3rd workshop on analysis, geometry and probability · 12.00 - 12.30 hassan jolany song-tian...
TRANSCRIPT
3rd Workshop on
Analysis, Geometry andProbability
September 28 - October 2, 2015
Ulm University, Germany
Contents
Conference Information 5
Program 11
Monday, September 28, 2015 . . . . . . . . . . . . . . . . . . . . . . 12
Tuesday, September 29, 2015 . . . . . . . . . . . . . . . . . . . . . . 13
Wednesday, September 30, 2015 . . . . . . . . . . . . . . . . . . . . 14
Thursday, October 1, 2015 . . . . . . . . . . . . . . . . . . . . . . . 15
Friday, October 2, 2015 . . . . . . . . . . . . . . . . . . . . . . . . . 16
Abstracts 17
Invited Lectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Contributed Talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
List of Participants 61
3
Conference Information
Following the workshops in Ulm (2013) and in Moscow (2014), this is the 3rdjoint workshop of Ulm University (UUlm, Germany) and Lomonosov MoscowState University (MSU, Russia) on Analysis, Geometry and Probability. Theworkshop is organized by the Institutes of Analysis, Mathematical Finance, PureMathematics, Number Theory and Probability Theory, Statistics, and Stochas-tics of Ulm University.
Con�rmed invited speakers:
• Eustasio del Barrio (Uni Valladolid)
• Victor Buchstaber (MSU)
• Alexander Bulinski (MSU)
• Henning Bruhn-Fujimoto (UUlm)
• Klaus Deckelnick (Uni Magdeburg)
• Shota Gugushvili (Uni Leiden)
• Dirk Lebiedz (UUlm)
• Markus Pauly (UUlm)
• Hans-Peter Sche�er (Uni Siegen)
• Albert Shiryaev (MSU)
• Robert Stelzer (UUlm)
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6 CONFERENCE INFORMATION
• Stanislav Stepin (MSU)
• Alexey Tuzhilin (MSU)
• Rico Zacher (UUlm)
Scienti�c committee:
• Alexander Bulinski (MSU)
• Anna Dall'Acqua (UUlm)
• Alexander Lindner (UUlm)
• Victor Buchstaber (MSU)
• Boris Kashin (MSU)
• Markus Pauly (UUlm)
• Albert Shiryaev (MSU)
• Volker Schmidt (UUlm)
• Evgeny Spodarev (UUlm)
• Ulrich Stadtmüller (UUlm)
• Robert Stelzer (UUlm)
• Stefan Wewers (UUlm)
In addition to the above mentioned members of the scienti�c committee fromUlm, the local organizing team includes:
• Patricia Alonso-Ruiz
• Dennis Dobler
• Michael Harder
• Renate Jäger
• Jürgen Kampf
7
• Frederik Klingler
• Bernadette Maiwald
• Eva Nacca
• Alexander Nerlich
• Matthias Neumann
• Judith Olszewski
• Stefan Roth
• Adrian Spener
• Christian Steck
Registration July 31, 2015Abstract submission July 31, 2015
Conference fee 50 eContact [email protected]
Conference Web-Page https://www.uni-ulm.de/index.php?id=64428
Conference venue :The �rst two days of the conference and the morning of the third day will takeplace in the Senatssaal at Helmholtzstraÿe 16. From Wednesday afternoon onthe conference will take place in room 2.20 of Helmholtzstraÿe 18.
Travel information :Christian Steck (0049 176 70797595, [email protected]) can pick youup, when you arrive at Ulm main station, and guide you to the B&B- or tothe Ibis-hotel. Please send him an email with your arrival time and your mobilephone number.
We recommend all participants to buy a DING ticket for the week from Septem-ber 28 till October 4 (local tra�c for Ulm and Neu-Ulm, price: 18.90 e). This
8 CONFERENCE INFORMATION
is available e.g. at the ticket machine in front of the main station (where busesand trams leave).
On Monday morning Michael Harder will pick you up at the B&B-hotel at 7:40and at the IBIS hotel at 7:55.
In order to get from the city center to the university take bus line 3. Leavethe bus at bus stop �Botanischer Garten�. There are buses at least every tenminutes most of the day. The schedule is available under http://www.ding.eu/fileadmin/content/schedules_ding/live/din_87003__j15.pdf. Af-ter leaving the bus, follow the road �Helmholtzstraÿe� (that is the road thatturns to the right in front of the bus stop) to its end. At the end of theHelmholtzstraÿe you will �nd the buildings number 16 and 18.
Reimbursement of travel costs :The guests from the MSU have to �ll in the forms concerning the reimbursementof travel costs on Monday morning. Please come to the o�ce of Renate Jäger,room 1.64, at the �rst �oor of Helmholtzstraÿe 18. Our administration needssome days to prepare the cash.
Internet access :WLAN with internet access is available in all buildings of the university. Par-ticipants who use the eduroam system can use it at Ulm University. Otherparticipants can use the following account during the workshop:
User name: [email protected]: x4Nu9qCDQ
Social Program :
• On Monday, September 28, there will be a get-together at 6 p.m. in frontof the Senatssaal.
• On Wednesday, September 30th, there will be a guided tour of the citylasting about 2 hours. We will visit Ulm's historic centre including itsgothic cathedral with the highest church tower in the world. The tour willstart at 5 p.m. sharply in front of the Stadthaus (the white building atthe southern end of the square in front of the cathedral).
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• On Thursday, October 1, at 7 p.m. there will be a conference dinner.It will take place at the restaurant �Drei Kannen� in Ulm's city center(Hafenbad 31/1). Coming from the university you have to leave the busat �Theater�. Then turn left into Olgastraÿe � you can take the tramif you like, but you already have to leave the tram at the next station�Justizgebäude�. Then turn right into Hafenbad � that is the �rst streetturning to the right after the tram station �Justizgebäude�. Soon you willsee the restaurant on the right side.The costs for a 3-course menu vary from 16.50 e to 22.60 e (withoutdrinks) depending on the choice of the main course (for invited speakersthis is included in the invitation). There are three ways of registering forthe conference dinner: You can �ll in the form on the conference webpage,you can write an e-mail to [email protected] and there will be a lastopportunity to register on Monday, September 28.
