finnish mathematical days 2020 - university of oulu · on early detection of high-risk prostate...

Finnish Mathematical Days 2020A collection of abstracts

Oulu2nd - 3rd of January 2020

1

Invited plenary lecturers

Christel Geiss University of JyväskyläSabrina Kombrink University of BirminghamAleksis Koski University of JyväskyläEveliina Peltola University of BonnStéphane Seuret Université Paris-Est CréteilSamuli Siltanen University of Helsinki

Other speakers

Simo Ali-Löytty Tampere UniversityGaëlle Brunet University of Eastern FinlandJean-Baptiste Casteras University of HelsinkiJosephine Dufitinema University of VaasaAnne-Maria Ernvall-Hytönen Åbo Akademi UniversityRagnar Freij-Hollanti Aalto UniversityJanne Gröhn University of Eastern FinlandPhilipp Guth University of MannheimAkseli Haarala University of HelsinkiAnni Hakanen University of TurkuAntti Hannukainen Aalto UniversityMiika Hannula University of HelsinkiPauliina Hirvi Aalto University

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Jani Hirvonen Tampere UniversityToni Hotanen University of TurkuJuha-Matti Huusko University of Eastern FinlandJokke Häsä University of HelsinkiKonstantin Izyurov University of HelsinkiJoonatan Jalonen University of TurkuJesse Jääsaari University of TurkuJarmo Jääskeläinen University of JyväskyläTerhi Kaarakka Tampere UniversityVesa Kaarnioja University of New South WalesLeena Kalliovirta University of HelsinkiIlmari Kangasniemi University of HelsinkiAnna Kausamo University of JyväskyläJukka Kemppainen University of OuluSampsa Kiiskinen University of JyväskyläJuha Kinnunen Aalto UniversityVille Kolehmainen University of Eastern FinlandJuho Kontio University of OuluJaakko Kultima University of OuluSaara Lehto University of HelsinkiKangwei Li Tianjin UniversitySauli Lindberg University of HelsinkiKerkko Luosto Tampere UniversityKamalakshya Mahatab University of Helsinki

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Mika Mattila Tampere UniversitySanteri Miihkinen Åbo Akademi UniversityTerhi Moisala University of JyväskyläAnton Nazarov Saint-Petersburg State UniversityThuan Nguyen University of JyväskyläAntti Niemi University of OuluIlmari Kangasniemi University of HelsinkiNeea Palojärvi Åbo Akademi UniversityJarkko Peltomäki University of TurkuKirsi Peltonen Aalto UniversityPetteri Piiroinen University of HelsinkiIstvan Prause University of Eastern FinlandJuha-Pekka Puska Aalto UniversityPaavo Raittinen Aalto UniversityJohanna Rantala University of JyväskyläJosé André Rodriguez Migueles University of HelsinkiLassi Roininen Lappeenranta University of TechnologyMatthew Romney University of JyväskyläJohanna Rämö University of HelsinkiMikko Salo University of JyväskyläTommi Sottinen University of VaasaGunnar Söderbacka Åbo Akademi UniversityOlli Tapiola University of JyväskyläVille Tengvall University of Helsinki

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Esko Turunen Tampere UniversityTeemu Tyni University of HelsinkiTopi Törmä University of OuluPauliina Uusitalo University of OuluAntti Valmari University of JyväskyläZhuang Wang University of JyväskyläHarri Varpanen JAMK University of Applied SciencesEsa Vesalainen Åbo Akademi UniversityLauri Viitasaari Aalto UniversityJoni Virta Aalto University/University of TurkuJani Virtanen University of Reading/University of HelsinkiHenrik Wirzenius University of Helsinki

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Sessions and talks

Plenary lectures, L1

Stéphane Seuret (Thursday 9:00 - 9:50)Function spaces in multifractal environment, and the Frisch-Parisiconjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Christel Geiss (Thursday 13:30 - 14:20)Regularity properties of backward stochastic differential equationsand their associated PDEs . . . . . . . . . . . . . . . . . . . . . . . 16

Eveliina Peltola (Thursday 14:30 - 15:20)On connections between critical models, SLE, and CFT . . . . . . . 16

Samuli Siltanen (Friday 9:00 - 9:50)Inverse Problems and the Nonlinear Fourier Transform . . . . . . . 17

Aleksis Koski (Friday 13:30 - 14:20)Sobolev Homeomorphic Extensions . . . . . . . . . . . . . . . . . . 18

Sabrina Kombrink (Friday 16:30 - 17:20)The Steiner formula - from convex bodies to fractals . . . . . . . . . 18

Sessions on Thursday 10:30 - 12:00

Harmonic analysis, L3

Kangwei LiSome weighted estimates on product spaces . . . . . . . . . . . . . 19

Olli TapiolaC

p

weights and the Coifman–Fefferman inequality . . . . . . . . . . 19Juha Kinnunen

Higher integrability for doubly nonlinear parabolic equations . . . . 20

6

Mathematics and arts, L4

Kirsi PeltonenAalto Math & Arts in Shanghai 2019 . . . . . . . . . . . . . . . . . 21

Saara LehtoDance in Mathematics Education . . . . . . . . . . . . . . . . . . . 21

Geometric analysis 1, L5

Terhi MoisalaRectifiability results in Carnot groups . . . . . . . . . . . . . . . . . 22

Zhuang WangTraces of first order Sobolev spaces on regular trees . . . . . . . . . 22

Matthew RomneyUniformization with infinitesimally metric measures . . . . . . . . . 23

Inverse problems 1, L6

Jaakko KultimaDirect and inverse scattering problems for quasi-linear biharmonicoperator in 3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Teemu TyniNonlinear inverse scattering for a biharmonic operator on the line . 24

Mikko SaloInverse problems for real principal type operators . . . . . . . . . . 25

Stochastics, L7

Konstantin IzyurovUniversality of spin correlations in the critical Ising model . . . . . 26

Thuan NguyenApproximation of certain stochastic integrals with jumps in weightedbounded mean oscillation spaces . . . . . . . . . . . . . . . . . . . . 27

Tommi SottinenTBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7

Coding theory, L8

Anni HakanenOn the Metric Dimension for Locating Multiple Objects . . . . . . 28

Anne-Maria Ernvall-HytönenOn the proximity of primes and elements in other sufficiently densesubsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Ragnar Freij-HollantiLifting a Code over a Simplicial Complex . . . . . . . . . . . . . . . 29


Partial differential equations, L3

Sauli LindbergConvex integration in magnetohydrodynamics . . . . . . . . . . . . 30

Jean-Baptiste CasterasTravelling wave solutions for a fourth order Schrödinger . . . . . . . 31

Akseli HaaralaOn the electrostatic Born-Infeld equations and the Lorentz meancurvature operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Matematiikan osaamisen tutkimusperustainen arviointi, L4

Terhi KaarakkaKäänteinen opetus ja arviointi - miten eteenpäin? . . . . . . . . . . 32

Jokke HäsäTodistamisajattelun tietokoneavusteinen kehittäminen ja arviointiyliopistomatematiikan johdantokurssilla . . . . . . . . . . . . . . . . 33

Johanna Rämö & Jokke HäsäArviointi 2020 – Ajankohtaista yliopistomatematiikan arvioinnissaja arviointitutkimuksessa . . . . . . . . . . . . . . . . . . . . . . . . 34

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Functional Analysis, L5

Jani VirtanenToeplitz operators on Fock spaces . . . . . . . . . . . . . . . . . . . 35

Santeri MiihkinenOn the Hilbert matrix operator on analytic function spaces . . . . . 35

Henrik WirzeniusCompact-by-approximable operators on Banach spaces failing theapproximation property . . . . . . . . . . . . . . . . . . . . . . . . 36

Number theory 1, L6

Mika MattilaThe connection between the cube semilattice structure and singu-larity of LCM-type matrices on GCD closed sets . . . . . . . . . . . 37

Neea PalojärviOn ⌧ -Li coefficients and explicit zero-free regions . . . . . . . . . . . 37

Topi TörmäGeneralized continued fraction expansions with constant partial de-nominators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Statistics 1, L7

Juho KontioScalable nonparametric dimension reduction method for searchinghigher-order interaction terms in high-dimensional regression models 38

Paavo RaittinenOn early detection of high-risk prostate cancer: applied discoveryand validation models using genotype information . . . . . . . . . . 39

Leena KalliovirtaScenario for structural development of livestock production in theBaltic littoral countries . . . . . . . . . . . . . . . . . . . . . . . . . 40

9

Discrete and symbolic dynamics, L8

Toni HotanenA new kind of measure-theoretic entropy concerning endomorphismsof measure-preserving dynamical systems . . . . . . . . . . . . . . . 41

Joonatan JalonenOne-sided vs. two-sided cellular automata . . . . . . . . . . . . . . 41

Jarkko PeltomäkiSymbolic Square Root Map . . . . . . . . . . . . . . . . . . . . . . 42

Computational mathematics 1, L9

Gaëlle BrunetCOMPUTATION OF PDE’S ON COMPACT MANIFOLDS . . . . 43

Antti HannukainenEigensolutions in Distributed Computing Environments . . . . . . . 44

Sampsa KiiskinenTowards a Formalization of Discrete Exterior Calculus . . . . . . . 45

Sessions on Friday 10:30 - 12:00


Anna KausamoThe Monge problem in optimal mass transportation: from two tomany marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

José Andrés Rodriguez MiguelesGeodesics on hyperbolic surfaces and volumes of link complementsin Seifert-fibered spaces . . . . . . . . . . . . . . . . . . . . . . . . . 46

Ilmari KangasniemiOn the entropy of uniformly quasiregular maps . . . . . . . . . . . 47

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Verkko-opetus ja opiskelijoiden etäosallistuminen, L5

Juha-Matti HuuskoMatematiikan verkkokurssin rakentamisen yksityiskohtia, haasteitaja ideoita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Harri VarpanenOhjelmallisia ratkaisuja yksilöityjen tehtävien toteuttamiseen verkko-opetuksessa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Simo Ali-LöyttySähköisten matematiikan tenttien esseekysymysten automaattinenarviointi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Mathematical physics, L6

Anton NazarovLimit shape of infinite tensor power decomposition in the infiniterank limit of Lie algebras so(2n+1) . . . . . . . . . . . . . . . . . . 50

Istvan PrauseRandom tilings, arctic curves and a Beltrami equation . . . . . . . 50

Jani VirtanenEntanglement entropy in quantum spin chain models . . . . . . . . 51

Number theory 2, L7

Kamalakshya MahatabJoint large values of Orthogonal L functions in Selberg Class . . . . 52

Jesse JääsaariSign changes of Hecke eigenvalues in GL(3) . . . . . . . . . . . . . . 52

Esa VesalainenOn Fourth and Higher Moments of Short Exponential Sums Relatedto Cusp Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

11


Ville KolehmainenData-driven regularization parameter selection in dynamic MRI . . 53

Juha-Pekka PuskaOptimal projection angles in X-ray tomography . . . . . . . . . . . 54

