Julkaisut

On gas giant planets the speed of sound is isotropic and goes to zero at the surface. Geometrically, this corresponds to a Riemannian manifold whose metric tensor has a conformal blow-up near the boundary. The blow-up is tamer than in asymptotically hyperbolic geometry: the boundary is at a finite distance. We study the differential geometry of such manifolds, especially the asymptotic behavior of geodesics near the boundary. We relate the geometry to the propagation of singularities of a hydrodynamic PDE and we give the basic properties of the Laplace–Beltrami operator. We solve two inverse problems, showing that the interior structure of a gas giant is uniquely determined by different types of boundary data.

We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen closed subset. The case of a Riemannian manifold with boundary with the boundary as the measurement set appears is a classical geometric inverse problem arising from Gel'fand's inverse boundary spectral problem. Examples of spaces satisfying our assumptions include some non-simple Riemannian manifolds, Euclidean domains with non-trivial topology, and metric trees.

The present article focuses on a unique continuation result for certain weighted ray transforms, utilizing the unique continuation property (UCP) of the fractional Laplace operator. Specifically, we demonstrate a conservative property for momentum ray transforms acting on tensors, as well as the antilocality property for both weighted ray and cone transforms acting on functions.

We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when $(X,E)$ is a tree and $B$ is the set of leaves of the tree, the graph $(X,E)$ can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph $(X,E)$, or one of those, which has a given number of vertices and the required distances between vertices in $B$. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $O(|X|^2)$ qubits, the same order as the number of elements in the adjacency matrix of $(X,E)$. It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider.

We prove that the geodesic X-ray transform is injective on $L^2$ when the Riemannian metric is simple but the metric tensor is only finitely differentiable. The number of derivatives needed depends explicitly on dimension, and in dimension $2$ we assume $g\in C^{10}$. Our proof is based on microlocal analysis of the normal operator: we establish ellipticity and a smoothing property in a suitable sense and then use a recent injectivity result on Lipschitz functions. When the metric tensor is $C^k$, the Schwartz kernel is not smooth but $C^{k-2}$ off the diagonal, which makes standard smooth microlocal analysis inapplicable.

We address the inverse problem of recovering the stiffness tensor and density of mass from the Dirichlet-to-Neumann map. We study the invariance of the Euclidean and Riemannian elastic wave equation under coordinate transformations. Furthermore, we present gauge freedoms between the parameters that leave the elastic wave equations invariant. We use these results to present gauge freedoms in the Dirichlet-to-Neumann map associated to the Riemannian elastic wave equation.

Dix formulated the inverse problem of recovering an elastic body from the measurements of wave fronts of point scatterers. We geometrize this problem in the framework of linear elasticity, leading to the geometrical inverse problem of recovering a Finsler manifold from certain sphere data in a given open subset of the manifold. We solve this problem locally along any geodesic through the measurement set.

It is known that a geometric measurement of the light cones of supernovae determines the conformal class of the visible part of the spacetime. The conformal factor is physically meaningful but cannot be determined geometrically by anything with zero mass, such as the photon. We show that measuring the neutrino cones in addition to light cones completely removes this gauge freedom. We describe the physical model in great detail, including why ultrarelativistic neutrinos are the only option.

If $G$ is a finite group, is a function $f:G\to\mathbb C$ determined by its sums over all cosets of cyclic subgroups of $G$? In other words, is the Radon transform on $G$ injective? This inverse problem is a discrete analogue of asking whether a function on a compact Lie group is determined by its integrals over all geodesics. We discuss what makes this new discrete inverse problem analogous to well-studied inverse problems on manifolds and we also present some alternative definitions. We use representation theory to prove that the Radon transform fails to be injective precisely on Frobenius complements. We also give easy-to-check sufficient conditions for injectivity and noninjectivity for the Radon transform, including a complete answer for abelian groups and several examples for nonabelian ones.

We establish spectral rigidity for spherically symmetric manifolds with boundary and interior interfaces determined by discontinuities in the metric under certain conditions. Rather than a single metric, we allow two distinct metrics in between the interfaces enabling the consideration of two wave types, like P- and S-polarized waves in isotropic elastic solids. Terrestrial planets in our solar system are approximately spherically symmetric and support toroidal and spheroidal modes. Discontinuities typically correspond with phase transitions in their interiors. Our rigidity result applies to such planets as we ensure that our conditions are satisfied in generally accepted models in the presence of a fluid outer core. The proof is based on a novel trace formula.

