Joint Mathematical Conference CSASC 2010iti.mff.cuni.cz › series › 2010 › 486.pdf · Austrian Mathematical Society in Vienna in November 2008. And nally CSASC 2010 is a full

Joint Mathematical Conference CSASC 2010(XI th Conference of Czech Mathematicians)

Prague, Czech Republic, January 22-27, 2010

organized by Czech Mathematical Society together withAustrian Mathematical Society, Catalan Mathematical Society,

Slovak Mathematical Society, and Slovenian Mathematical Society

Josef Cibulka, Bernard Lidický, Marek Tesař (eds.)

Preface

Welcome to Prague!

Dear participants of CSASC 2010 - the Joint Mathematics Conference co-organized by Catalan, Slovenian, Austrian, Slovak and Czech MathematicalSocieties. Welcome to Prague and to the first joint conference of this type inCentral Europe. The idea to organize such a conference originated severalyears ago and we see the upcoming event as a culmination of long lastingefforts.

The format of several parallel sessions each presenting its topic to all par-ticipants via a plenary talk is typical for joint meetings, and in the Euro-pean setting was propagated by the annual EMS Mathematical Weekends,one of which took part in Prague in 2004. Following the same pattern theCzech and Catalan Mathematical Societies organized their joint meetingsin 2005 and 2006 (the former in Prague and the latter in Barcelona). TheSlovak and Austrian Mathematical Societies meet in a similar way regu-larly already for several years. The idea of coordinating the efforts of moresociaties was discussed at the second Czech-Catalan meeting and during amemorable Slovak-Austrian conference in High Tatras in 2007. A DiscreteMathematics conference co-organized by our five societies was hosted by theAustrian Mathematical Society in Vienna in November 2008. And finallyCSASC 2010 is a full scale mathematics meeting covering topics rangingfrom Mathematical Physics through Algebra, Geometry, Analysis, Dyna-mical Systems, Statistics, and Topology to Discrete Mathematics, Security,and Theoretical Computer Science.

The choice of topics is not random. Every session is organized based onexisting strong collaboration among research groups in our societies. Strongresearch is not possible without international collaboration, yet it is pleasingto see all these links of joint research and friendship among our societies.

From the local mathematical society point of view this is the 11th Quadri-annual Conference of Czech Mathematicians which traditionally concludesa four year period of its executive committee. We would like to use this op-portunity to thank all its members for all the work they have done for oursociety in the past four years. Many of them are also session organizers orotherwise contributed to the organization of the conference. The thanks ofcourse extend to all the session organizers, and namely to the presidents of

2

the partner societies who helped so much to promote our joint idea amongtheir members.

The organzation at affordable costs has been made possible, apart fromthe contributions of the co-organizing societies, by generosity of variousresearch grants administered by the Czech session organizers, the Facultyof Mathematics and Physics of the Charles University in Prague which ishosting most of the sessions and all plenary talks, the Czech Technical Uni-versity who is hosting the session of Discrete Dynamical Systems, and theDIMATIA Center of Discrete Mathematics and Theoretical ComputerScience which is providing most of the logistics. We cordially thank all ofthem.

We wish all of you a pleasant and fruitful time in Prague and already lookforward to a next joint meeting, wherever and whenever should it be

Jiří Fiala, Jan Kratochvílfor the local organizers

3

Special sessions

Differential Geometry and Mathematical PhysicsXavier Gracia (Barcelona)Olga Krupková (Olomouc)

Discrete Algorithms and Computational ComplexityJan Kratochvíl (Prague)Oriol Serra (Barcelona)

Discrete Dynamical SystemsJosef Bobok (Prague)Armengol Gasull (Barcelona)

Enumerative and Analytic CombinatoricsMichael Drmota (Vienna)Martin Klazar (Prague)

Function Spaces and ApplicationsLuboš Pick (Prague)

Mathematical PhysicsPavel Exner (Prague)Gerald Teschl (Vienna)

Mathematics of Secret SharingFrantišek Matúš (Prague)Carles Padro (Barcelona)

Stochastic AnalysisBohdan Maslowski (Prague)Marta Sanz-Sole (Barcelona)

Topological, Geometric and Algebraic Graph TheoryRoman Nedela (Banska Bystrica)Tomaz Pisanski (Ljubljana)Pavel Valtr (Prague)

Triangulated CategoriesCarles Casacuberta (Barcelona)Jiří Rosický (Brno)

4

ProgramFriday January 22: Invited and contributed talks in parallel sessions

9:00-10:30 Enumerative and Analytic Combinatorics (S3, from 10:00)Discrete Algorithms and Computational Complexity (S4)Stochastic Analysis (S5)Function Spaces and Applications (S6)Discrete Dynamical Systems (ČVUT, Thákurova 7)

10:30-11:00 Coffee break11:00-12:30 Enumerative and Analytic Combinatorics (S3, from 10:00)

Discrete Algorithms and Computational Complexity (S4)Stochastic Analysis (S5)Function Spaces and Applications (S6)Discrete Dynamical Systems (ČVUT, Thákurova 7)Mathematical Physics (S8)

12:30-14:00 Lunch break14:00-15:30 Enumerative and Analytic Combinatorics (S3)

Differential Geometry and Mathematical Physics (S4)Stochastic Analysis (S5)Mathematical Physics (S8)Discrete Dynamical Systems (ČVUT, Thákurova 7)

15:30-16:00 Coffee break16:00-17:30 Enumerative and Analytic Combinatorics (S3)

Differential Geometry and Mathematical Physics (S4)Stochastic Analysis (S5)Discrete Dynamical Systems (ČVUT, Thákurova 7)

18:00-20:00 Business meeting of CMS

5

Saturday January 23: Plenary talks, all in S5

9:00-10:00 Marta Sanz-Solé (University of Barcelona)Hitting probabilities for solutions to stochastic partialdifferential equations

10:00-10:30 Coffee break10:30-11:30 Ilse Fischer (Vienna University)

Refined enumerations of alternating sign matrices11:30-12:00 Break12:00-13:00 Bohumír Opic (Academy of Sciences, Prague)

Embeddings of Bessel-potential-type spaces into generalizedHölder spacesinvolving k-modulus of smoothness

13:00-15:00 Lunch break15:00-16:00 Joan Porti (Universitat Autonoma de Barcelona)

Ricci flow and geometrization of three manifolds16:00-16:30 Coffee break16:30-17:30 Jiří Matoušek (Charles University, Prague)

Hardness of embedding simplicial complexes in Rd17:30-17:50 Break

17:50 Formal opening18:00-19:00 Jakob Yngvason (Schroedinger Institute in Vienna)

Quantum gases in fast rotation and vortices19:00-21:00 Banquet (down in the cafeteria)

Sunday January 24: Plenary talks, all in S5

13:00-14:30 The prize session14:30-15:00 Break15:00-16:00 Fernando Muro (University of Seville)

Representability of cohomology theories16:00-16:30 Break16:30-17:30 Ľubomír Snoha (Matej Bel University, Banska Bystrica)

Minimal sets in discrete dynamics - results, tools,open problems

17:30-18:00 Break18:00-19:00 Aleksander Malnič (University of Ljubljana)

Covering space techniques in graph theory

6

Monday January 25: Invited and contributed talks in parallel sessions

9:00-10:30 Topological, Geometric and Algebraic Graph Theory (S3)Differential Geometry and Mathematical Physics (S4)Triangulated Categories (S5)Mathematics of Secret Sharing (S6)Discrete Dynamical Systems (ČVUT, Thákurova 7)

10:30-11:00 Coffee break11:00-12:30 Topological, Geometric and Algebraic Graph Theory (S3)

Differential Geometry and Mathematical Physics (S4,till 13:05)

Triangulated Categories (S5)Mathematics of Secret Sharing (S6)Discrete Dynamical Systems (ČVUT, Thákurova 7)

12:30-14:00 Lunch break14:00-15:30 Topological, Geometric and Algebraic Graph Theory (S3)

Discrete Algorithms and Computational Complexity (S4)Triangulated Categories (S5)

15:30-16:00 Coffee break16:00-17:30 Topological, Geometric and Algebraic Graph Theory (S3)

Discrete Algorithms and Computational Complexity (S4)Triangulated Categories (S5)

7

Index

Plenary lecturesHitting probabilities for solutions to stochastic partialdifferential equationsMarta Sanz-Solé (Universitat de Barcelona) . . . . . . . . 16

Refined enumerations of alternating sign matricesIlse Fischer (University of Vienna) . . . . . . . . . . . . 16

Embeddings of Bessel-potential-type spaces intogeneralized Hölder spaces involving k-modulus ofsmoothnessBohumír Opic (Academy of Sciences, Prague) . . . . . . . 17

Ricci flow and geometrization of three manifoldsJoan Porti (Universitat Autonoma de Barcelona) . . . . . . 17

Hardness of embedding simplicial complexes in RdJiří Matoušek (Charles University, Prague) . . . . . . . . 18

Quantum gases in fast rotation and vorticesJakob Yngvason (University of Vienna) . . . . . . . . . . 18

Representability of cohomology theoriesFernando Muro (University of Seville) . . . . . . . . . . . 19

Minimal sets in discrete dynamics – results, tools, openproblemsĽubomír Snoha (Matej Bel University, Banská Bystrica) . . . 19

Covering space techniques in graph theoryAleksander Malnič (University of Ljubljana) . . . . . . . . 20

Invited and contributed lectures

Differential Geometry and Mathematical PhysicsThe Cartan form and its generalizations in the calculus ofvariationsDemeter Krupka (Masaryk University, Brno) . . . . . . . . 21

8

Symplectic and Poisson geometry of b-manifoldsEva Miranda (Universitat Politecnica de Catalunya) . . . . 21

Order reduction of the Euler-Lagrange equations of higherorder invariant variational problems on frame bundlesJán Brajerčík (University of Prešov) . . . . . . . . . . . 22

Some generalizations of the notion of Lie algebraJose Antonio Vallejo Rodriguez (State University of SanLuis Potosi (Mexico)) . . . . . . . . . . . . . . . . . . 22

