredient depends only on some homogeneous symbol terms (is local).This generalizes a result of Scott and Paycha for the closed manifold case, shown by rather differentmethods. In particular, logarithms of elliptic boundary value problems are investigated.
Hodge Theorems for Manifolds with Edge Metrics
Eugenie HunsickerPartially supported by the NSF through an ROA supplement to grant DMS-0204730
Consider a manifold M, with boundary, ∂M where ∂M is a compact fibre bundle with fibre F over acompact base, B. Endow M with a metric that makes a neighbourhood of ∂M quasi-isometric to a bundleover B whose fibres are truncated cones over F. We call such a metric an incomplete edge metric. We mayalso endow M with a metric for which a neighbourhood of any point on ∂M is quasi-isometric to a neighbourhoodof a point on the boundary of hyperbolic space crossed with a compact manifold. We call such a metric a completeedge metric.
Incomplete and complete edge metrics on M are conformally equivalent, which permits us to findHodge theorems for both using the same analytic techniques. Incomplete edge metrics have been studiedby Dai and Cheeger in the case that F is odd dimensional or has no middle degree cohomology. In thiscase, the Laplace operator on forms has a unique closed extension, and Cheeger has identified the spaceof L2 harmonic forms with respect to this extension with the (also unique) middle perversity intersectioncohomology of M. In the case that F is odd dimensional and has cohomology in its middle degree, the
7
Laplace operator on forms has several possible closed extensions. In addition, the middle perversityintersection cohomology of M is no longer unique.
In my talk, I will discuss Hodge theorems for three natural extensions in the case of a fibre with middledegree cohomology. This involves both soft analytic results, similar to those in the work of Cheeger, andharder analytic techniques based on Mazzeo’s edge calculus. Finally, I will state a Hodge theorem forcomplete edge metrics, which follows from the analysis in the incomplete case.
Estimates from below for the Spectral Function and for the Errorterm in Weyl’s Law
Dmitry JakobsonThis is a joint work with Iosif Polterovich and John TothThanks to NSERC, FQRNT and Dawson Fellowship
We obtain lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl’slaw on compact manifolds. In negative curvature, thermodynamic formalism is applied to improve theestimates. Our results can be considered pointwise versions (on a general manifold) of Hardy’s lowerbounds for the error term in the Gauss circle problem. Our work generalizes earlier results of Hejhal,Randol and Karnaukh. The results were announced in ERA-AMS 11 (2005), 71-77. The full text is availableat math.SP/0505400.
We next obtain lower bounds for the remainder in (global) Weyl’s law on compact negatively curvedmanifolds, generalizing earlier results of Hejhal and Randol on surfaces of constant negative curvature.Our approach uses wave trace asymptotics, equidistribution of closed geodesics and small-scale microlo-calization.
Continuation of Solutions to Boundary Problems
Mirza KaramehmedovicI cordially thank Prof. András Vasy for the many fruitful discussions regarding the matter presented here
Let eω be a complex manifold of complex dimension n, and eE, eF holomorphic vector bundles over eω suchthat ∀z ∈ eω dimC
eEz = dimCeFz = d. Let eP = OP (ep) be an elliptic holomorphic differential operator of
order m, mapping O eω, eE into O
eω, eF. Let ω be a Cω totally real submanifold of eω of real dimension
n, and assume eψ ∈ O ( eω) has a nondegenerate level set eω0 ⊂ eω noncharacteristic w.r.t. eP and such thatω0 = eω0 ∩ω 6= ∅ is an oriented hypersurface in ω. Let ω± be the two sides of ω0 in ω, and assume thatevery characteristic of eP intersecting ω− also intersects eω0. Let eG, eJ be holomorphic vector bundles overeω0 such that ∀z ∈ eω0 dimC
eG = md, and assume eB is a holomorphic differential operator from O eω0, eG
into O eω0, eJ. Set E, F to be the restrictions of eE, eF to ω and G, J to be the restrictions of eG, eJ to ω0, and
let P : Cω (ω, E) → Cω (ω, F), B : Cω (ω0, G) → Cω (ω0, J) be the obvious restrictions of eP and eB to ω
and ω0, respectively. Assume B is injective on the range of the usual Calderón projector associated withP and ω+.
