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• 1.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Max Planck Institute for Gravitational Physics, United Kingdom.
Hidden symmetries and decay for the wave equation on the Kerr spacetime2015In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 182, no 3, p. 787-853Article in journal (Refereed)

Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalisation of the vector-field method that allows the use of higher-order symmetry operators. In particular, our method makes use of the second-order Carter operator, which is a hidden symmetry in the sense that it does not correspond to a Killing symmetry of the spacetime.

• 2.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
On the shape of Bruhat intervals2009In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 170, no 2, p. 799-817Article in journal (Refereed)

Let (W, S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let J subset of S. Let W-J denote the set of minimal coset representatives modulo the parabolic subgroup W-J. For w is an element of W-J, let f(i)(w,J) denote the number of elements of length i below w in Bruhat order on W-J (with notation simplified to f(i)(w) in the case when W-J = W). We show that 0 <= i < j <= l(w)-i implies f(i)(w,J) <= f(j)(w,J). Also, the case of equalities f(i)(w) = f(l(w)-i)(w) for i = 1,..., k is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial P-e,P-w (q). We show that if W is finite then the number sequence f(0)(w), f(1)(w),... f(l(w))(w) cannot grow too rapidly. Further, in the finite case, for any given k >= 1 and any w is an element of W of sufficiently great length (with respect to k), we show f(l(w)-k)(w) >= f(l(w)-k+1)(w) >= ... >= f(l(w))(w). The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if (X) over bar (w) is a Schubert variety of dimension d = l(w), and lambda = c(1) (L) is an element of H-2 ((X) over bar (w)) is the restriction to (X) over bar (w) of the Chem class of an ample line bundle, then (lambda(k)) . : Hd-k((X) over bar (w)) -> Hd+k((X) over bar (w)) is injective for all k >= 0.

• 3. Borcea, Julius
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Polya-Schur master theorems for circular domains and their boundaries2009In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 170, no 1, p. 465-492Article in journal (Refereed)

We characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region Omega subset of C for arbitrary closed circular domains Omega (i.e., images of the closed unit disk under a Mobius transformation) and their boundaries. This provides a natural framework for dealing with several long-standing fundamental problems, which we solve in a unified way. In particular, for Omega = R our results settle open questions that go back to Laguerre and Polya-Schur.

• 4. Caffarelli, L. A.
KTH, Superseded Departments, Mathematics.
Regularity of a free boundary with application to the Pompeiu problem2000In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 151, no 1, p. 269-292Article in journal (Refereed)

In the unit ball B(0, 1), let u and Omega (a domain in R-N) salve the following overdetermined problem: Delta u = chi(Omega) in B(0, 1), 0 is an element of partial derivative Omega, u = \del u\ = 0 in B(0, 1) \ Omega, where chi(Omega) denotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement of Omega does not develop cusp singularities at the origin then we prove partial derivative Omega is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.

• 5.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Recovering the good component of the Hilbert scheme2014In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 179, no 3, p. 805-841Article in journal (Refereed)

We give an explicit construction, for a flat map X -> S of algebraic spaces, of an ideal in the n'th symmetric product of X over S. Blowing up this ideal is then shown to be isomorphic to the schematic closure in the Hilbert scheme of length n subschemesof the locusof n distinct points.This generalizes Haiman's corresponding result for the affine complex plane.However, our construction of the ideal is very different from that of Haiman, using the formalism of divided powers rather than representation theory.In the nonflat case we obtain a similar result by replacing the n'th symmetric product by the n'th divided power product.

• 6.
KTH, Superseded Departments, Mathematics.
Hodge integrals, partition matrices, and the lambda g conjecture2003In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 157, no 1, p. 97-124Article in journal (Refereed)

We prove a closed formula for integrals of the cotangent line classes against the top Chern class of the Hodge bundle on the moduli space of stable pointed curves. These integrals are computed via relations obtained from virtual localization in Gromov-Witten theory. An analysis of several natural matrices indexed by partitions is required.