Example of nonholonomic constraint of the second orderMartin Swaczyna (University of Ostrava) . . . . . . . . . 23

The nonholonomic variational principleOlga Krupková (Palacký University, Olomouc) . . . . . . . 23

Constraints in k-presymplectic field theoryXavier Gràcia (Technical University of Catalonia,Barcelona) . . . . . . . . . . . . . . . . . . . . . . . 23

Generalized Stäckel transform and its applications tointegrable systemsArtur Sergyeyev (Silesian University, Opava) . . . . . . . . 24

Maxwell equations with periodic coefficients in 1 + 1dimensions for an optical applicationVlasta Peřinová (Palacký University, Olomouc) . . . . . . . 25

Helmholtz conditions and their generalizationsRadka Malíková (University of Ostrava) . . . . . . . . . . 25

Asymptotic zero distributions of polynomials of the HeunequationMiloš Tater (Academy of Sciences, Řež) . . . . . . . . . . 26

Discrete Algorithms and Computational ComplexityGeneralizations of interval graphs motivated by theconstraint satisfaction problemPavol Hell (Simon Fraser University, Vancouver) . . . . . . 27

On efective representation of groups, monoids and posetsJaroslav Nešetřil (Charles University, Prague) . . . . . . . 27

9

Bounds on the k-th chromatic number of generalized starcoloringsGuillem Perarnau (Universitat Politecnica de Catalunya) . . 28

Complexity of the distance constrained labeling problemfor treesJiří Fiala (Charles University, Prague) . . . . . . . . . . 28

The k-in-a-path problem for claw-free graphsBernard Lidický (Charles University, Prague) . . . . . . . 28

On 3-Colorability of Pseudo-TriangulationsClemens Huemer (Universitat Politecnica de Catalunya) . . . 29

Unified Approach to Polynomial Algorithms on Graphs ofBounded (bi-)Rank-widthPetr Hliněný (Masaryk University, Brno) . . . . . . . . . 29

Multivariate complexity analysis of some voting problemsRolf Niedermeier (Friedrich-Schiller-Universität Jena) . . . . 30

What Makes Equitable Connected Partition EasyOndra Suchý (Charles University, Prague) . . . . . . . . . 30

Making graph minor theory constructiveIsolde Adler (Goethe University, Frankfurt am Main) . . . . 31

Testing Planarity of Partially Embedded GraphsJan Kratochvíl (Charles University, Prague) . . . . . . . . 32

Enumeration of perfect matchingsMartin Loebl (Charles University, Prague) . . . . . . . . . 33

Discrete Dynamical SystemsGeneralization of Sturmian WordsLubomíra Balková (Czech Technical University, Prague) . . . 34

On two and three periodic Lyness difference equationsArmengol Gasull (Universitat Autonoma de Barcelona) . . . 34

10

Global linearization of periodic recurrencesFrancesc Manosas (Universitat Autonoma de Barcelona) . . 35

About the connectivity of the escaping set for entiretranscendental mapsXavier Jarque (Universitat Rovira i Virgili (Tarragona)) . . . 35

The Lotka-Volterra map and some problems of the numbertheory.Peter Maličký (Matej Bel University, Banská Bystrica) . . . 36

Extensions without increasing the entropy as a tool forentropy bounds problemMatúš Dirbák (Matej Bel University, Banská Bystrica) . . . 36

On nowhere differentiable measure preserving mapsJozef Bobok (Czech Technical University, Prague) . . . . . 36

A strongly F -invariant pinched core strip that does notcontain any arc of curveLluis Alseda i Soler (Universitat Autonoma de Barcelona) . . 37

Global dynamics of discrete systems through LieSymmetriesAnna Cima Mollet (Universitat Autonoma de Barcelona) . . 37

Enumerative and Analytic CombinatoricsThe poset of bipartitionsChristian Krattenthaler (University of Vienna) . . . . . . . 38

Families of permutations with bounded VC-dimension havealmost exponential sizeJosef Cibulka (Charles University, Prague) . . . . . . . . . 38

Generalized Thue-Morse Sequences of SquaresJohannes Morgenbesser (Vienna University of Technology) . . 39

Convergence properties of q-binomial distributionsMartin Zeiner (TU Graz) . . . . . . . . . . . . . . . . 39

Parameters of integer partitions - a zoo of limit lawsStephan Wagner (Stellenbosch University) . . . . . . . . . 40

11

The degree distribution in Random Unlabelled SubcriticalGraph familiesVeronika Kraus (TU Wien) . . . . . . . . . . . . . . . 40

The degree of the nodes in random increasing k-treesGeorg Seitz (Vienna University of Technology) . . . . . . . 41

Function Spaces and ApplicationsGehring type results for generalized (Navier-)Stokes systemin 2D.Petr Kaplický (Charles University, Prague) . . . . . . . . 42

The mean value property of harmonic functions and theHeat equationYitzhak Weit (University of Haifa) . . . . . . . . . . . . 42

A note on bilinear multipliersSalvador Rodriguez (Universitat de Barcelona) . . . . . . . 42

Calderon-type interpolation theorems and applications tosharp Sobolev embeddingsGogatishvili Amiran (Academy of Sciences, Prague) . . . . . 43

Mathematical PhysicsDensity-Matrix Functionals: Asymptotic Behavior of theEnergyHeinz Siedentop (Ludwig-Maximilians-UniversitätMünchen) . . . . . . . . . . . . . . . . . . . . . . . 44

The Cauchy problem for the KdV equation with steplikefinite-gap initial dataEgorova Iryna (B. Verkin Institute for Low TemperaturePhysics, Kharkiv) . . . . . . . . . . . . . . . . . . . . 44

Long–time asymptotics for the Korteweg-de Vries equationKatrin Grunert (University of Vienna) . . . . . . . . . . 45

The Hardy inequality and the asymptotic behaviour of theheat equation in twisted domainsDavid Krejčiřík (Academy of Sciences, Prague) . . . . . . . 45

12

Spectrum of the Aharonov-Bohm Hamiltonian withconstant magnetic fields in the hyperbolic planeTakuya Mine (Kyoto Institute of Technology) . . . . . . . 46

1-D Schrödinger operators with local point interactionsOleksiy Kostenko (Dublin Institute of Technology) . . . . . 46

Mathematics of Secret SharingIdeal Hierarchical Secret Sharing SchemesCarles Padró (Universitat Politecnica de Catalunya) . . . . 48

Asymptotically good linear secret sharing with strongmultiplication over any finite fieldIgnacio Cascudo (University of Oviedo, Spain) . . . . . . . 49

Secret sharing on infinite structuresLászló Csirmaz (Central European University, Budapest) . . 49

Ideal secret sharing, matroids and relative entropyFrantišek Matúš (Academy of Sciences, Prague) . . . . . . 50

Stochastic AnalysisStein’s method and Malliavin calculus of PoissonfunctionalsFrederic Utzet (Universitat Autonoma de Barcelona) . . . . 51

Malliavin Calculus for Lévy Processes via DerivativeOperatorsJosep Lluís Solé i Clivillés (Universitat Autonoma deBarcelona) . . . . . . . . . . . . . . . . . . . . . . . 52

Stochastic bilinear differential equation with fractionalnoise in infinite dimensionJana Šnupárková (Charles University, Prague) . . . . . . . 52

SPDEs of reaction-diffusion type driven by Levy noiseErika Hausenblas (University of Salzburg) . . . . . . . . . 52

Moment inequalities for stochastic integrals in 2-smoothBanach spacesJan Seidler (Academy of Sciences, Prague) . . . . . . . . 53

Stochastic wave equationsMartin Ondreját (Academy of Sciences, Prague) . . . . . . 53

13

Delay differential equations with hereditary drift driven byfractional Brownian motion with Hurst parameter H > 1/2Carles Rovira Escofet (Universitat de Barcelona) . . . . . . 54

Relationships between fractional, bifractional andsubfractional Brownian motionsXavier Bardina (Universitat Autonoma de Barcelona) . . . . 54

Statistical inference for SPDEs driven by fractional noiseJan Pospíšil (University of West Bohemia, Pilsen) . . . . . 54

Stochastic porous medium equation with fractionalBrownian motionJan Bártek (Charles University, Prague) . . . . . . . . . . 55

Ornstein-Uhlenbeck bridgeJosef Janák (Charles University, Prague) . . . . . . . . . 55

Reversible Reactions in Reaction-Diffusion StochasticSimulation AlgorithmsJana Lipková (Charles University, Prague) . . . . . . . . . 56

LQ control for FBM-driven SPDEsBohdan Maslowski (Charles University, Prague) . . . . . . 56

Topological, Geometric and Algebraic Graph TheoryOn almost distance-regular graphsCristina Dalfó Simó (Universitat Politecnica deCatalunya) . . . . . . . . . . . . . . . . . . . . . . . 57

Applications of symmetries of combinatorial mapsTomaž Pisanski (University of Ljubljana) . . . . . . . . . 58

Vertex-transitive maps with Schläfli type 3,7Daniel Pellcer (University of Ljubljana) . . . . . . . . . . 58

Nowhere-zero flows in Cartesian bundles of graphsEdita Rollová (Comenius University, Bratislava) . . . . . . 59

Fundamental cycles and the maximum genus of a graphMichal Kotrbčík (Comenius University, Bratislava) . . . . . 59

Construction of Hamilton cycles in (2,s,3)-Cayley graphsKlavdija Kutnar (University of Primorska) . . . . . . . . . 60

14

Pseudo-distance-regular graphs around a set and the localspectrum of its subconstituentsMarc Cámara Vallejo (Universitat Politecnica deCatalunya) . . . . . . . . . . . . . . . . . . . . . . . 60

On the ascending subgraph decomposition problem forbipartite graphsJordi Moragas (Universitat Politecnica de Catalunya) . . . . 61

On cubic biabelian graphsIstván Kovács (FAMNIT, University of Primorska) . . . . . 61