Lassi RoininenPosterior Inference for Sparse Hierarchical Non-stationary Models . 54


Pauliina HirviGenerating head models for diffuse optical tomography . . . . . . . 55

Philipp GuthUncertainty quantification for PDE-constrained optimization usinga quasi-Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . 56

Vesa KaarniojaUncertainty quantification for partial differential equations usingperiodic random variables . . . . . . . . . . . . . . . . . . . . . . . 57

Dynamical systems and stochastics, PR101

Gunnar SöderbackaBifurcations of multiple attractors in a predator-prey system . . . . 58

Lauri ViitasaariStochastic heat equation revisited - quantitative approximation results 59

12


Analysis, L3

Janne GröhnConverse growth estimates for ODEs with slowly growing solutions 60

Ville TengvallLocal and global injectivity of branched coverings . . . . . . . . . . 60

Jarmo JääskeläinenImproved Hölder regularity for strongly elliptic PDEs . . . . . . . . 61

Yliopistomatematiikan kokeiluja ja käytännön vinkkejä, L4

Johanna RantalaKokemuksia automaattisen palautteen antamisesta MathChekillä . . 62

Simo Ali-LöyttyÄlyOppi matematiikan osahankkeen esittely . . . . . . . . . . . . . 64

Jani HirvonenYliopiston ensimmäiset insinöörimatematiikan kurssit flippaamalla . 65


Antti NiemiNumerical buckling analysis of circular cylindrical shell structures . 66

Pauliina UusitaloThe ABC of quantum waveguides of YZC . . . . . . . . . . . . . . 66

Jukka KemppainenPositivity of the fundamental solution for fractional diffusion andwave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

13

Logic, L6

Kerkko LuostoEmbedding-closed quantifiers . . . . . . . . . . . . . . . . . . . . . 68

Miika HannulaProbabilistic team semantics . . . . . . . . . . . . . . . . . . . . . . 69

Antti Valmari & Esko TurunenA Completeness Proof for A Predicate Logic with Undefined TruthValue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Statistics 2, L7

Josephine DufitinemaVolatility clustering, Risk-return relationship, and Asymmetric ad-justment in the Finnish Housing Market . . . . . . . . . . . . . . . 71

Petteri PiiroinenGeneralized modes and MAP estimators . . . . . . . . . . . . . . . 72

Joni VirtaFast tensorial independent component analysis . . . . . . . . . . . . 72

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Plenary lectures, L1

Stéphane Seuret (Thursday 9:00 - 9:50)

Function spaces in multifractal environment, and the Frisch-Parisi conjec-

ture

Multifractal properties of data, especially in turbulence, are now rigorously es-tablished. Unfortunately, the parameters measured on these data do not fit thosetheoretically obtained for the typical properties of functions in any standard func-tional spaces: Hölder, Sobolev, Besov... In this talk, we introduce new Besov-likespaces in which the typical functions possess very rich scaling properties, mim-icking those observed on data for instance. We obtain various characterizations ofthese function spaces, in terms of oscillations or wavelet coefficients. Combiningthis with a new construction of almost-doubling probability measures with pre-scribed multifractal properties, we bring a solution to the so-called Frisch-Parisiconjecture. This is a joint work with Julien Barral (Université Paris-Nord).

15

Christel Geiss (Thursday 13:30 - 14:20)

Regularity properties of backward stochastic differential equations and

their associated PDEs

Backward stochastic differential equations (BSDEs) are SDEs where instead of aninitial value a random terminal condition is given. If this terminal condition isdescribed by a functional of a solution to an SDE, we speak of forward-backwardSDEs.

This type of equations is intrinsically connected to semilinear 2nd order partialdifferential equations. For solutions to forward-backward SDEs driven by Brow-nian motion, Lévy noise or random walk we discuss smoothness (in the sense offractional Malliavin Sobolev spaces) and approximation properties and address therelation to the associated PDEs, integral-partial differential equations and finitedifference equations, respectively.

This talk is based on joint work with Stefan Geiss, Eija Laukkarinen, AnttiLuoto (Jyväskylä), Philippe Briand and Céline Labart (Université Savoie MontBlanc) and Alexander Steinicke (Montanuniversität Leoben).

Eveliina Peltola (Thursday 14:30 - 15:20)

On connections between critical models, SLE, and CFT

For a number of lattice models in 2D statistical physics, it has been proven thatthe scaling limit of an interface at criticality (with suitable boundary conditions)is a conformally invariant random curve, Schramm-Loewner evolution (SLE). Sim-ilarly, collections of several interfaces converge to collections of interacting SLEs.Connection probabilities of these interfaces encode crossing probabilities in thelattice models, which should also be related to correlation functions of appropri-ate fields in the corresponding conformal field theory (CFT); the latter, however,being mathematically ill-defined. I discuss results pertaining to make sense of thisrelationship.

16

Samuli Siltanen (Friday 9:00 - 9:50)

Inverse Problems and the Nonlinear Fourier Transform

Electrical impedance tomography (EIT) is an emerging medical imaging method.It is based on probing the human body with harmless electric currents fed throughelectrodes on the skin. The voltages appearing on the electrodes are measured, andthe aim of EIT is to recover the internal distribution of electric conductivity. Theresulting image can be used for diagnosing stroke or assessing the lung function ofcystic fibrosis patients.

The mathematical model of EIT is the inverse conductivity problem intro-duced by Alberto Calderón in 1980. It is a generic example of an ill-posed inverseboundary value problem, where one tries to reconstruct a PDE coefficient from aDirichlet-to-Neumann map. This reconstruction task is highly sensitive to mod-elling errors and measurement noise, and therefore requires regularised solution.

A mathematically satisfying regularisation approach is offered by a nonlin-ear Fourier transform, based on Complex Geometric Optics solutions introducedby John Sylvester and Gunther Uhlmann in 1987. A low-pass filter applied onthe nonlinear frequency domain enables robust real-time EIT imaging, with cut-off frequency determined by the amplitude of measurement noise. This imagingmethod is based on solving a D-bar equation and is connected to the theory ofpseudoanalytic functions.

There are further interesting possibilities arising from the use of the nonlinearFourier transform. An added one-dimensional Fourier transform leads to singu-larity propagation along two-dimensional leaves, according to the Duistermaat-Hörmander theory of complex principal type operators. This can be used in EITimaging for recovering boundaries between tissues and organs.

Furthermore, the nonlinear Fourier transform can be used for linearising theNovikov-Veselov equation, a (2+1) dimensional generalisation of the KdV equa-tion.

Based on these examples it is safe to say that the nonlinear Fourier transformis a versatile tool applicable to very different problems. It surely holds more secrets

17

yet to be revealed.

Aleksis Koski (Friday 13:30 - 14:20)

Sobolev Homeomorphic Extensions

In the mathematical models of nonlinear elasticity, Sobolev homeomorphisms areused to represent deformations between two elastic bodies (domains in Euclideanspace). Hence one of the most fundamental questions in this theory is whethertwo such bodies admit a Sobolev homeomorphism between them at all, possiblywith some fixed boundary values. Perhaps surprisingly, this question remains unan-swered in many of the important cases. In my talk, which is intended for a generalaudience, I will give an overview of the important questions and share some of therecent developments regarding the matter.

Sabrina Kombrink (Friday 16:30 - 17:20)

The Steiner formula - from convex bodies to fractals

For a given convex body A ⇢ Rd the Steiner formula states that the d-dimensionalvolume of the "-parallel set of A can be expressed as a polynomial in " of degree d.The coefficients of the polynomial provide important information on the geometryof the given set, such as its volume, its surface area or Euler characteristic. In thistalk we will investigate extensions of the Steiner formula to further classes of setsand discuss the meaning of the analogues of the coefficients. When investigatingthe class of fractal sets, do the analogues of the coefficients lead to notions of’fractal volume’, ’fractal surface area’ and ’fractal Euler characteristic’?

18


Harmonic analysis, L3

Kangwei Li

Some weighted estimates on product spaces

By extending a classical result due to Muckenhoupt and Wheeden to the productBMO setting, we establish the weighted estimates for general bilinear bi-parameterCalderon-Zygmund operators. We also demonstrate the Bloom type estimates inits full generality. This talk is based on the recent joint work with E. Airta, H.Martikainen and E. Vuorinen.

Olli Tapiola

Cp weights and the Coifman–Fefferman inequality

It is a long-standing open problem in harmonic analysis to characterize the weightsw that satisfy the Coifman–Fefferman inequality

kTfkL

p(w) CkMfkL

p(w),

where 0 < p < 1, T is a singular integral operator and M is the Hardy–Littlewoodmaximal operator. In the early 1980’s, Muckenhoupt showed that if the inequalityholds for the Hilbert transform, then the weight satisfies a generalized version ofthe A1 condition called the C

p

condition. He also conjectured that this conditionis sufficient for the inequality. In this talk, we discuss recent developments relatedto this conjecture and extensions of known results for rough homogeneous singularintegrals. This is a joint work with J. Canto, K. Li and L. Roncal.

19

Juha Kinnunen

Higher integrability for doubly nonlinear parabolic equations

We discuss a local higher integrability result for the spatial gradient of weak solu-tions to doubly nonlinear parabolic equations of the type

�|u|p�2u�t

� div

�|Du|p�2Du�= 0

in the range

max

⇢2n

n+ 2

, 1

�.

20

Mathematics and arts, L4

Kirsi Peltonen

Aalto Math & Arts in Shanghai 2019

Aalto Math & Arts in Shanghai Future Art Lab 2019 exhibition was a joint effort ofAalto School of Arts, Design and Architecture, Aalto School of Science and AaltoSchool of Engineering. The focus of the contribution of Aalto University was tointroduce the interdisciplinary Math & Arts program, especially its underlyingcourse Crystal Flowers in Halls of Mirrors: Mathematics, Arts and Architectureand related activities to the audience.

During our visit to Shanghai, it was also possible to test our LUMATIKKAand Aalto Scientist in School activities to local school teachers and students. Thistalk will discuss about the challenges and benefits of our experience from theperspective of school teachers.

Saara Lehto

Dance in Mathematics Education

For a mathematician, mathematics is fascinating, imaginative and inspiring. Re-search shows that we can make mathematics feel equally intriguing for studentsof all levels by using interactive and hands on teaching methods. One way to dothis is to introduce mathematics through art. Indeed, mathematics and art sharemany common characteristics.

Using dance in mathematics education has the added benefit of introducingmovement and embodied methods into teaching. Current research in medicine andcognitive science tells us that moving helps us learn. In our LUMATIKKA course“Kehollinen ja liikkuva matematiikka” we explore the importance of embodimentin mathematics education and give examples of teaching activities that combinemath and dance.

