Finnish Mathematical Days 2020 - A collection of abstracts Oulu 2nd - 3rd of January 2020

Finnish Mathematical Days 2020
      A collection of abstracts

                  Oulu
        2nd - 3rd of January 2020

                    1

Invited plenary lecturers
Christel Geiss               University of Jyväskylä
Sabrina Kombrink             University of Birmingham
Aleksis Koski                University of Jyväskylä
Eveliina Peltola             University of Bonn
Stéphane Seuret              Université Paris-Est Créteil
Samuli Siltanen              University of Helsinki

Other speakers
Simo Ali-Löytty              Tampere University
Gaëlle Brunet                University of Eastern Finland
Jean-Baptiste Casteras       University of Helsinki
Josephine Dutinema          University of Vaasa
Anne-Maria Ernvall-Hytönen   Åbo Akademi University
Ragnar Freij-Hollanti        Aalto University
Janne Gröhn                  University of Eastern Finland
Philipp Guth                 University of Mannheim
Akseli Haarala               University of Helsinki
Anni Hakanen                 University of Turku
Antti Hannukainen            Aalto University
Miika Hannula                University of Helsinki
Pauliina Hirvi               Aalto University

                             2

Jani Hirvonen         Tampere University
Toni Hotanen          University of Turku
Juha-Matti Huusko     University of Eastern Finland
Jokke Häsä            University of Helsinki
Konstantin Izyurov    University of Helsinki
Joonatan Jalonen      University of Turku
Jesse Jääsaari        University of Turku
Jarmo Jääskeläinen    University of Jyväskylä
Terhi Kaarakka        Tampere University
Vesa Kaarnioja        University of New South Wales
Leena Kalliovirta     University of Helsinki
Ilmari Kangasniemi    University of Helsinki
Anna Kausamo          University of Jyväskylä
Jukka Kemppainen      University of Oulu
Sampsa Kiiskinen      University of Jyväskylä
Juha Kinnunen         Aalto University
Ville Kolehmainen     University of Eastern Finland
Juho Kontio           University of Oulu
Jaakko Kultima        University of Oulu
Saara Lehto           University of Helsinki
Kangwei Li            Tianjin University
Sauli Lindberg        University of Helsinki
Kerkko Luosto         Tampere University
Kamalakshya Mahatab   University of Helsinki

                      3

Mika Mattila                    Tampere University
Santeri Miihkinen               Åbo Akademi University
Terhi Moisala                   University of Jyväskylä
Anton Nazarov                   Saint-Petersburg State University
Thuan Nguyen                    University of Jyväskylä
Antti Niemi                     University of Oulu
Neea Palojärvi                  Åbo Akademi University
Jarkko Peltomäki                University of Turku
Kirsi Peltonen                  Aalto University
Petteri Piiroinen               University of Helsinki
Istvan Prause                   University of Eastern Finland
Juha-Pekka Puska                Aalto University
Paavo Raittinen                 Aalto University
Johanna Rantala                 University of Jyväskylä
José André Rodriguez Migueles   University of Helsinki
Lassi Roininen                  Lappeenranta University of Technology
Matthew Romney                  University of Jyväskylä
Johanna Rämö                    University of Helsinki
Mikko Salo                      University of Jyväskylä
Tommi Sottinen                  University of Vaasa
Gunnar Söderbacka               Åbo Akademi University
Olli Tapiola                    University of Jyväskylä
Ville Tengvall                  University of Helsinki

                                4

Esko Turunen        Tampere University
Teemu Tyni          University of Helsinki
Topi Törmä          University of Oulu
Pauliina Uusitalo   University of Oulu
Antti Valmari       University of Jyväskylä
Zhuang Wang         University of Jyväskylä
Harri Varpanen      JAMK University of Applied Sciences
Esa Vesalainen      Åbo Akademi University
Lauri Viitasaari    Aalto University
Joni Virta          Aalto University/University of Turku
Jani Virtanen       University of Reading/University of Helsinki
Henrik Wirzenius    University of Helsinki

                    5

Sessions and talks
Plenary lectures, L1
  Stéphane Seuret    (Thursday 9:00 - 9:50)

       Function spaces in multifractal environment, and the Frisch-Parisi

       conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      15

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

       Regularity properties of backward stochastic dierential equations

       and 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 Li
       Some weighted estimates on product spaces           . . . . . . . . . . . . .   19

  Olli Tapiola
       Cp   weights and the CoifmanFeerman inequality . . . . . . . . . .            19

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

                                            6

Mathematics and arts, L4
  Kirsi Peltonen
       Aalto Math & Arts in Shanghai 2019 . . . . . . . . . . . . . . . . .          21

  Saara Lehto
       Dance in Mathematics Education        . . . . . . . . . . . . . . . . . . .   21

Geometric analysis 1, L5
  Terhi Moisala
       Rectiability results in Carnot groups . . . . . . . . . . . . . . . . .      22

  Zhuang Wang
       Traces of rst order Sobolev spaces on regular trees      . . . . . . . . .   22

  Matthew Romney
       Uniformization with innitesimally metric measures . . . . . . . . .          23

Inverse problems 1, L6
  Mikko Salo
       Inverse problems for real principal type operators      . . . . . . . . . .   24