We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with $C^{1,1}$ metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold. Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth.

The geodesic ray transform, the mixed ray transform and the transverse ray transform of a tensor field on a surface can all be seen as what we call mixing ray transforms, compositions of the geodesic ray transform and an invertible linear map on tensor fields. We provide an approach that uses a unifying concept of symmetry to merge various earlier transforms (including mixed, transverse, and light ray transforms) into a single family of integral transforms with similar kernels.

We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.

We provide new proofs based on the Myers-Steenrod theorem to confirm that travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to the natural gauge of a boundary fixing diffeomorphism. Our method of the proof leads to a Lipschitz-type stability estimate for the first two data sets in the class of simple metrics.

Consider the geometric inverse problem: There is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a Riemannian manifold with boundary? With a finite set of sources we can only hope to get an approximate reconstruction, and we indeed provide a discrete metric approximation to the manifold with explicit data-driven error bounds when the manifold is simple. This is the geometrization of a seismological inverse problem where we measure the arrival times on the surface of waves from an unknown number of unknown interior microseismic events at unknown times. The closeness of two metric spaces with a marked boundary is measured by a labeled Gromov–Hausdorff distance. If measurements are done for infinite time and spatially dense sources, our construction produces the true Riemannian manifold and the finite-time approximations converge to it in the metric sense.

We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differential structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance map determines the Finsler function in this set but not in its exterior. If the Finsler function is fiberwise real analytic, it is determined uniquely. We also discuss the smoothness of the distance function between interior and boundary points.

We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds $(M,g)$ with $g \in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This $C^{1,1}$-regularity is optimal on the Hölder scale. The bulk of the article is devoted to setting up a calculus of differential and curvature operators on the unit sphere bundle atop this non-smooth structure.

We prove that if $P(D)$ is some constant coefficient partial differential operator and $f$ is a scalar field such that $P(D)f$ vanishes in a given open set, then the integrals of $f$ over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.

We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace–Beltrami operator uniquely determines the length spectrum. The trace formula also applies for the toroidal modes of the free oscillations in the earth. We then prove that the length spectrum is rigid: Deformations preserving the length spectrum and spherical symmetry are necessarily trivial in any dimension, provided the Herglotz condition and a generic geometrical condition are satisfied. Combining the two results shows that the Neumann spectrum of the Laplace–Beltrami operator is rigid in this class of manifolds with boundary.

We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and vanishes on the rest. This is done in a geometry with non-positive sectional curvature and a strictly convex obstacle in any dimension. We give two proofs, both of which contain new features also in the absence of reflections. The result is new even for scalars in dimensions above two.

Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity holds if either a convex foliation condition is satisfied on a Cauchy surface on the manifold or the manifold is real analytic and null geodesics do not have cut points. Next, we consider the light ray transform on tensor fields of arbitrary rank in the more restrictive class of static Lorentzian manifolds and show that if the geodesic ray transform on tensors defined on the spatial part of the manifold is injective up to the natural gauge, then the light ray transform on tensors is also injective up to its natural gauge. Finally, we provide applications of our results to some inverse problems about recovery of coefficients for hyperbolic partial differential equations from boundary data.

If the integrals of a one-form over all lines meeting a small open set vanish and the form is closed in this set, then the one-form is exact in the whole Euclidean space. We obtain a unique continuation result for the normal operator of the X-ray transform of one-forms, and this leads to one of our two proofs of the partial data result. Our proofs apply to compactly supported covector-valued distributions.

We study uniqueness of the recovery of a time-dependent magnetic vector-valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet to Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold.

Within the field of seismic modelling for anisotropic media, dynamic ray tracing is a powerful technique for computation of amplitude and phase properties of the high-frequency Green's function. Dynamic ray tracing is based on solving a system of Hamilton-Jacobi perturbation equations, which may be expressed in different 3-D coordinate systems. We consider two particular coordinate systems; a Cartesian coordinate system with a fixed origin and a curvilinear ray-centred coordinate system associated with a reference ray. For each system we form the corresponding 6-D phase spaces, which encapsulate six degrees of freedom in the variation of position and momentum. The formulation of (standard) dynamic ray tracing in ray-centred coordinates is based on specific knowledge of the first-order transformation between the ray-centred and Cartesian phase spaces. Such transformation can also be used for defining initial conditions for dynamic ray tracing in Cartesian coordinates and for obtaining the coefficients involved in two-point traveltime extrapolation. As a step towards extending dynamic ray tracing in ray-centred coordinates to higher orders we establish detailed information about the higher-order properties of the transformation between the ray-centred and Cartesian phase spaces. By numerical examples, we 1) address the validity limits of the ray-centred coordinate system, and 2) demonstrate the transformation of higher-order derivatives of traveltime from Cartesian to ray-centred coordinates.