Program
11
12 PROGRAM
Monday, September 28
Location: Senatssaal, Helmholtzstraÿe 16
8.15 - 9.00 Declaration of travel expenses (in room 1.64 ofHelmholtzstraÿe 18; only for the guests from Moscow) (p. 8)
8.30 - 9.00 Registration (in front of the Senatssaal)
9.00 - 10.30 Victor BuchstaberFullerenes and polygonal partitions of surfaces (p. 22)
10.30 - 11.00 Co�ee break11.00 - 12.30 Alexey A. Tuzhilin
Proteins conformation and the database Protein Data Bank:geometrical validity testing (p. 32)
12.30 - 14.00 Lunch
14.00 - 15.30 Stanislav A. StepinShort-time asymptotics for semigroups from di�usionclass and beyond (p. 31)
15.30 - 16.00 Nikolay Yu. ErokhovetsFullerenes and combinatorics of 3-dimensional �agpolytopes (p. 39)
16.00 - 16.30 Co�ee break16.30 - 17.00 Nataliya V. Prudnikova
Endo-Kroto operation on the space of fullerenes (p. 50)17.00 - 17.30 Georgii Sechkin
Topology of dynamics of a nonhomogeneous rotationallysymmetric ellipsoid (p. 54)
17.30 - 18.00 Vsevolod SalnikovProbabilistic properties of topologies of �nite metric spaces'minimal �llings and rectilinear Steiner minimal trees (p. 52)
18.00 Get together (p. 8)
13
Tuesday, September 29
Location: Senatssaal, Helmholtzstraÿe 16
9.00 - 10.30 Albert N. ShiryaevQuickest detection problemsdiscrete and continuous time (p. 29)
10.30 - 11.00 Co�ee break11.00 - 12.30 Robert Stelzer
Lévy-driven CARMA Processes (p. 30)
12.30 - 14.00 Lunch
14.00 - 15.00 Peter Sche�erImplicit extremes and implicit max-stable laws (p. 28)
15.00 - 15.30 Georgiy ShevchenkoSmall ball probabilities and representations with respect toGaussian processes (p. 55)
15.30 - 16.00 Johannes Goldbachf -implicit max-in�nitely divisible distributions (p. 43)
16.00 - 16.30 Co�ee break16.30 - 17.00 Martin Drapatz
On spatial random recurrence equations (p. 38)17.00 - 17.30 Nikolai A. Slepov
Modi�cation of the MDR method related to generalizedpenalty function (p. 57)
17.30 - 18.00 Dennis DoblerInference procedures for cumulative incidence functions (p. 37)
14 PROGRAM
Wednesday, September 30
Location: Senatssaal, Helmholtzstraÿe 16
9.00 - 10.30 Alexander V. BulinskiMethods of variable selection (p. 23)
10.30 - 11.00 Co�ee break11.00 - 12.30 Markus Pauly
An excursion to randomization-type tests (p. 27)
12.30 - 14.00 Lunch
Location: Room 2.20, Helmholtzstraÿe 1814.00 - 15.00 Eustasio del Barrio
A contamination model for approximate stochastic order (p. 20)15.00 - 15.30 Dustin Kremer
Operator-stable and operator-self-similar random �elds (p. 48)15.30 - 16.00 Arnold Janssen
Preferences of goodness-of-�t tests: A survey aboutthe analysis of nonparametric power functions (p. 40)
17.00 Guided city tour (p. 8)
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Thursday, October 1
Location: Room 2.20, Helmholtzstraÿe 18
9.00 - 10.30 Rico ZacherNon-local in time subdi�usion equations (p. 33)
10.30 - 11.00 Co�ee break11.00 - 12.00 Klaus Deckelnick
Finite element methods for elliptic PDEs on surfaces (p. 24)
12.00 - 13.30 Lunch
13.30 - 14.30 Shota GugushviliNon-parametric Bayesian inference for multi-dimensionalcompound Poisson processes (p. 25)
14.30 - 15.00 Stefan RothAn inverse problem for in�nitely divisible random �eldswith integral representation (p. 51)
15.00 - 15.30 Co�ee Break15.30 - 16.00 Artem M. Savchuk
Uniform stability for the inverse Sturm-Liouville problem.Recovering of a potential from a �nite set of spectraldata (p. 53)
16.00 - 16.30 Mikhail A. TuzhilinRelation between the circular molecule and 4-dimensionalsingularity in the saddle-saddle case (p. 59)
16.30 - 17.00 Egor D.KosovEstimates of polynomials on spaces with logarithmicallyconcave measures (p. 46)
19.00 Conference dinner (p. 9)
16 PROGRAM
Friday, October 2
Location: Room 2.20, Helmholtzstraÿe 18
9.00 - 10.30 Henning Bruhn-FujimotoThe probabilistic method in discrete mathematics (p. 21)
10.30 - 11.00 Co�ee break11.00 - 11.30 Christian Steck
Models of curves and quotient singularities (p. 58)11.30 - 12.00 Iurii N. Shteinikov
On the divisibility of Fermat quotients (p. 56)12.00 - 12.30 Hassan Jolany
Song-Tian approach for solving the �nite generation ofcanonical ring of a projective variety (p. 42)
12.30 - 14.00 Lunch
14.00 - 14.30 Matthias NeumannStatistical modeling of three-phase structures withcompletely connected phases (p. 49)
14.30 - 15.00 Timothy BreretonEstimating the velocities of interacting charge carriersin disordered media (p. 36)
15.00 - 15.30 Co�ee break15.30 - 16.00 Jürgen Kampf
A functional central limit theorem for Lebesgue integralsof mixing random �elds (p. 44)
16.00 - 16.30 Alexey A. KozhevinVariable selection method with group overlapping (p. 47)
Abstracts
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18 ABSTRACTS
Invited Lectures
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20 ABSTRACTS
Eustasio del Barrio, University of Valladolid
A contamination model for approximate stochastic order
Stochastic ordering among distributions has been considered in a variety of sce-narios. Economic studies often involve research about the ordering of invest-ment strategies or social welfare. However, as noted in the literature, stochasticorderings are often a too strong assumption which is not supported by the dataeven in cases in which the researcher tends to believe that a certain variable issomehow smaller than other. Instead of considering this rigid model of stochas-tic order we propose to look at a more �exible version in which two distributionsare said to satisfy an approximate stochastic order relation if they are slightlycontaminated versions of distributions which do satisfy the stochastic ordering.The minimal level of contam- ination that makes this approximate model holdcan be used as a measure of the deviation of the original distributions from theexact stochastic order model. Our approach is based on the use of trimmings ofprobability measures. We discuss the connection between them and the approx-imate stochastic order model and provide theoretical support for its use in dataanalysis, including asymptotic distributional theory as well as non-asymptoticbounds for the error probabilities of our tests. We also provide simulation re-sults and a case study for illustration.
21
Henning Bruhn-Fujimoto, Ulm University
The probabilistic method in discrete mathematics
A common problem in any branch of mathematics is to prove the existenceof an object with certain, speci�ed properties. This could, for instance, besomething as mundane as showing that sin(x) has a root in [2, 4]. Often thesimplest proof consists in explicitly describing or constructing such an object,that is, we could verify that π is such a root. Sometimes, however, a directproof is unavailable, and more indirect existence proofs are called for. A prooftechnique that has turned out to be very powerful in discrete mathematics is theso called probabilistic method: devise a suitable probability space and show thatthe desired object occurs with positive probability, which means, in particular,that it has to exist.
In the talk I will give a �rst introduction to the probabilistic method and its usein graph theory.
22 ABSTRACTS
Victor Buchstaber, Moscow State University and Steklov Mathematical Institute
Fullerenes and polygonal partitions of surfaces
We consider polygonal partitions of a closed 2-surface M2, in which the in-tersection of any two polygons is either empty or their common edge. Such apolygonal partition is called simple if any its vertex is contained in exactly 3polygons. The combinatorics of such partitions is an area of research at theintersection of classical and most modern branches of mathematics and its ap-plications. It is known that the condition on the surface partition to be simpleallows to supplement classical Euler's formula, which connects the numbers f0,f1 and f2 of vertices, edges and polygons belonging to the surface partition, bya formula connecting the numbers p3, . . ., pk, . . ., where pk is the number ofk-gones belonging to this partition. Implications of this formula are non-trivialalready in the case of a sphere considered as the boundary of a convex three-dimensional body. For example, for a simple decomposition of a sphere involvingonly pentagons and hexagons, the number of pentagons should be 12.
Results on combinatorics of simple polygonal partitions have become very actualin connection with the discovery of remarkable molecules of carbon, namelyfullerenes (Nobel Prize in Chemistry, 1996, R. Curl, H. Kroto, R. Smalley). Themathematical model of a fullerene is a simple polygonal partition of a sphereon hexagons and pentagons. The big push intensifying the research in this �eldwas given by the discovery of such a carbon structure as graphene (Nobel Prizein Physics, 2010, A. Geim, K. Novoselov).
The mathematical model of graphene is the hexagonal simple partition of theplane. The graphene plane modulo a lattice de�nes a hexagonal partition of thetorus. In our focus will be the problems of combinatorial geometry and algebraictopology, motivated by problems of quantum physics and chemistry of fullerenes,graphene, and other carbon structures such as nanotubes and nanobuds. Thelecture is intended for a broad audience.