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

  Jaakko Kultima
       Direct and inverse scattering problems for quasi-linear biharmonic

       operator in 3D.   . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   25

Stochastics, L7
  Konstantin Izyurov
       Universality of spin correlations in the critical Ising model     . . . . .   26

  Thuan Nguyen
       Approximation of certain stochastic integrals with jumps in weighted

       bounded mean oscillation spaces . . . . . . . . . . . . . . . . . . . .       27

  Tommi Sottinen
       Integration-by-Parts Characterizations of Gaussian Processes . . . .          27

                                        7

Coding theory, L8
   Anni Hakanen
        On the Metric Dimension for Locating Multiple Objects           . . . . . .   28

   Anne-Maria Ernvall-Hytönen
        On the proximity of primes and elements in other suciently dense

        subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   28

   Ragnar Freij-Hollanti
        Lifting a Code over a Simplicial Complex . . . . . . . . . . . . . . .        29

Sessions on Thursday 16:00 - 17:30
Partial dierential equations, L3
   Sauli Lindberg
        Convex integration in magnetohydrodynamics          . . . . . . . . . . . .   30

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

   Akseli Haarala
        On the electrostatic Born-Infeld equations and the Lorentz mean

        curvature operator    . . . . . . . . . . . . . . . . . . . . . . . . . . .   31

Matematiikan osaamisen tutkimusperustainen arviointi, L4
   Terhi Kaarakka
        Käänteinen opetus ja arviointi - miten eteenpäin?       . . . . . . . . . .   32

   Jokke Häsä
        Todistamisajattelun tietokoneavusteinen kehittäminen ja arviointi

        yliopistomatematiikan johdantokurssilla . . . . . . . . . . . . . . . .       33

   Johanna Rämö & Jokke Häsä
        Arviointi 2020  Ajankohtaista yliopistomatematiikan arvioinnissa

        ja arviointitutkimuksessa . . . . . . . . . . . . . . . . . . . . . . . .     34

                                          8

Functional Analysis, L5
   Jani Virtanen
        Toeplitz operators on Fock spaces . . . . . . . . . . . . . . . . . . .           35

   Santeri Miihkinen
        On the Hilbert matrix operator on analytic function spaces . . . . .              35

   Henrik Wirzenius
        Compact-by-approximable operators on Banach spaces failing the

        approximation property          . . . . . . . . . . . . . . . . . . . . . . . .   36

Number theory 1, L6
   Mika Mattila
        The connection between the cube semilattice structure and singu-

        larity of LCM-type matrices on GCD closed sets . . . . . . . . . . .              37

   Neea Palojärvi
        On   τ -Li   coecients and explicit zero-free regions . . . . . . . . . . .      37

   Topi Törmä
        Generalized continued fraction expansions with constant partial de-

        nominators        . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   37

Statistics 1, L7
   Juho Kontio
        Scalable nonparametric dimension reduction method for searching

        higher-order interaction terms in high-dimensional regression models              38

   Paavo Raittinen
        On early detection of high-risk prostate cancer: applied discovery

        and validation models using genotype information . . . . . . . . . .              39

   Leena Kalliovirta
        Scenario for structural development of livestock production in the

        Baltic littoral countries . . . . . . . . . . . . . . . . . . . . . . . . .       40

                                             9

Discrete and symbolic dynamics, L8
   Toni Hotanen
       A new kind of measure-theoretic entropy concerning endomorphisms

       of measure-preserving dynamical systems . . . . . . . . . . . . . . .         41

   Joonatan Jalonen
       One-sided vs. two-sided cellular automata       . . . . . . . . . . . . . .   41

   Jarkko Peltomäki
       Symbolic Square Root Map        . . . . . . . . . . . . . . . . . . . . . .   42

Computational mathematics 1, L9
   Gaëlle Brunet
       COMPUTATION OF PDE'S ON COMPACT MANIFOLDS . . . .                             43

   Antti Hannukainen
       Eigensolutions in Distributed Computing Environments . . . . . . .            44

   Sampsa Kiiskinen
       Towards a Formalization of Discrete Exterior Calculus         . . . . . . .   45

Sessions on Friday 10:30 - 12:00
Geometric analysis 2, L4
   Anna Kausamo
       The Monge problem in optimal mass transportation: from two to

       many marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      46

   José Andrés Rodriguez Migueles
       Geodesics on hyperbolic surfaces and volumes of link complements

       in Seifert-bered spaces . . . . . . . . . . . . . . . . . . . . . . . . .    46

   Ilmari Kangasniemi
       On the entropy of uniformly quasiregular maps         . . . . . . . . . . .   47

                                        10

Verkko-opetus ja opiskelijoiden etäosallistuminen, L5
  Juha-Matti Huusko
       Matematiikan verkkokurssin rakentamisen yksityiskohtia, haasteita

       ja ideoita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    48

  Harri Varpanen
       Ohjelmallisia ratkaisuja yksilöityjen tehtävien toteuttamiseen verkko-

       opetuksessa     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   49

  Simo Ali-Löytty
       Sähköisten matematiikan tenttien esseekysymysten automaattinen

       arviointi   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   49

Mathematical physics, L6
  Anton Nazarov
       Limit shape of innite tensor power decomposition in the innite

       rank limit of Lie algebras so(2n+1)       . . . . . . . . . . . . . . . . . .   50

  Istvan Prause
       Random tilings, arctic curves and a Beltrami equation           . . . . . . .   50