Dynamic ray tracing is a robust and efficient method for computation of amplitude and phase attributes of the high-frequency Green's function. A formulation of dynamic ray tracing in Cartesian coordinates was recently extended to higher orders. It was demonstrated that the higher-order approach yields a much better extrapolation of traveltime and geometrical spreading into the paraxial region of a reference ray – for isotropic as well as anisotropic heterogeneous 3-D models of an elastic medium. This is of value in mapping, modelling, and imaging, where kernel operations are based on extrapolation or interpolation of Greens function attributes to densely sampled 3-D grids. As a next step, we introduce higher-order dynamic ray tracing in ray-centred coordinates, which has clear advantages: 1) Such coordinates fit naturally with the wave-propagation problems we study; 2) they lead to a reduction of the number of ordinary differential equations; 3) the initial conditions are simple and intuitive; 4) numerical errors due to redundancies are less likely to influence the results. In a numerical example, we demonstrate that paraxial extrapolation based on higher-order dynamic ray tracing in ray-centred coordinates yields results highly consistent with those obtained using Cartesian coordinates.

The broken scattering relation consists of the total lengths of broken geodesics that start from the boundary, change direction once inside the manifold, and propagate to the boundary. We show that if two reversible Finsler manifolds satisfying a convex foliation condition have the same broken scattering relation, then they are isometric. This implies that some anisotropic material parameters of the Earth can be in principle reconstructed from single scattering measurements at the surface.

We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

We show that the normal operator of the X-ray transform in $\mathbb R^d$, $d\geq2$, has a unique continuation property in the class of compactly supported distributions. This immediately implies uniqueness for the X-ray tomography problem with partial data and generalizes some earlier results to higher dimensions. Our proof also gives a unique continuation property for certain Riesz potentials in the space of rapidly decreasing distributions. We present applications to local and global seismology. These include linearized travel time tomography with half-local data and global tomography based on shear wave splitting in a weakly anisotropic elastic medium.

We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and in dimension three and higher we assume a foliation condition. We make no assumption regarding conjugate points or differentiability of the weight. This extends recent results for unweighted transforms.

We present a new computed tomography (CT) method for inverting the Radon transform in 2D. The idea relies on the geometry of the flat torus, hence we call the new method Torus CT. We prove new inversion formulas for integrable functions, solve a minimization problem associated to Tikhonov regularization in Sobolev spaces and prove that the solution operator provides an admissible regularization strategy with a quantitative stability estimate. This regularization is a simple post-processing low-pass filter for the Fourier series of a phantom. We also study the adjoint and the normal operator of the X-ray transform on the flat torus. The X-ray transform is unitary on the flat torus. We have implemented the Torus CT method using Matlab and tested it with simulated data with promising results. The inversion method is meshless in the sense that it gives out a closed form function that can be evaluated at any point of interest.

We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely determined by these measurements up to averaging over levelsets of the integrated attenuation. This leads to a generalized Laplace transform. We also discuss some numerical approaches and demonstrate the results with several examples.

With a Hamilton-Jacobi equation in Cartesian coordinates as a starting point, it is common to use a system of ordinary differential equations describing the continuation of first-order phase-space perturbation derivatives along a reference ray. Such derivatives can be exploited for calculation of geometrical spreading on the reference ray, and for establishing a framework for second-order extrapolation of traveltime to points outside the reference ray. The continuation of the first-order phase-space perturbation derivatives has historically been referred to as dynamic ray tracing. The reason for this is its importance in the process of calculating amplitudes along the reference ray. We extend the standard dynamic ray tracing scheme to include higher orders in the phase-space perturbation derivatives. The main motivation is to extrapolate and interpolate important amplitude and phase properties of high-frequency Green's functions with better accuracy. Principal amplitude coefficients, geometrical spreading factors, traveltimes, slowness vectors, and curvature matrices are examples of quantities for which we enhance the computation potential. This, in turn, has immediate applications in modelling, mapping, and imaging. Numerical tests for 3D isotropic and anisotropic heterogeneous models yield clearly improved extrapolation results for traveltime and geometrical spreading. One important conclusion is that the extrapolation function for geometrical spreading must be at least third order to be appropriate at large distances away from the reference ray.