Recognizing string graphs on surfaces in NPJan Kynčl (Charles University, Prague) . . . . . . . . . . 62

Archimedean solids of higher generaJán Karabáš (Matej Bel University, Banská Bystrica) . . . . 62

Skew-morphisms of groups and regular mapsRoman Nedela (Matej Bel University, Banská Bystrica) . . . 62

Coding and Counting Arrangements of PseudolinesPavel Valtr (Charles University, Prague) . . . . . . . . . . 63

Triangulated CategoriesCombinatorial model categoriesJiří Rosický (Masaryk University, Brno) . . . . . . . . . . 64

Representability theorems for well generated triangulatedcategoriesOriol Raventos (Universitat de Barcelona) . . . . . . . . . 64

Properness in the Bousfield Localisation of ModelCategoriesGeorge Raptis (University of Osnabrück) . . . . . . . . . 65

A homotopy theory for comodules and coalgebrasAlexandru Stanculescu (Masaryk University, Brno) . . . . . 65

Decompositions of p-local spacesDamir Franetic (University of Ljubljana) . . . . . . . . . 65

Cellularization of structures in triangulated categoriesJavier J. Gutierrez (Centre de Recerca Matematica) . . . . 66

15

Singly generated ideals and coideals in monoidaltriangulated categoriesCarles Casacuberta (Universitat de Barcelona) . . . . . . . 66

Fibrations up to an equivalence and homotopy colimitsLukáš Vokřínek (Masaryk University, Brno) . . . . . . . . 67

16

Plenary lectures

Hitting probabilities for solutions to stochasticpartial differential equations

Marta Sanz-SoléUniversitat de [email protected]

We will present some results on upper and lower bounds for hitting proba-bilities of random fields in terms of Hausdorff measure and Bessel-Rieszcapacity, respectively. Applications to several examples of stochastic partialdifferential equations will be discussed.

Refined enumerations of alternating signmatricesIlse Fischer

University of [email protected]

Alternating sign matrices are one of those fascinating combinatorial objectsthat admit an exceptional simple enumeration formula while at the sametime proving this formula is rather complicated. They were first definedand studied in the early 1980s by Robbins and Rumsey in connection withDodgson’s condensation method for computing determinants. The researchwas further stimulated after the discovery of the relation to a statisticalmechanics model (six-vertex model) for “square ice”. In the talk I will firstgive an introduction into this field and then present our approach to refinedenumerations of alternating sign matrices.

17

Embeddings of Bessel-potential-type spaces intogeneralized Hölder spaces involving k-modulus

of smoothnessBohumír Opic

Academy of Sciences, [email protected]

Classical Bessel potential spaces Hσ,p(IRn) = HσLp(IRn), introduced byN. Aronszajn, K. Smith and A. P. Calderón in 1961, have played a significantrole in mathematical analysis and in applications for many years. Thesespaces are modelled upon the scale of Lebesgue spaces Lp(IRn) and theycoincide with the Sobolev spaces W k,p(IRn) = W kLp(IRn) when σ = k ∈ INand p ∈ (1,+∞). However, it has gradually become clear that to handlesome situations (especially limiting ones) a more refined tuning is desirable.

For this purpose, we replace the Lebesgue spaces Lp(IRn) by rearrange-ment invariant Banach function spaces X(IRn) and we study embeddings ofBessel potential spaces HσX(IRn) with order of smoothness σ ∈ (0, n) intogeneralized Hölder spaces defined by means of the k-modulus of smoothness(k ∈ IN). We establish necessary and sufficient conditions for such embed-dings and we present applications of our results. In particular, we obtain newand sharp embeddings of Sobolev-Orlicz spaces Wm+1Ln/m(logL)α(IRn)and WmLn/m(logL)α(IRn) into generalized Hölder spaces, which improveknown results.

The lecture is based on a joint work with Amiran Gogatishvili and Júlio S.Neves.

Ricci flow and geometrization of three manifoldsJoan Porti

Universitat Autonoma de [email protected]

We give an overview of Thurston’s geometrization conjecture, the Ricci flowintroduced by Hamilton, and Perelman’s proof.

18

Hardness of embedding simplicial complexes inRd

Jiří MatoušekCharles University, [email protected]

It is well known that one can test in linear time whether a given graph isplanar. We consider the higher-dimensional generalization of this problem:given a k-dimensional simplicial complex K and a target dimension d, doesK embed into Rd? Surprisingly, rather little seems to be known about thealgorithmic aspects of this problem. (All the relevant topological notionswill be defined and explained during the talk, at least on an intuitive level.)Known results easily imply that the problem is solvable in polynomial timeif k = d = 2 or d = 2k ≥ 6. We show that the problem is algorithmicallyundecidable for k = d − 1 and d ≥ 5. This follows from a famous result ofNovikov on the unsolvability of recognizing the 5-sphere. Our main result isNP-hardness in the range d ≥ 4 and d ≥ k ≥ (2d− 2)/3. These dimensionsfall outside the so-called metastable range of a theorem of Haefliger andWeber, which characterizes embeddability in terms of the so-called deletedproduct obstruction. Our reductions are based on examples, due to Segal,Spiez, Freeman, Krushkal, Teichner, and Skopenkov, showing that outsidethe metastable range, the deleted product obstruction is insufficient. Jointwork with Martin Tancer and Uli Wagner.

Quantum gases in fast rotation and vorticesJakob Yngvason


A survey of recent mathematical results on quantum gases in fast rotationand the appearance and disappearance of vortices will be presented.

19

Representability of cohomology theoriesFernando Muro


Brown and Adams representability theorems for cohomology theories ori-ginated in algebraic topology. They have been extended to other algebraicand geometric contexts, where they have been successfully applied (Gro-thendieck duality in algebraic geometry, Auslander-Reiten theory in repre-sentation theory, motivic cohomology?). Nevertheless, there are still manyopen questions with potentially striking applications. We will introduce theaudience to what is known and report on recent advances.

Minimal sets in discrete dynamics – results,tools, open problems

Ľubomír SnohaMatej Bel University, Banská Bystrica

[email protected]

Discrete dynamical systems given by a continuous map on a topological(usually compact metrizable) space will be considered. Minimality of sucha system/map can be defined as the density of all forward orbits. Everycompact system contains at least one minimal set, i.e., a nonempty closedinvariant subset such that the restriction of the map to this subset is mi-nimal. A fundamental question in topological dynamics is the one on thetopological structure of minimal sets of continuous maps in a given space(and, in particular, whether the space itself admits a minimal map or not).

We will present some known as well as new results on topology of minimalsets, some tools used in the proofs, and some open problems.

20

Covering space techniques in graph theoryAleksander Malnič


Graph covers emerged forty years ago in the context of maps on surfaces,appropriately adapted to reflect the discrete nature of these objects;

the primary incentive was the final solution of the long standing Heawood’sMap Colour Problem.

Much around the same time these techniques became an indispensable toolin algebraic graph theory; here, the motivation came from problems rela-ted to classification of graphs and maps in terms of their symmetries. Animportant topic in this setting is the problem of lifting automorphisms, aproblem that has long been well understood at a very general level in alge-braic topology. These general results, however, are rather inadequate if weare to successfully cope with concrete examples and specific questions thatarise when studying symmetry from a combinatorial point of view in thesetting of graphs and maps. A number of papers dealing with the lifting pro-blem have been published in the past fifteen years. In terms of content theyrange from general considerations to more concrete applications in the fieldof graphs and maps on surfaces, dealing with enumeration, classification andconstruction of infinite families of graphs with specific symmetry properties,and compiling lists of such graphs up to a certain order.

In the talk I will review some results covering this topic.

21

Differential Geometry andMathematical Physics

The Cartan form and its generalizations in thecalculus of variationsDemeter Krupka

Masaryk University, [email protected]

We discuss possible extensions of the concept of the Cartan form of classicalmechanics to higher order mechanics on manifolds, higher order field the-ory on jet bundles and to parametric variational problems on slit tangentbundles and on bundles of non-degenerate velocities. We present a genera-lization of the Cartan form, known as a Lepage form, and basic propertiesof the Lepage forms. Both earlier and recent examples of differential formsgeneralizing the Cartan form are reviewed.

Symplectic and Poisson geometry of b-manifoldsEva Miranda

Universitat Politecnica de [email protected]

In this talk we will try to show the hidden geometry (symplectic and Pois-son) in what we call b-manifolds.

These manifolds were initially considered by Nest and Tsygan while study-ing formal deformations of symplectic manifolds with boundary and also byMelrose in the context of differential calculus and differential operators ofmanifolds with boundary.

Symplectic b-manifolds lie between the symplectic and Poisson world. Inparticular, it is possible to prove local and semiglobal normal forms via b-de Rham theory. This talk is based on joink work with Victor Guilleminand Ana Rita Pires.

22

Order reduction of the Euler-Lagrangeequations of higher order invariant variational

problems on frame bundlesJán Brajerčík

University of Preš[email protected]

Let µ : FX → X be a principal bundle of frames with the structuregroup Gln(R). It is shown that the variational problem, defined by Gln(R)-invariant Lagrangian on JrFX, can be equivalently studied on the associa-ted space of connections with some compatibility condition, which gives usorder reduction of the corresponding Euler-Lagrange equations.

Some generalizations of the notion of Lie algebraJose Antonio Vallejo Rodriguez

State University of San Luis Potosi (Mexico)[email protected]

There are several ways to generalize the notion of Lie algebra. For instance,one can lift the restriction of antisymmetry (obtaining the so called Loday-or Leibniz- algebras), or view a particular Lie algebra as a member of afamily indexed by the points of a manifold (a kind of moduli space calledLie algebroid). Alan Weinstein has shown how any finite-dimensional Liealgebra can be put in correspondence with a closed maximal subspace of acertain construction called omni-Lie algebra. In the talk, we will describe away of unifying those concepts through the notion of omni-Loday algebroid,presenting some simple examples.