  Jani Virtanen
       Entanglement entropy in quantum spin chain models             . . . . . . . .   51

Number theory 2, L7
  Kamalakshya Mahatab
       Joint large values of Orthogonal L functions in Selberg Class . . . .           52

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

  Esa Vesalainen
       On Fourth and Higher Moments of Short Exponential Sums Related

       to Cusp Forms       . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   52

                                          11

Inverse problems 2, L8
  Ville Kolehmainen
       Data-driven regularization parameter selection in dynamic MRI         . .   53

  Juha-Pekka Puska
       Optimal projection angles in X-ray tomography       . . . . . . . . . . .   54

  Lassi Roininen
       Posterior Inference for Sparse Hierarchical Non-stationary Models       .   54

Computational mathematics 2, L9
  Pauliina Hirvi
       Generating head models for diuse optical tomography        . . . . . . .   55

  Philipp Guth
       Uncertainty quantication for PDE-constrained optimization using

       a quasi-Monte Carlo method . . . . . . . . . . . . . . . . . . . . . .      56

  Vesa Kaarnioja
       Uncertainty quantication for partial dierential equations using

       periodic random variables   . . . . . . . . . . . . . . . . . . . . . . .   57

Dynamical systems and stochastics, PR101
  Gunnar Söderbacka
       Bifurcations of multiple attractors in a predator-prey system . . . .       58

  Lauri Viitasaari
       Stochastic heat equation revisited - quantitative approximation results 59

                                      12

Sessions on Friday 14:30 - 16:00
Analysis, L3
   Janne Gröhn
        Converse growth estimates for ODEs with slowly growing solutions             60

   Ville Tengvall
        Local and global injectivity of branched coverings     . . . . . . . . . .   60

   Jarmo Jääskeläinen
        Improved Hölder regularity for strongly elliptic PDEs . . . . . . . .        61

Yliopistomatematiikan kokeiluja ja käytännön vinkkejä, L4
   Johanna Rantala
        Kokemuksia automaattisen palautteen antamisesta MathChekillä . .             62

   Simo Ali-Löytty
        ÄlyOppi matematiikan osahankkeen esittely        . . . . . . . . . . . . .   64

   Jani Hirvonen
        Yliopiston ensimmäiset insinöörimatematiikan kurssit ippaamalla .           65

Computational mathematics 3, L5
   Antti Niemi
        Numerical buckling analysis of circular cylindrical shell structures     .   66

   Pauliina Uusitalo
        The ABC of quantum waveguides of YZC           . . . . . . . . . . . . . .   66

   Jukka Kemppainen
        Positivity of the fundamental solution for fractional diusion and

        wave equations   . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   67

                                        13

Logic, L6
   Kerkko Luosto
        Embedding-closed quantiers       . . . . . . . . . . . . . . . . . . . . .   68

   Miika Hannula
        Probabilistic team semantics . . . . . . . . . . . . . . . . . . . . . .      69

   Antti Valmari & Esko Turunen
        A Completeness Proof for A Predicate Logic with Undened Truth

        Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   69

Statistics 2, L7
   Josephine Dutinema
        Volatility clustering, Risk-return relationship, and Asymmetric ad-

        justment in the Finnish Housing Market        . . . . . . . . . . . . . . .   71

   Petteri Piiroinen
        Generalized modes and MAP estimators          . . . . . . . . . . . . . . .   72

   Joni Virta
        Fast tensorial independent component analysis . . . . . . . . . . . .         72

                                         14

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 t those

theoretically obtained for the typical properties of functions in any standard func-

tional spaces: Hölder, Sobolev, Besov... In this talk, we introduce new Besov-like

spaces in which the typical functions possess very rich scaling properties, mim-

icking those observed on data for instance. We obtain various characterizations of

these function spaces, in terms of oscillations or wavelet coecients. Combining

this with a new construction of almost-doubling probability measures with pre-

scribed multifractal properties, we bring a solution to the so-called Frisch-Parisi

conjecture. This is a joint work with Julien Barral (Université Paris-Nord).

                                        15

Christel Geiss (Thursday 13:30 - 14:20)
Regularity properties of backward stochastic dierential equations and
their associated PDEs
Backward stochastic dierential equations (BSDEs) are SDEs where instead of an

initial value a random terminal condition is given. If this terminal condition is

described by a functional of a solution to an SDE, we speak of forward-backward

SDEs.

    This type of equations is intrinsically connected to semilinear 2nd order partial

dierential equations. For solutions to forward-backward SDEs driven by Brow-

nian motion, Lévy noise or random walk we discuss smoothness (in the sense of

fractional Malliavin Sobolev spaces) and approximation properties and address the

relation to the associated PDEs, integral-partial dierential equations and nite

dierence equations, respectively.

    This talk is based on joint work with Stefan Geiss, Eija Laukkarinen, Antti

Luoto (Jyväskylä), Philippe Briand and Céline Labart (Université Savoie Mont

Blanc) 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 that

the 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 the

lattice models, which should also be related to correlation functions of appropri-

ate elds in the corresponding conformal eld theory (CFT); the latter, however,

being mathematically ill-dened. I discuss results pertaining to make sense of this

relationship.

