Tsukuba Conference on Integral Geometryand Harmonic Analysis
Date: August 7 (Monday) – August 10 (Thursday), 2006
Place: Laboratory of Advanced Research B, University of Tsukuba
1-1-1, Tennodai, Tsukuba-shi, Ibaraki, 305-8577, Japan (cf. WEB site at the bottom)

PROGRAM
August 7 (Monday)
13:00-13:10 Opening remarks
Chairperson: Toshio Oshima (University of Tokyo)
13:10-14:10 Sigurdur Helgason (Massachusetts Institute of Technology)
The X-ray Transform
Abstract: In this lecture we discuss the X ray transform on a symmetric space, compact andnon compact. For the noncompact case the theory is fairly complete, injectivity, support
theorems and inversion formulas. We will discuss recent progress in the area.
14:10-14:30 Break
14:30-15:30 Satoshi Ishikawa (University of Tokyo)
The Radon Transform for the double fibrations of noncompact symmetric spaces
Abstract: I present the results on the operational properties of the Radon Transform for thedouble fibrations of noncompact symmetric spaces associated with the inclusion incidencerelations. Our approach is based on the projection slice theorem which relates the Radontransform and the Fourier transform on symmetric spaces and the underlying geometry ofthe double fibrations also plays an important role.
15:30-15:50 Break
15:50-16:50 Yutaka Matsui (University of Tokyo)
Topological Radon transforms and their applications (joint with Kiyoshi Takeuchi)
Abstract : Topological Radon transforms of (subanalytically) constructible functions aredefined by taking the topological Euler characteristics of hyperplane sections of subanalyticsets. We study topological Radon transforms of constructible functions on projective spaces
or Grassmann manifolds by means of the combinatorial Schubert calculus and the microlocal
theory of sheaves developed by Kashiwara-Schapira. In particular, we introduce inversion
formulas of them and the characterization of their images. Some applications to algebraic
geometry will be also given.

August 8 (Tuesday)
Chairperson: Gestur Olafsson (Louisiana State University)
10:00-11:00 Hiroshi Isozaki (University of Tsukuba)
Inverse boundary value problems and hyperbolic spaces
Abstract: We present a new approach to the inverse problem based on the structure ofhyperbolic space. There are 3 main issues :• Solving the inverse boundary value problems in Rn (n ≥ 3) by using hyperbolic manifolds• Inverse boundary value problems in the horosphere• Detection of location of inclusions from the local DN map