23
Alexander V. Bulinski, Moscow State University
Methods of variable selection
In a number of stochastic models a studied variable (response) Y depends onsome collection of (random) factorsX = (X1, . . . , Xn). For example, in medicaland biological research Y can characterize the health state of a patient whereasthe components of X describe genetic and nongenetic factors. In this regardwe refer to the genome-wide association studies (GWAS). One of the challengingproblems concerning the response analysis consists in identi�cation of signi�cant(or in�uential) collection of factors (Xi1 , . . . , Xir ), 1 ≤ i1 < . . . < ir ≤ n, suchthat Y depends on it in essential way, in a certain sense. Many complemen-tary methods for solving this problem were introduced. They can be classi�edinto three categories according to their strategies, namely, exhaustive, stochasticand heuristic search. Note that probabilistic and statistical techniques, machinelearning and computer simulation are combined in such investigations. We con-centrate on multi-step designs. Main attention will be paid to MDR (multifactordimensionality reduction) method introduced by M.Ritchie et al. in 2001. Thismethod was applied and further developed in more than 200 publications. Thelecture is based on a cycle of recent author's papers. We consider statisticalestimation of the error functional for the response forecast based on i.i.d. ob-servations, a penalty function estimate and a cross-validation procedure. Thismethod permits to comprise nonbinary response. The regularized estimates ofthe mentioned functional are de�ned and a central limit theorem is establishedfor them. We also discuss the results of computer simulation showing the e�ec-tiveness of our approach.
The work is supported by RSF, project 14-21-00162.
24 ABSTRACTS
Klaus Deckelnick, Otto von Guericke University of Magdeburg
Finite element methods for elliptic PDEs on surfaces
Partial di�erential equations on �xed or moving surfaces appear in a varietyof applications, e.g. in multiphase �ows or di�usion induced grain boundarymotion. Quite often the corresponding mathematical models involve a couplingbetween surface and bulk equations giving rise to complicated systems. For thisreason the development and analysis of numerical methods plays an importantrole in understanding the behaviour of solutions.The talk will focus on the following model problem:
−∆Γu+ u = f on Γ,
where Γ is a given compact hypersurface without boundary and ∆Γ denotes theLaplace�Beltrami operator. In order to discretize this problem with the help ofthe �nite element method the equation is �rst reformulated in variational formafter which the space of test functions is replaced by a suitable �nite�dimensional(�nite element) space. In addition, a full discretization of the problem requiresthe approximation of Γ by a suitable discrete hypersurface Γh. Roughly speak-ing, one can distinguish the following two approaches:
a) The �nite element space is constructed on a direct piecewise polynomial ap-proximation Γh of Γ;b) The �nite element space is induced by an outer triangulation of an openneighbourhood of Γ, while Γh itself is represented in implicit form.
The talk will introduce corresponding numerical schemes and discuss their anal-ysis.
25
Shota Gugushvili, Leiden University
Non-parametric Bayesian inference for multi-dimensional compound Pois-
son processes
Joint work with Frank van der Meulen and Peter Spreij
Given a sample from a discretely observed multi-dimensional compound Poissonprocess, we study the problem of non-parametric estimation of its jump sizedensity r0 and intensity λ0. We take a non-parametric Bayesian approach tothe problem and determine posterior contraction rates in this context, which,under some assumptions, we argue to be optimal posterior contraction rates. Inparticular, our results imply existence of Bayesian point estimates that convergeto the true parameter pair (r0, λ0) at these rates. We will also present somesimulations to illustrate the practical performance of our method.
References:
[1] Gugushvili, S., van der Meulen, F. and Spreij, P. (2015): NonparametricBayesian inference for multidimensional compound Poisson processes, ModernStochastics: Theory and Applications, 2:1-15.
[2] Gugushvili, S., van der Meulen, F. and Spreij, P. (2015): A non-parametricBayesian approach to decompounding from high frequency data. arXiv:1507.03263[math.ST].
26 ABSTRACTS
Dirk Lebiedz, Ulm University
From Microscopic to Macroscopic Dynamics in Kinetic Multi-Scale Mod-
eling
Joint work with Pascal Heiter and Jonas Unger
We will elucidate analytic, geometric and statistical issues related to the com-putation of slow invariant attracting manifolds (SIAM) in kinetic multiple timescale ordinary di�erential equation (ODE) models. Based on a Riemannian ge-ometry view on the phase space of ODE vector �elds we introduce a variationalproblem formulation for SIAM characterization and computation. Both analyticand numerical aspects of the problem are addressed, numerical real-time opti-mization techniques can be exploited for highly e�cient computation of SIAMsin applications to high-dimensional models of chemical reaction kinetics. Westart by motivating our research by addressing the challenging problem of mod-eling and numerical simulation of detailed reactive �ow models as arising e.g.in combustion and chemical engineering. We then introduce fundamental con-cepts from chemical reaction kinetics. Finally we point out the development andimplications of our variational approach to SIAM computation.
References:
[1] Lebiedz, D., Unger, J. (2015) On unifying concepts for trajectory-based slowinvariant attracting manifold computation in multi-scale models of chemicalkinetics. (submitted), arXiv:1405.1856v2 [math.DS]
[2] Lebiedz, D., Siehr, J. (2013) A continuation method for the e�cient solutionof parametric optimization problems in kinetic model reduction. SIAM J. Sci.Comput. 35, A1584�A1603
[3] Lebiedz, D., Siehr, J. Unger, J. (2011) A Variational principle for computingslow invariant manifolds in dissipative dynamical systems. SIAM J. Sci. Comput.33, 703�720
27
Markus Pauly, University of Ulm
An Excursion to Randomization-Type Tests
Joint work with Edgar Brunner, Lutz Dümbgen, Arnold Janssen and Frank Koni-etschke
In a nutshell, the main purpose of statistical hypothesis testing is to decide (witha small pre-speci�ed error probability) that the unknown underlying distributionof a given sample belongs to a speci�c class of distributions (the so called alter-native). When turning from parametric hypotheses (class of possibly underlyingdistributions is described by a �nite dimensional space) to larger nonparametrichypotheses classical test procedures (as e.g. the two sample t-test) are in generalnot valid anymore. To this end, the current talk will propagate the applicationof di�erent kinds of randomization-type tests as powerful solutions. Therefore,the �rst part will introduce this class of tests and explain the correspondingbasic �nite sample theory. Afterwards more recent considerations for even largerclasses of alternatives will be discussed in several examples.
Some References:
Chung, E. and Romano, J. P. (2013): Exact and asymptotically robust permu-tation tests. The Annals of Statistics, 41(2), 484-507.
Dümbgen, L. and Pauly, M (2015): Randomization-type tests for homogeneityand proportionality of covariance matrices. Under preperation.
Janssen, A. (1999): Testing nonparametric statistical functionals with applica-tions to rank tests. Journal of Statistical Planning and Inference, 81(1), 71-93.
Janssen, A. and Pauly, M. (2009): Asymptotics and e�ectiveness of conditionaltests with applications to randomization tests. Technical Report, University ofDuesseldorf.
Konietschke, F. and Pauly, M. (2014): Bootstrapping and Permuting pairedt-test type statistics. Statistics and Computing, 24, p. 283 - 296.
Pauly, M., Brunner, E. and Konietschke, F. (2015): Asymptotic PermutationTests in General Factorial Designs. Journal of the Royal Statistical Society -Series B 77, 461-473.
28 ABSTRACTS
Peter Sche�er, University of Siegen
Implicit Extremes and Implicit Max-Stable Laws
Joint work with Stilian Stoev, University of Michigan, Ann Arbor
Let X1, · · · , Xn be iid Rd-valued random vectors and f ≥ 0 be a non�negativefunction. Let also k(n) = Argmaxi=1,··· ,nf(Xi). We are interested in thedistribution of Xk(n) and their limit theorems. In other words, what is thedistribution the random vector where a function of its components is extreme.This question is motivated by a kind of inverse problem where one wants todetermine the extremal behavior of X when only explicitly observing f(X). Weshall refer to such types of results as to implicit extremes. It turns out that, asin the usual case of explicit extremes, all limit implicit extreme value laws areimplicit max�stable. We characterize the regularly varying implicit max�stablelaws in terms of their spectral and stochastic representations. We also developa theory of regular variation on general cones.