                                         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 through

electrodes on the skin. The voltages appearing on the electrodes are measured, and

the aim of EIT is to recover the internal distribution of electric conductivity. The

resulting image can be used for diagnosing stroke or assessing the lung function of

cystic brosis 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 inverse

boundary value problem, where one tries to reconstruct a PDE coecient from a

Dirichlet-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 oered by a nonlin-

ear Fourier transform, based on Complex Geometric Optics solutions introduced

by John Sylvester and Gunther Uhlmann in 1987. A low-pass lter applied on

the nonlinear frequency domain enables robust real-time EIT imaging, with cut-

o frequency determined by the amplitude of measurement noise. This imaging

method is based on solving a D-bar equation and is connected to the theory of

pseudoanalytic functions.

        There are further interesting possibilities arising from the use of the nonlinear

Fourier 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 EIT

imaging for recovering boundaries between tissues and organs.

        Furthermore, the nonlinear Fourier transform can be used for linearising the

Novikov-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 transform

is a versatile tool applicable to very dierent 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 are

used to represent deformations between two elastic bodies (domains in Euclidean

space). Hence one of the most fundamental questions in this theory is whether

two such bodies admit a Sobolev homeomorphism between them at all, possibly

with some xed boundary values. Perhaps surprisingly, this question remains unan-

swered in many of the important cases. In my talk, which is intended for a general

audience, I will give an overview of the important questions and share some of the

recent 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-dimensional
volume of the   ε-parallel set of A can be expressed as a polynomial in ε of degree d.
The coecients of the polynomial provide important information on the geometry

of the given set, such as its volume, its surface area or Euler characteristic. In this

talk we will investigate extensions of the Steiner formula to further classes of sets

and discuss the meaning of the analogues of the coecients. When investigating

the class of fractal sets, do the analogues of the coecients lead to notions of

'fractal volume', 'fractal surface area' and 'fractal Euler characteristic' ?

                                          18

Sessions on Thursday 10:30 - 12:00

Harmonic analysis, L3
Kangwei Li
Some weighted estimates on product spaces
By extending a classical result due to Muckenhoupt and Wheeden to the product

BMO setting, we establish the weighted estimates for general bilinear bi-parameter

Calderon-Zygmund operators. We also demonstrate the Bloom type estimates in

its 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 CoifmanFeerman inequality
It is a long-standing open problem in harmonic analysis to characterize the weights

w   that satisfy the CoifmanFeerman inequality

                              kT f kLp (w) ≤ CkM f kLp (w) ,

where   0 < p < ∞, T   is a singular integral operator and     M   is the HardyLittlewood

maximal operator. In the early 1980's, Muckenhoupt showed that if the inequality

holds for the Hilbert transform, then the weight satises a generalized version of

the   A∞   condition called the   Cp   condition. He also conjectured that this condition

is sucient for the inequality. In this talk, we discuss recent developments related

to this conjecture and extensions of known results for rough homogeneous singular

integrals. 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−2 u t − div |Du|p−2 Du = 0
                                  
                  


in the range                            
                                    2n             2n
                         max           ,1 < p <          ,
                                   n+2          (n − 2)+
where   n∈N    is the spatial dimension. We show that a gradient of a nonnegative

weak solution to a doubly nonlinear equation belongs locally to a slightly higher

Sobolev space than assumed a priori with a reverse Hölder type estimate. The

key ingredient in the proof of our main result is an appropriate intrinsic geometry

that depends on the the solution as well as its spatial gradient and thus allows us

to rebalance the mismatch between the function and its gradient in the equation.

Related results and open questions are also discussed.

                                          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 eort of

Aalto School of Arts, Design and Architecture, Aalto School of Science and Aalto

School of Engineering. The focus of the contribution of Aalto University was to

introduce the interdisciplinary Math & Arts program, especially its underlying

course Crystal Flowers in Halls of Mirrors: Mathematics, Arts and Architecture

and related activities to the audience.

    During our visit to Shanghai, it was also possible to test our LUMATIKKA

and Aalto Scientist in School activities to local school teachers and students. This

talk will discuss about the challenges and benets of our experience from the

perspective 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 students

of all levels by using interactive and hands on teaching methods. One way to do

this is to introduce mathematics through art. Indeed, mathematics and art share

many common characteristics.

    Using dance in mathematics education has the added benet of introducing

movement and embodied methods into teaching. Current research in medicine and

cognitive science tells us that moving helps us learn. In our LUMATIKKA course

Kehollinen ja liikkuva matematiikka we explore the importance of embodiment

in mathematics education and give examples of teaching activities that combine

math and dance.

                                          21

Geometric analysis 1, L5
Terhi Moisala
Rectiability results in Carnot groups
In the mid fties De Giorgi proved in his groundbreaking work that the reduced

boundary of a set of nite perimeter is countably rectiable. This result has far

reaching consequences in Geometric Measure Theory and its generalization into

more general metric measure spaces has been object of several studies in the last

decades. In this talk I will consider the rectiability problem in Carnot groups,

which are certain kinds of sub-Riemannian Lie groups with a rich metric-measure-

space structure. In Euclidean spaces rectiability can be equivalently described in

terms of   C 1 -hypersurfaces,   Lipschitz-graphs, or a geometric condition which we

call a "cone property". In Carnot groups these dierent types of rectiability have

their natural counterparts, but their equivalence is still unknown. I will describe

results regarding all of the rectiability types mentioned above and give examples

of new classes of Carnot groups where a rectiability result is obtained. This talk

is based on joint work with Sebastiano Don, Enrico Le Donne and Davide Vittone.

