the xxist rolf nevanlinna colloquiumsugawa/nevanlinna/abstract.pdf · 2015-05-11 · the xxist rolf...

The XXIst Rolf Nevanlinna Colloquium

Clock Tower Centennial HallKyoto University

September 7 - 11, 2009

Program and Abstracts as of September 6, 2009

Scientific Committee :

Hiroaki Aikawa (Hokkaido)Kari Astala (Helsinki)Brian Bowditch (Warwick)Peter Buser (EPFL, Lausanne)Ruth Kellerhals (Fribourg)Sadayoshi Kojima (TokyoTech), ChairKyoji Saito (IPMU, Kashiwa)Mika Seppala (Tallahassee/Helsinki)Mitsuhiro Shishikura (Kyoto)

Organizing Committee :

Peter Buser (EPFL, Lausanne)Michihiko Fujii (Kyoto)Ruth Kellerhals (Fribourg)Masashi Kisaka (Kyoto)Sadayoshi Kojima (TokyoTech), ChairYohei Komori (OCU, Osaka)Takashi Kumagai (Kyoto)Kyoji Saito (IPMU, Kashiwa)Mika Seppala (Tallahassee/Helsinki)Toshiyuki Sugawa (Tohoku, Sendai)

Host Organizations :

Department of Mathematics, Kyoto UniversityResearch Institute for Mathematical Sciences, Kyoto UniversityMathematical Society of Japan

Sponsoring Organizations :

Japan Society for the Promotion of ScienceSwitzerland National Science FoundationSakae Stunzi FoundationKyoto University FoundationGlobal COE Program of Kyoto UniversityGlobal COE Program of Tokyo Institute of TechnologyTopology Project

Schedule Overview

Monday, September 7th

9:45- Registration

Hall I & II & III10:15-10:20 Opening Remark10:20-11:20 Noguchi(Tea at Room III)11:45-12:45 Saksman(Lunch)

Hall I Hall II Hall III14:00-14:45 Ohshika Yamanoi Tyson15:00-15:45 Kapovich Peltonen Tsukamoto(Tea at Room III)16:15-17:00 Duchin Murata –17:15-18:00 Canary Shahgholian –18:15-20:00 Opening Reception

Tuesday, September 8th

Room IV9:30 - Poster Session

Hall I & II & III9:30-10:30 Minsky(Tea at Room III)11:00-12:00 Buser(Lunch)

Hall I Hall II Hall III14:00-14:45 Calegari Koch to be announced15:00-15:45 Shackleton Favre to be announced(Tea at Room III)

Hall I & II & III (mini course)16:15-17:00 Fukaya, I17:15-18:00 Fukaya, II

Wednesday, September 9th

Hall I & II & III9:30-10:30 Kinnunen(Tea at Room III)11:00-12:00 Lyubich12:00- Photo(Lunch)14:00- Excursion18:30- Banquet

Thursday, September 10th

Hall I & II & III9:30-10:30 Wilking(Tea at Room III)11:00-12:00 Kleiner(Lunch)

Hall I Hall II Hall III14:00-14:45 Souto Fujikawa Hencl15:00-15:45 Schleimer Markovic Hirata(Tea at Room III)

Hall I & II & III (mini course)16:15-17:00 Kobayashi, I17:15-18:00 Fukaya, III

Friday September 11th

Hall I & II & III9:30-10:30 Long(Tea at Room III)11:00-12:00 Kumagai(Lunch)

Hall I Hall II Hall III14:00-14:45 Kida Silhol Kim15:00-15:45 Shalen Shiga Peyerimhoff(Tea at Room III)

Hall I & II & III (mini course)16:15-17:00 Kobayashi, II17:15-18:00 Kobayashi, III18:00-18:05 Concluding Remark

Title of Talks :

Monday, September 7th

Hall I & II & III10:20-11:20 Junjiro Noguchi (Univ Tokyo)

Nevanlinna theory in higher dimensions and related Diophantineproblems

11:45-12:45 Eero Saksman (Helsinki)On random qc-maps

Hall I14:00-14:45 Ken’ichi Ohshika (Osaka Univ)

Classification of geometric limits and its applications15:00-15:45 Misha Kapovich (UC Davis)

Noncoherence of rank 1 lattices16:15-17:00 Moon Duchin (Ann Arbor)

The space of flat metrics17:15-18:00 Dick Canary (Ann Arbor)

Moduli spaces of hyperbolic 3-manifolds

Hall II14:00-14:45 Katsutoshi Yamanoi (Kumamoto Univ)

Some topics in Nevanlinna theory concerning derivatives ofmeromorphic functions

15:00-15:45 Kirsi Peltonen (TKK)Rational dynamics and horizontal conformal structures inthe Heisenberg group

16:15-17:00 Minoru Murata (TokyoTech)Structure of nonnegative solutions for parabolic equations andperturbation theory for elliptic operators

17:15-18:00 Henrik Shahgholian (KTH)An unstable free boundary problem

Hall III14:00-14:45 Jeremy Tyson (UI Urbana-Champaign)