Our continuation result is now as follows:
8
1.11.1 Theorem
If g ∈ Cω (ω0, J) can be extended holomorphically to eω0 then there exists a function u ∈ Cω (ω, E)such that Pu = 0 in ω+ and Bγ+u = g in ω0, where γ+ is the usual trace operator on ω0 fromω+.
Dimensional Reduction of Functional Determinants
Klaus Kirsten
It is well known that functional determinants of Sturm-Liouville type operators in one variable caneasily be evaluated. It is therefore highly desirable, if possible, to express the determinants of operators inseveral variables in terms of one-dimensional ones. In this talk it is shown how the dimensional reductionfor a Laplace-type operator L = −∆ + V(r) in RD with a spherically symmetric potential V(r) can beperformed. In detail, the determinant of L will be expressed in terms of determinants related to the(one-dimensional) radial differential operator
Ll = − d2
dr2 −D− 1
rddr
+l(l + D− 2)
r2 + V(r),
which is obtained after separation of variables and where l is the angular momentum. For a given specifiedpotential V(r) the result can be used effectively to provide an explicit answer for the determinant.
Spectral Gaps for Periodic Schrödinger Operators with MagneticWells
Yuri A. KordyukovSupported in part by Russian Foundation of Basic Research, grant 04-01-00190
Let M be a noncompact oriented manifold of dimension n ≥ 2 equipped with a properly disconnectedaction of a finitely generated, discrete group Γ such that M/Γ is compact. Suppose that H1(M, R) = 0, i.e.any closed 1-form on M is exact. Let g be a Γ-invariant Riemannian metric and B a real-valued Γ-invariantclosed 2-form on M. Assume that B is exact and choose a real-valued 1-form A on M such that dA = B.
Consider a Schrödinger operator
Hh = (ih d + A)∗(ih d + A)
with magnetic potential A as a self-adjoint operator in the Hilbert space L2(M) (here h > 0 is a semiclas-sical parameter). Our main goal is to find sufficient conditions on the magnetic field, which ensure theexistence of a gap (or, even more, an arbitrarily large number of gaps) in the spectrum of Hh for any h > 0small enough. Here by a gap in the spectrum σ(T) of a self-adjoint operator T we mean an interval (a, b)such that (a, b) ∩ σ(T) = ∅.
For any x ∈ M denote by B(x) the anti-symmetric linear operator on the tangent space Tx M associatedwith the 2-form B:
gx(B(x)u, v) = Bx(u, v), u, v ∈ Tx M.
9
Recall that the intensity of the magnetic field is defined as
Tr+(B(x)) =12
Tr([B∗(x) · B(x)]1/2).
Letb0 = minTr+(B(x)) : x ∈ M.
Assume that there exist a (connected) fundamental domain F and ε0 > 0 such that(1.13.0.6)
Tr+(B(x)) ≥ b0 + ε0, x ∈ ∂F .
For any ε1 ≤ ε0, letUε1 = x ∈ F : Tr+(B(x)) < b0 + ε1.
Thus Uε1 is an open subset of F such that Uε1 ∩ ∂F = ∅ and, for ε1 < ε0, Uε1 is compact and includedin the interior of F . Any connected component of Uε1 with ε1 < ε0 can be understood as a magnetic well(attached to the effective potential h · Tr+(B(x))).
Consider the set U+ε0
, which consists of all x ∈ Uε0 such that the rank of B(x) is constant in an openneighborhood of x. Assume that(1.13.0.7)
Tr+B is not locally constant on U+ε0
.
This holds for any B, satisfying the assumption (1.13.0.6), if the dimension n equals 2 or 3.
1.13.1 Theorem
Under the assumptions (1.13.0.6) and (1.13.0.7), for any natural N, there exists h0 > 0 such that,for any h ∈ (0, h0], the spectrum of Hh contained in [0, h(b0 + ε0)] has at least N gaps.
Now assume that(1.13.0.8)
b0 = 0 ⇐⇒ there exists at least one zero of B,
and, in addition, that the magnetic field behaves quite regularly near the bottoms of wells. More precisely,suppose that, for some integer k > 0, if B(x0) = 0, then there exists a positive constant C such that for allx in some neighborhood of x0
(1.13.0.9)C−1|x− x0|k ≤ Tr+(B(x)) ≤ C|x− x0|k .