23

Example of nonholonomic constraint of thesecond orderMartin SwaczynaUniversity of Ostrava

[email protected]

The nonholonomic variational principleOlga Krupková

Palacký University, [email protected]

A variational principle for mechanical systems and fields subject to nonholo-nomic constraints is found, providing Chetaev-reduced equations as equati-ons for extremals. Two cases are studied: first, when the constrained systemarises from an unconstrained Lagrangian system defined in a neighbourhoodof the constraint, and second, more general, when an ”internal” constrainedsystem on the constraint manifold is given.

Constraints in k-presymplectic field theoryXavier Gràcia

Technical University of Catalonia, [email protected]

Field theories have several geometric formulations. An especially simple oneis the k-symplectic formalism, since only tangent and cotangent bundlesare needed in its description. Its defining elements have a close relationshipwith those in the symplectic formulation of mechanics. The purpose of thistalk is to show that this relationship also stands in the presymplectic case,and more particularly to show that one can mimick the presymplectic con-

24

straint algorithm to obtain a constraint algorithm that can be applied tok-presymplectic field theory.

Generalized Stäckel transform and itsapplications to integrable systems

Artur SergyeyevSilesian University, Opava

[email protected]

In this work we introduce multiparameter generalized Stäckel transform –a noncanonical transformation of a special kind relating the sets of (notnecessarily commuting) Hamiltonians and leaving the phase space coordi-nates intact. It is shown that under certain conditions the transformationin question preserves Liouville integrability, noncommutative integrabilityand superintegrability. For instance, when applied to an n-tuple of commu-ting Hamiltonians, this transformation yields a (new) n-tuple of commutingHamiltonians. Thus, one can construct new integrable systems from the oldones and also fi

nd new links among known integrable systems using the transformationunder study. In particular, we show that integrable systems associated withthe so-called k-hole deformations of the separation curve of Benenti type arerelated to the ones associated with the original separation curve of Benentitype through an appropriately chosen multiparameter generalized Stäckeltransform. The corresponding transformation for the equations of motionproves to be nothing but a reciprocal transformation of a special form, andwe briefly discuss the properties of this class of reciprocal transformations,including applications to hydrodynamic-type systems.

This is joint work with Maciej Blaszak. For more details see A. Sergyeyev, M.Blaszak, Generalized Stäckel transform and reciprocal transformations forfinite-dimensional integrable systems, J. Phys. A: Math. Theor. 41 (2008),paper 105205.

25

Maxwell equations with periodic coefficients in 1+ 1 dimensions for an optical application

Vlasta PeřinováPalacký University, [email protected]

On using the Fourier transformation, the time included in the Maxwellequations can be replaced by the frequency of a component, but a simplifi-cation does not come about, in spite of the parametric approximation, whena field in a nonlinear medium is described. We encounter spatially periodiccoefficients. We deal with a decomposition of a solution into the surfaceand volume components in the case, when the nonlinear media of differentcoefficients of nonlinearity alternate.

Helmholtz conditions and their generalizationsRadka Malíková


In the paper, properties of morphisms in the variational sequence are in-vestigated. The Euler–Lagrange morphism E1 is well-understood. It is alsoknown that the kernel of the Helmholtz morphism E2 consists of locallyvariational dynamical forms, and is characterized by Helmholtz conditions.We study the image of E2 and the kernel of the next morphism E3, andsolve the corresponding local and global inverse problem when a three-formcomes (via a variational map) from a dynamical form, i.e., corresponds to asystem of differential equations. We find identities, that are a generalizationof the Helmholtz conditions to this situation, and show that the problemis closely related to the question on existence of a closed three-form. Theobtained results extend known results on Lagrangins and locally variationaldynamical forms to general dynamical forms, and open a new possibility tostudy non-variational equations by means of closed three-forms, as a paral-lel to extremal problems (variational equations) that are studied by meansof closed two-forms (Cartan forms, symplectic geometry).

26

Asymptotic zero distributions of polynomials ofthe Heun equation

Miloš TaterAcademy of Sciences, Řež[email protected]

We solve the classical (multi)spectral problem for the Heun equation sugges-ted by E. Heine and T. Stieltjes in the 19th century. We study asymptoticbehaviour of roots of both Van Vleck and Stieltjes polynomials. Existenceof the asymptotic measures is proved and their supports are characterized.

27

Discrete Algorithms andComputational Complexity

Generalizations of interval graphs motivated bythe constraint satisfaction problem

Pavol HellSimon Fraser University, Vancouver

[email protected]

Recently, interval graphs have naturally arisen in the context of constra-int satisfaction problems. Progress in understanding what structures makeCSP’s tractable suggests generalizations and analogues of interval graphs.I will discuss obstruction characterizations and recognition algorithms forthese generalizations. Applications to the CSP dichotomy problem will alsobe mentioned. This is joint work with Tomas Feder, Jing Huang, and ArashRafiey.

On efective representation of groups, monoidsand posetsJaroslav Nešetřil

Charles University, [email protected]

We discuss the title of the lecture in the context of graph classes. Some ofthe results are joint with Jan Hubička and Patrice Ossona de Mendez.

28

Bounds on the k-th chromatic number ofgeneralized star colorings

Guillem PerarnauUniversitat Politecnica de [email protected]

A graph has star chromatic number k if it can be colored with k colors insuch a way that each color class is a stable set and the subgraph inducedby any two color classes is a forest of stars. The star chromatic number canbe generalized to χp(G), p ≥ 2, by requiring that any i ≤ p classes inducea graph with treedepth at most i. Nešetřil and Ossona de Mendez provedthat, for each p ≥ 1, χp(G) is bounded in the class of bounded expansiongraphs. By using the Local Lovász Lemma we give some specific bounds forthe subclass of graphs with maximum degree ∆.

Complexity of the distance constrained labelingproblem for trees

Jiří FialaCharles University, [email protected]

We show a full complexity dichotomy for the distance constrained labelingproblem on the class of trees.

The k-in-a-path problem for claw-free graphsBernard Lidický


Testing whether there is an induced path in a graph spanning k given ver-tices is already NP-complete in general graphs when k = 3. We show howto solve this problem in polynomial time on claw-free graphs, when k is not

29

part of the input but an arbitrarily fixed integer. This is a joint work withJiří Fiala, Marcin Kaminski and Daniel Paulusma.

On 3-Colorability of Pseudo-TriangulationsClemens Huemer


Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, e.g. triangulati-ons, polynomial time algorithms exist.

We consider the family of pseudo-triangulations, which are a generalizationof triangulations, and prove NP-completeness for this class. This also holds,if we bound their face degree to four or consider exclusively pointed pseudo-triangulations with maximum face-degree five. In contrast to these com-pleteness results we show that pointed pseudo-triangulations with maxi-mum face-degree four are always 3-colorable. An according 3-coloring canbe found in linear time.

This is a joint work with Oswin Aichholzer, Franz Aurenhammer, ThomasHackl, Alexander Pilz and Birgit Vogtenhuber.

Unified Approach to Polynomial Algorithms onGraphs of Bounded (bi-)Rank-width

Petr HliněnýMasaryk University, [email protected]

In this paper we develop new algorithmic machinery for solving hard pro-blems on graphs of bounded rank-width and on digraphs of bounded bi-rank-width in polynomial (XP, to be precise) time. These include, particularly,graph colouring and chromatic polynomial problems, the Hamiltonian pathand c-min-leaf outbranching, the directed cut, and more generally MSOL-partitioning problems on digraphs. Our focus on a formally clean and unified

30

approach for the considered algorithmic problems is in contrast with manyprevious published XP algorithms running on graphs of bounded clique-width, which mostly used ad hoc techniques and ideas. The new contributi-ons include faster algorithms for computing the chromatic number and thechromatic polynomial on graphs of bounded rank-width, and new algori-thms for solving the defective colouring, the min-leaf outbranching, and thedirected cut problems.

Joint work with Robert Ganian and Jan Obdržálek.

Multivariate complexity analysis of some votingproblems

Rolf NiedermeierFriedrich-Schiller-Universität [email protected]

We survey some recent results on the parameterized computational comple-xity of NP-hard problems in the context of voting systems. These problemsinclude Kemeny, Dodgson, and Young voting. Moreover, we consider NP-hard possible winner problems for scoring protocols.

What Makes Equitable Connected PartitionEasy

Ondra SuchýCharles University, [email protected]

We study the Equitable Connected Partition problem: partitioningthe vertices of a graph into a specified number of classes, such that eachclass of the partition induces a connected subgraph, so that the classeshave cardinalities that differ by at most one. We examine the problem fromthe parameterized complexity perspective with respect to various (aggre-gate) parameterizations involving such secondary measurements as: (1) thenumber of partition classes, (2) the treewidth, (3) the pathwidth, (4) theminimum size of a feedback vertex set, (5) the minimum size of a vertex

31

cover, (6) and the maximum number of leaves in a spanning tree of thegraph. In particular, we show that the problem is W[1]-hard with respectto the first four combined, while it is fixed-parameter tractable with respectto each of the last two alone. The hardness result holds even for planar gra-phs. The problem is in XP when parameterized by treewidth, by standarddynamic programming techniques. Furthermore, we show that the closelyrelated problem of Equitable Coloring (equitably partitioning the ver-tices into a specified number of independent sets) is FPT parameterized bythe maximum number of leaves in a spanning tree of the graph.

(joint work with Rosa Enciso, Michael R. Fellows, Jiong Guo, Iyad Kanj,and Frances Rosamond)

Making graph minor theory constructiveIsolde Adler

Goethe University, Frankfurt am [email protected]

By the Graph Minor Theorem (previously Wagner’s Conjecture), every classof finite graphs that is closed under taking minors has a characterization by*finitely* many excluded minors. Moreover, every minor closed graph classhas a cubic time membership test. This membership test uses the excludedminor characterization.

The proof of the Graph Minor Theorem is non-constructive, hence, un-fortunately, for many minor closed classes we do not know the correspon-ding excluded minors! Some work has been done to overcome this non-constructiveness, but still many problems remain unsolved. In this talk wepresent a method to compute excluded minors from certain given infor-mation of a minor ideal. Our method uses graph minor theory, logic andautomata. In particular, we show: Given two minor closed classes C and Dby their excluded minor characterizations, we can effectively compute anupper bound for the size of the excluded minors of C ∪D.