We show that the existence of a function in $L^{1}$ with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions of constant geodesic X-ray transform always exist on manifolds with rotational symmetry.

We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to also prove similar results when the underlying equation is the quasilinear $p$-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation $\operatorname{div}(\sigma\lvert\nabla u\rvert^{p-2}\nabla u)=0$ where the measurable conductivity $\sigma\colon\Omega\to[0,\infty]$ is zero or infinity in large sets and $1 < p < \infty$.

We study the problem of recovering a function on a pseudo-Riemannian manifold from its integrals over all null geodesics in three geometries: pseudo-Riemannian products of Riemannian manifolds, Minkowski spaces and tori. We give proofs of uniqueness anc characterize non-uniqueness in different settings. Reconstruction is sometimes possible if the signature $(n_1,n_2)$ satisfies $n_1\geq1$ and $n_2\geq2$ or vice versa and always when $n_1,n_2\geq2$. The proofs are based on a Pestov identity adapted to null geodesics (product manifolds) and Fourier analysis (other geometries). The problem in a Minkowski space of any signature is a special case of recovering a function in a Euclidean space from its integrals over all lines with any given set of admissible directions, and we describe sets of lines for which this is possible. Characterizing the kernel of the null geodesic ray transform on tori reduces to solvability of certain Diophantine systems.

We recover the higher order terms for the acoustic wave equation from measurements of the modulus of the solution. The recovery of these coefficients is reduced to a question of stability for inverting a Hamiltonian flow transform, not the geodesic X-ray transform encountered in other inverse boundary problems like the determination of conformal factors. We obtain new stability results for the Hamiltonian flow transform, which allow to recover the higher order terms.

The X-ray transform on the periodic slab $[0,1]\times\mathbb T^n$, $n\geq0$, has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless $n=0$. We characterize the kernel of the geodesic X-ray transform for $L^2$-regular $m$-tensors for any $m\geq0$. The characterization extends to more general manifolds, twisted slabs, including the Möbius strip as the simplest example.

We study ray transforms on spherically symmetric manifolds with a piecewise $C^{1,1}$ metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of $L^2$ functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds with a $C^{1,1}$ metric. To make these problems tractable in low regularity, we introduce and study a class of generalized Abel transforms and study their properties. This low regularity setting is relevant for geophysical applications.

We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to $S^1$ nor to $S^3$. This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Radon transform as a family of symmetric operators indexed by nontrivial homomorphisms from $S^1$.

We consider the broken ray transform on Riemann surfaces in the presence of an obstacle, following earlier work of Mukhometov. If the surface has nonpositive curvature and the obstacle is strictly convex, we show that a function is determined by its integrals over broken geodesic rays that reflect on the boundary of the obstacle. Our proof is based on a Pestov identity with boundary terms, and it involves Jacobi fields on broken rays. We also discuss applications of the broken ray transform.

We discuss a quantum mechanical indirect measurement method to recover a position dependent Hamilton matrix from time evolution of coherent quantum mechanical states through an object. A mathematical formulation of this inverse problem leads to weighted X-ray transforms where the weight is a matrix. We show that such X-ray transforms are injective with very rough weights. Consequently, we can solve our quantum mechanical inverse problem in several settings, but many physically relevant problems we pose also remain open. We discuss the physical background of the proposed imaging method in detail. We give a rigorous mathematical treatment of a neutrino tomography method that has been previously described in the physical literature.

We show injectivity of the X-ray transform and the $d$-plane Radon transform for distributions on the $n$-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvière. We also show solenoidal injectivity of the X-ray transform on the $n$-torus for tensor fields of any order, allowing the tensors to have distribution valued coefficients. These imply new injectivity results for the periodic broken ray transform on cubes of any dimension.

We reduce the broken ray transform on some Riemannian manifolds (with corners) to the geodesic ray transform on another manifold, which is obtained from the original one by reflection. We give examples of this idea and present injectivity results for the broken ray transform using corresponding earlier results for the geodesic ray transform. Examples of manifolds where the broken ray transform is injective include Euclidean cones and parts of the spheres $S^n$. In addition, we introduce the periodic broken ray transform and use the reflection argument to produce examples of manifolds where it is injective. We also give counterexamples to both periodic and nonperiodic cases. The broken ray transform arises in Calderón's problem with partial data, and we give implications of our results for this application.