In this case, the existence of an arbitrarily large number of gaps in the spectrum of Hh as h → 0 can beshown for any interval of size O(hα) with some α > 1.
1.13.2 Theorem
Under the assumptions (1.13.0.8) and (1.13.0.9), there exists an increasing sequence λmm∈N,λm → ∞ as m → ∞, such that for any a and b with λm < a < b < λm+1 for some m,
[ah2k+2k+2 , bh
2k+2k+2 ] ∩ σ(Hh) = ∅
for any h > 0 small enough.
References(1) B. Helffer, Yu. A. Kordyukov, Semiclassical asymptotics and gaps in the spectra of periodic Schrödinger operators
with magnetic wells, preprint math.SP/0601366.
(2) Yu. A. Kordyukov, Spectral gaps for periodic Schrödinger operators with strong magnetic fields, Commun. Math.Phys. 253 (2005), no. 2, 371–384.
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Relative Pairing in Cyclic Cohomologyy and Divisor Flows
Matthias LeschThis talk is based on a recent joint paper with Henri Moscovici and Markus Pflaum
We show that Melrose’s divisor flow and its generalizations by Lesch and Pflaum are invariants ofK-theory classes for algebras of parametric pseudodifferential operators on a closed manifold. Theseinvariants are obtained by means of pairing the relative K-theory modulo the symbols with the cycliccohomological characters of odd relative cycles, constructed out of the regularized operator trace togetherwith its symbolic boundary. This representation gives a clear and conceptual explanation to all the es-sential features of the divisor flow - its homotopy nature, additivity and integrality. It also provides acohomological formula for the spectral flow along a smooth path of self-adjoint elliptic first order differ-ential operators, between any two invertible such operators on a closed manifold.
Zeta Functions and Heat Kernel Asymptotics on Conic Manifolds
Paul LoyaThis is about a joint work with Klaus Kirsten and Jinsung Park
We discuss exotic phenomena and a complete classification of the meromorphic structure of zeta-functions associated to self-adjoint extensions of Laplace-type operators over conic manifolds. Simpleexamples of such operators are the (standard Euclidean) Laplacian in R3 restricted to a cone and a moreprimitive example is the Laplacian in R2 \ 0 written in polar coordinates. We show that these zeta func-tions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusuallocations of poles with arbitrarily large multiplicity. The corresponding heat kernel and resolvent traceexpansions also exhibit exotic behaviors with logarithmic terms of arbitrary positive and negative mul-tiplicity. We also give an explicit algebraic combinatorial formula to compute these singularities anddemonstrate such singularities in the simple examples mentioned above.
Bergman Kernels and Geometric Quantization
Xiaonan Ma
In this talk, we explain our recent results on the asymptotic expansion of the Bergman kernel and itsrelation to the geometric quantization [8]. The interested readers may find complete references in [3], [5],[8], especially in the forthcoming book [7].
Let (X, ω) be a compact symplectic manifold of real dimension 2n. Assume that there exists a Hermitianline bundle L over X endowed with a Hermitian connection ∇L with the property that
√−1
2π RL = ω, whereRL = (∇L)2 is the curvature of (L,∇L). Let (E, hE) be a Hermitian vector bundle on X equipped with aHermitian connection ∇E and RE denotes the associated curvature.
Let gTX be a Riemannian metric on X. Let J be an almost complex structure such that gTX(Ju, Jv) =gTX(u, v), ω(Ju, Jv) = ω(u, v), and that ω(·, J·) defines a metric on TX. The associated spinc Dirac
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operator Dp acts on Ω0,•(X, Lp ⊗ E) =Ln
q=0 Ω0,q(X, Lp ⊗ E), the direct sum of spaces of (0, q)–formswith values in Lp ⊗ E. We denote by D+
p the restriction of Dp on Ω0,even(X, Lp ⊗ E).Let G be a compact connected Lie group with Lie algebra g and dim G = n0. Suppose that G acts on X
and its action on X lifts on L and E. Moreover, we assume the G-action preserves the above connectionsand metrics on TX, L, E and J. Then Ind(D+
p ) is a virtual representation of G. Denote by (Ker Dp)G ,theG-trivial components of Ker Dp.