References:
[1] Sche�er, P. and Stoev, S. (2015): Implicit Extremes and Implicit Max-StableLaws, preprint.
29
Albert N. Shiryaev, Steklov Mathematical Institute, Moscow State University
Quickest detection problems � discrete and continuous time
We observe a random process X = (Xt) on an interval [0, T ], T ≤ ∞. Thefollowing objects θ and τ which are introduced and considered below are essentialthroughout the lectures:
• θ is a parameter or a random variable, it is a hidden, nonobservable char-acteristic; for example, it can be a time when observed process X = (Xt)changes its character of behavior or its characteristics;
• τ is a stopping (Markov) time which serves as the time of �alarm�, it warnsof the coming of the time θ.
The following observed process X = (Xt) plays the crucial role:
Xt = µ(t− θ)+ + σBt,
where B = (Bt)t≥0 is a Brownian motion. We formulate four variants of thequickest detection problem. One of them is Variant A: to �nd
A(c) = infτ
[PG(τ < θ) + cEG(τ − θ)+
],
d where PG is a distribution with respect to a priori distribution G.
We formulate also other problems (solved and unsolved).
The work is supported by RSF, project 14-21-00162. References:
[1] Besag, J., York, J. and Mollié, A. (1991): Bayesian image restoration, withtwo applications in spatial statistics (with discussion), Annals of the Institute ofStatistical Mathematics, 43, 1�59.
30 ABSTRACTS
Robert Stelzer, Ulm University
Lévy-driven CARMA Processes
Joint work with Tina Marquardt, Eckhard Schlemm, Florian Fuchs and ZywillaFechner.
We present an outline of the theory of certain Lévy-driven, multivariate stochas-tic processes, where the processes are represented by rational transfer functions(Continuous-time AutoRegressive Moving Average or CARMA models) and theirapplications in non-Gaussian time series modelling. We discuss in detail theirde�nition, their spectral representation, the equivalence to linear state spacemodels and further properties like the second order structure and the tail be-haviour under a heavy-tailed input. Furthermore, we study the estimation of theparameters using quasi-maximum likelihood estimates for the auto-regressiveand moving average parameters, as well as how to estimate the driving Lévyprocess.
31
Stanislav A. Stepin, Moscow State University
Short-time asymptotics for semigroups from di�usion class and beyond
A relationship between Feynman-Kac formula and parametrix representation forSchroedinger semigroup proves to be useful for evaluation of the regularizedtrace and calculation of the heat invariants, i.e. coe�cients of the heat ker-nel short-time asymptotics. In view of the asymptotic analysis to be carriedout for evolutionary semigroups beyond the di�usion type class we revisit pathintegral approach to the study of heat kernel asymptotics and heat trace estima-tions. Within this approach for the case of di�usion with a drift the heat kernelasymptotic properties are speci�ed. Making use of parametrix expansion andBorn approximation instead of path integrals we investigate semigroups gener-ated by potential perturbations of bi-Laplacian. Namely short-time asymptoticsfor the corresponding Schwartz kernel and regularized trace are derived.
32 ABSTRACTS
Alexey A. Tuzhilin, Moscow State University
Proteins conformation and the database Protein Data Bank: geometrical
validity testing
The investigation of polypeptides conformation (3D-structure) is well-known andvery important problem. In particular, its great importance is related to the factthat conformation changes of proteins in living cells caused by some mutationsmay lead to serious diseases. Notice that experimental determination of suchmutations consequences is time consuming and expensive process. Therefore, itis naturally and important to work out theory and software for performing "thoseexperiments" by means of computer. Unfortunately, the modeling problem itselfturns out to be very hard and, in spite of many e�orts of the worldwide com-munity, is very far from �nal solution. Notice that this problem stands at theinterfaces of many sciences such as biology, chemistry, physics, geometry, vari-ational calculus, probability, computational mathematics. Thus, it is necessaryto attract attention of specialists from very di�erent branches of science and tocreate a possibility for the interested novices to get fast abreast of and to starttheir own experiments with the data collected. However, it turned out that themodern state of biological information collected in the main internet databaseProtein Data Bank (PDB) devoted to the conformation of biomolecules is veryfar from perfection. That is why one needs to workout e�ective methods forveri�cation and extracting correct information from the database.
In our lecture we present several geometrical and statistical methods to analyzethe information in PDB for consistency and general theory �tting. In addition,we include a list of examples demonstrating the necessity of careful analysis andsampling of the PDB items. In particular, we show some examples visualizingthe distribution of disturbance level of the famous Pauling plane law, and ofCIS-con�gurations rareness. In addendum, we discuss a method for automaticrecognition of helices in a protein, based on discrete analogues of curvature andtorsion of a smooth curve.
The work was supported by Russian Science Foundation (RSF), project 14-50-00029.
33
Rico Zacher, Ulm University
Non-local in time subdi�usion equations
Partially joint work with Jukka Kemppainen, Juhana Siljander, and VicenteVergara
We will consider a class of non-local in time di�usion equations, where insteadof the usual time derivative one has an integro-di�erential operator w.r.t. time.Important special cases are time-fractional and ultraslow di�usion equations,which have seen much interest during the last years, mostly due to their appli-cations in the modeling of anomalous di�usion. I will discuss several aspects ofthese equations, such as well-posedness, regularity, and the long-time behaviourof solutions. Among others, I will show that the decay behaviour is markedlydi�erent from the heat equation case.
References:
[1] Kemppainen, J., Siljander, J., Vergara, V. and Zacher, R. (2014): Decayestimates for time-fractional and other non-local in time subdi�usion equationsin Rd, Preprint.
[2] Kochubei, A.N. (2008): Distributed order calculus and equations of ultraslowdi�usion, J. Math. Anal. Appl., 340, 252�281.
[3] Metzler, R., Klafter, J. (2000): The random walk's guide to anomalousdi�usion: a fractional dynamics approach, Phys. Rep., 339, 1�77.
[4] Prüss, J. (1993): Evolutionary Integral Equations and Applications, Mono-graphs in Mathematics, 87, Birkhäuser, Basel.
[5] Vergara, V., Zacher, R. (2015): Optimal decay estimates for time-fractionaland other non-local subdi�usion equations via energy methods, SIAM J. Math.Anal., 47, 210�239.
[6] Zacher, R. (2013): A De Giorgi-Nash type theorem for time fractional di�u-sion equations, Math. Ann., 356, 99�146.
[7] Zacher, R. (2009): Weak solutions of abstract evolutionary integro-di�erentialequations in Hilbert spaces, Funkcialaj Ekvacioj, 52, 1�18.
Contributed Talks
35
36 ABSTRACTS
Timothy Brereton, Ulm University
Estimating the velocities of interacting charge carriers in disordered
media
Joint work with Aaron Smith, Rafal Kulik, and Volker Schmidt
Charge transport in disordered materials (such as organic semiconductors) can bemodeled by considering charge carriers as moving on the vertices of a geometricgraph with periodic boundary conditions. The vertices have energies assigned tothem. The dynamics of the charge carriers are then described by a continuoustime Markov chain, whose transition rates are determined by the di�erences inenergies between vertices of the graph.
One of the key quantities of interest when considering charge transport is theaverage speed of a charge carrier as it travels through the material under thein�uence of an external electric �eld. Often, Monte Carlo methods need to beused to estimate this speed. However, Monte Carlo simulation is di�cult becauserealistic models of disordered materials have many traps (i.e., low energy regionswith large rates leading in to them and small rates leading out of them) in whichcharge carriers become stuck for long periods of time. This poses a signi�cantproblem for simulation, as almost all computational e�ort is spent on simulatingthe charge carriers hopping around in the trap regions. We propose a method for�coarsening� the state space in such a manner that we do not need to simulatethe exact dynamics of the charge carriers in the trap regions whilst still extractingall the information we need in order to estimate their speed consistently.