We reduce boundary determination of an unknown function and its normal derivatives from the (possibly weighted and attenuated) broken ray data to the injectivity of certain geodesic ray transforms on the boundary. For determination of the values of the function itself we obtain the usual geodesic ray transform, but for derivatives this transform has to be weighted by powers of the second fundamental form. The problem studied here is related to Calderón's problem with partial data.

Given a bounded $C^1$ domain $\Omega\subset{\mathbb R}^n$ and a nonempty subset $E$ of its boundary (set of tomography), we consider broken rays which start and end at points of $E$. We ask: If the integrals of a function over all such broken rays are known, can the function be reconstructed? We give positive answers when $\Omega$ is a ball and the unknown function is required to be uniformly quasianalytic in the angular variable and the set of tomography is open. We also analyze the situation when the set of tomography is a singleton.

We present a poem written in the honour of Anders Johan Lexell (1740–1784), a mathematician of Finnish origin, who became a collaborator and successor of Leonhard Euler. The poem was composed in Latin by Fredrik Pryss (1741–1767) in the honour of the 18-year-old promising young man in 1759. We discuss the poem itself and its connections to ancient poetic tradition as well as the foresight of Pryss in seeing the career that lay ahead of Lexell. We find that the poem is of excellent quality as a piece of art following ancient style in form, language and content. Discussing Lexell's life in light of the poem reveals that Pryss did see that Lexell would rise to fame, but not how.

We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.

These are lecture notes for the course "MATS4120 Geometry of geodesics" given at the University of Jyväskylä in Spring 2020. Basic differential geometry or Riemannian geometry is useful background but is not strictly necessary. Exercise problems are included, and problems marked important should be solved as you read to ensure that you are able to follow.

These are lecture notes for the course "MATS4300 Analysis and X-ray tomography" given at the University of Jyväskylä in Fall 2017. The course is a broad overview of various tools in analysis that can be used to study X-ray tomography. The focus is on tools and ideas, not so much on technical details and minimal assumptions. Only very basic functional analysis is assumed as background. Exercise problems are included.

This PhD thesis studies the broken ray transform, a generalization of the geodesic X-ray transform where geodesics are replaced with broken rays that reflect on a part of the boundary. The fundamental question is whether this transform is injective. We employ four different methods to approach this question, and each of them gives interesting results. Direct calculation can be used in a ball, where the geometry is particularly simple. If the reflecting part of the boundary is (piecewise) flat, a reflection argument can be used to reduce the problem to the usual X-ray transform. In some geometries one can use broken rays near the boundary to determine the values of the unknown function at the reflector, and even construct its Taylor series. One can also use energy estimates – which in this context are known as Pestov identities – to show injectivity in the presence of one convex reflecting obstacle. Many of these methods work also on Riemannian manifolds. We also discuss the periodic broken ray transform, where the integrals are taken over periodic broken rays. The broken ray transform and its periodic version have applications in other inverse problems, including Calderón's problem and problems related to spectral geometry. (More detailed abstract in the PDF file. The PDF only contains the introductory part of the thesis.)

We introduce the coherent quasiparticle approximation (cQPA), a model in thermal quantum field theory which describes various effects of temporally varying thermal medium on particle propagation. We present the cQPA Feynman rules and develop related calculational tools. Using these methods we calculate neutrino self energies in the Standard Model and derive an equation of motion for neutrino propagation in a very general framework. Some immediate implications of this equation are discussed. (More detailed abstract in the PDF file.)

The theory of neutrino oscillations has turned out to be the most reasonable explanation to the observed violations in lepton number conservation of solar and atmospheric neutrino fluxes. A derivation of the most important results of this theory is first given using a plane wave treatment and subsequently using a three-dimensional shape-independent wave packet approach. Both methods give the same oscillation patterns, but only the latter one serves as a decent starting point for analyzing coherence in neutrino oscillations. A numerical analysis of the oscillation patterns on various distance scales is also given to graphically illustrate the phenomenon of neutrino oscillation and loss of coherence in it. Several coherence conditions related to wave packet separation and the uncertainties of energy and momentum in the mass states produced in a weak charged current reaction are derived. In addition, a new limit is obtained for neutrino flux, beyond which the oscillation pattern may be washed out due to the overlap of the wave packets describing neutrinos originating from different reactions. Whether or not any phenomena will take place in the case of very high flux remains uncertain, because the flux limit is beyond the scope of any modern neutrino experiment.