The G-invariant Bergman kernel is PGp (x, x′) (x, x′ ∈ X), the smooth kernel of PG
p , the orthogonal projec-tion from (Ω0,•(X, Lp ⊗ E), 〈 〉) on (Ker Dp)G , with respect to the Riemannian volume form dvX(x′). Thepurpose of this paper is to study the asymptotic expansion of the G-invariant Bergman kernel PG
p (x, x′)as p → ∞, and we will relate it to the asymptotic expansion of the Bergman kernel on the symplecticreduction XG .
Let µ : X → g∗ be the associated moment map. We suppose that 0 ∈ g∗ is a regular value of µ.
1.16.1 Theorem
For any open G-neighborhood U of µ−1(0) in X, ε0 > 0, l, m ∈ N, there exists Cl,m > 0 (dependon U, ε0) such that for p ≥ 1, x, x′ ∈ X, d(Gx, x′) ≥ ε0 or x, x′ ∈ X \U,
|PGp (x, x′)|C m ≤ Cl,m p−l .(1.16.0.10)
where C m is the C m-norm induced by ∇L,∇E, ∇TX , hL, hE, gTX .
From Theorem 1.16.1, we only need to study the asymptotic for PGp (x, x′) for x, x′ near µ−1(0). As one
of the main results, we establish the full asymptotic expansion for PGp (x, x′) for x, x′ near µ−1(0). We will
also explain some geometric consequences of our asymptotic expansion.The basic philosophy developed in [3], [5], [7] is that the spectral gap properties for the operators proved
in [2], [4] implies the existence of the asymptotic expansion for the corresponding Bergman kernels, byusing the analytic localization technique inspired by §11 in [1]. The key observation here is that the G-invariant Bergman kernel is exactly smooth kernel of the orthogonal projection onto the zero space of adeformation of D2
p by the Casimir operator (i.e., to consider D2p − pCas) which has a spectral gap. Thus
the above philosophy applies to the proof of the existence of the asymptotic expansion.
References(1) J.-M. Bismut and G. Lebeau, Complex immersions and Quillen metrics, Inst. Hautes Études Sci. Publ. Math.
(1991), no. 74, ii+298 pp. (1992).
(2) J.-M. Bismut and E. Vasserot, The asymptotics of the Ray–Singer analytic torsion associated with high powers of apositive line bundle, Commun. Math. Phys.125 (1989), 355–367.
(3) X. Dai, K. Liu, and X. Ma, On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006),1–41; announced in C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 193–198.
(4) X. Ma and G. Marinescu, The Spinc Dirac operator on high tensor powers of a line bundle, Math. Z. 240 (2002),no. 3, 651–664.
(5) , Generalized Bergman kernels on symplectic manifolds, C. R. Math. Acad. Sci. Paris 339 (2004), no. 7,493–498. The full version: math.DG/0411559.
(6) , Toeplitz operators on symplectic manifolds, Preprint.
(7) , Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, Birkhäuser Boston,Inc., Boston, MA. Award winning monograph of the 2006 Ferran Sunyer i Balaguer Prize.
(8) X. Ma and W. Zhang, Bergman kernels and symplectic reduction, C. R. Math. Acad. Sci. Paris 341 (2005),297-302.
(9) Y. Tian and W. Zhang, An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg, Invent.Math. 132 (1998), no. 2, 229–259.
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The Eta Invariant for Quantum Graphs
Patrick McDonaldThis is about at joint work with Paul Loya
Let G be a compact quantum graph and E an Hermitian vector bundle over G. Let C∞(G, E) be thefunctions φ : G → E which are smooth on the interior of each edge and which have a smooth extensionto each closed edge. Given a complex structure on E, the length associated to each edge gives rise to aDirac operator, D : C∞(G, E) → C∞(G, E). We parameterize self-adjoint extensions of D using a collectionof unitary matrices determined by E and the given complex structure. For each self-adjoint extension ofthe Dirac operator we analyze the eta function and compute the eta invariant using the associated unitarymatrix.