This is joint work with Martin Grohe and Stephan Kreutzer

32

Testing Planarity of Partially Embedded GraphsJan Kratochvíl


We pose and study the following question: Given a (planar graph) G and aplanar embedding of its subgraph H, can this be extended to a noncrossingembedding of the entire graph G? This approach follows the paradigm ofcompleting a partial solution of a particular problem, which has been stu-died in many different situations before. Unlike in many cases, when thepresence of a partial solution in the input makes an otherwise easy pro-blem hard, we show that the planarity question remains polynomial-timesolvable. Our algorithm is based on several combinatorial lemmas whichshow that planarity of partially embedded graphs performs the ”oncas” be-haviour - obvious necessary conditions for planarity are also sufficient. Inparticular, a 2-connected graph allows an extension of an embedding of itssubgraph H if and only the skeleton of each node of its SPQR-tree has anembedding compatible with the give embedding of H. This implies that nodynamic programming is needed for a decision algorithm, the nodes of theSPQR-tree can be processed independently in parallel. It should be notedthat though 2-connected graphs form the core situation, nontrivial steps areneeded to handle the less connected cases. By refining the techniques andusing appropriately adjusted data structures we manage to achieve a lineartime algorithm.

On the other hand we consider several generalizations of the problem, e.g.minimizing the number of edges of the partial embedding that need to bererouted, and argue that they already become NP-hard.

The talk is based on a joint paper with atrizio Angelini, Giuseppe Di Bat-tista, Fabrizio Frati, Vít Jelínek, Maurizio Patrignani, and Ignaz Rutter.

33

Enumeration of perfect matchingsMartin Loebl


Some recent results will be presented.

34

Discrete Dynamical Systems

Generalization of Sturmian WordsLubomíra Balková

Czech Technical University, [email protected]

Sturmian words are aperiodic words with the lowest possible complexity. Nowonder they belong to the most studied infinite words and many equivalentdefinitions of Sturmian words havee been found out. In our contribution,we will consider the generalizations of their combinatorial definitions andproperties to multiliteral alphabets. We will in particular explain the relati-ons between such properties and we will mention interesting open questions.This is a joint work with Štěpán Starosta.

On two and three periodic Lyness differenceequations

Armengol GasullUniversitat Autonoma de Barcelona

[email protected]

We describe the sequences xn given by the non-autonomous second orderLyness difference equations xn+2 = (an + xn+1)/xn, where an is either a 2-periodic or a 3-periodic sequence of positive values and the initial conditionsx1, x2 are as well positive. We also show an interesting phenomenon of thediscrete dynamical systems associated to some of these difference equations:the existence of one oscillation of their associated rotation number functi-ons. This behavior does not appear for the autonomous Lyness differenceequations.

35

Global linearization of periodic recurrencesFrancesc Manosas


We deal with m-periodic, n-th order difference equations and study whe-ther they can be globally linearized. We give an affirmative answer whenm = n + 1 and for most of the known examples appearing in the litera-ture. Our main tool is a refinement of the Montgomery-Bochner Theorem.We also prove some nice features of the (n+ 3)-periodic Coxeter differenceequations.

About the connectivity of the escaping set forentire transcendental maps

Xavier JarqueUniversitat Rovira i Virgili (Tarragona)

Since Eremenko’s conjecture (all connected components of the escaping setfor transcendental entire maps are unbounded) the study of the connecti-vity (or not) of the escaping set has been studied in different frameworks.Recently two main results has been proved: On one hand it has been foundan example showing that the conjecture is not true even for entire maps inclass B for which wall singular values are contained in a compact set (L.Rempe et al). On the other under certain hypotheses (precisely strongly sub-hyperbolicity) it has been shown that the escaping set is disconnected (H.Mihaljevic-Brandt). We finally discuss recent work about the connectivityof the escaping set for the complex exponential family.

36

The Lotka-Volterra map and some problems ofthe number theory.Peter Maličký

Matej Bel University, Banská [email protected]

The Lotka-Volterra map is a transformation of the triangle with vertices(0,0),(4,0) and (4,0) given by formula F (x, y) = (x(4 − x − y), xy). Weshow that properties of interior periodic orbits are closely related to someproblems of the number theory. Some of these problems are open.

Extensions without increasing the entropy as atool for entropy bounds problem

Matúš DirbákMatej Bel University, Banská Bystrica

[email protected]

We show how extensions without increasing the entropy can be used to de-termine the infimum of topological entropies of all maps with a given pro-perty on a given space and present known as well as new results in thisdirection.

On nowhere differentiable measure preservingmaps

Jozef BobokCzech Technical University, Prague

[email protected]

We present several known results on nowhere differentiable measure preser-ving interval maps and discuss their properties with respect to the topolo-gical entropy. Some open problems will also be stated.

37

A strongly F -invariant pinched core strip thatdoes not contain any arc of curve

Lluis Alseda i SolerUniversitat Autonoma de Barcelona

[email protected]

In [1] the authors define the concept of pinched core strip. So far it has notbeen given an example of such an object that is strongly invariant under aquasi-periodic triangular function and it is not a curve. In this talk we willdescribe how to construct such an example.

[1] R. Fabbri, T. Jäger, R. Johnson and G. Keller, A Sharkovskii-type theo-rem for minimally forced interval maps, Topological Methods in NonlinearAnalysis, Journal of the Juliusz Shauder Center, 2005(26), 163–188.

Global dynamics of discrete systems through LieSymmetries

Anna Cima MolletUniversitat Autonoma de Barcelona

[email protected]

We use the existence of Lie Symmetries to study the discrete dynamics ofsome rational maps in Rn.

38

Enumerative andAnalytic Combinatorics

The poset of bipartitionsChristian Krattenthaler


Bipartitional relations were introduced by Foata and Zeilberger in their cha-racterization of relations which give rise to equidistribution of the associatedinversion statistic and major index. I shall consider the natural partial orderon bipartitional relations given by inclusion. This partial order gives in factrise to a lattice which has the remarkable property that the Möbius functionof any interval is 0, 1, or -1. However, to prove the latter fact, is surprisinglydifficult to prove. It requires Robin Forman’s Discrete Morse Theory, themain ideas of which I shall outline. This is joint work with Gabor Hetyei.

Families of permutations with boundedVC-dimension have almost exponential size

Josef CibulkaCharles University, [email protected]

A family P of n-permutations has VC-dimension k if k is the largest num-ber such that the elements of P induce all k! permutations on some k-tupleof indices. An example of a family with VC-dimension k are permutati-ons avoiding some fixed (k + 1)-permutation. Marcus and Tardos provedthe Stanley–Wilf conjecture, which says that the number of n-permutationsavoiding an arbitrary given permutation grows only exponentially in n. Razshowed that if a family of n-permutations has VC-dimension 2, then its sizeis at most exponential in n.

In contrary to these two results, we find a family of 2Ω(n log(α(n))) permu-tations with VC-dimension 3. On the other hand, we show the upper bound

39

of 2O(n log?(n)) for any family of permutations with constant VC-dimension.

We also study a related problem of the maximum number of 1-entries in ann×n (0, 1)-matrix with no k-tuple of columns containing all k-permutationmatrices. It is known that this number is linear in n if k ≤ 3. For any fixedk ≥ 4, we show bounds Ω(nα(n)) and O(n2α

O(1)(n)).

This is a joint work with Jan Kynčl.

Generalized Thue-Morse Sequences of SquaresJohannes Morgenbesser

Vienna University of [email protected]

Let (tn) = (0110100110010110...) be the Thue-Morse sequence. Then theletters 0 and 1 are equally likely. Interestingly, this property persists forsubsequences like linear progressions (tan+b). Recently, Mauduit and Rivatcould settle the long standing conjecture (attributed to Gelfond) that (tp),p prime, and (tn2) have the same property.

In our work, we consider compact group generalizations T (n) of the Thue-Morse sequence. We prove that the subsequence T (n2) is uniformly distri-buted with respect to a proper measure, that is absolutely continuous withrespect to the Haar measure. The proof is based on a generalization of theFourier based method of Mauduit and Rivat to group representations. Asan application, we show a result on the frequency of letters of subsequencesof special automatic sequences.

Convergence properties of q-binomialdistributionsMartin Zeiner

TU [email protected]

We investigate in two q-analogues of the binomial distribution which areconnected with basic hypergeometric series (or q-series). We study Kemp’s

40

q-binomial distribution (joint work with Stefan Gerhold) and the q-deformedbinomial distribution which have applications in biology and physics. Seve-ral convergence results involving the classical binomial, the Heine, the Euler,the discrete normal, the Poisson, and the exponential distribution are es-tablished. Some of them are q-analogues of classical convergence properties(e.g. there are q-analogues of the convergence of the classical binomial dis-tribution with constant mean to the Poisson distribution or of the binomialdistribution to the normal distribution).

Parameters of integer partitions - a zoo of limitlaws

Stephan WagnerStellenbosch [email protected]

Limit laws for various parameters of integer permutations and their gene-ralisations (where the parts are restricted to certain sets, e.g. the set of allsquares or all primes) are presented. Various classical distributions (Gaus-sian, Rayleigh, Gumbel, Geometric) as well as more ”exotic” distributionsoccur as limit laws in this context. For certain parameters, interesting phasetransitions can be observed as well.

The degree distribution in Random UnlabelledSubcritical Graph families

Veronika KrausTU Wien

[email protected]

We study the distribution of node degrees in subcritical families of gra-phs. These are families which are block-stable, that is they are determi-ned only by their 2-connected components, and the generating function of2-connected components fulfills certain analytic conditions, which we callsubcriticality conditions. Important subcritical families are outerplanar andseries-parallel graphs. We extend recent results by Drmota, Gimenez and

41

Noy in the labelled case, focusing mainly on the unlabelled framework. Weuse a generating functions approach, taking into account symmetries evol-ving from unlabelling and therefore using Polya’s theory on cycle indexsums. By methods of singularity analysis we can prove a central limit lawfor the number of nodes of given degree k in a random unlabelled graph ofsize n in any subcritical family of graphs. Therefore we obtain a degree dis-tribution. This results holds generally in the subcritical framework, withoutany further knowledge on the structure of 2-connected components.