Zhuang Wang
Traces of rst order Sobolev spaces on regular trees
In this talk, I will give the characterizations for the existence of traces for rst

order Sobolev spaces dened on regular trees. Three denitions of trace are given

and we will also discuss the equivalences between them.

                                            22

Matthew Romney
Uniformization with innitesimally 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 metric

2-spheres due to uniformization theorems of BonkKleiner, Rajala, and Lytchak

Wenger. On the other hand, the fractal case is much more dicult. In this talk,

we present the notion of innitesimally metric measures as a tool for approaching

the fractal problem. This is joint work with K. Rajala and M. Rasimus.

                                       23

Inverse problems 1, L6
Mikko Salo
Inverse problems for real principal type operators
We consider inverse boundary value problems for general real principal type dier-

ential operators. The rst results state that the Cauchy data set uniquely deter-

mines the scattering relation of the operator and bicharacteristic ray transforms of

lower order coecients. We also give two dierent boundary determination meth-

ods for general operators, and prove global uniqueness results for determining co-

ecients in nonlinear real principal type equations. The article presents a unied

approach 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) and

Gunther Uhlmann (Washington / IAS HKUST).

Teemu Tyni
Nonlinear inverse scattering for a biharmonic operator on the line
We consider an inverse scattering problem for a dierential operator of order four

on the line with two coecients which may be complex-valued. We let the two

unknown coecients depend non-linearly on the total eld. Such operators arise

for example in the theory of vibrations of beams and the study of elasticity. In

this talk we show that the Born approximation can be used eectively to recover

essential information about a combination of the coecients from the knowledge

just one datum, the reection coecient at high frequencies.

    This is a joint work with V. Serov.

                                          24

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 rst and

zero order perturbations in 3D. Perturbations are assumed to be non-linear func-

tions depending on the absolute value of the total eld. We start by providing the

unique solvability of this problem in some suitable Sobolev space. As a main result

we present Saito's formula as well as two of its corollaries, namely uniqueness and

representation formula for the solution of the inverse problem.

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

                                        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 surfaces

can be approximated by square grid. Recently, we were able to prove convergence

of correlations on isoradial graphs, that is, on rhombi tilings. The proof is based

on a new observation in discrete complex analysis which also simplies the original

proof 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 integrals

driven 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 of

the driving process. To deal with this situation, we introduce another approxima-

tion scheme where one uses additionally random time nets that capture the big

jumps, whereas the expected cardinality of these additional random time nets can

be controlled.

    Exploiting features of weighted BMO spaces, we show that one can signi-

cantly improve distributional estimates for the error process and our results also

allow changes of the underlying measure, while keeping the error estimates, if the

change of measure satises a reverse Hölder inequality.

    We also provide some illustrative examples in exponential Lévy models. It

turns out that the convergence rate of the error process depends on an interplay

between the smoothness of the terminal condition and the intensity of small jumps

of the underlying Lévy process.

Tommi Sottinen
Integration-by-Parts Characterizations of Gaussian Processes
The Stein's lemma characterizes the Gaussian distribution via an integration-by-

parts formula. We show that a similar integration-by-parts formula characterizes

a wide class of Gaussian processes, the so-called Gaussian Fredholm processes.

    The talk is based on joint work with Ehsan Azmoodeh (U Bochum), Ciprian

A. Tudor (U Lille 1) and Lauri Viitasaari (Aalto U)

                                       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 also

connected to error-correcting codes and covering codes.

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

vertices in a graph one at a time. In this presentation, two variants of resolving sets

are considered, namely,          {`}-resolving sets   and   `-solid-resolving sets. These variants
can be used to locate multiple vertices simultaneously (i.e. determine the elements

of a set of vertices).

         This presentation is based on joint work with Ville Junnila, Tero Laihonen

and María Luz Puertas.

Anne-Maria Ernvall-Hytönen
On the proximity of primes and elements in other suciently dense subsets
Minjia Shi, Florian Luca and Patrick Sole considered the q-ary representations of

primes and interpreted these as vectors or as code words. They showed that the

Hamming distance of some two elements is at most two. In this project, we try to

loosen the conditions, namely to study the smallest possible density of the set so

that it still has this property. This gives some necessary characterization for a set

of n-digit numbers which can be used as a code of Hamming distance greater than

two. The arguments are very elementary.

                                                   28

Ragnar Freij-Hollanti
Lifting a Code over a Simplicial Complex
We dene the lift of a linear code over an abstract simplicial complex on the same

ground set, to be the smallest code whose projection to any simplex agrees with

that of the original code. We show that this is not a matroid invariant, and provide

some support for the conjecture that it is matroid invariant for generic codes.

