# Publicationes

Index publicationum Anglice scriptus est, sed aliquot textus in eo occurentes Finnice scripti sunt. Publicationes legi possunt etiam per RSS et nonnihil de eis invenitur in pagina mea in Google Scholar.

## Coauctores

- Tommi Brander
- Maarten de Hoop
- Lasse Franti
- Hans Hartikainen
- Tiia Haverinen
- Vitaly Katsnelson
- Manas Kar
- Anna-Leena Kähkönen
- Mika Latva-Kokko
- Jere Lehtonen
- Anssi Lindell
- Topi Löytäinen
- François Monard
- Heikki Mäntysaari
- Gabriel Paternain
- Vesa Pitkänen
- Mikko Salo
- Johan C.-E. Stén
- Gunther Uhlmann
- Matti Väisänen
- Alden Waters

[1] | Integral geometry on manifolds with boundary and applications , 2018.
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[arXiv] [bibtex] [pdf]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. |

[2] | Broken ray tensor tomography with one reflecting obstacle , 2018.
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[arXiv] [bibtex] [pdf]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. |

[3] | Spectral rigidity for spherically symmetric manifolds with boundary , 2017.
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[arXiv] [bibtex] [pdf]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. |

[4] | On Radon transforms on finite groups , 2014.
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[arXiv] [bibtex] [pdf]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. |

[5] | Tensor tomography in periodic slabs , Journal of Functional Analysis, 2017. (To appear.)
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[arXiv] [bibtex] [pdf] [doi]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. |

[6] | Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds , Mathematical Proceedings of the Cambridge Philosophical Society, 2017. (To appear.)
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[arXiv] [bibtex] [pdf]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. |

[7] | Recovery of the sound speed for the Acoustic wave equation from phaseless measurements , Communications in Mathematical Sciences, 2015. (To appear.)
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[arXiv] [bibtex] [pdf] [cites]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. |

[8] | X-ray transforms in pseudo-Riemannian geometry , Journal of Geometric Analysis, volume 28, number 1, pp. 606–626, 2018.
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[arXiv] [bibtex] [pdf] [doi] [cites]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. |

[9] | Superconductive and insulating inclusions for linear and non-linear conductivity equations , Inverse Problems and Imaging, volume 12, number 1, pp. 91–123, 2018.
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[arXiv] [bibtex] [pdf] [doi] [cites]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 $1 < p < \infty$. |

[10] | Abel transforms with low regularity with applications to X-ray tomography on spherically symmetric manifolds , Inverse Problems, volume 33, number 12, pp. 124003, 2017. (Special issue "100 Years of the Radon Transform".)
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[arXiv] [bibtex] [pdf] [doi]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. |

[11] | Coherent Quantum Tomography , SIAM Journal on Mathematical Analysis, volume 48, number 5, pp. 3039–3064, 2016.
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[arXiv] [bibtex] [pdf] [doi] [cites]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. |

[12] | Broken ray transform on a Riemann surface with a convex obstacle , Communications in Analysis and Geometry, volume 24, number 2, pp. 379–408, 2016.
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[arXiv] [bibtex] [pdf] [doi] [cites]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. |

[13] | On Radon transforms on compact Lie groups , Proceedings of the American Mathematical Society, volume 144, number 2, pp. 681–691, 2016.
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[arXiv] [bibtex] [pdf] [doi] [cites]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 $S^1$. |

[14] | A reflection approach to the broken ray transform , Mathematica Scandinavica, volume 117, number 2, pp. 231–257, 2015.
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[arXiv] [eprint] [bibtex] [pdf] [doi] [cites]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. |

[15] | On Radon transforms on tori , Journal of Fourier Analysis and Applications, volume 21, number 2, pp. 370–382, 2015.
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[arXiv] [bibtex] [pdf] [doi] [cites]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. |

[16] | Boundary reconstruction for the broken ray transform , Annales Academiae Scientiarum Fennicae Mathematica, volume 39, number 2, pp. 485–502, 2014.
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[arXiv] [MathSciNet] [eprint] [bibtex] [pdf] [doi] [cites]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. |