Simplicial Complexes and Pseudodifferential Operators
Richard B. Melrose
The general question of attaching an alebra of pseudodifferential operators to a simplicial complex, ormore generally a ‘tied manifold’ is considered. For the algebra of b-pseudodifferential operators it isshown that the Atiyah-Singer index map extends to this more general setting. The adiabatic limit undersmooth approximation is also discussed.
b-complex Manifolds
Gerardo A. Mendoza
A b-complex manifold is a manifold M with boundary together with an involutive subbundle bT1,0Mof the complexification of its b-tangent bundle, bTM, such that bT1,0M+ bT1,0M = C bTM as a directsum. The bundle bT0,1M = bT1,0M gives rise to a b-elliptic complex(*)
· · · → C∞(M; bV0,qM)
b∂−→ C∞(M; bV0,q+1M) → · · ·
on the exterior powers bV0,qM of the dual, b
V0,1M, of bT0,1M. After fixing a defining function r we getvia Mellin transform and for each σ ∈ C, an elliptic complex(**)
· · · → C∞(M;VqV∗) D(σ)−−−→ C∞(M;
Vq+1V∗) → · · ·
on the dual of an involutive subbundle V of CT∂M determined by bT0,1M. Let N be a component of ∂Mand let Hq
D(σ)(N) be the q-th cohomology group of the complex (**) restricted to N. The q-th boundary
spectrum of the complex (*) at N is defined as the set
specqb,N(b∂) =
nσ ∈ C : Hq
D(σ)(N) 6= 0
o.
There is a canonical real nowhere vanishing vector field T which is a section of V . Assuming N compactand the existence of a T -invariant Riemannian metric on N, I’ll present fairly precise theorems on the
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location of specqb,N(b∂) and also relate specq
b,N(b∂) with vanishing theorems (such as Kodaira’s theorem)for ∂-cohomology of holomorphic line bundles over compact complex manifolds.
The Index of Dirac Operators on Manifolds with Non-exact CuspMetrics
Sergiu MoroianuThis is about a joint work with Jean-Marc Schlenker
We consider 3-manifolds with hyperbolic metric and cone singularities along a graph. We show thatthe cone-hyperblic metrics are infinitesimally rigid if all the cone angles are smaller than π. Similarresults have been proved by Hodgson-Kerckhoff, Bromberg, and Weiss. The novelty here is that weconsider cone singularities along infinite half-lines stretching to infinity. From a Physical point of view,these singularities give models for interacting massive spinless point particles. We also show that thedeformations of the metric (modulo trivial deformations) are parametrized by the cone angles and themarked conformal structure on the conformal infinity.
We use the method of Weil and Garland to prove our rigidity result. The tangent space to the modulispace of deformations modulo trivial deformations is isomorphic to a certain twisted de Rham cohomol-ogy group. The Laplacian on this bundle is strictly positive on compactly-supported sections. We firstnote, essentially following Weiss, that the Dirac operator is essentially self-adjoint in L2. These facts implythat the L2 cohomology vanishes. Most of the work is devoted to understanding the normalization frominfinity which gives L2 representatives for deformations which preserve both the cone angles and themarked conformal structure at infinity.
Hodge Theory for Manifolds with Cusps
Werner MüllerThis is about a joint work with Jörn Müller
Manifolds with cusps are generalizations of locally symmetric spaces of rank one. Let M be such amanifold. We show that the de Rham cohomology H∗(M) of M has a canonical splitting
H∗(M) = H∗(2)(M)⊕H∗
Eis(M),
where H∗(2)(M) is the space of L2-harmonic forms and H∗
Eis(M) is given by special values of Eisensteinforms. In particular, each cohomology class in H∗
Eis(M) can be represented by a harmonic form (which isnot square integrable).
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Pseudo Riemannian Foliations
Bernd MümkenPartially supported by DFG, SFB 478
Pseudo Riemannian foliations are a natural globalization of a real vector space with a general symmetricbilinear form. We investigate a kind of geodesic random walk on an n-dimensional pseudo Riemannianfoliation of a smooth manifold M of dimension n + D.