21


Terhi Moisala

Rectifiability results in Carnot groups

In the mid fifties De Giorgi proved in his groundbreaking work that the reducedboundary of a set of finite perimeter is countably rectifiable. This result has farreaching consequences in Geometric Measure Theory and its generalization intomore general metric measure spaces has been object of several studies in the lastdecades. In this talk I will consider the rectifiability problem in Carnot groups,which are certain kinds of sub-Riemannian Lie groups with a rich metric-measure-space structure. In Euclidean spaces rectifiability can be equivalently described interms of C1-hypersurfaces, Lipschitz-graphs, or a geometric condition which wecall a "cone property". In Carnot groups these different types of rectifiability havetheir natural counterparts, but their equivalence is still unknown. I will describeresults regarding all of the rectifiability types mentioned above and give examplesof new classes of Carnot groups where a rectifiability result is obtained. This talkis based on joint work with Sebastiano Don, Enrico Le Donne and Davide Vittone.

Zhuang Wang

Traces of first order Sobolev spaces on regular trees

In this talk, I will give the characterizations for the existence of traces for firstorder Sobolev spaces defined on regular trees. Three definitions of trace are givenand we will also discuss the equivalences between them.

22

Matthew Romney

Uniformization with infinitesimally metric measures

The uniformization problem asks when a metric space homeomorphic to the 2-sphere must also be quasiconformally or quasisymmetrically equivalent to the 2-sphere. This problem is fairly well understood in the case of non-fractal metric2-spheres due to uniformization theorems of Bonk–Kleiner, Rajala, and Lytchak–Wenger. On the other hand, the fractal case is much more difficult. In this talk,we present the notion of infinitesimally metric measures as a tool for approachingthe fractal problem. This is joint work with K. Rajala and M. Rasimus.

23


Jaakko Kultima

Direct and inverse scattering problems for quasi-linear biharmonic operator

in 3D.

We consider a direct scattering problem for a biharmonic operator with first andzero order perturbations in 3D. Perturbations are assumed to be non-linear func-tions depending on the absolute value of the total field. We start by providing theunique solvability of this problem in some suitable Sobolev space. As a main resultwe present Saito’s formula as well as two of its corollaries, namely uniqueness andrepresentation formula for the solution of the inverse problem.

This is a joint work with M. Harju and V. Serov.

Teemu Tyni

Nonlinear inverse scattering for a biharmonic operator on the line

We consider an inverse scattering problem for a differential operator of order fouron the line with two coefficients which may be complex-valued. We let the twounknown coefficients depend non-linearly on the total field. Such operators arisefor example in the theory of vibrations of beams and the study of elasticity. Inthis talk we show that the Born approximation can be used effectively to recoveressential information about a combination of the coefficients from the knowledgejust one datum, the reflection coefficient at high frequencies.

This is a joint work with V. Serov.

24

Mikko Salo

Inverse problems for real principal type operators

We consider inverse boundary value problems for general real principal type differ-ential operators. The first results state that the Cauchy data set uniquely deter-mines the scattering relation of the operator and bicharacteristic ray transforms oflower order coefficients. We also give two different boundary determination meth-ods for general operators, and prove global uniqueness results for determining co-efficients in nonlinear real principal type equations. The article presents a unifiedapproach for treating inverse boundary problems for transport and wave equations,and highlights the role of propagation of singularities in the solution of related in-verse problems.

This is joint work with Lauri Oksanen (UCL), Plamen Stefanov (Purdue) andGunther Uhlmann (Washington / IAS HKUST).

25

Stochastics, L7

Konstantin Izyurov

Universality of spin correlations in the critical Ising model

Convergence of spin correlations in the Ising model on the square lattice to confor-mally covariant limits has been proven a few years ago by D. Chelkak, C. Hongler,and K. I. Extending this result to other lattices is interesting in its own and rele-vant for the study of the model on Riemann surfaces, since few Riemann surfacescan be approximated by square grid. Recently, we were able to prove convergenceof correlations on isoradial graphs, that is, on rhombi tilings. The proof is basedon a new observation in discrete complex analysis which also simplifies the originalproof in the case of the square grid. Joint work with D. Chelkak and R. Mahfouf.

26

Thuan Nguyen

Approximation of certain stochastic integrals with jumps in weighted bounded

mean oscillation spaces

In this talk we discuss the approximation problem for certain stochastic integralsdriven by a semimartingale with jumps, where the induced error process is mea-sured in weighted bounded mean oscillation (BMO) spaces.

In our setting, it is impossible to use deterministic time nets in the Rie-mann approximation of the stochastic integral because of possibly big jumps ofthe driving process. To deal with this situation, we introduce another approxima-tion scheme where one uses additionally random time nets that capture the bigjumps, whereas the expected cardinality of these additional random time nets canbe controlled.

Exploiting features of weighted BMO spaces, we show that one can signifi-cantly improve distributional estimates for the error process and our results alsoallow changes of the underlying measure, while keeping the error estimates, if thechange of measure satisfies a reverse Hölder inequality.

We also provide some illustrative examples in exponential Lévy models. Itturns out that the convergence rate of the error process depends on an interplaybetween the smoothness of the terminal condition and the intensity of small jumpsof the underlying Lévy process.

Tommi Sottinen

TBA

TBA

27

Coding theory, L8

Anni Hakanen

On the Metric Dimension for Locating Multiple Objects

Resolving sets have been widely studied in recent years. Resolving sets have ap-plications in robot navigation and network discovery, for example. They are alsoconnected to error-correcting codes and covering codes.

Let G be a connected graph with vertices V . A set R ✓ V is a resolving setof G if for all u, v 2 V there exists a vertex r 2 R such that d(r, u) 6= d(r, v).The distance array of a vertex v 2 V with respect to the set R = {r1, r2, . . . , rk}is D

R

(v) = (d(r1, v), d(r2, v), . . . , d(rk, v)). If the set R is a resolving set of G,then each vertex has a unique distance array. Resolving sets can be used to locatevertices in a graph one at a time. In this presentation, two variants of resolving setsare considered, namely, {`}-resolving sets and `-solid-resolving sets. These variantscan be used to locate multiple vertices simultaneously (i.e. determine the elementsof a set of vertices).

This presentation is based on joint work with Ville Junnila, Tero Laihonenand María Luz Puertas.

Anne-Maria Ernvall-Hytönen

On the proximity of primes and elements in other sufficiently dense subsets

Minjia Shi, Florian Luca and Patrick Sole considered the q-ary representations ofprimes and interpreted these as vectors or as code words. They showed that theHamming distance of some two elements is at most two. In this project, we try toloosen the conditions, namely to study the smallest possible density of the set sothat it still has this property. This gives some necessary characterization for a setof n-digit numbers which can be used as a code of Hamming distance greater thantwo. The arguments are very elementary.

28

Ragnar Freij-Hollanti

Lifting a Code over a Simplicial Complex

We define the lift of a linear code over an abstract simplicial complex on the sameground set, to be the smallest code whose projection to any simplex agrees withthat of the original code. We show that this is not a matroid invariant, and providesome support for the conjecture that it is matroid invariant for generic codes.

29


Partial differential equations, L3

Sauli Lindberg

Convex integration in magnetohydrodynamics

In their groundbreaking 2009 paper, de Lellis and Székelyhidi used convex integra-tion to construct bounded weak solutions of Euler equations that are compactlysupported in space-time. In particular, the solutions violate the conservation of ki-netic energy. The method of convex integration has since been applied to numerousother equations of fluid dynamics, in particular to Navier-Stokes by Buckmasterand Vicol. It was eventually used by Isett to solve the Onsager conjecture.

In the talk, I discuss my joint work with Faraco and Székelyhidi on convexintegration in 3D magnetohydrodynamics (MHD). MHD combines Euler equationswith Maxwell equations to study the interplay between a plasma and a magneticfield e.g. in solar physics. Convex integration is substantially harder to use in MHDthan, e.g., the Euler equations, mainly because an integral quantity called magnetichelicity is conserved even for L3 solutions without any further regularity. However,recently, Beekie-Buckmaster-Vicol and Faraco-L.-Székelyhidi independently man-aged to run convex integration in 3D MHD. The results complement each othernicely: Beekie, Buckmaster and Vicol show that magnetic helicity is not conservedby all L2 solutions, while the solutions of Faraco, I and Székelyhidi are bounded(and, therefore, in the intergability regime where magnetic helicity is conserved).

30

Jean-Baptiste Casteras

Travelling wave solutions for a fourth order Schrödinger

In this talk, we will be interested in standing and travelling wave solutions to afourth order nonlinear Schrödinger equation having second and fourth order dis-persion terms. This kind of equation naturally appears in nonlinear optics. In a firsttime, we will establish the existence of ground-state and renormalized solutions.We will then be interested in their qualitative properties.

Akseli Haarala

On the electrostatic Born-Infeld equations and the Lorentz mean curvature

operator

In 1930’s Born and Infeld proposed a new model of nonlinear electrodynamics.In the electrostatic case the Born-Infeld equations lead to the study of a certainquasilinear, non-uniformly elliptic operator that comes with a natural gradientconstraint. The same operator appears also as the mean curvature operator ofspacelike surfaces in the Lorentz-Minkowski space, the setting of special relativity.We will explain both of these contexts to motivate the mathematical study of saidoperator.

Our main focus will be on the regularity of the solutions of the electrostaticBorn-Infeld equations. We will talk about some now classical results as well assome recent developments.

31

Matematiikan osaamisen tutkimusperustainen arviointi,

L4

Terhi Kaarakka

Käänteinen opetus ja arviointi - miten eteenpäin?

Tampereen yliopiston Hervannan kampuksella syksyllä 2019 aloittavista insinööri-opiskelijoista hieman vajaa kolmasosa eli noin 250 opiskelijaa opiskelee ensim-mäisen vuoden matematiikan kaikki neljä opintojaksoa käänteisen opetuksen ide-ologian mukaisesti niin sanotusti flippaamalla.

Useimmilla yliopiston matematiikan opintojaksoilla opiskelijat saavat opinto-jakson päätyttyä arvosanan. Perinteisesti arviointi on perustunut tenttiin, joitakinlisäpisteitä on voinut saada harjoitustehtävien tekemisestä. Viime vuosina on kuitenkinnoussut esiin kysymyksiä siitä, onko tentti sittenkään paras mittari osaamisenarviointiin. Kun opetustavat uudistuvat ja monipuolistuvat, tenttiin perustuvaarviointi jättää monta esimerkiksi työelämän kannalta relevanttia osaamisaluettaarvioinnin ulkopuolelle. Tentti kuitenkin puolustaa paikkaansa sillä, että arviointion tällöin yhdenvertaista.

Tentti sijoittuu opintojakson loppuun, jolloin sillä ei voi ohjata oppimistaopintojakson aikana. Jatkuvan arvioinnin tarkoituksena on loppuarvioinnin lisäksiohjata opiskelijan oppimisprosessia. Tällä pyritään motivoimaan opiskelijaa jatku-vaan pitkäkestoiseen syväsuuntautuneeseen oppimiseen. Isoilla opetusryhmillä jatkuvayksilökeskeinen arviointi on kuitenkin haasteellista ja kallista.