Dimension comparison in sub-Riemannian geometry15:00-15:45 Masaki Tsukamoto (Kyoto Univ)

Instanton approximation, periodic ASD connections, andmean dimension

Tuesday, September 8th

Hall I & II & III9:30-10:30 Yair Minsky (Yale)

Dynamics on character varieties for free groups11:00-12:00 Peter Buser (EPFL)

Simple closed geodesics on a Riemann surface

Hall I14:00-14:45 Danny Calegari (CalTech)

Scl, sails and surgery15:00-15:45 Ken Shackleton (IPMU)

On the coarse geometry of Weil-Petersson’s metric onTeichmuller space

Hall II14:00-14:45 Sarah Koch (Harvard)

Thurston’s pullback map15:00-15:45 Charles Favre (Paris VII)

Integrability of rational maps

Hall III14:00-14:45 to be announced

15:00-15:45 to be announced

Hall I & II & III (mini course)16:15-17:00 Kenji Fukaya (Kyoto Univ)

Lagrangian Floer theory, I17:15-18:00 Kenji Fukaya (Kyoto Univ)

Lagrangian Floer theory, II

Wednesday, September 9th

Hall I & II & III9:30-10:30 Juha Kinnunen (TKK)

The De Giorgi measure and an obstacle problem related to minimalsurfaces in metric spaces

11:00-12:00 Mikhail Lyubich (SUNY Stony Brook)On the problem of local connectivity of the Mandelbrot set

Thursday, September 10th

Hall I & II & III9:30-10:30 Burkhard Wilking (Munster)

Sharp estimates for the Ricci flow11:00-12:00 Bruce Kleiner (NYU)

A new proof of Gromov’s theorem on groups of polynomial growth

Hall I14:00-14:45 Juan Souto (Ann Arbor)

(Non)-actions of the mapping class group15:00-15:45 Saul Schleimer (Warwick)

Canonical triangulations of Dehn fillings

Hall II14:00-14:45 Ege Fujikawa (Chiba)

Dynamics of holomorphic self-embeddings of Teichmuller spaces15:00-15:45 Vlad Markovic (Warwick)

Teichmuller theory of large surfaces and hyperbolic three manifolds

Hall III14:00-14:45 Stanislav Hencl (Charles Univ)

Homeomorphisms of bounded variation and homeomorphisms inthe Sobolev space

15:00-15:45 Kentaro Hirata (Akita)Boundary behavior of superharmonic functions satisfying nonlinearinequalities

Hall I & II & III (mini course)16:15-17:00 Ryoichi Kobayashi (Nagoya Univ)

Nevanlinna-Galois theory for pseudo-algebraic minimal surfaces, I17:15-18:00 Kenji Fukaya (Kyoto Univ)

Lagrangian Floer theory, III

Friday, September 11th

Hall I & II & III9:30-10:30 Darren Long (UCSB)

Commensurators of infinite volume hyperbolic groups11:00-12:00 Takashi Kumagai (Kyoto Univ)

Random walks on disordered media and their scaling limits

Hall I14:00-14:45 Yoshikata Kida (Kyoto Univ)

Measure equivalence rigidity of amalgamated free products15:00-15:45 Peter Shalen (UIC)

Trace fields, Margulis numbers and exponential growth

Hall II14:00-14:45 Robert Silhol (Montpellier)

Hyperbolic transformations and algebraic transformations onmoduli spaces

15:00-15:45 Hiroshige Shiga (TokyoTech)Holonomies and the slope inequalities of Lefschetz fibrations

Hall III14:00-14:45 Inkang Kim (KIAS)

Rigidity and flexibility of surface group representationsin semisimple Lie groups

15:00-15:45 Norbert Peyerimhoff (Durham)Spectral properties of Laplacians on the trihexagonal tiling

Hall I & II & III (mini course)16:15-17:00 Ryoichi Kobayashi (Nagoya Univ)

Nevanlinna-Galois theory for pseudo-algebraic minimal surfaces, II17:15-18:00 Ryoichi Kobayashi (Nagoya Univ)

Nevanlinna-Galois theory for pseudo-algebraic minimal surfaces, III

Title of Posters (Tuesday through Thursday at Room IV):

Laura Astola (Eindhoven University of Technology)Elementary Multi-scale Riemann-Finsler Geometry and Applications to Tensor ValuedMedical Image Analysis

Albert Clop (Universitat Autonoma de Barcelona)Stability of Calderon’s inverse conductivity problem in the plane

Iulia Elena Hirica(University of Bucharest)On semi-symmetric Riemann spaces

Katsuya Ishizaki (Nippon Institute of Technology)Entire functions of small order of growth

Riikka Kangaslampi (Helsinki University of Technology)Lattes-type uniformly quasiregular mappings and their Julia sets

Yu Kawakami (Kyusyu University)Ramification estimates for the Gauss map of various surfaces

Yohei Komori (Osaka City University)On holomorphic sections of certain holomorphic family of Riemann surfaces of genus two

Hideki Miyachi (Osaka University)The earthquake measure map is a homeomorphism

Gou Nakamura (Aichi Institute of Technology)Examples of the coordinates of genus 2-surfaces with regular fundamental octagon