Denote by h the parameter of the foliated geodesic flow. For small values of h > 0 we define the N-steppartition function ZN
ϕ,h as an integral over closed broken geodesics depending on a smooth weight ϕ. Weshow that there is an asymptotic expansion
ZNϕ,h ∼ h−D
∞XL=0
hLZNϕ (L),
h 0, the coefficients of which are computable from local data. Next we define transfer operators Mkϕ,h
acting on complex valued tangential k-forms with compact support, 0 ≤ k ≤ n. We have the trace formula
ZNϕ,h =
nXk=0
(−1)ktrace(Mkϕ,h
N).
Moreover, differential operators appear in special cases by a variant of Pizzetti’s formula. Then we usethe zeta function
ζϕ,h(z) := exp∞X
N=1
zN
NZN
ϕ,h =nY
k=0
hYµk
ϕ,h
(1− zµkϕ,h)
i(−1)k+1
to prove that the spectra µkϕ,h of the transfer operators determine the asymptotic behavior of the partition
functions for large values of N by
lim supN→∞
ZNϕ,h −
nXk=0
(−1)kX
|µkϕ,h |>ε
µkϕ,h
N1/N
≤ ε
for all ε > 0.Let M be compact. If the foliation is Riemannian or Lorentzian and there is a timelike foliated vector
field, the tangential Hodge theory of Álvarez López and Kordyukov [1] relates the zeta function to thereduced tangential cohomology of M.
References(1) Álvarez López, J.A., Kordyukov, Y.A.: Long time behavior of leafwise heat flow for Riemannian foliations.
Compositio Math. 125(2), 129–153 (2001)
(2) Mümken, B.: Pseudo Riemannian Foliations. Preprint (2006)
Spectrum of Dirac operator over Cofinite Quotients of PSL(2, R)
Jinsung ParkThis is a partial result of the ongoing project with Paul Loya and Sergiu Moroianu
We introduce a recent result on the behavior of the discrete and continuous spectrum of the Diracoperator over the cofinite quotients of PSL(2, R) under the collapsing the S1-fiber.
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Applications of Perturbation Theory to the Spectral Theory ofHyperbolic Manifolds
Yiannis PetridisThis is about a joint work with Morten S. Risager
The main problem in the spectral theory of a hyperbolic surface M of finite area (not compact) is theexistence of eigenvalues embedded in the continuous spectrum. While this happens for arithmetic reasons,it is expected that a generic surface should have at most finitely many eigenvalues. Phillips and Sarnak in1985 formulated a condition that guarantees that an embedded eigenvalue sj(1− sj) becomes a scatteringpole under perturbation. Fermi’s Golden Rule (Phillips-Sarnak 1992) gives the rate at which this happens:
Re s(2) = Red2s(0)
dε2 ∝ −|〈∆(1)uj, E(·, sj)〉|2.
Here uj is the corresponding eigenfunction and E(z, s) is the generalized eigenfunction. What happensif the right-hand-side is 0? We need to investigate formulas for higher approximation of the embeddedeigenvalues. Define the infinitesimal variations of the generalized eigenfunctions E(z, s, ε) as:
Em(z, s) =dm
dεm E(z, s, 0).
1.24.1 Theorem (Y. Petridis., M. S. Risager 2006)
Assume Ek(z, s) are holomorphic at sj for k < n. Then
Re s(2n) = Red2ns(0)
dε2n = −cn(sj)Ress=sj En(z, s)
2 .
The infinitesimal variations Em(z, s) also give distribution results for additive homomorphisms of π1(M)(Gaussian distribution laws).
Groups with Torsion, Bordism and rho-Invariants
Paolo Piazza
I will report on joint work with Thomas Schick. Let Γ be a discrete group, and let M be a closed spinmanifold of dimension m > 3 with π1(M) = Γ. We assume that M admits a Riemannian metric of positivescalar curvature. We discuss how to use the L2 rho invariant ρ(2) of Cheeger-Gromov and the delocalizedeta invariant η<g> of Lott associated to the Dirac operator on M in order to get information about thespace of metrics with positive scalar curvature. Our results improve earlier results by Gromov-Lawson,Botvinnik-Gilkey and others; we employ index theory for the spin Dirac operator in a crucial way.