Flippauskokeilussamme päädyimme pelkän tentin asemasta painottamaan jatku-vaa arviointia. Flippauksen osallistuvat opiskelijat keräävät 7 viikkoa kestävänopintojakson aikana maksimissaan 700 pistettä alkutasotestistä, käsitteenmuodostus-, laskuharjoitus- ja ryhmätehtävistä, tehtävien itse- ja vertaisarvioinneista, omanosaamisen itsearvioinneista sekä opettajan vetämien keskusteluja oppimistilaisuuk-sien eli prime time -tilaisuuksien keskusteluista. Lopuksi järjestettävä tentti antaamaksimissaan 300 pistettä. Opintojakson suoritukseen vaaditaan noin 500 pistettäja arvosanat annetaan noin 100 pisteen välein.

32

Flipatuilla opintojaksoillamme ryhmätehtävillä ja prime time-keskusteluillaon oppimiselle suuri merkitys. Opiskelijat oppivat keskustelemaan matematiikasta,neuvomaan toisiaan ja yhdessä toimien päätymään yhteiseen ratkaisuun. Tehtäv-inä voi olla laskutehtäviä, todistuksia, käsitteenmuodostustehtäviä tai kielentämis-tehtäviä. Vaikka arviointimme on melko monipuolista, niin esimerkiksi ryhmässätoimimalla oppiminen on arvioinnissa jäänyt melko pieneen rooliin.

Haluamme herättää keskustelua, saada ideoita ja keksiä yhdessä uusia tapojaarviointiin, kun oppiminen ja opetusmenetelmät uudistuvat ja monipuolistuvat.Kuinka pystyisimme arvioimaan juuri niitä taitoja, joita on opittu?

Jokke Häsä

Todistamisajattelun tietokoneavusteinen kehittäminen ja arviointi yliopis-

tomatematiikan johdantokurssilla

Helsingin yliopiston kurssille Johdatus yliopistomatematiikkaan kehitettiin tätäsyksyä varten digitaalisia tehtäviä, joiden tarkoitus on johdatella opiskelijoita todis-tusten lukemiseen ja kirjoittamiseen. Tehtävät perustuvat Annie ja John Seldenintutkimuksiin todistamisajattelun opettamisesta yliopiston johdantokursseilla. Tehtävienvaikutuksen arvioimiseksi opiskelijoilla teetettiin kurssin alussa ja lopussa kysely,jolla mitattiin heidän tunteitaan, asenteitaan ja pystyvyysuskoaan todistamiseenliittyen. Lisäksi heiltä kysyttiin kurssin lopussa heidän kokemuksiaan ja käsityk-siään kurssille kehitetyistä tehtävistä. Tutkimusaineistoon kuuluu myös opiskeli-joiden kurssilla laatimat ratkaisut perinteisiin todistustehtäviin. Esitelmässä ku-vailemme kehitettyjä tehtäviä ja kerromme alustavien analyysien tuloksia.

33

Johanna Rämö & Jokke Häsä

Arviointi 2020 – Ajankohtaista yliopistomatematiikan arvioinnissa ja arvioin-

titutkimuksessa

Osio koostuu alustuksesta ja yhteisestä keskustelussa. Alustuksessa käsitellääntutkimuksen ja esimerkkien kautta, millaista voi olla oppimista tukeva arviointimatematiikan opetuksessa. Mikä on arvioinnin tarkoitus? Miksi arviointiin ontärkeä kiinnittää huomiota? Miten arviointikulttuuri on muuttumassa? Tämänjälkeen keskustellaan arvioinnin nykytilasta ja tulevaisuudesta.

34

Functional Analysis, L5

Jani Virtanen

Toeplitz operators on Fock spaces

I discuss the status of the theory of Toeplitz operators on various types of Fockspaces and compare it with what is known about these operators on Hardy andBergman spaces. I also present some recent results on Fredholmness of Toeplitzoperators on generalized Fock spaces (which were introduced by Schuster andVarolin), even with small exponents, for which reason we also need to characterizethe dual of these generalized Fock spaces. Joint work with Zhangjian Hu (HuzhouUniversity).

Santeri Miihkinen

On the Hilbert matrix operator on analytic function spaces

The infinite Hilbert matrix H can be interpreted as a linear operator on spaces ofanalytic functions in the open unit disc of the complex plane by its action on theirTaylor coefficients. The boundedness of H on the Hardy spaces Hp for 1 < p < 1and Bergman spaces Ap for 2 < p < 1 was established by Diamantopoulos andSiskakis. The exact value of the norm of H acting on the Bergman spaces Ap for4 p < 1 was shown to be ⇡

sin(2⇡/p) by Dostanic, Jevtic and Vukotic in 2008. Thecase 2 < p < 4 was an open problem until in 2018 it was shown by Bozin andKarapetrovic that the norm has the same value also on the scale 2 < p < 4. In thistalk, we review some of the old results and consider the still partly open problemregarding the value of the norm on weighted Bergman spaces. The talk is basedon a joint work with Lindström and Wikman (Åbo Akademi).

35

Henrik Wirzenius

Compact-by-approximable operators on Banach spaces failing the approx-

imation property

Let K(X) denote the algebra of compact operators acting on a Banach space X

and A(X) = F(X) the closure of the bounded finite rank operators. In this talk Iwill describe recent work on the quotient algebra A

X

= K(X)/A(X) of compact-by-approximable operators, which is non-trivial only within the class of Banachspaces X failing the approximation property. I will discuss the size of A

X

andpresent examples where A

X

contains non-trivial closed ideals. This is a joint workwith Hans-Olav Tylli.

36

Number theory 1, L6

Mika Mattila

The connection between the cube semilattice structure and singularity of

LCM-type matrices on GCD closed sets

TBA

Neea Palojärvi

On ⌧ -Li coefficients and explicit zero-free regions

In this talk I will give an introduction to ⌧ -Li coefficients and my results consideringthe coefficients and explicit zero-free regions. The ⌧ -Li coefficients are members ofan infinite sequence of real numbers which can be used to determine whethercertain functions satisfy the Generalized Riemann Hypothesis or not. In the talk,I describe how finitely many ⌧ -Li coefficients can be used to determine whethercertain functions have certain zero-free regions inside the critical strip or not.

Topi Törmä

Generalized continued fraction expansions with constant partial denomi-

nators

We study generalized continued fraction expansions of the form

a1N +

a2N +

a3N + · · · ,

where N is a fixed positive integer and the partial numerators ai

are positiveintegers for all i. We call these expansions dn

N

expansions and show that everypositive real number has infinitely many dn

N

expansions for each N . In particularwe consider the dn

N

expansions of rational numbers and quadratic irrationals andprove some results regarding their periodicity. Finally we show that every positivereal number has for each N a dn

N

expansion with bounded partial numerators.

37

Statistics 1, L7

Juho Kontio

Scalable nonparametric dimension reduction method for searching higher-

order interaction terms in high-dimensional regression models

In many applications, including interaction terms into a regression model char-acterizes the relationships between the response and explanatory variables moreaccurately than individual variables can additively. Interaction terms importantto the response are typically identified through enumeration via exhaustive searchalgorithms. An immediate problem is that the number of higher-order interactionsgrows rapidly infeasible imposing a serious computational challenge. Ideally, thedimension of a feature space could be reduced before enumeration based on strongmarginal associations with the response. Unfortunately, individual explanatoryvariables contributing to the response through their interaction are not identifi-able by simple linear pre-screening methods unless they exhibit linear associationswith the response as well. However, this is rarely the case in many real-life prob-lems. The only way of identifying individual determinants of interaction terms fromthe marginal associations is to use more complex nonparametric/nonlinear meth-ods. A Gaussian process (GP) based automatic relevance determination (ARD) isknown to be theoretically among the best alternatives for such purpose. However,the estimation of GP models is feasible only for low-dimensional datasets ( 200variables) preventing the GP-based ARD method to be applied broadly. We havedeveloped a nonparametric pre-screening method* which reduces the computa-tionally expensive GP-based ARD method into a simple linear kernel regressionproblem. The proposed method preserves all the major benefits of the GP-basedARD and extends its scalability to high-dimensional datasets with tens of thou-sands of explanatory variables. Some examples will be presented to illustrate theefficiency of this method and its usability in genetic association studies.

38

Paavo Raittinen

On early detection of high-risk prostate cancer: applied discovery and val-

idation models using genotype information

Prostate cancer incidence rate is extremely high and on the rise, counting over 1.2million new cases annually and causing 350 000 deaths in 2018. While the prognosisis typically good, approximately 20dire consequences. Moreover, the initial prostatecancer diagnosis always reflects as worry and quality of life impairment. The initialprostate cancer determination is based on prostate specific antigen (PSA) measure,which cannot distinguish between low-risk and high-risk cases. After the PSA de-termination, the tumor state is characterized with various invasive methods suchas Gleason score and T-stage classification. However, both methods display inac-curacy and puts patient under infection risk. Our take on this challenge is to useinflammation-related gene single nucleotide polymorphisms (SNP) as predictors ofhigh-risk prostate cancer. SNP is a low-cost, non-invasive, and stable biomarker.We have explored inflammation SNP association with high-risk prostate cancer ina genotyped part of Finnish Randomized Screening for Prostate Cancer cohort (n= 2715) and found several statistically significant associations. Furthermore, ourvalidation model using unknown prostate cancer cohort collected during hospitalvisits (n = 888) is in concordance with our discovery model. Remarkably, few SNPsincrease early high-risk prostate cancer detection over PSA alone.

39

Leena Kalliovirta

Scenario for structural development of livestock production in the Baltic

littoral countries

Livestock production in developed countries has undergone profound changes overrecent decades, a development that seems to continue apace. One consequence isthat manure is being — and will be — produced on fewer but larger farms. Eurostatpublishes the bulk of manure nutrients from each country, but it is not known howit is distributed across farms of different sizes. This study 1) gives an estimate forthe distribution of main manure nutrients production between farms of differentsizes, 2) gives an estimate how this deviation will change in the near future and3) discusses the land use effects of this development. Results based on stationaryMarkov chain model on the farm size development suggest that by the year 2030farms housing more than 500 livestock units will produce more than two-thirds ofall manure phosphorus, whereas the proportion in 2010 was one-third. With theNitrates Directive limiting the use of organic nitrate of manure, growing farmsneed to acquire, or conclude contracts for the use of, 4.9 million hectares fromexiting farms or the open market in order to fulfil manure spreading requirements.This shift will involve 64 % of the total spreading area of 2010 and 15 % of thetotal utilized agricultural area of the regions studied. In light of these predictions,international nutrient policies should consider the evolution of farm structure andespecially manure phosphorus agglomeration. Also salient is improved co-operationbeyond the farm level to ensure the functionality of crop-livestock systems.