37
Dennis Dobler, Ulm University
Inference Procedures for Cumulative Incidence Functions
Joint work with Jan Beyersmann and Markus Pauly
The analysis of cumulative incidence functions (and more generally of transitionprobability matrices of non-homogeneous Markov processes) is of great impor-tance (especially in medical applications) and it constantly gives rise to newstatistical developments. While observations may be incomplete, e.g. due torandom left-truncation and right-censoring, estimation of these matrices is con-ducted by employing the Aalen-Johansen estimator which is based on countingprocesses. However, results of weak convergence towards a Gaussian processcannot be utilized straightforwardly since the complicated limiting covariancestructure depends on unknown quantities. Already established tests, which arepopular in the literature, have a simple limit distribution but may have poorpower.
In order to construct asymptotically valid and powerful inference procedures, weintroduce a large variety of resampling techniques using a martingale representa-tion of this estimator. A new aspect to this approach is given by the possibilityto choose these multipliers dependent on the data, covering, for instance, theWild bootstrap as well as the Weird bootstrap. In doing so, we gain conditionalweak convergence towards a Gaussian process with correct covariance functionsresulting in consistent tests and con�dence bands.
For small samples the performance in the simple competing risks set-up is as-sessed via simulation studies illustrating the type I error control and analyzingthe power of the developed tests and con�dence bands.
38 ABSTRACTS
Martin Drapatz, Ulm University
On spatial random recurrence equations
We consider random recurrence equations
Y (i, j) = A0(i− 1, j)Y (i− 1, j) +A1(i, j − 1)Y (i, j − 1) + c(i, j),
where i, j,∈ Z and (A0(i− 1, j), A1(i, j − 1), c(i, j))i,j∈Z is an independentand identical distributed (i.i.d.) sequence of random variables. Relations totime series models (more precisely random �eld models) and percolation theorywill be discussed.
39
Nikolay Yu. Erokhovets, Moscow State University
Fullerenes and combinatorics of 3-dimensional �ag polytopes
The talk is based on the joint work with V.M. Buchstaber.
Let P be a simple convex 3-polytope. A k-belt is a cyclic sequence (F1, . . . , Fk)of 2-faces, such that Fi1∩· · ·∩Fir 6= ∅ if and only if {i1, . . . , ir} ∈ {{1, 2}, . . . ,{k−1, k}, {k, 1}}. P is a �ag polytope if it is not a simplex and does not contain3-belts.
Theorem 1. A simple 3-polytope P is �ag if and only if it is combinatoriallyequivalent to a polytope obtained from the cube by a sequence of edge trun-cations and truncations along two incident edges lying in a k-gonal face withk ≥ 6.
A fullerene is a simple convex 3-polytope with all facets being pentagons andhexagons.
Theorem 2. Any fullerene P is a �ag polytope. It contains no 4-belts andhas 12 + k �ve-belts, where 12 belts surround pentagons and k belts consist ofhexagons with any hexagon intersecting neighbours by opposite edges. More-over, if k > 0 then P consists of two dodecahedral caps and k hexagonal 5-beltsbetween them.
We give applications of these results to the toric topology.
The work is supported by the Russian President grant MK-600.2014.1 and theRFBR grant 14-01-31398-a.
40 ABSTRACTS
Arnold Janssen, Heinrich Heine University of Duesseldorf
Preferences of goodness-of-�t tests: A survey about the analysis of non-
parametric power functions
Goodness of �t tests are usually constistent for nonparametric models. How-ever, they do not meet the power of oracle tests (Neyman Person tests) for localalternatives when the distributions would be known. The statistician likes todistinguish and to compare the power of di�erent competing tests. It is shownthat under certain circumstances every test has a preference for a �nite dimen-sional space of alternatives. Apart from this space, the power function is almost�at on balls of alternatives. There exists no test which pays equal attention toan in�nite number of orthogonal alternatives. The results are not surprising.Every statistician knows that it is impossible to seperate an in�nite sequenceof di�erent parameters simultaneously if only a �nite number of observations isavailable.The conclusions of the results are two-fold.
1. The statistician should analyze the goodness of �t tests of his computerpackage in order to get some knowledge and an impression about theirpreferences.
2. A well-re�ected choice of tests requires some knowledge about preferencesconcerning alternatives which may come from the practical experiment.A guide to the construction of tests is given. A principle componentdecomposition of goodness of �t tests has been studied in Janssen (1995,2000, 2003).
Global power functions of one-sided Kolmogorov-Smirnov tests for a restrictedclass of alternatives were obtained by And¥l (1967) and Hájek and �idák (1967).We refer also to the early work of Neuhaus (1976), Milbrodt and Strasser (1990)and Rahnenfuehrer (2003).
References
J. And¥l (1967) Local asymptotic power and e�ciency of tests of Kolmogorov-Smirnov type, Ann. Math. Statist. 38, 1705�1725.
J. Hájek, Z.�idák (1967) Theory of Rank Tests. Academic Press, New York.
41
A. Janssen (1995) Principal component decomposition of non-parametric tests,Probab. Theory Related Fields 101, 193�209.
A. Janssen (2000) Global power functions of goodness of �t tests, Ann. Statist.28, No. 1, 239�253.
A. Janssen (2003) Which power of goodness of �t tests can really be expected:intermediate versus contiguous alternatives, Statistics and Decisions 21, 301�325.
H. Milbrodt, H. Strasser (1990) On the asymptotic power of the two-sidedKolmogorov-Smirnov test, J. Statist. Plann. Inference 26, 1�23.
G. Neuhaus (1976) Asymptotic power properties of the Cramér-von Mises testunder contiguous alternatives, J. Multivariate Anal. 6, 95�110.
J. Rahnenfuehrer (2003) On preferences of general two-sided tests with ap-plications to Kolmogorov-Smirnov-type tests, Statistics and Decisions 21,149�170.
42 ABSTRACTS
Hassan Jolany, University of Lille
Song-Tian approach for solving the �nite generation of canonical ring of
a projective variety
Existence of canonical metric on a projective variety was a long standing con-jecture which the major part of this conjeture is about varieties which do nothave de�nite �rst Chern class. There is a program which is known as Song-Tianapproach for �nding canonical metric on canonical model of a projective varietyand its connection with Weil-Petersson metrics along a divisor with conical andcusp singularities. I give a logarithmic version of this program and strengthenthe previous results of Song-Tian. In �nal I explain the analogus method ofSong-Tian for Sasakian manifolds.
References:
[1] Jian Song; Gang Tian, The Kahler-Ricci �ow on surfaces of positive Kodairadimension, Inventiones mathematicae. 170 (2007), no. 3, 609�653.
[2] Jian Song; Gang Tian, Canonical measures and Kähler-Ricci �ow, J. Amer.Math. Soc. 25 (2012), no. 2, 303�353.
[3] Hassan Jolany, Weil-Petersson geometry on metrics with Poincaré singulari-ties, In prepration.
[4] Hassan Jolany, Song-Tian theory for Sasakian setting, In prepration.
43
Johannes Goldbach, University of Siegen
f-Implicit Max-In�nitely Divisible Distributions
Joint work with Hans-Peter Sche�er
Let f : Rd → [0,∞) be continuous and 1-homogeneous with f(x) = 0 ⇔x = 0. In applications f will be referred to as the loss function. We equipRd with a speci�c non-commutative but associative (inner) binary operation∨f , called the f -max operation. It turns out that (Rd,∨f ) is a semigroup butnot necessarily a topological one. This yields, in a natural way, to the conceptof so called f -implicit max-in�nitely divisible (f -imid) distributions. Our aimis to establish a theory of this new concept extending the theory of f -implicitmax-stable distributions, analogous to the case of in�nitely divisible and stabledistributions regarding the +-operation and the ∨-operation, respectively. Undersome mild assumptions on x 7→ P(f(X) ≤ x) for a given random vector X wethen show that X is f -imid. For the moment the general case still remains as anopen question that might be solved using the theory of so called ∨f -compoundpoisson processes.