In particular we prove that, if Γ contains torsion and m ≡ 3 (mod 4) then M admits infinitely manydifferent bordism classes of metrics with positive scalar curvature. This implies that there exist infinitelymany concordance classes; we show that this is true even up to diffeomorphism.
If Γ has certain special properties, e.g. if it contains polynomially growing conjugacy classes of finiteorder elements, then we obtain more refined information about the “size” of the space of metric of positivescalar curvature, and these results also apply if the dimension is congruent to 1 mod 4.
These results are explained in the Preprint 2006 "Group with torsion, bordism and rho-invariants".At the end of the talk I will also speak about:
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• the L2 rho-invariant for the signature operator and its use in distinguishing homotopy-equivalentmanifolds that are non-diffeomorphic (always under the assumption that the fundamental grouphas torsion)
• the radically different behavior of these invariants when the fundamental group is torsion free. Thisis the content of a Preprint 2005 "Bordism, rho-invariants and the Baum-Connes conjecture", alwaysin collaboration with Thomas Schick.
Small Time Heat Expansion for the Laplace-Beltrami on anAnalytic Hypersurface with an Isolated Singularity
Demetrios Pliakis
We will present the small time heat expansion for the Laplace-Beltrami on an analytic hypersurface withan isolated singularity. First we obtain a parametrization of the singularity by the flow out of suitable ap-proximating smooth algebraic submanifolds through a perturbation of the geodesic equations. The latterbeing highly singular require a careful study of their degeneracies, the existence and the asymptotics ofthe flow is achieved by an application of the Nash-Moser implicit function theorem. The model opera-tors that are obtained then, are studied through essential modifications of standard ideas from singularanalysis. Generalizations to more complicated singularities will be discussed.
Eta Invariant, Boundaries and the Determinant Bundle
Frédéric RochonThis is a joint work with Richard B. Melrose
We will discuss the definition of the eta invariant on manifolds with boundary using cusp suspendedoperators. This will be used to show that the (exponentiated) eta invariant of a family of elliptic oper-ators trivializes the determinant bundle of the associated family of operators on the boundary, giving apseudodifferential generalization of a result of Dai and Freed.
An Approximate Spectral Projection of the Laplace Operator
Yuri SafarovThis is about a joint work with Andrew McKeag
Let −∆ be the Laplace operator on a compact Riemannian manifold and ν be a self-adjoint classicalpseudodifferential operator (PDO) of order zero.
Denote by χλ the characteristic function of the interval (−∞, λ) and define ψλ,µ := χλ ∗ ρµ, whereρµ(τ) := µ−1 ρ(µ−1τ) and ρ is an arbitrary C∞
0 -function on R such thatR∞−∞ ρ(τ) dτ = 1. If µ is relatively
17
small then the operator ψλ,µ(√−∆ + ν) can be thought of as an approximate spectral projection of
√−∆ +
ν corresponding to the interval (−∞, λ).
Assuming that λ, µ → +∞ and λ−1µ → 0, we construct a calculus of parameter dependent PDOswhich includes operators of the form ψλ,µ(
√−∆ + ν). In this calculus, the integral kernels of PDOs are
represented by oscillatory integrals modulo smooth functions vanishing with all their derivatives fasterthan any power of µ as µ → ∞. In particular, our results imply that the Hilbert–Schmidt norm of thecomposition χλ(
√−∆ + ν)
I − χλ+µ(
√−∆)
vanishes faster than any power of µ as µ → ∞.
If µ = λκ with κ > 1/2 then the above results can be obtained by means of the standard techniqueof PDOs. However, in the general case the corresponding PDOs cannot be defined with the use of localcoordinates. We use instead the coordinate free approach suggested in [1].
References(1) Y. Safarov. Pseudodifferential operators and linear connections. Proc. London Math. Soc., 74 (1997), 379–417.
On the Homotopy Classification of Elliptic Operators onManifolds with Corners
Anton SavinThis is about a joint work with Vladimir Nazaikinskii and Boris SterninAMS subject classification: 58J40, 19K33
We define a natural dual object for manifolds with corners. The dual object is a stratified manifold andwe show how pseudodifferential operators on manifolds with corners can be considered as local operatorsin the sense of Atiyah on the dual stratified manifold. This enables us to establish the stable homotopyclassification of elliptic pseudodifferential operators on manifolds with corners and show that the set ofelliptic operators modulo stable homotopy is isomorphic to the K-homology group of the dual stratifiedmanifold. By way of application, generalizations of some recent results due to Monthubert and Nistorand the computation of K-groups of the algebras of pseudodifferential operators are given.