The starting point is the following observation. Given a boundary value problem (−∆+q)u =(2−n)/20 in Ω ⊂ Rn , let v = xn
u. Then v satisfies (−∆g +V )v = 0, where V = x2n q −n(n−2)/422and ∆g = xn ∂n − (n − 2)xn ∂n + x2n ∆x , which is just the Laplace-Beltrami operator on Hn .
(We denote the points in Rn by (x, xn ), x ∈ Rn−1 , and the domain Ω is regarded to sit in theregion {xn > 0}). This means that the inverse boundary value problem in Rn and the one
in Hn are equivalent. To study the latter, the main tool we use is the Green function of thegauge-transformed operator H0 (θ) = −e−ix·θ ∆g eix·θ , θ ∈ Cn−1 , which is written inp terms of0 )·ξ
i(x−x0n−1)/2modified Bessel functions : e(yy )Iν (ζ(ξ, θ)y)Kν (ζ(ξ, θ)y), ζ(ξ, θ) = (ξ + θ)2 .
R0This is a counter part of Faddeev’s Green function ei(x−x )·ξ (ξ 2 + 2ζ · ξ)−1 dξ, a fundamental
tool in the inverse scattering theory for Schrödinger operators on the Euclidean space.
(1) Inverse boundary value problems. To solve the inverse boundary value problem in Hn ,we pass it to the quotient space Hn /Γ, where Γ is a lattice in Rn−1 of rank n − 1 which istaken so large that the fundamental domain E = Rn−1 /Γ contains the region Ω completelyinside. In this case, θ varies over the fundamental domian of the dual lattice of Γ and isregarded as a Floquet parameter in the theory of periodic Schrödinger operators. From theDirichlet-Neumann map for the boundary value problem, one can construct the scattering
amplitude. This latter is analytic in some region of complexified Floquet parameters θ. By
passing to this parameter to infinity (complex Born approximation), one can reconstruct q
from the DN map.
(2) Horosphere boundary value problem and Barber-Brown algorithm. Consider a boundary
value problem for the Schrödinger operator −∆+q(x) in a ball Ω : (x1 +R)2 +x22 +(x3 −r)2 <r2 , whose boundary we regard as a horosphere in the hyperbolic space H3 realized in theupper half space. Let S = {|x| = R, x3 > 0} be a hemisphere, which is generated by afamily of geodesics in H3 . By imposing a suitable boundary condition on ∂Ω in terms ofa pseudo-differential operator, we compute the integral mean of q(x) over S ∩ Ω from theassociated (generalized) Robin-to-Dirichlet map for −∆ + q(x). The potential q(x) is thenreconstructed by virtue of the inverse Radon transform on hyperbolic space. This justifies
the well-known Barber-Brown algorithm in electrical impedance tomography.
(3) Detection of inclusions - Applications to numerical computation. This hyperbolic spaceapproach can also be used to detect the location of non-smooth part of conductivity γ(x) of abody Ω in Rd , d = 2, 3. Suppose for the sake of simplicity that we know the DN map Λ0 for
the case that γ is a constant, and that the conductivity is different from this constant on a² = {x ∈ ∂Ω; |x−x | >subset Ω1 ⊂ Ω. Take x0 from outsied of the convex hull of Ω and let Sout
0²R + ²}, Sin = {x ∈ ∂Ω; |x − x0 | < R − ²}. Then one can construct the boundary data fτ (x)²² ), f (x)depending on a large paramter τ > 0 having the following properties : On Sout(Sinτ
is exponentially decreasing (increasing) in τ . Let Λ be the DN map for γ. If R < dis(x0 , Ω1 ),then 0 ≤ (Λ − Λ0 )fτ , fτ ) < Ce−δτ , and if R > dis(x0 , ∂Ω1 ), then ((Λ − Λ0 )fτ , fτ ) > C 0 eδτ .
This means that one can detect the location of inclusions from the boundary data which areessentially localized on a part of the boundary.
References[Is1] H. Isozaki, Inverse spectral problems on hyperbolic manifolds and their applications to inverse boundary value
problems in Euclidean space, Amer. J. Math. 126 (2004), 1261-1313.
[Is2] H. Isozaki, Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical
impedance tomography, preprint.
[IINSU] T. Ide, H. Isozaki, S. Nakata, S. Siltanen and G. Uhlmann, Probing for electrical inclusions with complex
spherical waves, to appeai in C.P.A.M..

11:00-11:20 Break (group picture)
11:20-12:20 Eric Todd Quinto (Tufts University)
Support Theorems for the Spherical Radon Transform on Manifolds
Abstract: Let M be a real-analytic manifold and let S be a real-analytic hypersurface. We
prove a local support theorem for the spherical Radon transform that integrates over spheres

centered at points on S. Our theorem is valid for distributions supported on one side of”tangent surfaces” to S. The proof involves the microlocal analysis of the sphere transformand a microlocal Holmgren theorem of Kawai, Kashiwara, and Hoermander. Generalizations
and possible applications to sonar and the wave equation will be outlined.
12:20-14:30 Lunch Time
Chairperson: Eric Grinberg (University of New Hampshire)
14:30-15:30 Boris Rubin (Louisiana State University)
Integral geometry and fractional calculus
Abstract: Fractional calculus is a branch of analysis which studies different kinds of fractional
integrals, their modifications and generalizations. These include Riemann-Liouville integrals,
Riesz potentials, analytic families of cosine transforms, and many others. Diverse (well knownand new) fractional integrals arise in a natural way in integral geometry. I am planning totalk about application of fractional calculus to Radon transforms of different types and theBusemann-Petty problem on sections of convex bodies.
15:30-15:50 Break
15:50-16:50 Takaaki Nomura (Kyushu University)
Tube domain and an orbit of a complex triangular group (joint with Hideyuki Ishi)
Abstract: Let w be a complex symmetric matrix of order r, and ∆1 (w), . . . , ∆r (w) theprincipal minors of w. If w belongs to the Siegel right half space, then we haveRe∆k (w)> 0 (k = 1, . . . , r).
∆k−1 (w)(∗)