References:
[1] Stoev, S. and Taqqu, M.S. (2005): Extremal stochastic integrals: a parallelbetween max-stable processes and α-stable processes, Extremes, Vol. 8, 237-266.
[2] Resnick, S.I. (2007): Extreme Values, Regular Variation, and Point Pro-cesses, Springer.
[3] Sche�er, H.-P. and Stoev, S. (2014): Implicit Extremes and Implicit max-stable Laws, arXiv:1411.4688.
[4] Meerschaert, M.M. and Sche�er, H.-P. (2001): Limit Distributions for Sumsof Independent Random Vectors, John Wiley & Sons Inc.
44 ABSTRACTS
Jürgen Kampf, Ulm University
A Functional Central Limit Theorem for Lebesgue Integrals of Mixing
Random Fields
Joint work with Evgeny Spodarev
Central limit theorems for random �elds have been studied for a long time. In1984 Gorodetskii [1] proved a �rst central limit theorem for integrals of the form∫Wn
X(t) dt, where (X(t))t∈Rd is a random �eld and (Wn)n∈N is a sequenceof integration domains tending to Rd in an appropriate way. A central limittheorem for
∫Wn
f(X(t)) dt with a map f : R → R is an easy corollary ofthis. Meschenmoser and Shashkin [2] showed a functional central limit theoremfor Lebesgue measures of excursion sets of random �elds, i.e. for integrals ofthe form
∫Wn
1[u,∞)(X(t)) dt, u ∈ R, where the stochastic process is indexedby u. We will extend the result of [2] to a functional central limit theoremfor
∫Wn
f(X(t)) dt, where the stochastic process is indexed by the functionf : R → R, which is assumed to be Lipschitz continuous. While replacingindicator functions by Lipschitz continuous functions is straight-forward, we needan entirely di�erent approach, since the index set of the stochastic process ismuch bigger now (it is the real line in [2] and the space of Lipschitz continuousfunction in this talk).
The functional central limit theorem is derived for a stationary and measurablerandom �eld which ful�lls some α-mixing condition as well as some integrabilityconditions. We show that the sequence (Φn)n∈N of stochastic processes de�nedby
Φn(f) :=
∫Wn
f(X(t)) dt− λd(Wn) · Ef(X(0))√λd(Wn)
converges in distribution to a centered Gaussian process Φ with
Cov(Φ(f),Φ(g)
)=
∫Rd
Cov(f(X(0)), g(X(t))
)dt.
As topology we use the weak topology on the dual space of the space of Lipschitzcontinuous functions. The proof is essentially based on a result from Oppel [3],which tells us that it su�ces to show that the �nite-dimensional distributionsconverge and that the limiting process has a version with linear and continuouspaths.
45
We will also discuss the question whether the variance of the limiting processis strictly positive. For Gaussian random �elds with non-negative covariancefunction we show that the asymptotic variance is strictly positive except forsome trivial counterexamples. On the other hand, based on Poisson-Voronoimosaics, we can construct an example of a random �eld for which the varianceof Φ vanishes for all functions f : R→ R simultanously.
References:
[1] Gorodetskii, V.V. (1984): The central limit theorem and an invariance prin-ciple for weakly dependent random �elds, Soviet Mathematics (Doklady), 29(3),529�532.
[2] Meschenmoser, D. and Shashkin, A. (2011): Functional central limit theoremfor the volume of excursion sets generated by associated random �elds, Statistics& Probability Letters, 81(6), 642�646.
[3] Oppel, U. (1973): Schwache Konvergenz kanonischer zufälliger Funktionale,Manuscripta Mathematica, 8, 323�334.
46 ABSTRACTS
Egor D.Kosov, Moscow State University
Estimates of polynomials on spaces with logarithmically concave mea-
sures
We will discuss several new estimates for polynomials on spaces with logarith-mically concave measures. One of the important examples of logarithmicallyconcave measures is Gaussian measure and the results are new for this mea-sure too. One of such results is an estimate of L1-norm of a polynomial f byL1-norm of its restriction to a set U of positive measure:∫
fdµ ≤ C∫U
fdµ,
where constant C depends only on the degree of the polynomial and measureof the set U . Some distributional inequalities for polynomials will also be con-sidered. Moreover, we will discuss some approaches on which proofs of thepresented results are based.
This work was supported by the Russian Science Foundation grant N 14-11-00196.
47
Alexey A. Kozhevin, Moscow State University
Variable selection method with group overlapping
In our talk we consider model from genome-wide association studies (GWAS) inwhich binary or continuous response variable depends on predictors X1, . . . , Xp
where Xj takes values in {0, 1, 2} for all j. The aim is to �nd all the predic-tors that essentially in�uence on response. This task leads to high-dimensionalproblem where number of predictors exceeds the number of observations anddependence between the predictors due to linkage disequilibrium (LD) may ex-ist.
There are di�erent methods for this task solution. One of them proposed in[1] takes into account certain dependence between predictors. Authors em-ploy three-step approach which is based on combining the predictors into non-overlapping groups, estimation of the number of groups and variable selectionby group LASSO.
We modify method introduced in [1] by another hierarchical clustering whichallows overlaps between groups of the predictors.
This work is partially supported by RFBR grant 13-01-00612.
References:
[1] Dehman, A., Ambroise, C. and Neuvial, P. Performance of a blockwise ap-proach in variable selection using linkage disequilibrium information, BMC Bioin-formatics (2015) 16:148.
48 ABSTRACTS
Dustin Kremer, University of Siegen
Operator-stable and operator-self-similar random �elds
Joint work with Hans-Peter Sche�er
There is an increasing interest in so called operator-self-similar random �elds dueto applications and as they provide practical dependence structures expressed bylinear operators. Underlining the theoretical importance of this concept, we willshow that (under mild assumptions) any random �eld is operator-self-similar ifand only if its domain of attraction (which we shall de�ne) is not empty. On theother hand, operator-stable distributions seem to be an attractive and - in thecontext of multivariate limit theory - appropriate choice for the correspondingmarginals. Combining both ideas, we will give comprehensive examples of such�elds in terms of a harmonic and a moving-average representation, respectively,as proposed in [1] using stochastic integral representations and homogeneousfunctions (see [2]).
References:
[1] Yi, L. and Xiao, Y. (2011): Multivariate Operator-Self-Similar RandomFields, Stochastic Processes and their Applications, 121, 1178�1200.
[2] Sche�er, H.-P., Meerschaert, M. M. and Biermé, H. (2007): MultivariateOperator Scaling Stable Random Fields, Stochastic Processes and their Appli-cations, 117, 312�332.
49
Matthias Neumann, Ulm University
Statistical modeling of three-phase structures with completely connected
phases
Joint work with Jakub Stan¥k, Omar Pecho, Lorenz Holzer, Viktor Bene², VolkerSchmidt
In this talk a new parametric model for three-phase structures in Rd, d ≥ 3 ispresented, where each phase is completely connected with probability 1. In orderto obtain complete connectivity in the model, it is based on three connectedrandom graphs, so-called beta-skeletons, see [2], as backbones of the threephases. Each phase is then given by those points in Rd which are closer (withrespect to the Euclidean distance) to the corresponding graph than to the othertwo graphs. For this model a relationship between the expected total edgelengths of the three beta-skeletons per unit volume and the volume fractions ofthe phases is established by a simulation study for d = 3. Furthermore, we derivea formula for the expected total edge length of a beta-skeleton per unit volumein arbitrary dimensions, see also [1] for d = 2. This makes it possible to usethe model for statistical description of three phase structures with completelyconnected phases, where the volume fractions can be adjusted by the modelparameters.
For �tting the model to three-dimensional tomographic data from materialsscience we suggest parameter estimation by minimizing the di�erence betweenrelevant microstructural characteristics of experimental tomographic data andsimulated structures with the Nelder-Mead method, see [3]. A generalizationof the model has shown its potential to describe image data of a three-phasemicrostructure occurring in solid oxide fuel cells.