40

Discrete and symbolic dynamics, L8

Toni Hotanen

A new kind of measure-theoretic entropy concerning endomorphisms of

measure-preserving dynamical systems

In this talk we define a new kind of measure-theoretic entropy concerning endo-morphisms of measure-preserving dynamical systems, where the action is takenover discrete and countable amenable groups. Intuitively our entropy gives therate of the information rate of the endomorphism by the size of a given subgroup,when we know the behaviour of the group action of said subgroup. We also definea generalization for Lyapunov exponents of one-dimensional cellular automata fortopological dynamical systems over zero-dimensional compact metric spaces andderive a connection to our notion of entropy.

Joonatan Jalonen

One-sided vs. two-sided cellular automata

In this talk we give a short introduction to cellular automata theory by comparingone-sided and two-sided cellular automata. First we discuss computability. It isstraightforward to simulate Turing machines with two-sided cellular automata, andwith a small trick, even with reversible two-sided cellular automata. We discusswhy universal computing with reversible one-sided cellular automata necessarilyrequires some more surprising ideas. Due to the difficulties in doing computationwith reversible one-sided cellular automata there are almost no undecidabilityresults for them. Secondly we discuss some topological dynamical properties, inparticular expansivity. Here more is known for one-sided cellular automata than fortwo-sided; in particular it is known that for one-sided cellular automata expansivityimplies pseudo-orbit tracing property, while the same problem is open for two-sidedcellular automata.

As mentioned, this talk serves as an introduction, and as such, no prior knowl-

41

edge about cellular automata is assumed.

Jarkko Peltomäki

Symbolic Square Root Map

Let a � 1 and b � 0 be fixed integers. Consider the following six binary words:

S1 = 0, S4 = 10

a,

S2 = 010

a�1, S5 = 10

a+1(10

a)

b,

S3 = 010

a, S6 = 10

a+1(10

a)

b+1.

Let w be an infinite word that is expressible as a concatenation of squares of thesewords, that is, w = X2

1X22 · · · where X

i

2 {S1, S2, . . . , S6} for all i. Define theinfinite word

pw, the square root of w, as the word X1X2 · · · obtained from w by

removing half of each square Xi

.If X is the set of all infinite words for which the square map is defined, then

the square root map is a continuous map X ! X with respect to the producttopology on the set of infinite words over the alphabet {0, 1}.

In the talk, I will briefly describe what is known about the dynamics ofthe square root map. In particular, I will introduce certain interesting invariantsubsets of X such as Sturmian subshifts and so-called SL-subshifts. I will con-sider fixed points, periodic points, and asymptotic behavior in these subshifts.The preceding subshifts are constructed using solutions to the word equationX2

1 · · ·X2n

= (X1 · · ·Xn

)

2 where Xi

2 {S1, S2, . . . , S6} for all i. A complete charac-terization of the solutions was recently obtained by A. Saarela and the author.

42


Gaëlle Brunet

COMPUTATION OF PDE’S ON COMPACT MANIFOLDS

Killing vector fields are important in differential geometry because their flowsgenerate isometries on Riemannian manifolds. Equations for Killing fields is anoverdetermined system of PDEs which can be hard to solve explicitly. This prob-lem can be reduced to a symmetric eigenvalue problem where Killing fields aregenerated by the eigenvectors corresponding to zero eigenvalue. The method itselfis valid in any dimension, but numerical results are computed only in two dimen-sional case. To solve numerically this problem we used finite element method. Ona manifold one has to use in general several coordinate systems to describe theproblem, and the technical difficulty is then how to patch these coordinate systemstogether. We propose to solve this eigenvalue problem on the sphere with severallocal coordinate systems. This method of constructing operators on manifolds canalso be used to study other PDE systems.

43

Antti Hannukainen

Eigensolutions in Distributed Computing Environments

Let ⌦ ⇢ R3 and V ⇢ H10 (⌦) be the standard first order finite element space

over a tetrahedral partition of ⌦. In this presentation, we consider the eigenvalueproblem: Find (�, u) 2 R+ ⇥ V \ {0} such that for each w 2 V

Z

⌦

ru ·rw dx = �

Z

⌦

uw dx and kukL

2(⌦) = 1. (1)

Our aim is to compute the eigenvalues and the correspoding eigenfunctions inthe spectral interval of interest (0,⇤) to a user specified accuracy. We focus onproblems whose solution using a single workstation is impossible due to severaleigenvalues that belong to (0,⇤), need for high accuracy, or complicated geome-try that requires the use of a fine mesh. We solve problem (1) approximately byRitz projection to a subspace eV that is constructed from several local subspaces.These local subspaces can be independently constructed without any intermediatecommunication. Hence, the proposed method is well suited to distributed comput-ing environments. As an example, we describe implementation using 20 standarddesktop computers at our home institute.

44

Sampsa Kiiskinen

Towards a Formalization of Discrete Exterior Calculus

Discrete exterior calculus (dec) is a mathematical formalism for the numericalsolution of second-order boundary value problems. Since this class of problemscovers many partial differential equations that appear in physics, dec has quite afew potential practical applications. Previously, a member of our research group haswritten a C++ implementation of dec and we have used it in various collaborations.However, a growing demand to extend the implementation has turned out to beproblematic for several reasons. Upon closer inspection, the problems seem tobe just another special case that demonstrates the fundamental shortcomings inthe way we develop software for computational sciences. In this talk, I present anapproach to get around these problems by leveraging recent progress in type theoryand category theory. In particular, I propose formalizing a category-theoreticalmodel of dec in the Coq proof assistant, extracting executable code from the proofsand linking the extracted code with existing implementations. Doing this via cleveruse of type classes and nothing but constructive axioms should let us express themost salient features of dec with no runtime performance penalty. While the projectis still in its infancy, the prototypes I have built so far are very promising.

45



Anna Kausamo

The Monge problem in optimal mass transportation: from two to many

marginals

I will briefly introduce the standard deterministic optimal mass transportation(OT) problem, also known as the Monge problem. Then I move on to the multi-marginal optimal transportation. In this generalization of the (OT) problem, in-stead of minimizing the cost of moving mass from the first marginal to the secondone, we try to find the optimal way of coupling a finite number of marginal mea-sures. Optimality here is defined by the minimimality of the integral of a given costfunction with respect to the measure that couples the marginals. I will discuss thedifficulties related to solving the Monge problem in the multi-marginal framework,and present some results obtained on the topic in collaboration with Tapio Rajalaand Augusto Gerolin.

José Andrés Rodriguez Migueles

Geodesics on hyperbolic surfaces and volumes of link complements in

Seifert-fibered spaces

Let � be a link in a Seifert-fibered space M over a hyperbolic surface ⌃ thatprojects injectively to a collection of closed geodesics � in ⌃. When � is filling,the complement M� of � in M admits a hyperbolic structure of finite volume. Wegive bounds of its volume in terms of topological invariants of (�,⌃).

46

Ilmari Kangasniemi

On the entropy of uniformly quasiregular maps

Uniformly quasiregular (UQR) maps are a higher-dimensional generalization ofholomorphic dynamics based on quasiconformal analysis. In this talk, I discuss ajoint work with Yusuke Okuyama, Pekka Pankka and Tuomas Sahlsten, where westudy the entropy of UQR maps. The question indicated by holomorphic dynamicsis whether the topological entropy of a UQR map on a closed manifold equalslog(deg f); our results show that this is true when the ambient manifold is not arational cohomology sphere.

47

Verkko-opetus ja opiskelijoiden etäosallistuminen,

L5

Juha-Matti Huusko

Matematiikan verkkokurssin rakentamisen yksityiskohtia, haasteita ja ideoita

Millainen prosessi on rakentaa matematiikan verkkokurssi, joka sisältää teoriaa,kuvia, tehtäviä, videoita sekä tenttejä?

Sujuvin tapa kirjoittaa matemaattista tekstiä lienee LaTeX, jolla kirjoitetutkaavat saa toimimaan selaimessa MathJaxin avulla. Toiminnallisuutta ja muo-toiluja voi rakentaa käyttämällä verkkosivujen perusominaisuuksia, kuten CSS jaJavaScript.

Mikä on paras tapa piirtää kuvat? Käsin, ohjelmistolla, TikZ/LaTeX-komennoilla,SVG-koodina tai ehkä JSXGraph-kirjastolla?

Teoriatekstiä lukevaa opiskelijaa voi aktivoida lyhyillä kysymyksillä ja laskute-htävillä, joiden ratkaisut voi laittaa näytä/piilota-napin taakse. Videoita esimerkki-laskuista voi tehdä tabletin ja mikrofonin avulla.

Moodleen integroitu STACK-lisäosa mahdollistaa tehtävien lukuarvojen sat-unnaistamisen ja opiskelijan vastauksen sieventämisen. JSXGraph mahdollistaagraafisten tehtävien tekemisen.

Verkkokurssin voi perustaa esimerkiksi Moodleen tai DigiCampukseen. Kuinkakurssin materiaalit voi tehokkaasti ladata alustalle? Kurssin materiaalit kannataavarmuuskopioida muun muassa muistitikulle. Muistitikulle voi myös rakentaa toimi-van version kurssista.

Käydään läpi prosessiin liittyviä yksityiskohtia, haasteita ja oivalluksia. Koke-mukset perustuvat siihen kuinka neljän opintopisteen laajuinen yliopiston ensim-mäisen vuoden matematiikan kurssi rakennettiin verkkokurssiksi syksyllä 2019.

48

Harri Varpanen

Ohjelmallisia ratkaisuja yksilöityjen tehtävien toteuttamiseen verkko-opetuksessa

Googlen ohjelmallisten rajapintojen avulla luodaan opiskelijoille yksilöityjä kyse-lyjä, taulukoita ja dokumentteja. Näytän koodeja ja kerron kokemuksistani JAMK:nsyksyn 2019 opintojaksoilta Tilastotieteen perusteet (verkkokurssi) sekä Optimointija verkkomallit.

Simo Ali-Löytty

Sähköisten matematiikan tenttien esseekysymysten automaattinen arviointi

Tässä esitelmässä teemme katsauksen, kuinka matematiikan opetuksessa voidaanhyödyntää sähköisiä vaihtoehtoja osaamisen arvioinnissa. Tehtävien, joiden ratkaisuvoidaan kirjoittaa symbolisen laskennan ohjelmaksi, pystytään lähtökohtaisesti ar-vostelemaan esimerkiksi Moodlessa toimivilla STACK-tehtävillä.

Matematiikassa monet tehtävät ovat todistuksia ja käsitteenmuodostustehtäviä.Näiden arvostelemiseen ei ole vielä luotettavaa automaattista menetelmää ja mon-esti nämä tehtävät arvostellaan kokonaan manuaalisesti. Esittelemme erilaisia läh-estymistapoja siitä, kuinka koneoppimista voidaan hyödyntää matematiikan lyhyi-den esseetehtäviä arvioinnissa. Esitelmässä käymme lävitse alustavia tuloksia siitä,kuinka menetelmät toimivat yliopistomatematiikan tehtävissä.