[17] | Broken ray tomography in the disc , Inverse Problems, volume 29, number 3, pp. 035008, 2013.
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[arXiv] [MathSciNet] [bibtex] [pdf] [doi] [cites]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. |

[18] | A Eulogy in Honour of Anders Johan Lexell, an 18th Century Finnish Mathematician , In Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture. Bridges Finland (E. Torrence, B. Torrence, C. H. Séquin, D. McKenna, K. Fenyvesi, R. Sarhangi, eds.), Phoenix: Tessellations Publishing, pp. 545–548, 2016.
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[bibtex] [pdf]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. |

[19] | Analysis and X-ray tomography , 2017.
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[arXiv] [bibtex] [pdf]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. |

[20] | On the broken ray transform , PhD thesis, University of Jyväskylä, Department of Mathematics and Statistics, Report 140, 2014. (advisor: Mikko Salo)
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[arXiv] [eprint] [bibtex] [pdf] [cites]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.) |

[21] | Neutrino transport in coherent quasiparticle approximation , Master's thesis, University of Jyväskylä, Department of Physics, 2012. (advisor: Kimmo Kainulainen)
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[eprint] [bibtex] [pdf]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.) |

[22] | Metrisoituvuuden yleistämisestä , Master's thesis, University of Jyväskylä, Department of Mathematics and Statistics, 2011. (advisor: Raimo Näkki)
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[bibtex] [pdf]Tavallisesti joukolla $X$ määritellään metriikka kuvauksena $X\times X\to{\mathbb R}$. Tässä työssä tutkitaan, mitä käy kun reaaliakseli korvataan jollain toisenlaisella järjestetyllä ryhmällä. |

[23] | Coherence in neutrino oscillations , Bachelor's thesis, University of Jyväskylä, Department of Physics, 2011. (advisor: Jukka Maalampi)
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[eprint] [bibtex] [pdf] [cites]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. |

[24] | Suomelle hopeaa ja pronssia vuoden 2017 Kansainvälisissä fysiikkaolympialaisissa , Dimensio, volume 2017, number 5, pp. 8–12, 2017.
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[bibtex] [pdf]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. |

[25] | Suomelle kaikkien aikojen palkintosaalis vuoden 2016 Kansainvälisissä fysiikkaolympialaisissa , Dimensio, volume 2016, number 5, pp. 9–15, 2016.
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[bibtex] [pdf]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. |

[26] | Suomalaisnuoret menestyivät 46. kansainvälisissä fysiikkaolympialaisissa Intiassa , Dimensio, volume 2015, number 5, pp. 31–34, 2015.
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[bibtex] [pdf]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. |

[27] | Kansainvälisiset fysiikkaolympialaiset Kazakstanissa , Dimensio, volume 2014, number 5, pp. 24–27, 2014.
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[bibtex] [pdf]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. |

[28] | De problematis inversis , Melissa, volume 180, pp. 4–5, 2014.
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[bibtex] [pdf]Haec introductio tironis in problemata inversa etiam pars dissertationis doctoralis divulgabitur. Admodum simpliciter scripta est, quo facilius a Latinistis mathematices non peritis legeretur. |

[29] | Suomi menestyi kansainvälisissä fysiikkaolympialaisissa , Dimensio, volume 2013, number 5, pp. 24–27, 2013.
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[bibtex] [pdf]Järjestyksessään 44. kansainväliset fysiikkaolympialaiset pidettiin Tanskassa Kööpenhaminassa 7.–15.7.2013. Kilpailuun osallistui 374 kilpailijaa 83 maasta, ja Suomi menestyi kilpailussa erinomaisesti. |

[30] | Suomi menestyi kansainvälisissä fysiikkaolympialaisissa , Dimensio, volume 2012, number 5, pp. 10–13, 2012.
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[bibtex] [pdf]43. kansainväliset fysiikkaolympialaiset pidettiin Virossa Tallinnassa ja Tartossa 15.–24.7.2012. |

[31] | One loop neutrino self energies in coherent quasiparticle approximation , 2011. (research training report)
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[bibtex] [pdf]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. |