The degree of the nodes in random increasingk-treesGeorg Seitz

Vienna University of [email protected]

Increasing k-trees are labelled graphs which are recursively defined as follows:The k-clique formed by the nodes 1, . . . , k is the only increasing k-tree withk nodes. An increasing k-tree with n > k nodes is constructed by connectinga new node n to all nodes of a k-clique in an increasing k-tree with n − 1nodes.

For n ≥ l, we study the probability distribution of the degree of node l + kin random increasing k-trees of size n + k. Using a generating functionsapproach, we derive the exact distribution for fixed n and l. Furthermore,we fully characterize the limiting behaviour for n→∞, both for l fixed andfor l →∞. As a corollary, we obtain a weak form of the well-known powerlaw for the degree distribution in increasing k-trees for k ≥ 2.

42

Function Spaces and Applications

Gehring type results for generalized(Navier-)Stokes system in 2D.

Petr KaplickýCharles University, Prague

[email protected]

The mean value property of harmonic functionsand the Heat equation

Yitzhak WeitUniversity of Haifa

[email protected]

A class of radial measures µ on Rn is defined so that integrable harmonicfunctions are characterized as the solutions of fµ = f . These results areapplied to the investigation of some periodic solutions of the Heat equation.

A note on bilinear multipliersSalvador Rodriguez

Universitat de [email protected]

The classical result of De Leeuw (Ann. Math., 1965) states that for 1 < p <∞, given a continuous and bounded function such that defines a Fouriermultiplier on Lp, the sequence defined by its restriction to the integers de-fines a Fourier multiplier in Lp in the periodic case.

43

A similar result for bilinear multipliers was obtained by D. Fan and S. Sato(J. Aust. Math. Soc., 2001).

We present an abstract version of a De Leeu’s type result for bilinear mul-tipliers that allows D. Fan and Sato’s result to be recovered and to generalizeG. Diestel and Grafakos’ (Nagoya Math. Journal 2007) result also.

Calderon-type interpolation theorems andapplications to sharp Sobolev embeddings

Gogatishvili AmiranAcademy of Sciences, [email protected]

We present a new Calderon-type interpolation theorems for operators sa-tisfying nonstandard endpoint estimates. Using this new interpolation the-orems we give a characterization of sharp Sobolev embeddings for spacesbased on rearrangement invariant spaces on domains with a sufficientlysmooth boundary. We show that each optimal sobolev embedding can beobtained by the real interpolation of the well-known endpoint embeddings.

44

Mathematical Physics

Density-Matrix Functionals: AsymptoticBehavior of the Energy

Heinz SiedentopLudwig-Maximilians-Universität München

[email protected]

We will show that a class of density matrix functionals introduced by Müllerand others which describe correlated systems yield atomic energies which di-ffer only by o(Z5/3), Z the atomic number, from the infimum of the spectrumof the corresponding Schrödinger operator.

The Cauchy problem for the KdV equation withsteplike finite-gap initial data

Egorova IrynaB. Verkin Institute for Low Temperature Physics, Kharkiv

[email protected]

Let p±(x, t) be two arbitrary finite-gap solutions of the the Korteweg–deVries equation qt = −qxxx + 6qqx.

By means of the inverse scattering transform we solve the initial valueproblem for the KdV equation with smooth initial data q(x, 0), which areasymptotically close to p±(x, 0) in the following sense:

(q(s)(x, 0)− p(s)± (x, 0))|x|m ∈ L1(R±) as ≤ 0 ≤ s ≤ n.

If the perturbation is sufficiently smooth (n ≥ 13) and has a sufficientnumber of moments finite, 8 ≤ m ≤ n− 5, then the classical solution q(x, t)of the KdV equation exists, is unique, and belongs to the class

(q(s)(x, t)− p(s)± (x, t))|x|bm

2 c−2 ∈ L1(R±) with 0 ≤ s ≤ n−m− 2.

45

In particular, this implies that the solution remains in the Schwartz class ofperturbations provided the initial condition is in this class.

Joint work with Katrin Grunert and Gerald Teschl.

Long–time asymptotics for the Korteweg-deVries equationKatrin Grunert


One of the most famous examples of completely integrable wave equations isthe Korteweg–de Vries equation, qt(x, t) = 6q(x, t)qx(x, t)−qxxx(x, t), whichcan be solved via the inverse scattering method. We will consider decayinginitial data and show how the inverse scattering problem can be reformu-lated as a Riemann–Hilbert problem. Based on this we will present howto extract long-time asymptotics by using the nonlinear steepest descentmethod.

The Hardy inequality and the asymptoticbehaviour of the heat equation in twisted

domainsDavid Krejčiřík


In this talk we revise a recently established Hardy inequality in twisted tu-bes on the background of transience of the Brownian motion. We beginby recalling the classical Hardy inequality and its relation to geometric,spectral, stochastic and other properties of the underlying Euclidean space.After discussing the complexity of the problem when reformulated for quasi-cylindrical subdomains, we focus on the prominent class of tubes. As themain result, we show that the geometric deformation of twisting yields animproved decay rate for solutions of the heat equation in three-dimensionaltubes of uniform cross-section. This is a joint work with Enrique Zuazua.

46

Spectrum of the Aharonov-Bohm Hamiltonianwith constant magnetic fields in the hyperbolic

planeTakuya Mine

Kyoto Institute of [email protected]

We consider the Schrödinger operator H in the hyperbolic plane with themagnetic field given by the sum of a constant times the surface form anda finite number of the Dirac δ measures. Since the corresponding classicalsystem contains both bounded orbits and unbounded ones, the spectrum ofH contains both the continuous spectrum and infinitely degenerated eigen-values (Landau levels). In the presence of the δ magnetic fields, there alsoexist a finite number of eigenvalues of finite multiplicity. We give an upperbound for the number of eigenvalues (counting multiplicity) between twosuccessive Landau levels.

1-D Schrödinger operators with local pointinteractionsOleksiy Kostenko

Dublin Institute of [email protected]

Spectral properties of 1-D Schrödinger operators

HX,α := − d2

dx2+∑xn∈X

αnδ(x− xn)

with local point interactions on a discrete set X = xn∞n=1 are well studiedwhen d∗ := infn,k∈N |xn − xk| > 0 (numerous results as well as a com-prehensive list of references may be found in the monograph of Albeverio,Gesztesy, Hoegh-Krohn, Holden, Solvable Models in Quantum Mechanics,AMS Chelsea Publ., 2005; see also a survey of recent result given by Exnerin Appendix K).

47

In the case d∗ = 0, it is only known that the operator HX,α may be sym-metric with nontrivial deficiency indices (the example of Shubin Christ andStolz, J. Math. Anal. Appl. 184 (1994)).

The main aim of our talk is the spectral analysis of the operators HX,α inthe case d∗ = 0. We show that spectral properties of the operators HX,α

correlate with the corresponding spectral properties of certain classes ofunbounded Jacobi matrices. We exploit this connection to investigate self-adjointness, lower semiboundedness, and discreteness of the operators withlocal point interactions.

The talk is based on the joint work with Mark Malamud.

The work is supported by IRCSET Post-Doctoral Fellowship Program.

48

Mathematics of Secret Sharing

Ideal Hierarchical Secret Sharing SchemesCarles Padró


he search of efficient constructions of ideal schemes for families of non-threshold access structures that may have useful applications has attracteda lot of attention. Several proposals have been made for access structureswith hierarchical properties, in which the participants are distributed intolevels that are hierarchically ordered.

Here, we study hierarchical secret sharing in all generality by providing anatural definition for the family of the hierarchical access structures. Speci-fically, an access structure is said to be hierarchical if every two participantscan be compared according to the following natural hierarchical order: whe-never a participant in a qualified subset is substituted by a hierarchicallysuperior participant, the new subset is still qualified.

We present a complete characterization of the ideal hierarchical access structu-res, that is, the ones admitting an ideal secret sharing scheme. We use thewell known connection between ideal secret sharing and matroids and, inparticular, the fact the every ideal access structure is a matroid port. In ad-dition, we use recent results on ideal multipartite access structures and theconnection between multipartite matroids and discrete polymatroids. Weprove that every ideal hierarchical access structure is the port of a repre-sentable matroid and, more specifically, we prove that every ideal structurein this family admits ideal linear secret sharing schemes over fields of allcharacteristics. This generalizes previous results on weighted threshold ac-cess structures. Finally, we use our results to find a new characterization ofthe ideal weighted threshold access structures that is more precise than theexisting one.

This is a joint work with Oriol Farràs

49

Asymptotically good linear secret sharing withstrong multiplication over any finite field

Ignacio CascudoUniversity of Oviedo, Spain

[email protected]

Linear secret sharing schemes(LSSS) with an extra algebraic property knownas t-strong multiplication have important cryptographic applications: mul-tiparty computation protocols can be based upon such schemes (Cramer,Damgaard, Maurer, Eurocrypt 2000); and more recently, Ishai, Kushilevitz,Ostrovsky and Sahai have found important applications in zero knowledge(STOC 07) and correlation extractions (FOCS 09) as well. For all these usesit is crucial to study the existence of asymptotically good families of LSSSsover fixed finite fields, i.e., families of LSSSs with an unbounded numberof shares and strong multiplication for a constant fraction of the shares. Aseries of results, which relate this problem to interesting questions in codetheory and towers of algebraic function fields, were given by Chen and Cra-mer in Crypto 2006 and Cascudo, Chen, Cramer, Xing in Crypto 2009 andin subsequent work and will be presented in this talk.