References:
[1] Aldous D. J. and Shun J. (2010): Connected spatial networks over randompoints and a route-length statistic. Statistical Science, 25, 275�288.
[2] Hirsch C., Neuhäuser D., and Schmidt V. (2013): Connectivity of randomgeometric graphs related to minimal spanning forests. Advances in AppliedProbability, 45, 289�298.
[3] Nelder J. A. and Mead R. (1965): A simplex method for function minimiza-tion. Computer Journal, 7, 308�313.
50 ABSTRACTS
Nataliya V. Prudnikova, Moscow State University
Endo-Kroto operation on the space of fullerenes
The talk is based on the joint work with V.M. Buchstaber.
A mathematical fullerene is a simple closed convex three-dimensional polytopewith only pentagonal and hexagonal faces. We consider the space of combinato-rial types of fullerenes F =
⋃Fn where n is the number of hexagons. This paper
deals with Endo-Kroto operation (E-K). The fullerene is named admissible if hehas a patch consisting of two non-adjacent pentagons joined by a hexagon. In-admissible fullerene is a �xed point of the operator E-K. Endo-Kroto operationtransfers an admissible fullerene into admissible and increases the number ofhexagons by 1. Thus we have an in�nite orbit. The fullerene without hexagonsF0 - dodecahedron - is a �xed point E-K. There is no fullerene with one hexagon,and there is only one fullerene with two hexagons; it is called a Barrel. Startingfrom Barrel and applying the operator E-K it is possible to obtain all the pointsof F4 and F5. Also, there are fullerenes, which can not be received by applyinga sequence of Endo-Kroto operations - IPR-fullerenes, where none of the pen-tagons have common edge. The most well-known of these, C60 is inadmissible.Moreover, it can be veri�ed that if the fullerene has a symmetry group and weapply the operator E-K on the patch and on its image under symmetry, then wewill receive the fullerenes with the same combinatorial type.
The paper describes the results of investigation of applying the Endo-Krotooperation on the space of fullerenes.
The work is supported by the RNF grant 14-11-00414.
51
Stefan Roth, Ulm University
An Inverse Problem for In�nitely Divisible Random Fields with Integral
Representation
Joint work with Evgeny Spodarev
Given n observations X(t1), . . . , X(tn) of a d-dimensional, stationary in�nitelydivisible random �eld X = {X(t) =
∫Rd f(x − t)Λ(dx)}, we provide three
nonparametric estimation approaches for the Lévy characteristics of the inde-pendently scattered random measure Λ. Existence and uniqueness of a solutionfor this inverse problem will be discussed in the case of a simple function f usinga one to one correspondence between the Lévy triplets of X and Λ. We alsogive L2-error bounds for the estimated Lévy measures.
References:
[1] Balram S. Rajput and Jan Rosinski. Spectral representations of in�nitelydivisible processes. Probab. Th. Rel. Fields, (82):451�487, 1989.
[2] J. Rosinski. Bilinear random integrals. Dissertationes Mathematicae CCLIX,1987.
[3] M. H. Neumann and M. Reiÿ. Nonparametric estimation for Lévy processesfrom low-frequency observations. Bernoulli, 15(1):223�248, 02 2009.
[4] S. Gugushvili et al. Nonparametric inference for discretely sampled Lévyprocesses. In Annales de l' Institut Henri Poincaré, Probabilités et Statistiques,volume 48, pages 282�307. Institut Henri Poincaré, 2012.
[5] F. Comte and V. Genon-Catalot. Nonparametric estimation for pure jumpLévy processes based on high frequency data. Stochastic Processes and theirApplications, 119(12):4088�4123, 2009.
52 ABSTRACTS
Vsevolod Salnikov, Lomonosov Moscow State University
Probabilistic properties of topologies of �nite metric spaces' minimal �ll-
ings and rectilinear Steiner minimal trees
Both Steiner minimal tree problem and minimal �lling problem [1] are proved tobe NP-hard. That is why even approximate solutions are of a great interest. Herewe want to divide the task into two di�erent parts: �rst to �nd an underlyingtree structure or so-called topology � how nodes are connected and then to�nd a particular representative of the topology which solves the problem. Thelater part is comparably fast for both tasks, meaning that there are e�cientalgorithms, but for the �rst part it is not the case. That's why we want tointroduce a probability measure on the space of topologies, which favorites suchtopologies, that are likely to contain an optimal solution.
One of the main contributions of this work is that we introduce such a proba-bility measure for minimal �llings of �nite additive metric spaces as well as analgorithm for its computation. We also perform various numerical experimentsto check the consistency of this measure and make some analytical computationsof asymptotic behavior for families of graph structures.
Another important contribution is that we start a discussion of probabilisticproperties for the Steiner problem itself, in particular for the Manhattan metricsand get analytical results for the small amount of terminals (nodes for whichthe Steiner problem is considered).
References:
[1] Alexander O. Ivanov, Alexey A. Tuzhilin (2013): Gromov minimal �llingsfor �nite metric spaces, Publications de l' Institut Mathematique. 01/2013;94(108). DOI: 10.2298/PIM1308003I
53
Artem M. Savchuk, Moscow State University
Uniform stability for the inverse Sturm-Liouville problem. Recovering of
a potential from a �nite set of spectral data
Joint work with A.A. Shkalikov.
We deal with two classical inverse problems for the Sturm-Liouville operator
Ly = −y′′ + q(x)y, x ∈ [0, π], q ∈W−12 [0, π].
The �rst one was originated by Borg. The problem is to recover a potential
q(x)⇐= {λn}∞1 ∪ {µn}∞1
by given two spectra of this operator with Dirichet-Neumann and Dirichletboundary conditions. The second inverse problem was launched by Gelfand,Levitan and Marchenko. This problem is to recover a potential q(x) by spectralfunction of operator L with Direchlet boundary conditions. In our case x ∈ [0, π]the spectral function is fully determined by eigenvalues and norming constants.
Given only �nitely many, say 2N , eigenvalues (or N eigenvalues and N normingconstants) do not permit to recover the potential q explicitly. However, wecan associate with such a �nite set of spectral data the function q2N which wecall the 2N -approximation of the potential q. Our main result can be roughlyformulated as follows:
‖q − qN‖κ−1 6 CNκ−θ,
provided that κ ∈ (0, θ) and we a priori know that q ∈W θ2 .
54 ABSTRACTS
Georgii Sechkin, Moscow State University
Topology of dynamics of a nonhomogeneous rotationally symmetric el-
lipsoid on a smooth plane
Let us consider an ellipsoid of revolution moving on a smooth horizontal planeunder the action of gravity. We construct topological invariants for this systemand classify corresponding Liouville foliations up to Liouville equivalence. Twosystems are called equivalent if they have the same closure of integral trajectoriesof systems solutions. Suppose that the mass distribution in the ellipsoid is suchthat it has an axis of dynamical symmetry coinciding with the axis of geometricsymmetry. Moments of inertia about principal axes of inertia perpendicular tosymmetry axis are equal to each other. We also assume that the center ofmass lies on this symmetry axis (as in the Lagrange top) at distance s from thegeometric center of the body.
A free rigid body has six degrees of freedom. We need three coordinates todescribe the position of an arbitrary point in the body (e.g., the center of mass)with respect to a �xed space frame, and three more coordinates to describe theorientation of principal axes.
In our case, there is one holonomic constraint: the height of the center of massabove the plane is determined by the orientation of principal axes. Thus, thenumber of degrees of freedom is reduced to �ve. Let us write the equationin Euler's form using f ′ = {f ;H}, where H is the Hamiltonian, and {,} isthe Poisson bracket on e(3)∗. Then in standard (S, R) coordinates we get the
following �rst integrals: H = 12
∑3i=1
S2i
Ai+ U , where U is the potential energy
and A is a constant, and K = S3.Using the Fomenko-Zieshang invariants, we prove the following result.