49

Mathematical physics, L6

Anton Nazarov

Limit shape of infinite tensor power decomposition in the infinite rank

limit of Lie algebras so(2n+1)

Representation theory of simple Lie algebras plays central role in the descriptionof symmetries in physical systems, in particular in quantum field theory and inexactly solved models of statistical mechanics. We consider simple Lie algebras ofthe series so(2n+1) in the limit when n goes to infinity. We present limit shapetheorem for the probability measure defined by the decomposition into irreduciblesof tensor powers of fundamental representation. This theorem generalizes resultsof A.M. Vershik, S.V. Kerov, P.Biane for Plancherel and Schur-Weyl measures forthe permutation group and Lie algebras sl(n) to Lie algebras so(2n+1) and Braueralgebras. We illustrate our result with Monte-Carlo simulations and discuss theentropy of the measure as well as possible generalizations to character measuresand series so(2n) and sp(2n) of Lie algebras.

Istvan Prause

Random tilings, arctic curves and a Beltrami equation

Random tilings exhibit an intriguing arctic curve phenomenon - the system displaysdistinct phases (ordered and disordered) with a phase-separating (often algebraic)curve. First, I will review how one can study such phenomena through a singularvariational problem with a gradient constraint. Then, I will address how one cananalyse the variational problem with the help of a Beltrami equation. Especially, Iwill explain why arctic curves in the dimer model are algebraic and describe theiruniversal geometry.

The talk is based on joint work with K. Astala, E. Duse and X. Zhong.

50

Jani Virtanen

Entanglement entropy in quantum spin chain models

I discuss entanglement entropy of bipartite systems using the von Neumann en-tropy as measure of entanglement. Some of the most widely studied systems includeone-dimensional quantum critical systems, such as quantum spin chains, which intheir simplest setting consist of N spins. Of particular interest is the XX spin chainmodel with zero magnetic field and the study of the von Neumann entropy of thesubsystem P of spins on lattice sites {1, 2, . . . ,m} [ {2m + 1, 2m + 2, . . . , 3m},which can be analyzed using certain integral representations. For a single blocksubsystem, the integral representation involves Toeplitz determinants and the en-tropy can be calculated using the Fisher-Hartwig asymptotic expansion of thesedeterminants. In this talk, we consider a subsystem that consists of two blocks ofspins separated by one spin and compute the mutual information between the twointervals explicitly and rigorously using the Riemann-Hilbert approach. Joint workwith György Gehér (Reading University) and Alexander Its (Indiana University-Purdue University Indianapolis).

51

Number theory 2, L7

Kamalakshya Mahatab

Joint large values of Orthogonal L functions in Selberg Class

We will show that two orthogonal L-functions in the Selberg class will take jointlarge values of size exp(c(logT )1��/ log log T ) for some constant c > 0 and for everyvertical line segment {� + it : T t 2T} where 1/2 < � < 1. This is a jointwork with A. Vatwani and L. Pankowski.

Jesse Jääsaari

Sign changes of Hecke eigenvalues in GL(3)

I will discuss recent work concerning sign changes of Hecke eigenvalues related toself-dual GL(3) Maass cusp forms.

Esa Vesalainen

On Fourth and Higher Moments of Short Exponential Sums Related to

Cusp Forms

We discuss our recent upper bounds for fourth and higher moments of short ex-ponential sums related to cusp forms. These are joint work with A.-M. Ernvall-Hytönen.

52


Ville Kolehmainen

Data-driven regularization parameter selection in dynamic MRI

Dynamic magnetic resonance imaging (MRI) is used to study microvascular struc-ture and function by contrast agent or stimulus related changes in a time seriesof MRI images. Sufficient time resolution can often only be obtained using animaging protocol which produces undersampled data for each image in the timeseries. This has lead to the popularity of compressed sensing (CS) based image re-construction approaches, where all the images in the time series are reconstructedsimultaneously using a joint reconstruction formulation, which uses sparsity pro-moting regularization for the spatial and temporal features of the image series.One of the main problems in the CS approaches is how to determine the regular-ization parameters, which control the balance between the data fidelity term andspatial and temporal regularization terms. In this work, we propose a data-drivenapproach for selection of the regularization parameters such that the reconstruc-tions yield expected levels of sparsity in the domains of the regularization. Weconsider two different formulations. The first one is based on a 2D search wherewe seek a pair of regularization parameters which produce expected sparsity inboth the temporal and spatial regularization domains. The second approach is anapproximation of the first, where the sparsity based selection is split to two 1Dsearches using the S-curve method. The approaches are evaluated using simulatedand experimental golden angle dynamic contrast-enhanced (DCE) MRI data froma small animal brain study, and both approaches are demonstrated to lead tohighly feasible values for the regularization parameters

53

Juha-Pekka Puska

Optimal projection angles in X-ray tomography

This talk considers (sequential) optimization of projection angles in (parallel beam)X-ray tomography. Assuming additive Gaussian noise model and Gaussian prior,the next projection angle is chosen based on the A or D optimality conditionof Bayesian experimental design. In other words, either the determinant of thetrace of the posterior covariance matrix after the projection is minimized. Theoptimization process can be carried out before the measurements based only onthe available prior information.

Lassi Roininen

Posterior Inference for Sparse Hierarchical Non-stationary Models

Gaussian processes are valuable tools for non-parametric modelling, where typi-cally an assumption of stationarity is employed. While removing this assumptioncan improve pre- diction, fitting such models is challenging. In this work, hierarchi-cal models are constructed based on Gaussian Markov random fields with stochas-tic spatially varying parameters. Im- portantly, this allows for non-stationaritywhile also addressing the computational burden through a sparse banded repre-sentation of the precision matrix. In this setting, efficient Markov chain MonteCarlo (MCMC) sampling is challenging due to the strong coupling a posterioriof the parameters and hyperparameters. We develop and compare three adaptiveMCMC schemes and make use of banded matrix operations for faster inference.Furthermore, a novel extension to multi-dimensional settings is proposed throughan additive structure that retains the flexibility and scalability of the model, whilealso inheriting interpretability from the additive approach. A thorough assessmentof the efficiency and accuracy of the methods in nonstationary settings is presentedfor both simulated experiments and a computer emulation problem.

54


Pauliina Hirvi

Generating head models for diffuse optical tomography

Diffuse optical tomography (DOT) is a novel method for imaging cortical brainfunction. The safe and convenient setup is suitable for imaging awake toddlers,since the child does not have to be very still or alone. This presentation explainsthe process of creating head models for a study on affective touch processing intwo-year-olds with DOT.

In DOT, the head is illuminated with near-infrared light from the wavelengthrange of 650–950 nm, which opens an optical window into the cerebral cortex dueto the relatively small absorption of light. The DOT image formation is an inverseproblem, where detected changes in the amplitude of transmitted light are matchedwith a three-dimensional map of changes in the total hemoglobin concentration.The problem can be linearized by using the varying sensitivities of source-detector–pairs to different regions in the brain. The sensitivity profiles can be computedby simulating the propagation of photons in a model of the optical properties ofthe subject’s head. This presentation demonstrates the usage of age-appropriateaverage head models in place of the subject’s own magnetic resonance images(MRI) to model the head anatomy.

The first step in the process is to reconstruct target points from the sub-ject’s head using photogrammetry. The accuracy of the method is confirmed withcomparisons to computed tomography (CT) images of a doll phantom. As thesecond step, a publicly available two-year-old’s atlas model is deformed to matchits exterior surface with the target points. A grid search is implemented to findthe optimal affine transformation parameters for the registration from iterativelydensening grids of test points.

55

Philipp Guth

Uncertainty quantification for PDE-constrained optimization using a quasi-

Monte Carlo method

In this work we analyze the application of a quasi-Monte Carlo (QMC) methodto an optimal control problem constrained by an elliptic partial differential equa-tion (PDE) equipped with an uncertain diffusion coefficient. A deterministic re-formulation of the problem is to minimize the expected value of a tracking typecost functional with an additional penalty on the control. The uncertain coeffi-cient in the PDE is parametrized by a countably infinite number of terms viaa Karhunen–Loève expansion (KLE) and the expected value is considered as aninfinite-dimensional integral in the corresponding parameter space.

We discretize the optimization problem by truncating the KLE after s terms,approximating the expected value by an n-point QMC rule in s dimensions andapproximating the solution of the PDE using finite elements (FE). Numericalexperiments confirming our theoretical convergence results will be presented.

This talk is based on joint work with Vesa Kaarnioja (UNSW Sydney), FrancesY. Kuo (UNSW Sydney), Claudia Schillings (University of Mannheim), and IanH. Sloan (UNSW Sydney).

Reference: P. A. Guth, V. Kaarnioja, F. Y. Kuo, C. Schillings, and I. H. Sloan.A quasi-Monte Carlo method for an optimal control problem under uncertainty.Preprint, arXiv:1910.10022 [math.NA].

56

Vesa Kaarnioja

Uncertainty quantification for partial differential equations using periodic

random variables

Many studies in uncertainty quantification concentrate on the analysis of partialdifferential equations, where one or more of the coefficients are assumed to be ran-dom or uncertain. A popular model for representing such input random fields isthe so-called “affine model” which is assumed to depend on independent, uniformlydistributed random variables in a linear manner. In this talk, we consider an al-ternative model for the input random field, where the random variables enter theinput field as periodic functions instead. We show that it is possible to constructsimple lattice quasi-Monte Carlo rules within this framework that obtain higherorder cubature convergence rates. We apply our method to elliptic partial differ-ential equations with random coefficients, and present numerical experiments thatverify our theoretical results.

This is a joint work with Frances Y. Kuo (UNSW Sydney) and Ian H. Sloan(UNSW Sydney).

Reference: V. Kaarnioja, F. Y. Kuo, and I. H. Sloan: Uncertainty quantifica-tion using periodic random variables, preprint arXiv:1905.07693v2 [math.NA].

57

Dynamical systems and stochastics, PR101

Gunnar Söderbacka

Bifurcations of multiple attractors in a predator-prey system

This is a review of most recent results on bifurcation and chaos in a knownpredator-prey system and thus an update of the papers in the references. Thesystem of n competing predators feeding on the same prey is of the type

X 0i

= pi

'i

(S)Xi

� di

Xi

, i = 1, ..., n, (2a)

S 0= H(S)�

nX

i=1

qi

'i

(S)Xi

, (2b)

where the variable S represents the prey and the variables Xi

represent thepredators. They are, of course, non-negative. The function '

i

is assumed non-decreasing.

We consider the case where

H(S) = r S

✓1� S

K

◆, '

i

(S) =S

S + Ai

, (3)

and where the parameters r, K and Ai

are positive.The dynamics in the coordinate planes representing one of the predators and

the prey is well known and there is no more than one cycle. The system has noequilibrium, where predators coexist (in non-degenerate cases). But the predatorscan coexist in a cyclic and complicated way. There exists multiple attractors ofcyclic and different chaotic chaos including "spiral-like" chaos. This happens evenin cases, where the populations do not become unrealistic low. We present newdiscovered phenomena and discuss the possible bifurcations of these and contoursfrom where they could develop.