The property (∗) is true for general symmetric right half spaces if we consider ∆k in theframework of Jordan algebras. Now principal minors are basic relative invariants for the
triangular group, and in this way we can consider (∗) even in the nonsymmetric case. For
nonsymmetric right half spaces, the above property (∗) is in general false. But we will
present a series of nonsymmetric right half spaces with the property (∗). The proof of (∗)
for symmetric spaces consists of two parts. One part is generalized to nonsymmetric cases,
while the other is generally false there. This will be the final part of the talk.

18:00-20:00

Reception (Tremont Hotel)

August 9 (Wednesday)
Chairperson: Fulton Gonzalez (Tufts University)
10:00-11:00 Toshiyuki Kobayashi (Kyoto University)
Conformally invariant Hilbert structure on the solution space of the ultrahyperbolic Laplace
equation on Rp+q
Abstract: The indefinite orthogonal group O(p+1, q+1) acts as the Möbius group of conformaltransformations on Rp+q , and preserves the space of solutions of the ultrahyperbolic Laplace
equation on Rp+q . Inspired by the idea of Sato’s hyperfunctions, we construct in an intrinsic
and natural way a Hilbert space of solutions by the integration of the data of solutions overa hypersurface, and give a new construction of its minimal unitary representation for p + q
even.
11:00-11:20 Break
11:20-12:20 Eric Grinberg (University of New Hampshire)
TBA
12:20-14:30 Lunch Time

Chairperson: Toshiyuki Kobayashi (Kyoto University)
14:30-15:30 Fulton Gonzalez (Tufts University)
The range of the matrix Radon transform (joint with Tomoyuki Kakehi)
Abstract: In this talk I will describe a range characterization of the matrix Radon transformby invariant differential operators, in the case when the source manifold (in this case Mn,k , thevector space of n × k real matrices) has lower dimension than the target manifold, the space
Ξ of affine matrix planes. This generalizes analogous results, given by the group-invariant
form of John’s equations, characterizing the range of the d-plane transform in Rn .
15:30-15:50 Break
15:50-16:50 Gen Mano (Kyoto University)
The unitary inversion operator for the minimal representation of O(p,q)
Abstract: The L2 -model of the minimal representation of the indefinite orthogonal groupO(p, q) (p + q: even, greater than 4) was established by Kobayashi-Orsted (2003). In thistalk, we present an explicit formula for the unitary inversion operator, which plays a keyrole in this L2 -model. Our proof uses the analysis on the Radon transform of distributions
supported on the light cone.

August 10 (Thursday)
Chairperson: Sigurdur Helgason (Massachusetts Institute of Technology)
9:00-10:00 Hideko Sekiguchi (University of Tokyo)
Radon-Penrose transform for the quantization of elliptic orbits
Abstract: The Radon-Penrose transform is defined on the quantization of elliptic coadjoint
orbits as intertwining operators of the transformation group. In this talk, I will give thecharacterization of the Penrose transform in the degenerate case for real symplectic groups.
10:00-10:20 Break
10:20-11:20 Gestur Olafsson (Louisiana State University)
Holomorphic Aspects of the Abel Transform
11:20-11:40 Break
11:40-12:40 Toshio Oshima (University of Tokyo)
Differential equations attached to generalized flag manifolds
Abstract: Generalized flag manifolds are homogeneous spaces of real reductive Lie groups
where the degenerate principal series representations of the groups. We give explicit expressions of the annihilators of the representations and then the images of equivariant integral
transformations, such as Poisson, Radon and Penrose transformations etc., satisfy differential
equations corresponding to the annihilators.

For the details of the conference, please visit the following site:
http://www.math.tsukuba.ac.jp/˜kakehi/integral-geometry/conference.html

Organizers:
Toshio Oshima (University of Tokyo)
oshima@ms.u-tokyo.ac.jp
Fulton Gonzalez (Tufts University)fulton.gonzalez@tufts.edu
Tomoyuki Kakehi (University of Tsukuba)
kakehi@math.tsukuba.ac.jp