                                        29

Sessions on Thursday 16:00 - 17:30

Partial dierential 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 compactly

supported in space-time. In particular, the solutions violate the conservation of ki-

netic energy. The method of convex integration has since been applied to numerous

other equations of uid dynamics, in particular to Navier-Stokes by Buckmaster

and 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 convex

integration in 3D magnetohydrodynamics (MHD). MHD combines Euler equations

with Maxwell equations to study the interplay between a plasma and a magnetic

eld e.g. in solar physics. Convex integration is substantially harder to use in MHD

than, e.g., the Euler equations, mainly because an integral quantity called magnetic

helicity 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 other

nicely: Beekie, Buckmaster and Vicol show that magnetic helicity is not conserved

by 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 a

fourth order nonlinear Schrödinger equation having second and fourth order dis-

persion terms. This kind of equation naturally appears in nonlinear optics. In a rst

time, 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 certain

quasilinear, non-uniformly elliptic operator that comes with a natural gradient

constraint. The same operator appears also as the mean curvature operator of

spacelike surfaces in the Lorentz-Minkowski space, the setting of special relativity.

We will explain both of these contexts to motivate the mathematical study of said

operator.

    Our main focus will be on the regularity of the solutions of the electrostatic

Born-Infeld equations. We will talk about some now classical results as well as

some 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 ippaamalla.

    Useimmilla yliopiston matematiikan opintojaksoilla opiskelijat saavat opinto-

jakson päätyttyä arvosanan. Perinteisesti arviointi on perustunut tenttiin, joitakin

lisäpisteitä on voinut saada harjoitustehtävien tekemisestä. Viime vuosina on kuitenkin

noussut esiin kysymyksiä siitä, onko tentti sittenkään paras mittari osaamisen

arviointiin. Kun opetustavat uudistuvat ja monipuolistuvat, tenttiin perustuva

arviointi jättää monta esimerkiksi työelämän kannalta relevanttia osaamisaluetta

arvioinnin ulkopuolelle. Tentti kuitenkin puolustaa paikkaansa sillä, että arviointi

on tällöin yhdenvertaista.

    Tentti sijoittuu opintojakson loppuun, jolloin sillä ei voi ohjata oppimista

opintojakson aikana. Jatkuvan arvioinnin tarkoituksena on loppuarvioinnin lisäksi

ohjata opiskelijan oppimisprosessia. Tällä pyritään motivoimaan opiskelijaa jatku-

vaan pitkäkestoiseen syväsuuntautuneeseen oppimiseen. Isoilla opetusryhmillä jatkuva

yksilökeskeinen arviointi on kuitenkin haasteellista ja kallista.

    Flippauskokeilussamme päädyimme pelkän tentin asemasta painottamaan jatku-

vaa arviointia. Flippauksen osallistuvat opiskelijat keräävät 7 viikkoa kestävän

opintojakson aikana maksimissaan 700 pistettä alkutasotestistä, käsitteenmuodostus-

, laskuharjoitus- ja ryhmätehtävistä, tehtävien itse- ja vertaisarvioinneista, oman

osaamisen itsearvioinneista sekä opettajan vetämien keskustelu- ja oppimistilaisuuk-

sien eli prime time -tilaisuuksien keskusteluista. Lopuksi järjestettävä tentti antaa

maksimissaan 300 pistettä. Opintojakson suoritukseen vaaditaan noin 500 pistettä

ja arvosanat annetaan noin 100 pisteen välein.

                                         32

Flipatuilla opintojaksoillamme ryhmätehtävillä ja prime time-keskusteluilla

on 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 ryhmässä

toimimalla oppiminen on arvioinnissa jäänyt melko pieneen rooliin.

     Haluamme herättää keskustelua, saada ideoita ja keksiä yhdessä uusia tapoja

arviointiin, 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 tätä

syksyä varten digitaalisia tehtäviä, joiden tarkoitus on johdatella opiskelijoita todis-

tusten lukemiseen ja kirjoittamiseen. Tehtävät perustuvat Annie ja John Seldenin

tutkimuksiin todistamisajattelun opettamisesta yliopiston johdantokursseilla. Tehtävien

vaikutuksen arvioimiseksi opiskelijoilla teetettiin kurssin alussa ja lopussa kysely,

jolla mitattiin heidän tunteitaan, asenteitaan ja pystyvyysuskoaan todistamiseen

liittyen. 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 käsitellään

tutkimuksen ja esimerkkien kautta, millaista voi olla oppimista tukeva arviointi

matematiikan opetuksessa. Mikä on arvioinnin tarkoitus? Miksi arviointiin on

tärkeä kiinnittää huomiota? Miten arviointikulttuuri on muuttumassa? Tämän

jälkeen 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 Fock

spaces and compare it with what is known about these operators on Hardy and

Bergman spaces. I also present some recent results on Fredholmness of Toeplitz

operators on generalized Fock spaces (which were introduced by Schuster and

Varolin), even with small exponents, for which reason we also need to characterize

the dual of these generalized Fock spaces. Joint work with Zhangjian Hu (Huzhou

University).