58

References:

1. Osipov, A, V., Söderbacka, G. Poincaré map construction for some classictwo predators - one prey systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg27, 1750116, 9 pp, (2017).

2. Osipov, A, V., Söderbacka, G. Review of results on a system of type manypredators- one prey, Nonlinear Systems and Their Remarkable MathematicalStructures, pp 520-540, CRC Press, 2018.

Lauri Viitasaari

Stochastic heat equation revisited - quantitative approximation results

In this talk we discuss d-dimensional stochastic heat equations driven by a Gaus-sian noise which is white in time and has a spatial covariance given by the Rieszkernel. Basic theory and properties of the solutions are discussed. As a main result,we present a quantitative central limit theorem stating that the spatial average ofthe solution over an Euclidean ball is close to a Gaussian distribution, when the ra-dius of the ball tends to infinity. Our central limit theorem is described in the totalvariation distance, using Malliavin calculus and Stein’s method. We also providea functional central limit theorem and analogous results in the case of space-timewhite noise. Extensions and further open questions are discussed.

59


Analysis, L3

Janne Gröhn

Converse growth estimates for ODEs with slowly growing solutions

Let f1, f2 be linearly independent solutions of f 00+ Af = 0, where the coeffi-

cient A is an analytic function in the open unit disc D of the complex plane.Many properties of this differential equation can be described in terms of thesubharmonic auxiliary function u = � log (f1/f2)

#. For example, the case whensup

z2D |A(z)|(1� |z|2)2 < 1 and f1/f2 is normal, is characterized by the conditionsup

z2D |ru(z)|(1� |z|2) < 1. Different types of Blaschke-oscillatory equations arealso described in terms of harmonic majorants of u.

Even if f1, f2 are bounded linearly independent solutions of f 00+ Af = 0,

it is possible that sup

z2D |A(z)|(1 � |z|2)2 = 1 or f1/f2 is non-normal. Theseresults relate to the sharpness discussion of recent results in the literature, and aresucceeded by a detailed analysis of differential equations with bounded solutions.

Instead of considering solutions with prescribed zeros, it is also possible toconstruct a bounded solution of f 00

+ Af = 0 in such a way that it solves aninterpolation problem natural to bounded analytic functions, while |A(z)|2(1 �|z|2)3 dm(z) remains to be a Carleson measure.

Ville Tengvall

Local and global injectivity of branched coverings

We discuss local and global injectivity of branched coverings, i.e., continuous, open,and discrete mappings.

60

Jarmo Jääskeläinen

Improved Hölder regularity for strongly elliptic PDEs

We show that solutions to an autonomous Beltrami equation enjoy a quantitativeimproved degree of Hölder regularity, higher than what is given by the classicalexponent 1/K. As a consequence we have improved Schauder regularity propertiesfor solutions to the Leray-Lions divergence type equation and the nonlinear Bel-trami equation in the plane. The talk is based on a joint work with Kari Astala,Albert Clop, Daniel Faraco and Aleksis Koski.

61

Yliopistomatematiikan kokeiluja ja käytännön vinkkejä,

L4

Johanna Rantala

Kokemuksia automaattisen palautteen antamisesta MathChekillä

Jyväskylän yliopistossa (ja aiemmin Tampereella) on kehittetty ja käytetty MathCheck-ohjelmistoa tukemaan matematiikan opettamista tietotekniikan opiskelijoille. Ohjelmistomahdollistaa automatisoidun palautteen antamisen opiskelijoille heidän ratkaisu-istaan välivaiheineen. Tällä hetkellä MathCheck tukee seitsemää tehtävätyyppiä.Sillä voi teettää tehtäviä esimerkiksi perinteisiin logiikkaan ja yhtälöihin liittyen,mutta myös harvinaisempia lausekepuihin ja yhteysriippumattomiin kielioppeihinliittyviä tehtävätyyppejä on kehitetty.

Ohjelmistoa on käytetty Jyväskylässä yhteensä neljällä kurssitoteutuksella,ja opiskelijoilta on kerätty palautetta sen käytöstä. Diskreetit rakenteet -kurssinpääpaino on propositio- ja predikaattilogiikassa. Lisäksi käsiteltiin matemaattis-ten rakenteiden hahmottamista ja jäsentelyä, joukko-oppia ja matematiikan pe-rusasioita. Automaatit ja kieliopit -kurssi puolestaan käsittelee tietotekniikan teo-riaa, sillä käytetään paljon matemaattisia merkintätapoja ja se sisältää matemaat-tista pohdintaa sekä päättelyä. Molemmilla kursseilla MathCheck-ohjelmistoa onkäytetty kahdella eri kurssitoteutuksella eri vuosina.

Predikaattilogiikkaan liittyvät tehtävät osoittivat, että opiskelijat ymmärsiväteron lausekkeiden 8i; 1 i n : '(A, i) ja 9i; 1 i n : '(A, i) välillä, kun'(A, i) on jokin taulukon A[1 . . . n] lauseke. Monet osasivat myös tulkita lausek-keen 3 n ^ '(A, 3), mutta muotoa 1 i n ^ '(A, i) oleva lauseke onkin johankalampi. Sama ilmiö näkyy myös ohjelmointitehtävissä: monet opiskelijat os-aavat kirjoittaa aliohjelman, joka tulostaa kolmen kertotaulun. Vaikeuksia tuottaasama tehtävä silloin, kun opiskelijan tulisi kirjoittaa aliohjelma, jolle kerrottavatuodaan parametrinä.

Monissa matematiikan pohjataitoihin liittyvissä asioissa toistojen määrä ontärkeää. Käytännössä tätä voi olla usein vaikea toteuttaa. Oman kokemuksemme

62

mukaan MathCheck-ohjelmistoa käytettäessä opiskelijat ovat taipuvaisempia tekemäänenemmän töitä MathCheck-tehtävillä verrattuna perinteisiin kynällä ja paper-illa tehtäviin tehtäviin. Lisäksi ohjelman antama palaute mahdollistaa sen, etteiopiskelija jää jumiin heti tehtävän alussa, ja että ratkaistuista tehtävistä vain on-gelmakohdat tarvitsee käydä yhdessä läpi.

Syksyllä 2018 järjestetyllä Automaatit ja kieliopit -kurssilla opiskelijoilla teetet-tiin yhteysriippumattomiin kielioppeihin johdatteleva tehtäväsarja, ja tehtävienläpikäyntitilaisuudessa opiskelijat täyttivät siihen liittyvän kyselylomakkeen. Vas-taukset viittasivat vahvasti siihen, että vaikka myös perinteiset opetusmetoditkoettiin hyvinä, MathCheck-tehtäviä pidettiin sekä miellyttävämpinä että opet-tavaisempina.

Seuraavana syksynä järjestetyllä Automaatit ja kieliopit -kurssilla MathCheck-tehtäviä teetettiin opiskelijoilla viikottain perinteisten tehtävien ohella. Opiske-lijoilta kurssin lopussa kerätyssä avoimessa palautteessa MathCheckin antamapalaute oli useimmin mainittu opiskelua edistävä asia. Vaikeuksiksi puolestaanmainittiin puutteellinen matemaattinen tausta, matemaattisten merkintöjen käyttöja oma riittämätön panostus kurssiin.

63

Simo Ali-Löytty

ÄlyOppi matematiikan osahankkeen esittely

Älykkäät oppimisympäristöt ja niiden sisällöntuotanto (ÄlyOppi)-hankkeen tarkoituk-sena on kehittää ja laajentaa yliopistokohtaisia sähköisiä oppimisympäristöjä ko-rkeakoulujen väliseen käyttöön. Älykkäillä oppimisympäristöillä tarkoitetaan muunmuassa automaattista tarkastamista, visualisaatioita ja simulaatioita. Näillä pyritääntarjoamaan opiskelijoille vuorovaikutteista oppimismateriaalia ja tukea itsenäistätyöskentelyä, sekä mallintaa ja analysoida alakohtaista oppisisältöä.

ÄlyOppi-hankkeen tarkoitus on hyödyttää opiskelijan lisäksi opettajia ja kurssienjärjestäjiä rakentamalla yhtenäisiä ympäristöjä korkeakoulujen välille ja mahdol-listamalla järjestelmien käytön avoimesti kaikissa oppilaitoksissa. Opettajille muo-dostetaan avoimet alakohtaiset verkostot, joiden avulla mahdollistetaan sisältöjenjakaminen ja tarjotaan tukea käyttöönottoon.

Hanke jakautuu kolmeen osa-alueeseen: ohjelmointiin, matematiikkaan ja fysi-ikkaan. Tässä esitelmässä keskitymme matematiikan osahankkeeseen. Matemati-ikan osahanke laajentaa vuonna 2015 perustettua Abacus-materiaalipankkia(https://abacus.aalto.fi/). Hankkeessa keskitytään automaattisesti arvioitavien matem-atiikan tehtävien laadintaan, sähköiseen tenttimiseen sekä kurssiaihioiden raken-tamiseen. Hankkeessa laaditut tehtävät ovat pääosin Moodlen STACK-tehtävätyypillätoteutettuja. Sähköisessä tenttimisessä ollaan pilotoitu Moodle-tehtävien käyttöäEXAM-tenteissä useissa korkeakouluissa samaa Moodle-palvelinta käyttäen. Kurssi-aihioita on laadittu hyvin kattavasti tekniikan yliopistojen ensimmäisen vuodenkursseille, lukuisille ammattikorkeakoulun kursseille sekä joillekin maisterivaiheenkursseille.

64

Jani Hirvonen

Yliopiston ensimmäiset insinöörimatematiikan kurssit flippaamalla

Kolmasosa Tampereen yliopiston Hervannan kampuksella syksyllä 2019 aloittaneistaopiskelijoista on opiskellut ensimmäiset kaksi matematiikan opintojaksoaan kään-teisellä opetuksella eli flippaamalla. Syksyn opintojaksojen lisäksi sama opiskeli-jaryhmä osallistuu myös keväällä flipatuille opintojaksoille.

Flipatuilla opintojaksoilla opiskelijat ovat tutustuneet kurssimateriaaleihin it-senäisesti tai ryhmissä, ilman luento-opetusta. Apua käsitteiden ymmärtämiseensekä laskentaan on ollut tarjolla laskutupa-tyyppisissä tilaisuuksissa sekä laskuhar-joituksissa. Näillä opintojaksoilla käsiteltävien tehtävien lukumäärä on ollut paljonsuurempi kuin perinteisellä luento-opetuksella toteutetuissa, erityisesti käsitteen-muodostukseen liittyviä automaattisesti arvioitavia tehtäviä on lisätty aiempiintoteutuksiin verrattuna. Laskuharjoitustilaisuuksien lisäksi opiskelijat ovat osallis-tuneet viikoittaisiin prime time –tilaisuuksiin, joissa opettajat ovat käyneet heidänkanssaan pienryhmäkeskusteluja alle kymmenen hengen ryhmissä viikon aiheista.Omaa keskusteluvuoroaan odottelevat pienryhmät ovat tehneet ryhmätehtäviä.