Theorem. The Liouville foliation associated with the above-described prob-lem can be embedded in the foliation corresponding to the Zhukovsky systemdescribing a heavy gyrostat.
Note that N. E. Zhukovsky (1899) found a generalization of Euler's integrable
case, with Hamiltonian H = 12
∑3i=1
(Si+λi)2
Ai. The additional integral is the
same as in the Euler's case: K = S21 + S2
2 + S23 .
55
Georgiy Shevchenko, Taras Shevchenko National University of Kyiv
Small ball probabilities and representations with respect to Gaussian pro-
cesses
The talk will be devoted to stochastic integral representations of the form
ξ =
∫ T
0
ψ(s)dX(s), (1)
where φ is an adapted process, and X is a Hölder continuous process of orderµ > 1/2. Studying such representations is motived by applications in �nan-cial mathematics, where φ plays a role of a risky component of self-�nancingportfolio.
A su�cient condition for representation (1) will be formulated in terms of theHölder exponent γ and small ball exponents for X. In view of this, some newresults on small ball probabilities for Gaussian processes will be given.
56 ABSTRACTS
Iurii N. Shteinikov, Moscow State University
On the divisibility of Fermat quotients
For a prime p and an integer a with gcd(a, p) = 1 the Fermat quotient qp(a) isde�ned as qp(a) = ap−1−1
p . For p let lp be the smallest a for which p does notdivide qp(a). I will consider the tasks of estimating lp for
1) all primes p,
2) all p, with the exception of primes from a set with relative zero density.
The recent estimates are based on results about the distribution of elements ofmultiplicative subgroups of residue rings and exponential sums.
This work is supported by the RSF grant 14-11-00433.
57
Nikolai A. Slepov, Moscow State University
Modi�cation of the MDR method related to generalized penalty function
We are interested in models described by a binary response variable Y (e.g.,indicating the state of a patient's health) depending on huge number of factorsX = (X1, . . . , Xn) with values in a �nite set. The problem is to �nd the mostsigni�cant collection of factors and an appropriate function F depending onthese factors to predict Y . The accuracy of such prediction is characterized byan error functional Err(F ) involving some penalty function, see [1].
To �nd the signi�cant (in a sense) factors, we use a modi�cation of the mul-tifactor dimensionality reduction (MDR) method introduced in [2]. Namely,following [1] we employ a penalty function to evaluate Err(F ). However, weuse prediction algorithms such that one has to specify various thresholds forP(Y = 1|Xα = xα) where Xα = (Xi1 , . . . , Xir ) and 1 ≤ i1 < . . . < ir ≤ n.The advantages of this approach are demonstrated.
This work is partially supported by RFBR grant 13-01-00612.
References:
[1] Bulinski A.V. To the foundation of the dimensionality reduction method forexplanatory variables. Probability and statistics. Part 18, Zap. Nauchn. Sem.POMI, 408, POMI, St. Petersburg, 84�101 (2012).
[2] Ritchie M.D., Hahn L.W., Roodi N., Bailey L.R., Dupont W.D., Parl F.F.,Moore J.H. Multifactor-dimensionality reduction reveals high-order interactionsamong estrogen-metabolism genes in sporadic breast cancer. Am. J. HumGenet. 69(1), 138-147 (2001).
58 ABSTRACTS
Christian Steck, Ulm University
Models of curves and quotient singularities
This talk is devoted to resolution of tame cyclic quotient singularities on integralNoetherian schemes. In dimension 2 such singularities can be resolved by a pro-cedure resembling the classical Hirzebruch-Jung resolution on complex surfaces.The methods are then applied to compute regular models of curves over local�elds.
59
Mikhail A. Tuzhilin, Moscow State University
Relation between the circular molecule and 4-dimensional singularity in
the saddle-saddle case
Many dynamical systems are integrable Hamilton systems with two degrees offreedom, for example, the Lagrange top or the humidistat in the gravity �eld.Two integrals of such systems generate Momentum map. Classical Liouvilletheorem describes the foliation on trajectories in preimages of su�ciently smallneighborhoods of regular values of the Momentum map. For singular valuesthe situation is much more di�cult. A.T. Fomenko has suggested an e�ectivemethod to describe the boundary of a singular value neighborhood preimage bymeans of so-called marked circular molecule. The next step is to investigatewhen such boundary determine completely the foliation of whole preimage.
It is well-known that this situation holds for the singular values of types center-saddle, center-center and focus-focus, but in the saddle-saddle case it is nottrue. To demonstrate this, Grabezhnoy constructed two examples of di�erentsingularities with coincided marked circular molecules. These two examples thereare the only ones known nowadays. In our talk we present an in�nite seriesof di�erent singularities with coincided circular molecules. Besides that, wedescribe some new relations between the circular molecule and the correspondingfoliation.
I would like to thank academision A.T. Fomenko for stating the problem andpermanent attention to my work.
List of Participants
Alkadour, Abdulkahar, Ulm [email protected]
Alonso-Ruiz, Patricia, Ulm [email protected]
Arslan, Hülya, Ulm [email protected]
Bartholomäus, Lukas, Ulm [email protected]
del Barrio, Eustasio, University of [email protected]
Bitter, Annemarie, Ulm [email protected]
Brandes, Dirk, Ulm [email protected]
Brereton, Timothy, Ulm [email protected]
Bruhn-Fujimoto, Henning, Ulm [email protected]
Buchstaber, Victor, Moscow State University, Steklov Mathematical [email protected]
Bulinski, Alexander, Moscow State [email protected]
61
62 LIST OF PARTICIPANTS
Curato, Imma, Ulm [email protected]
Dall'Acqua, Anna, Ulm [email protected]
Deckelnick, Klaus, University of [email protected]
Dobler, Dennis, Ulm [email protected]
Drapatz, Martin, Ulm [email protected]
Erokhovets, Nikolay, Moscow State [email protected]
Goldbach, Johannes, University of [email protected]
Gugushvili, Shota, Leiden [email protected]
Harder, Michael, Ulm [email protected]
Janssen, Arnold, University of Dü[email protected]
Jolany, Hassan, University of Lille1, Princeton [email protected]
Kampf, Jürgen, Ulm [email protected]
Kashin, Boris, Moscow State [email protected]
Kosov, Egor, Moscow State [email protected]
Kozhevin, Alexey, Moscow State [email protected]
63
Kremer, Dustin, University of [email protected]
Lebiedz, Dirk, Ulm [email protected]
Lindner, Alexander, Ulm [email protected]
Nerlich, Alexander, Ulm [email protected]
Neumann, Matthias, Ulm [email protected]
Olszewski, Judith, Ulm [email protected]
Pan, Lei, Ulm [email protected]
Pauly, Markus, Ulm [email protected]
Prudnikova, Nataliya, Moscow State [email protected]
Roth, Stefan, Ulm [email protected]
Salnikov, Vsevolod, University of [email protected]
Savchuk, Artem, Moscow State [email protected]
Sche�er, Hans-Peter, University of Siegensche�[email protected]
Schmidt, Volker, Ulm [email protected]
Sechkin, Georgii, Moscow State [email protected]
64 LIST OF PARTICIPANTS
Shevchenko, Georgiy, University of [email protected]
Shiryaev, Albert, Moscow State [email protected]
Shteinikov, Iurii, Moscow State [email protected]
Slepov, Nikolai, Moscow State [email protected]
Spener, Adrian, Ulm [email protected]
Spodarev, Evgeny, Ulm [email protected]
Stadtmüller, Ulrich, Ulm [email protected]
Steck, Christian, Ulm [email protected]
Stelzer, Robert, Ulm [email protected]
Stepin, Stanislav, Moscow State [email protected]
Sto�ers, Frederic, Ulm Universityfrederic.sto�[email protected]
Tolozova, Tatyana, Moscow State [email protected]
Tuzhilin, Alexey, Moscow State [email protected]
Tuzhilin, Mikhail, Moscow State [email protected]
Zacher, Rico, Ulm [email protected]