Santeri Miihkinen
On the Hilbert matrix operator on analytic function spaces
The innite Hilbert matrix    H    can be interpreted as a linear operator on spaces of

analytic functions in the open unit disc of the complex plane by its action on their

Taylor coecients. The boundedness of        H    on the Hardy spaces   Hp   for   1

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 nite rank operators. In this talk I

will describe recent work on the quotient algebra    AX = K(X)/A(X)       of compact-

by-approximable operators, which is non-trivial only within the class of Banach

spaces   X   failing the approximation property. I will discuss the size of   AX   and

present examples where    AX   contains non-trivial closed ideals. This is a joint work

with 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 τ -Li coecients and explicit zero-free regions
In this talk I will give an introduction to     τ -Li coecients and my results considering
the coecients and explicit zero-free regions. The        τ -Li   coecients are members of

an innite sequence of real numbers which can be used to determine whether

certain functions satisfy the Generalized Riemann Hypothesis or not. In the talk,

I describe how nitely many         τ -Li   coecients can be used to determine whether

certain 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

                                      a1 a2 a3
                                                   ,
                                      N +N +N +···
where    N   is a xed positive integer and the partial numerators           ai   are positive

integers for all   i.   We call these expansions dnN expansions and show that every

positive real number has innitely many dnN expansions for each             N . In particular
we consider the dnN expansions of rational numbers and quadratic irrationals and

prove some results regarding their periodicity. Finally we show that every positive

real number has for each       N   a dnN 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 more

accurately than individual variables can additively. Interaction terms important

to the response are typically identied through enumeration via exhaustive search

algorithms. An immediate problem is that the number of higher-order interactions

grows rapidly infeasible imposing a serious computational challenge. Ideally, the

dimension of a feature space could be reduced before enumeration based on strong

marginal associations with the response. Unfortunately, individual explanatory

variables contributing to the response through their interaction are not identi-

able by simple linear pre-screening methods unless they exhibit linear associations

with 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 from

the marginal associations is to use more complex nonparametric/nonlinear meth-

ods. A Gaussian process (GP) based automatic relevance determination (ARD) is

known to be theoretically among the best alternatives for such purpose. However,

the estimation of GP models is feasible only for low-dimensional datasets ( 200

variables) preventing the GP-based ARD method to be applied broadly. We have

developed a nonparametric pre-screening method* which reduces the computa-

tionally expensive GP-based ARD method into a simple linear kernel regression

problem. The proposed method preserves all the major benets of the GP-based

ARD and extends its scalability to high-dimensional datasets with tens of thou-

sands of explanatory variables. Some examples will be presented to illustrate the

eciency 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 1.2

million new cases annually and causing 350 000 deaths in 2018. While the prognosis

is typically good, approximately 20dire consequences. Moreover, the initial prostate

cancer diagnosis always reects as worry and quality of life impairment. The initial

prostate cancer determination is based on prostate specic 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 such

as Gleason score and T-stage classication. However, both methods display inac-

curacy and puts patient under infection risk. Our take on this challenge is to use

inammation-related gene single nucleotide polymorphisms (SNP) as predictors of

high-risk prostate cancer. SNP is a low-cost, non-invasive, and stable biomarker.

We have explored inammation SNP association with high-risk prostate cancer in

a genotyped part of Finnish Randomized Screening for Prostate Cancer cohort (n

= 2715) and found several statistically signicant associations. Furthermore, our

validation model using unknown prostate cancer cohort collected during hospital

visits (n = 888) is in concordance with our discovery model. Remarkably, few SNPs

increase 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 over

recent decades, a development that seems to continue apace. One consequence is

that manure is being  and will be  produced on fewer but larger farms. Eurostat

publishes the bulk of manure nutrients from each country, but it is not known how

it is distributed across farms of dierent sizes. This study 1) gives an estimate for

the distribution of main manure nutrients production between farms of dierent

sizes, 2) gives an estimate how this deviation will change in the near future and

3) discusses the land use eects of this development. Results based on stationary

Markov chain model on the farm size development suggest that by the year 2030

farms housing more than 500 livestock units will produce more than two-thirds of

all manure phosphorus, whereas the proportion in 2010 was one-third. With the

Nitrates Directive limiting the use of organic nitrate of manure, growing farms

need to acquire, or conclude contracts for the use of, 4.9 million hectares from

exiting farms or the open market in order to full manure spreading requirements.

This shift will involve 64 % of the total spreading area of 2010 and 15 % of the

total utilized agricultural area of the regions studied. In light of these predictions,

international nutrient policies should consider the evolution of farm structure and

especially manure phosphorus agglomeration. Also salient is improved co-operation

beyond 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 dene a new kind of measure-theoretic entropy concerning endo-

morphisms of measure-preserving dynamical systems, where the action is taken

over discrete and countable amenable groups. Intuitively our entropy gives the

rate 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 dene

a generalization for Lyapunov exponents of one-dimensional cellular automata for

topological dynamical systems over zero-dimensional compact metric spaces and

derive 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 comparing

one-sided and two-sided cellular automata. First we discuss computability. It is

straightforward to simulate Turing machines with two-sided cellular automata, and

with a small trick, even with reversible two-sided cellular automata. We discuss

why universal computing with reversible one-sided cellular automata necessarily

requires some more surprising ideas. Due to the diculties in doing computation

with reversible one-sided cellular automata there are almost no undecidability

results for them. Secondly we discuss some topological dynamical properties, in

particular expansivity. Here more is known for one-sided cellular automata than for

two-sided; in particular it is known that for one-sided cellular automata expansivity

implies pseudo-orbit tracing property, while the same problem is open for two-sided

cellular 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 xed integers. Consider the following six binary words:

                                 S1 = 0,            S4 = 10a ,
                                 S2 = 010a−1 , S5 = 10a+1 (10a )b ,
                                 S3 = 010a ,        S6 = 10a+1 (10a )b+1 .