Secret sharing on infinite structuresLászló Csirmaz

Central European University, [email protected]

Traditionally, secret sharing is restricted to finite access structures. This po-ses no problem in the main field of applications. For finite structures we havea complete picture, at least in the case of perfect secret sharing. When onetries to extend the results for infinitely many participants, technical difficul-ties arise immediately. How to define the size of the share relative to that ofthe secret, as entropy, in general, can only be defined for discrete randomvariables. Even the existence of appropriate (measurable) random variablesare far from being trivial. One can easily get mislead by the analogy to thefinite case. For example, if F is a finite field, then shifting the field by a

50

random element makes a perfect homeomorphic (with respect to addition)hiding. However when the field is the set of reals, then any probability dis-tribution leaks some information (as taught in statistics classes). One ofthe first geometrical secret sharing construction using projective planes byBlakley and Swanson is also problematic. While the distribution is homo-genous, when taking the conditional distribution given several shares, theresult is concentrated on a zero measure set.

We overcome some of the difficulties mentioned above by defining secretsharing on an infinite structure as the limit of its finite restrictions. Wejustify the soundness of this notion, and give examples showing the non-triviality of the notion.

Ideal secret sharing, matroids and relativeentropy

František MatúšAcademy of Sciences, [email protected]

In an information-theoretical setting, a secret sharing scheme (sss) can bedescribed by a discrete random vector with finitely many values which sa-tisfies the constraints on Shannon entropies of subvectors corresponding tocryptographic requirements on the access to the secret. A sss is ideal if eachparticipant has at most as much information as the dealer distributing thesecret. Ideal sss’s with the same access structure are known to correspondto special representations of a matroid. We will review the results and openproblems on the resulting classes of representations and matroids. They areclosely related to entropic polymatroids, their limits and information theo-retical inequalities among Shannon entropies. Ideal sss’s with a given accessstructure and the size of secret will be interpreted as the statistically leastfavorable cases of the maximum likelihood estimation problems in familiesof Gibbs distributions.

51

Stochastic Analysis

Stein’s method and Malliavin calculus ofPoisson functionalsFrederic Utzet


Joint work with Giovanni Peccati, Université Paris Ouest – Nanterre; Mu-rad Taqqu, Boston University; Josep Lluís Solé, Universitat Autonoma deBarcelona

In a recent series of papers by Nourdin, Peccati, Réveillac and Reiner it hasbeen shown that Stein’s method can be effectively combined with Malliavincalculus on a Gaussian space in order to obtain explicit bounds for thenormal and non-normal approximation of smooth functionals of Gaussianfields, mainly multiple Wiener-Itô integrals.

In this talk we will present how these results can be extended to the appro-ximation (in the Wasserstein distance) of functionals of Poisson measures.In particular we apply these ideas to a sequence of multiple Wiener-Itôintegrals with respect to a Poisson measure.

As in the Gaussian case, the main ingredients of the analysis are the following:

1. A set of Stein differential equations, relating the normal approximationin the Wasserstein distance to first order differential operators.

2. A derivative operator D, acting on square-integrable random variables.

3. An integration by parts formula, involving the adjoint operator of D, theSkorohod integral.

4. A pathwise representation of D which, in the Poisson case, involves adifference operator.

52

Malliavin Calculus for Lévy Processes viaDerivative OperatorsJosep Lluís Solé i Clivillés


We develop a Malliavin calculus for Lévy processes based on derivative ope-rators. This approach include the classical Malliavin calculus for Gaussianprocesses and the one for the Poisson process introduced by Carlen andPardoux. As an application we analyze sufficient conditions for the abso-lute continuity of the law of some stochastic differential equations driven byLévy processes.

Stochastic bilinear differential equation withfractional noise in infinite dimension

Jana ŠnupárkováCharles University, Prague

[email protected]

We will study the infinite-dimensional stochastic bilinear equation with one-dimension fractional noise in the singular case H < 1/2.

SPDEs of reaction-diffusion type driven by Levynoise

Erika HausenblasUniversity of Salzburg

[email protected]

The topic of the talk are Spdes driven by Poisson random measure of re-action diffusion type. In the first part I will shortly outline Poisson randommeasures, then I will dwell shortly on the connection between Levy pro-

53

cesses and Poisson random measures. Moreover, I will introduce space timeLevy noise respective space time Poissonian noise. Then, the stochastic in-tegral with respect to Poisson random measures in Banach spaces will betreated and some inequalities pointed out.

In the second part of the talk Stochastic partial Differential Equations drivenby Possion random measures will be considered. Here, I want to present someresult concerning SPDEs driven by only continuous coefficients. In this case,one can show only the existstence of martingale solutions.

The topic of the talk rests on the following papers.

Brzezniak and Hausenblas, Maximale inequality of the stochastic convolu-tion process, PTRF, 2008.

Brzezniak and Hausenblas, Uniqueness of the Ito integral driven by Poissonrandom measure, to appear in the Ascona proceedings, 2009.

Brzezniak and Hausenblas, Uniqueness of the stochastic convolution processdriven by Poisson random measure, submitted, 2009.

Brzezniak and Hausenblas, SPDEs of reaction diffusion type Poisson randommeasure driven by Poisson random measure, submitted, 2009.

Moment inequalities for stochastic integrals in2-smooth Banach spaces

Jan SeidlerAcademy of Sciences, [email protected]

Stochastic wave equationsMartin Ondreját


A survey of basic and recent results in the theory of wave equations driven

54

by Wiener processes.

Delay differential equations with hereditarydrift driven by fractional Brownian motion with

Hurst parameter H > 1/2Carles Rovira EscofetUniversitat de [email protected]

We prove an existence and uniqueness result of solution for stochastic diffe-rential delay equations with hereditary drift driven by a fractional Brownianmotion with Hurst parameter H > 1/2. We show that when the delay goesto zero the solutions to these equations converge, almost surely and in Lp,to the solution for the equation without delay.

We consider also the case of stochastic differential delay equations withreflection.

Relationships between fractional, bifractionaland subfractional Brownian motions

Xavier BardinaUniversitat Autonoma de Barcelona

[email protected]

Statistical inference for SPDEs driven byfractional noiseJan Pospíšil

University of West Bohemia, [email protected]

We will discuss a parameter estimation problem for a diagonalizable stochas-

55

tic evolution equation driven by additive noise that is white in space andfractional in time. Some asymptotic properties of the maximum likelihoodestimator, as the number of the Fourier coefficients of the solution increases,will be investigated. Necessary and sufficient conditions for consistency andasymptotic normality will be presented.

Stochastic porous medium equation withfractional Brownian motion

Jan BártekCharles University, [email protected]

In this talk we study the relation between solutions to a certain type of non-linear partial differential equation and solutions to its stochastic analogydriven by a fractional Brownian motion with Hurst index H > 1/2. Obta-ined formula is used to study properties of the solution to the stochasticporous medium equation.

Ornstein-Uhlenbeck bridgeJosef Janák


Fractional Ornstein-Uhlenbeck processes in finite-dimensional spaces willbe studied. First, known results on existence and uniqueness on solutionsto linear stochastic differential equations, which define Ornstein-Uhlenbeckprocesses, will be recalled. Then we will introduce general Gaussian bridgesand their representations. Finally, we will derive a formula for a Gaussianbridge driven by an Ornstein-Uhlenbeck process.

56

Reversible Reactions in Reaction-DiffusionStochastic Simulation Algorithms

Jana LipkováCharles University, [email protected]

Introduction to some mathematical models used in modelling of chemicalreactions. Drawback of these models when reversible reactions are conside-red. Introduction to Andrew and Bray modelling of reversible reactions andrelated problems. Introduction and closer look on stochastic approach withimplementation of probability.

LQ control for FBM-driven SPDEsBohdan Maslowski


The linear-quadratic control problem is studied for stochastic equation ininfinite dimensions that is driven by the fractional Brownian motion. Theexistence and uniqueness of the optimal control is proved and, under morerestrictive assumptions, the optimal conrtol is given in the feedback form.The results are applicable to parabolic and hyperbolic stochastic PDEs withdistributed noise and to parabolic systems with boundary/pointwise noiseand control.

57

Topological, Geometric andAlgebraic Graph Theory

On almost distance-regular graphsCristina Dalfó Simó


Distance-regular graphs have been a key concept in Algebraic Combinatoricsand have given place to several generalizations, such as association schemesand almost distance-regular graphs, which are the topic of this talk. Rou-ghly speaking, almost distance-regular graphs have some kind of regularitythat is characteristic of distance-regular graphs. An example is the conceptof m-walk regularity, which is a generalization of walk-regularity, the latterdefined in terms of the invariance of the number of l-walks between verti-ces at a given distance at most m. Other concepts are those of m-partiallydistance-polynomial and m-partially distance-regular graphs, characterizedby the existence of polynomials with a certain degree in the adjacency ma-trix of the graphs, which give its corresponding distance matrices. In thistalk, we give some algebraic and combinatorial characterizations of suchalmost distance-regular graphs, based on the so-called local spectrum andpredistance polynomials of the graph. The main results can be seen as ageneralization for almost distance-regular graphs of the spectral excess the-orem for distance-regular graphs, which provides a quasi-spectral characte-rization in terms of the excess (number of vertices at maximum distance)of each of its vertices.

(joint work with E.R. van Dam, M.A. Fiol, E. Garriga, and B.L. Gorissen).

58

Applications of symmetries of combinatorialmaps

Tomaž PisanskiUniversity of Ljubljana

[email protected]

In 1980 combinatorial maps were introduced by A. Vince in his thesis. Theyare a useful tool for investigation of various combinatorial structures suchas maps or abstract polytopes. Each combinatorial map admits a uniquequotient ”type graph”. Using elementary methods related to type graphswe derive the 14 types of edge-transitive maps first studied by J. Graver andM. Watkins in 1997. Applications to abstract polytopes are also mentioned.