Flippaus prime time –keskustelujen kanssa toteutettuna päästää opettajanlähemmäs opiskelijaa kuin yliopiston perinteinen matematiikan luento-opetus. Jatku-van arvioinnin sekä opiskelijoiden tekemien viikoittaisten itsearviointien avullavoidaan jo hyvissä ajoin ennen opintojakson loppumista saada parempi kuva yk-sittäisten opiskelijoiden suoriutumisesta, sekä tarpeen vaatiessa eriyttää opetustatai ohjata tukitoimiin.

Flipatut opintojaksot ovat vaatineet normaalia enemmän opettajaresurssejaerityisesti pienryhmäkeskustelujen takia. Syksyn molempien opintojaksojen primetime –tilaisuuksia on ollut vetämässä neljä opettajaa. Toteuttamamme opetusyleisestikin kuormittaa opettajaa jonkin verran enemmän kuin perinteinen opetus.Tämän takia vastuunkantoa opintojakson sujumisesta on jaettu hiukan kaikkienneljän opettajan kesken, vaikka jokaisella yksittäisellä opintojaksolla varsinaisenavastuuopettajana ja käsiteltävien aiheiden suunnittelijana onkin ollut vain yksineljästä.

65


Antti Niemi

Numerical buckling analysis of circular cylindrical shell structures

Elastic buckling of thin shell structures is a mysterious phenomenon. The criticalload of the real shell is known to be extremely sensitive to geometric imperfectionsand deviations in boundary conditions. It is not straightforward to account forthese uncertainties in linear or nonlinear stability theory. As a result, theoreticaland experimental results do not agree well in many loading scenarios. In anycase, even the linear stability theory provides us useful information regarding thebuckling behavior of thin shells. Moreover, it serves as the first step in the analysisof geometrically non-linear large deformations. In this work, the buckling problemof circular cylindrical shells is analyzed from the point of view of finite elementanalysis. In particular, the asymptotic behavior of the buckling modes for thinshells and the associated numerical problem of locking are discussed.

Pauliina Uusitalo

The ABC of quantum waveguides of YZC

We consider waveguides with the shapes of letters Y, Z and C. We study the lowfrequency range of the spectrum (discrete spectrum) of the Dirichlet Laplacian.

At the beginning, we show that in each for these waveguides there exists atleast one bound state, i.e. an eigenvalue belonging to the discrete spectrum. More-over, we study how the number of eigenvalues in the discrete spectrum may changewhen the geometrical parameters (width, height or angle) vary in the waveguide.We also pay attention to the monotonicity of the eigenvalue as the geometric pa-rameters change.

The numerical computations corresponding to these problems are made byusing the finite element method with Matlab PDE toolbox.

66

Jukka Kemppainen

Positivity of the fundamental solution for fractional diffusion and wave

equations

We study the question of positivity of the fundamental solution for fractionaldiffusion and wave equations of the form, which may be of fractional order bothin space and time. We give a complete characterization of the positivity of thefundamental solution in terms of the order of the time derivative ↵ 2 (0, 2), theorder of the spatial derivative � 2 (0, 2], and the spatial dimension d. It turnsout that the fundamental solution fails to be positive for all ↵ 2 (1, 2) and either� 2 (0, 2] and d � 2 or � < ↵ and d = 1, whereas in the other cases it remainspositive.

In contrast to the heat kernel, the fundamental solution of the fractional dif-fusion and wave equations leads to the study of a quite general class of specialfunctions called as the Fox H-functions, which are defined via Mellin-Barnes typeintegrals. In this talk we briefly describe how we end up to these type of spe-cial functions and give the main ideas used in the proof of the positivity of thefundamental solution.

67

Logic, L6

Kerkko Luosto

Embedding-closed quantifiers

This work is joint with Jevgeni Haigora.

Many intresting properties of mathematical structures of various kinds areembedding-closed, or complements thereof. Examples include linearly ordered sets,planar graphs, torsion-free graphs and infinite-dimensional vector spaces. Manyof this kind of natural properties are also not definable in the first-order logic.It is well-known in the logic community that the minimal way of gaining theexpressibility of the desired property is to add a (generalized) quantifier expressingthat very property. In the other end, we might consider adding all the embedding-closed quantifiers. This is exactly what we do, and we undertake the study of thelogic L1!

(Qemb) where Qemb is the collection of all embedding-closed quantifiers.In order to study the logic L1!

(Qemb), we present a game that characterizes itsequivalene ⌘emb. We show that even if the logic L1!

(Qemb) appears to be extremelystrong, it admits quantifier elimination in quasi-homogeneous structures. Amongother results, we also show that for every finite set of quantifiers Q ✓ Qemb, thelogic L!

1!

(Q) where the number of variables is limited to finite, has the so-called0–1-law.

68

Miika Hannula

Probabilistic team semantics

Team semantics is a framework for the study of dependence and independence con-cepts ubiquitous in many areas such as database theory and statistics. In recentworks team semantics has been generalized to accommodate also probabilistic de-pendencies. This gives rise to new logical formalisms capable of describing variousprobability properties. In this talk we outline some recent developments of proba-bilistic independence logic which is a new logical language for analyzing conditionalindependence in discrete probability distributions.

Antti Valmari & Esko Turunen

A Completeness Proof for A Predicate Logic with Undefined Truth Value

Undefined expressions, such as division by zero, are ubiquitous in mathematicsand computer science. To bring them to the scope of exact formal reasoning,computer scientists have developed several logics with two or three truth values.Unfortunately, none of these logics matches how undefined expressions are dealtwith in everyday mathematics. For instance, when solving 2x�6

|x|�3 = x, an intuitivelyreasonable next step is

�x < 0 ^ 2x�6

�x�3 = x� _ �

x � 0 ^ 2x�6x�3 = x

�and last step is

x = �6 _ x = 2, but this cannot be justified in these logics.Therefore, we have developed a logic which matches how undefined expressions

were dealt with when solving equations in the 1970s in Finnish high schools. Ourlogic is an extension of Kleene’s or Łukasiewicz’s ternary logic to predicate logic.It assumes a formula

⌅f⌥

for every function symbol f which specifies when thefunction is defined. The formula

⌅f⌥

itself must always be defined. For instance,in the case of division x

y

, this formula is y 6= 0. Given these and any formula ', aformula

⌅'⌥

is obtained by a simple recursive algorithm, specifying when ' yieldstrue or false (instead of undefined). For instance, if t and t0 are terms, then

⌅t = t0

⌥

is⌅t⌥ ^ ⌅

t0⌥. Furthermore,

⌅' ^ ⌥ is (

⌅'⌥ ^ ⌅

⌥) _ (

⌅'⌥ ^ ¬') _ (

⌅ ⌥ ^ ¬ ).

Variables are never undefined, but terms may be. If a term contains an un-defined sub-term, then the term as a whole is undefined. A relation yields the

69

undefined truth value if and only if it contains at least one undefined term. There-fore, x = x yields true but x

0 =

x

0 yields undefined.To ensure that our logic is on a healthy basis, we have designed a natural

deduction proof system for it and proven that the proof system is sound and com-plete. The completeness proof relies on the Henkin construction. The universe ofthe constructed model contains, for every formula ', precisely one of the following:', ¬', or ¬⌅'⌥.

The proof system has five important differences to the binary logic case. Thelaw of the excluded middle has been replaced by ; ` ' _ ¬' _ ¬⌅'⌥ (each claimis either true, false or undefined). There is an extra law {'} ` ⌅

'⌥

(if somethingis true, then it is defined). Reflexivity of equality is assumed only on definedterms: {⌅t⌥} ` t = t. If f is an n-ary function symbol and 1 i n, then{t

i

= t0i

,⌅f(t1, . . . , tn)

⌥} ` f(t1, . . . , tn) = f(t1, . . . , ti�1, t0i

, ti+1, . . . , tn). If t is free

for x in ', then {⌅t⌥, 8x : '(x)} ` '(t).

70

Statistics 2, L7

Josephine Dufitinema

Volatility clustering, Risk-return relationship, and Asymmetric adjustment

in the Finnish Housing Market

The purpose of the paper is to examine whether the apartment types house pricesin Finland share financial characteristics with assets such as stocks. The studiedregions are fifteen main regions in Finland over the period of 1988:Q1 to 2018:Q4.These regions are divided geographically into forty-five cities and sub-areas ac-cording to their postcode numbers. Both Portmanteau and Lagrange Multipliertests are used to test for clustering effects (ARCH effects). For cities and sub-areaswith significant clustering effects, the Generalized Autoregressive Conditional Het-eroscedasticity (GARCH)-in-mean model is used to determine the potential im-pact that the conditional variance may have on returns. Moreover, the ExponentialGARCH (EGARCH) model is employed to examine the possibility of asymmet-ric effects of shocks on volatility. For each apartment type, individual models areestimated, enabling different house price dynamics, and variation of signs andmagnitude of different effects across cities and sub-areas. Results reveal that clus-tering effects exist in over half of the cities and sub-areas in all studied types ofapartments. Moreover, mixed results on the sign of the significant risk-return re-lationship are observed across cities and sub-areas in all three apartment types.Furthermore, the evidence of the asymmetric impact of shocks on housing volatil-ity is noted in almost all the cities and sub-areas housing markets. These studiedvolatility properties are further found to differ across cities and sub-areas, and byapartment types. The existence of these volatility patterns has essential implica-tions, such as investment decision making and portfolio management. The studyoutcomes will be used in a forecasting procedure of the volatility dynamics of thestudied types of dwellings.

71

Petteri Piiroinen

Generalized modes and MAP estimators

In this talk, we consider a generalization of nonparametric modes to address cer-tain shortcomings in different definitions of modes. We will discuss what propertiesmodes should have and how different definitions are related, we will consider vari-ational ways of finding these generalized modes and discuss some applications tononparametric Bayesian analysis, namely maximum a posteriori (MAP) estimatesfor priors that do not admit continuous densities.

Joni Virta

Fast tensorial independent component analysis

A novel method of tensorial independent component analysis is proposed basedon TJADE and k-JADE, two recently proposed generalizations of the classicalJADE algorithm. The new method achieves consistency and the limiting distri-bution of TJADE under mild assumptions, and at the same time offers notableimprovement in computational speed. A trade-off between computational speedand assumptions is controlled by a tuning parameter which has a natural interpre-tation as the maximal kurtosis multiplicity. Simulations and timing comparisonsdemonstrate the method’s gain in speed and show that the desired efficiency isobtained approximately also for finite samples. The method is applied successfullyto large-scale video data, for which neither TJADE nor k-JADE is feasible. Fi-nally, an experimental procedure is proposed to select the values of a set of tuningparameters.

72

finnish mathematical days 2020 - university of oulu · on early detection of high-risk prostate...

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