Let   w be an innite word that is expressible as a concatenation of squares of these
words, that is,         w = X12 X22 · · ·   where    Xi ∈ {S1 , S2 , . . . , S6 }   for all   i.   Dene the
                    √
innite word            w, the square root of w, as the word X1 X2 · · ·               obtained from   w   by

removing half of each square           Xi .
      If   X   is the set of all innite words for which the square map is dened, then

the square root map is a continuous map                    X → X           with respect to the product

topology on the set of innite words over the alphabet                       {0, 1}.
      In the talk, I will briey describe what is known about the dynamics of

the square root map. In particular, I will introduce certain interesting invariant

subsets of      X    such as Sturmian subshifts and so-called SL-subshifts. I will con-

sider xed points, periodic points, and asymptotic behavior in these subshifts.

The preceding subshifts are constructed using solutions to the word equation

X12 · · · Xn2 = (X1 · · · Xn )2   where     Xi ∈ {S1 , S2 , . . . , S6 }   for all i. A complete charac-

terization of the solutions was recently obtained by A. Saarela and the author.

                                                      42

Computational mathematics 1, L9
Gaëlle Brunet
COMPUTATION OF PDE'S ON COMPACT MANIFOLDS
Killing vector elds are important in dierential geometry because their ows

generate isometries on Riemannian manifolds. Equations for Killing elds is an

overdetermined system of PDEs which can be hard to solve explicitly. This prob-

lem can be reduced to a symmetric eigenvalue problem where Killing elds are

generated by the eigenvectors corresponding to zero eigenvalue. The method itself

is valid in any dimension, but numerical results are computed only in two dimen-

sional case. To solve numerically this problem we used nite element method. On

a manifold one has to use in general several coordinate systems to describe the

problem, and the technical diculty is then how to patch these coordinate systems

together. We propose to solve this eigenvalue problem on the sphere with several

local coordinate systems. This method of constructing operators on manifolds can

also be used to study other PDE systems.

                                       43

Antti Hannukainen
Eigensolutions in Distributed Computing Environments
Let   Ω ⊂ R3    and   V ⊂ H01 (Ω)     be the standard rst order nite element space

over a tetrahedral partition of   Ω.    In this presentation, we consider the eigenvalue

problem: Find    (λ, u) ∈ R+ × V \ {0} such that for each w ∈ V
               Z                   Z
                  ∇u · ∇w dx = λ uw dx          and      kukL2 (Ω) = 1.               (1)
                 Ω                      Ω

Our aim is to compute the eigenvalues and the correspoding eigenfunctions in

the   spectral interval of interest (0, Λ)   to a user specied accuracy. We focus on

problems whose solution using a single workstation is impossible due to several

eigenvalues that belong to    (0, Λ),   need for high accuracy, or complicated geome-

try that requires the use of a ne mesh. We solve problem (1) approximately by

Ritz projection to a subspace     V
                                  e   that is constructed from several local subspaces.

These local subspaces can be independently constructed without any intermediate

communication. Hence, the proposed method is well suited to distributed comput-

ing environments. As an example, we describe implementation using           20   standard

desktop 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 numerical

solution of second-order boundary value problems. Since this class of problems

covers many partial dierential equations that appear in physics, dec has quite a

few potential practical applications. Previously, a member of our research group has

written 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 be

problematic for several reasons. Upon closer inspection, the problems seem to

be just another special case that demonstrates the fundamental shortcomings in

the way we develop software for computational sciences. In this talk, I present an

approach to get around these problems by leveraging recent progress in type theory

and category theory. In particular, I propose formalizing a category-theoretical

model of dec in the Coq proof assistant, extracting executable code from the proofs

and linking the extracted code with existing implementations. Doing this via clever

use of type classes and nothing but constructive axioms should let us express the

most salient features of dec with no runtime performance penalty. While the project

is still in its infancy, the prototypes I have built so far are very promising.

                                          45

Sessions on Friday 10:30 - 12:00

Geometric analysis 2, L4
Anna Kausamo
The Monge problem in optimal mass transportation: from two to many
marginals
I will briey 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 rst marginal to the second

one, we try to nd the optimal way of coupling a nite number of marginal mea-

sures. Optimality here is dened by the minimimality of the integral of a given cost

function with respect to the measure that couples the marginals. I will discuss the

diculties related to solving the Monge problem in the multi-marginal framework,

and present some results obtained on the topic in collaboration with Tapio Rajala

and Augusto Gerolin.

José Andrés Rodriguez Migueles
Geodesics on hyperbolic surfaces and volumes of link complements in
Seifert-bered spaces
Let   Γ   be a link in a Seifert-bered space          M   over a hyperbolic surface            Σ   that

projects injectively to a collection of closed geodesics          Γ   in   Σ.   When      Γ   is lling,

the complement     MΓ   of   Γ   in   M   admits a hyperbolic structure of nite volume. We

give bounds of its volume in terms of topological invariants of                 (Γ, Σ).

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Ilmari Kangasniemi
On the entropy of uniformly quasiregular maps
Uniformly quasiregular (UQR) maps are a higher-dimensional generalization of

holomorphic dynamics based on quasiconformal analysis. In this talk, I discuss a

joint work with Yusuke Okuyama, Pekka Pankka and Tuomas Sahlsten, where we

study the entropy of UQR maps. The question indicated by holomorphic dynamics

is whether the topological entropy of a UQR map on a closed manifold equals

log(deg f );   our results show that this is true when the ambient manifold is not a

rational cohomology sphere.

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