• 7.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Heisenberg uniqueness pairs and the Klein-Gordon equation2011In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 173, no 3, p. 1507-1527Article in journal (Refereed)

A Heisenberg uniqueness pair (HUP) is a pair (Γ,Λ), where Γ is a curve in the plane and Λ is a set in the plane, with the following property: any finite Borel measure μ in the plane supported on Γ, which is absolutely continuous with respect to arc length, and whose Fourier transform μˆ vanishes on Λ, must automatically be the zero measure. We prove that when Γ is the hyperbola x1x2=1 %, and Λ is the lattice-cross Λ=(αZ×{0})∪({0}×βZ), where α,β are positive reals, then (Γ,Λ) is an HUP if and only if αβ≤1; in this situation, the Fourier transform μˆ of the measure solves the one-dimensional Klein-Gordon equation. Phrased differently, we show that eπiαnt,eπiβn/t,n∈Z, span a weak-star dense subspace in L∞(R) if and only if αβ≤1. In order to prove this theorem, some elements of linear fractional theory and ergodic theory are needed, such as the Birkhoff Ergodic Theorem. An idea parallel to the one exploited by Makarov and Poltoratski (in the context of model subspaces) is also needed. As a consequence, we solve a problem on the density of algebras generated by two inner functions raised by Matheson and Stessin.

• 8.
KTH, Superseded Departments, Mathematics.
Discrete orthogonal polynomial ensembles and the Plancherel measure2001In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 153, no 1, p. 259-296Article in journal (Refereed)
• 9. Krishnapur, Manjunath
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Nodal length fluctuations for arithmetic random waves2013In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 177, no 2, p. 699-737Article in journal (Refereed)

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus ("arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is nonuniversal. Our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

• 10.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
On the distribution of matrix elements for the quantum cat map2005In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 161, no 1, p. 489-507Article in journal (Refereed)

For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable. In this paper we study the fluctuations of the matrix elements for the desymmetrized quantum cat map. We present a conjecture for the distribution of the normalized matrix elements, namely that their distribution is that of a certain weighted sum of traces of independent matrices in SU(2). This is in contrast to generic chaotic systems where the distribution is expected to be Gaussian. We compute the second and fourth moment of the normalized matrix elements and obtain agreement with our conjecture.

• 11. Marklof, Jens
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems2010In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 172, no 3, p. 1949-2033Article in journal (Refereed)

The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the Boltzmann-Grad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.

• 12.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Strong cosmic censorship in T-3-Gowdy spacetimes2009In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 170, no 3, p. 1181-1240Article in journal (Refereed)

Einstein's vacuum equations can be viewed as an initial value problem, and given initial data there is one part of spacetime, the so-called maximal globally hyperbolic development (MGHD), which is uniquely determined up to isometry. Unfortunately, it is sometimes possible to extend the spacetime beyond the MGHD in inequivalent ways. Consequently, the initial data do not uniquely determine the spacetime, and in this sense the theory is not deterministic. It is then natural to make the strong cosmic censorship conjecture, which states that for generic initial data, the MGHD is inextendible. Since it is unrealistic to hope to prove this conjecture in all generality, it is natural to make the same conjecture within a class of spacetimes satisfying some symmetry condition. Here, we prove strong cosmic censorship in the class of T-3-Gowdy spacetimes. In a previous paper, we introduced a set G(i,c) of smooth initial data and proved that it is open in the C-1 x C-0-topology. The solutions corresponding to initial data in G(i,c) have the following properties. First, the MGHD is C-2-inextendible. Second, following a causal geodesic in a given time direction, it is either complete, or a curvature invariant, the Kretschmann scalar, is unbounded along it (in fact the Kretschmann scalar is unbounded along any causal curve that ends on the singularity). The purpose of the present paper is to prove that G(i,c) is dense in the C-infinity-topology.

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