Tavallisesti joukolla $X$ määritellään metriikka kuvauksena $X\times X\to{\mathbb R}$. Tässä työssä tutkitaan, mitä käy kun reaaliakseli korvataan jollain toisenlaisella järjestetyllä ryhmällä.

Järjestyksessä 49. kansainväliset fysiikkaolympialaiset järjestettiin tänä vuonna Portugalin pääkaupungissa Lissabonissa. Kokeellisen ja teoreettisen fysiikan osaamisessa kilpaili noin 400 lukioikäistä opiskelijaa 87 eri maasta.

Järjestyksessä jo 48. kansainväliset fysiikkaolympialaiset järjestettiin tänä vuonna Jaavan saarella Indonesiassa. Kokeellisen ja teoreettisen fysiikan osaamisessa kilpaili noin 400 lukioikäistä opiskelijaa 86:stä eri maasta.

Järjestyksessä jo 47. kansainväliset fysiikkaolympialaiset järjestettiin tänä vuonna Sveitsin Zürichissä 11.–17.7.2016. Kilpailussa kokeellisen ja teoreettisen fysiikan osaamistaan esitteli noin 400 lukioikäistä opiskelijaa 87:stä eri maasta.

Järjestyksessään jo 46. kansainväliset fysiikkaolympialaiset järjestettiin Intian Mumbaissa 5.–13.7.2015. Kilpailussa kokeellisen ja teoreettisen fysiikan osaamistaan esitteli 382 lukiolaista 83 eri maasta. Suomen viisihenkinen joukkue menestyi erinomaisesti. Joukkue, johon kuuluivat Arttu Tolvanen Järvenpään lukiosta, Tuomas Oikarinen Ounasvaaran lukiosta sekä Joonatan Bergholm, Iiro Sallinen ja Timo Takala Olarin lukiosta, saivat tuliaisiksi kaikkiaan neljä pronssimitalia (ks Kuva 1). Suomen edustajat ovat aikaisemmin yltäneet samaan mitalimäärään ainoastaan vuonna 1982 Länsi-Saksassa.

Haec introductio tironis in problemata inversa etiam pars dissertationis doctoralis divulgabitur. Admodum simpliciter scripta est, quo facilius a Latinistis mathematices non peritis legeretur.

Järjestyksessään jo 45. kansainväliset fysiikkaolympialaiset järjestettiin Kazakstanin pääkaupungissa Astanassa 13.–21.7.2014. Kilpailussa kokeellisen ja teoreettisen fysiikan osaamistaan esitteli 374 lukiolaista 85 eri maasta, ja Suomen viisihenkinen joukkue menestyi erinomaisesti: kaikki pääsivät palkintosijoille ja Suomi oli jälleen paras pohjoismaa. Uusina valtioina kilpailussa oli mukana Saudi-Arabia ja Latvia.

Järjestyksessään 44. kansainväliset fysiikkaolympialaiset pidettiin Tanskassa Kööpenhaminassa 7.–15.7.2013. Kilpailuun osallistui 374 kilpailijaa 83 maasta, ja Suomi menestyi kilpailussa erinomaisesti.

43. kansainväliset fysiikkaolympialaiset pidettiin Virossa Tallinnassa ja Tartossa 15.–24.7.2012.

In this research training report we calculate leading order corrections to neutrino self energies in coherent quasiparticle approximation (cQPA). These corrections are needed for a treatment of neutrino oscillation in matter in finite temperature; such a treatment in cQPA will take coherence into account more carefully than the standard approach. We first briefly review cQPA, and to this end will briefly discuss the role of coherence in quantum mechanics and the different formulations and phenomena of thermal field theory. Using the Feynman rules for evaluating self energy corrections in cQPA, we identify the relevant diagrams and calculate the corrections. Finally we discuss the corrections and compare them to the standard approach to neutrino oscillations in matter, which does not similarly take into account nonlocal coherence and the fermionic nature of neutrinos. Application of the obtained results to neutrino oscillations are unfortunately beyond the scope of this work, and will hopefully be discussed in a further study.