Vertex-transitive maps with Schläfli type 3,7Daniel Pellcer


When classifying maps with certain degree of symmetry on surfaces of smallEuler characteristic the Hurwitz bounds play an important role. The equive-lar maps with highest Hurwitz bounds are those whose vertices are 3-valentand whose faces are heptagons, as well as their duals. If these maps are clas-sified, the computer time to compute the remaining equivelar maps reducesconsiderably. In this talk we present a procedure to classify all 3-valentvertex-transitive maps with heptagonal faces and their duals by means ofsmaller regular or 2-orbit maps.

59

Nowhere-zero flows in Cartesian bundles ofgraphs

Edita RollováComenius University, [email protected]

We extend the results of Imrich and Skrekovski [J. Graph Theory 43 (2003),93-98] about nowhere-zero flows in Cartesian product graphs to ”twisted”Cartesian products – Cartesian bundles. Our main result states that everyCartesian bundle of two graphs with positive minimum valency has a nowhere-zero 4-flow.

Fundamental cycles and the maximum genus ofa graph

Michal KotrbčíkComenius University, [email protected]

We study the interplay between the maximum genus of a graph and the cyclebasis with respect to a given spanning tree via the corresponding intersectiongraph. We show that different spanning trees may lead to intersection gra-phs with different matching numbers. This fact disproves the main resultof [Sci China Ser A 52: 1920–1926 (2009)] and invalidates the algorithmfor the maximum genus of a graph based on that theorem. We give somefurther examples to show that extending a graph with a pair of edges whosefundamental cycles intersect need not increase the maximum genus by one,and that a spanning tree with n pairs of intersecting fundamental cyclesneed not guarantee a 2-cell embedding of a graph in an orientable surfaceof genus n, contrary to what is claimed in op. cit.

60

Construction of Hamilton cycles in(2,s,3)-Cayley graphsKlavdija Kutnar


A path (cycle) containing every vertex in a graph is called a Hamilton path(Hamilton cycle, respectively). A graph is called vertex-transitive if for anypair of vertices u and v there exists an automorphism mapping u to v. In1969, Lovász asked whether every finite connected vertex-transitive graphhas a Hamilton path. With the exception of the complete graph on twovertices, only four connected vertex-transitive graphs that do not have aHamilton cycle are known to exist. These four graphs are the Petersengraph, the Coxeter graph and the two graphs obtained from them by repla-cing each vertex by a triangle. The fact that none of these four graphs isa Cayley graph has led to a folklore conjecture that every Cayley graphhas a Hamilton cycle. (A Cayley graph is a graph whose automorphismgroup admits a regular subgroup.) Both of these two problems are stillopen. However, a considerable amount of partial results are known. In thistalk a special emphasis will be given to recent results concerning the exis-tence of Hamilton cycles in cubic Cayley graphs arising from groups having(2,s,3)-presentation.

Pseudo-distance-regular graphs around a setand the local spectrum of its subconstituents

Marc Cámara VallejoUniversitat Politecnica de Catalunya

[email protected]

Pseudo-distance-regularity around a set of vertices generalizes the notion ofcompletely regular code to non-necessary regular graphs. In this talk we willstudy some quasi spectral characterizations of pseudo-distance-regularityaround a set of vertices. These characterizations are obtained through thestudy of the local spectrum of a set of vertices, which play a similar role to

61

that of the spectrum of the whole graph when it is seen from that set ofvertices. We will also discuss the relation between the local spectrum of aset of vertices and that of its subconstituents, paying special attention toits antipodal set.

On the ascending subgraph decompositionproblem for bipartite graphs

Jordi MoragasUniversitat Politecnica de Catalunya

[email protected]

The Ascending Subgraph Decomposition (ASD) Conjecture asserts thatevery graph G with

(n+1

2

)edges admits an edge decomposition G = H1 ⊕

· · · ⊕Hn such that Hi has i edges and is isomorphic to a subgraph of Hi+1,i = 1, . . . , n − 1. In this talk we summarize some known results about thisconjecture and give necessary and sufficient conditions for a bipartite graphso that it admits an ascending subgraph decomposition in which each partis a star forest.

On cubic biabelian graphsIstván Kovács

FAMNIT, University of [email protected]

A biabelian graph is a finite simple graph which admits a semiregular au-tomorphism group with two

orbits of equal size. In particular we speak of a bicirculant graph if the semi-regular automorphism group is cyclic. A classification of cubic bicirculantgraphs was given recently by T. Pisanski [A classification of cubic bicircu-lants, Discrete Math. 307 (2007), 567-578]. In this talk we discuss similarresults for cubic biabelian graphs.

62

Recognizing string graphs on surfaces in NPJan Kynčl


We show that graphs representable as intersection graphs of strings in anyfixed surface S can be recognized in NP. We also show that the crossingnumber of a minimal string representation in S is bounded by 2cn

k

for someconstants c, k where k does not depend on the genus of S. This impro-ves the double exponential upper bound given by Schaefer, Sedgwick andŠtefankovič and also generalizes their result for the planar case.

Archimedean solids of higher generaJán Karabáš


We will deal with vertex-transitive and polyhedral maps embeded into ori-entable surfaces of higher genera. These maps naturally generalise the sphe-rical maps associated with classical Archimedean solids. Therefore we callthem Archimedean maps. The main idea is based on the fact that eachArchimedean map projects onto one-vertex (or two-vertex) quotient map.For given genus g there are just finitely many quotients and all Archime-dean maps of genus g can be reconstructed from these quotients. We willalso show the sketch of method of construction of unoriented ”Archimedeanmaps” of higher genera.

Skew-morphisms of groups and regular mapsRoman Nedela


63

Skew-morphism of a group is a generalization of a group automorphism.Skew-morphisms were investigated in connection of regular embeddings ofgraphs. In the present talk we survey some results on this interesting re-lationship. Further, we present some results on skew-morphisms of cyclicgroups.

Coding and Counting Arrangements ofPseudolinesPavel Valtr


Arrangements of lines and pseudolines are important and appealing ob-jects for research in discrete and computational geometry. We show thatthere are at most 20.66n2

simple arrangements of n pseudolines in the plane.This improves on previous work by Knuth who proved an upper bound of

3(n2) ∼= 20.79n2

in 1992 and S. Felsner who obtained 20.69n2in 1997. The

argument uses surprisingly little geometry. The main ingredient is a lemmathat was already central to the argument given by Knuth. Joint work withStefan Felsner.

64

Triangulated Categories

Combinatorial model categoriesJiří Rosický

Masaryk University, [email protected]

The talk will survey some results and open problems concerning combinato-rial model categories. Above all, the conjecture of J. H. Smith about modelstructures determined by a set of morphisms and my problem of generali-zed Brown representability will be discussed. In addition, the concept of aclass-combinatorial model category (introduced in my joint work with B.Chorny) will be mentioned.

Representability theorems for well generatedtriangulated categories

Oriol RaventosUniversitat de [email protected]

Let S be a triangulated subcategory of a triangulated category T . A con-travariant functor from S to the category of abelian groups is representableif it is the restriction to S of a functor of the form Hom(−, X), for X inT . We discuss some conditions on T and S that ensure that cohomologicalfunctors as above that send coproducts in S to products are representable.

65

Properness in the Bousfield Localisation ofModel CategoriesGeorge Raptis

University of Osnabrü[email protected]

The purpose of this talk will be to discuss the relevance of the property of(left) properness in the (left) Bousfield localisation of model categories. TheBousfield localisation of a model category at a set of morphisms is known toexist for the left proper cellular model categories and the left proper com-binatorial model categories. The main result of the talk will say that theBousfield localisation of a well-behaved cofibrantly generated model cate-gory exists iff the candidate class of trivial cofibrations is cofibrantly closed.This result will be presented as an application of the various known factsabout combinatorial model categories, that will be briefly reviewed.

A homotopy theory for comodules andcoalgebras

Alexandru StanculescuMasaryk University, [email protected]

We construct a Quillen model category structure on the categories of como-dules and comonoids in certain monoidal model categories.

Decompositions of p-local spacesDamir Franetic


A ring is called semi-perfect if the identity 1 can be (uniquely) decomposed

66

as a sum of orthogonal local idempotents. When G is a p-local H-space,there is a unit-reflecting homomorphism from [G,G] into some semi-perfectring. Unique decomposition of 1 as a sum of local idempotents then impliesunique decomposition of G as a product of indecomposable p-local H-spaces.Also, a straightforward dualization of this argument gives a decompositionfor p-local coH-spaces. This proof is much shorter than the original proofgiven by C. Wilkerson.

Cellularization of structures in triangulatedcategories

Javier J. GutierrezCentre de Recerca Matematica

[email protected]

We describe the formal properties of cellularization and nullification func-tors in triangulated categories. We give sufficient conditions for cellulari-zation functors to preserve ring structures and module structures in stablehomotopy categories. As an example, we compute some cellularizations ofEilenberg-Mac Lane objects in the homotopy category of spectra.

Singly generated ideals and coideals in monoidaltriangulated categoriesCarles Casacuberta

Universitat de [email protected]

In joint work with Gutiérrez and Rosický, we show that Vopěnka’s Principleimplies that localizing ideals are coreflective and colocalizing coideals arereflective in homotopy categories of stable monoidal combinatorial modelcategories. We also prove that every localizing ideal is singly generated un-der Vopěnka’s Principle. In this talk we will outline the arguments in ourproofs and give some evidence that the statement that colocalizing coidealsare singly generated might fail to be true in general.

67

Fibrations up to an equivalence and homotopycolimits

Lukáš VokřínekMasaryk University, [email protected]

Both quasifibrations and local homology fibrations behave nicely with re-spect to homotopy colimits. We give conditions on a class H of continuousmaps (with examples being weak equivalences and homology equivalencesas above) that determine a theory of fibrations up to an equivalence from Hthat has similar properties. Slight variations of these, universal H-fibrations,behave nicely with respect to pullbacks. We will also give a classificationresult for H-fibrations up to a natural notion of their equivalence.

68

Documents

Joint Mathematical Conference CSASC 2010iti.mff.cuni.cz › series › 2010 › 486.pdf · Austrian Mathematical Society in Vienna in November 2008. And nally CSASC 2010 is a full