A Continuous Field of C∗-algebras and the Tangent Groupoid forManifolds with Boundary
Elmar SchroheAMS subject classification: 58J32, 58H05, 35S15, 46L80
For a smooth manifold X with boundary we construct a semigroupoid T −X and a continuous fieldC∗r (T −X) of C∗-algebras which extend Connes’ construction of the tangent groupoid.
The main difference lies in the fact that instead of gluing the tangent space TX to X × X, we glue thehalf-tangent space T−X, consisting of all those tangent vectors v in m ∈ X for which expm(−hv) lies in Xfor small h.
18
We show that for h 6= 0, the fiber C∗r (T −X(h)) consists of the compact operators. The fiber over h = 0on the other hand is the C∗-closure of C∞
c (T−X) with respect to two representations: The representationπ on L2(TX), where the operators act by convolution, and the representation π∂ on L2(T+X|∂X) actingby half-convolution.
As an application we show the asymptotic multiplicativity of h-scaled truncated pseudodifferentialoperators with smoothing symbols and compute the K-theory of the associated symbol algebra.
Asymptotics on Manifolds with Corners
B.-W. Schulze
The solutions to elliptic equations on a manifold with corners have a specific local behaviour near thesingularities, namely, to belong to weighted corner spaces and to have asymptotics in the various singular"half-axis directions", with some interplay between the ingredients from the different strata. Asymp-totics are determined by global meromorphic operator functions (higher analogues of conormal symbols)belonging to various cones transversal to corresponding singularities. The expressions for the singularfunctions are determined by anisotropic descriptions of edge spaces in terms of the action of some groupsof isomorphisms on spaces on infinite cones. The nature of asymptotics is related to that of analoguesof Green’s functions with respect to the lower-dimensional strata and to the structure of extra edge con-ditions of trace and potential type. The latter ones are influenced by vanishing or non-vanishing of atopological obstruction, analogously to the one of Atiyah and Bott in the case of classical boundary valueproblems.
High Energy Limits of Laplace-type and Dirac-type Eigenfunctionsand Frame Flows
Alexander StrohmaierThis is about a joint work with Dmitry Jakobson
We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the correspond-ing manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriatesense for those operators. Observables for the corresponding quantum systems are matrix-valued pseu-dodifferential operators and therefore the system remains non-commutative in the high-energy limit. Wediscuss to what extent the space of stationary high-energy states behaves classically.
19
Maximal Blowup of Modes and Quasimodes
John A. TothThis is about a joint work with Christopher Sogge and Steve Zelditch
In this talk, I will present some recent work in which we improve on previous results on maximalLaplace eigenfunction growth. In addition, we show that our results our sharp by establishing severalconverse theorems.
Some Remarks about Heat Kernel Coefficients
Gregor Weingart
The integrand of the local index theorem for Dirac operators is the only possible integrand allowed bythe asymptotic heat kernel expansion, which can not be used to construct non–trivial (universal) polyno-mials in the curvature tensor integrating to zero over every compact Riemannian manifold. Taking thenon–existence of such polynomials for granted we can see this statement as a “philosophical proof” of thelocal index theorem. In the talk I want to discuss this strange characterization of the integrand of the localindex theorem besides some remarks about the asymptotic expansion of its gradient off the diagonal.
Complex Zeros of Real Ergodic Eigenfunctions
Steve Zelditch
This talk concerns the distribution of nodal hypersurfaces Zλ = φλ = 0 of eigenfunctions φλ of ∆ ona compact real analytic Riemannian manifold (M, g) of eigenvalue λ2. It seems a distant goal to determinethe asymptotics of integrals
RZλ
f dHn−1 for any smooth function f (including f = 1). However, I will
show that if M is complexified, if φλ is analytically continued to the complexification and if ZCλ is the
complex zero set, then one can determine the asymptotics of (almost all)R
ZCλ
f when the geodesic flow of
(M, g) is ergodic.