Isamu Ohnishi (Hiroshima University)Qualitative analysis to distribution of types of DNA knots by use of topological invariants

Yusuke Okuyama (Kyoto Institute of Technology)Nonlinearity of morphisms in non-Archimedean and complex dynamics

Hiroki Sumi (Osaka University)Cooperation Principle in Random Complex Dynamics and Singular Functionson the Complex Plane

Hiroki Takahasi (Kyoto University)Toward a rigorous measure estimate of the stochastic parameter setfor the quadratic family

Kohei Ueno (Kyoto University)Weighted Green functions of polynomial skew products on C2

Xiangdong Xie (Georgia Southern University)Quasisymmetric maps on the ideal boundary of some negatively curved solvable Lie groups

Plenary Lectures

Simple closed geodesics on a Riemann surface

Peter Buser

Ecole Polytechnique Federale de Lausanne (EPFL)SB-IGAT-GEOM, Station 8, 1015 Lausanne, Switzerland

peter.buser@epfl.ch

It is well known that on a compact manifold of negative curvature the closed geodesics aredense and their number grows exponentially as a function of their lengths. In dimensiontwo most of these geodesics have self-intersections, and those without, i.e. the simpleones, are much more sparse in various ways. The lecture will overview old and newresults, some theoretical, some algorithmic. There will also be some pictures such asthe one below which shows 72 simple closed geodesics of the same lengths drawn on afundamental domain of the Fuchsian group F48.

The De Giorgi measure and an obstacle problem related to minimal surfacesin metric spaces

Juha Kinnunen

Helsinki University of TechnologyInstitute of Mathematics, P.O. Box 1100,

FI-02015 Helsinki University of Technology, Finlandjuha.kinnunen@tkk.fi

We discuss the existence of a set with minimal perimeter that separates two disjoint setsin a metric measure space equipped with a doubling measure and supporting a Poincareinequality. A measure constructed by De Giorgi (see [1] and [2]) is used to state a relaxedproblem, whose solution coincides with the solution to the original problem for measuretheoretically thick sets. Moreover, we show that the De Giorgi measure is comparableto the Hausdorff measure of codimension one using a metric space version of the boxinginequality in [3]. The theory of functions of bounded variation in metric spaces is usedextensively.

Joint work with Riikka Korte (University of Helsinki), Nageswari Shanmugalingam (University of Cincin-nati) and Heli Tuominen (University of Jyvaskyla), see [4].

References

[1] E. De Giorgi. Problemi di superfici minime con ostacoli: forma non cartesiana. (Italian) Boll. Un.Mat. Ital. (4) 8 (1973), suppl. no. 2, 80–88.

[2] E. De Giorgi, F. Colombini and L.C. Piccinini. Frontiere orientate di misura minima e questionicollegate. Scuola Normale Superiore, Pisa, 1972.

[3] J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen. Lebesgue points and capacities viaboxing inequality in metric spaces. Indiana Univ. Math. J. 57(1) (2008), 401–430.

[4] J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen. The De Giorgi measure and anobstacle problem related to minimal surfaces in metric spaces. Submitted.

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A new proof of Gromov’s theorem on groups of polynomial growth

Bruce Kleiner

Courant Institute of Mathematical Sciences, New York University251 Mercer Street

New York, NY 10012-1185bkleiner@cims.nyu.edu

In 1981 Gromov showed that any finitely generated group of polynomial growth contains afinite index nilpotent subgroup. This has a variety of applications, ranging from dynamics

to probability theory. Gromov’s proof was based in part on a beautiful rescaling argument,and the Montgomery-Zippin solution to Hilbert’s fifth problem on topological groups.The purpose of the lecture is to describe a new, much shorter, proof of Gromov’s theo-rem, based on harmonic maps instead of the Montgomery-Zippin theory. I will begin byreviewing the history of Gromov’s theorem, and some of its applications.

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Random walks on disordered media and their scaling limits

Takashi Kumagai

Department of Mathematics, Kyoto UniversityKyoto 606-8502, Japan

kumagai@math.kyoto-u.ac.jp

One of the motivations of random walks on disordered media comes from the work ofmathematical physicists. They analyzed properties of disordered media such as the struc-ture of polymers and networks, growth of crystals – see [2] for a survey and bibliography.In this talk, we will summarize some mathematical work on the behavior of random walkson disordered media and their scaling limits. Especially, we will consider the followingobjects.

1 Random walks on fractal graphs

2 Random walks on stochastic models such as percolation clusters(cf. [1, 3])

We will estimate the following quantities and observe ‘anomalous’ behavior of the randomwalks.

i) Average of the exit time from the ball of radius R

ii) Spectral dimension (Roughly speaking, random walk Xn has spectral dimensionds at x if P x(X2n = x) ³ n−ds/2.)

Note that for simple random walk on Zd, i) is R2 and ds = d for ii).We will also discuss a well-known Alexander-Orbach conjecture that asserts ds = 4/3 forrandom walk on the critical percolation cluster on Zd.

References

[1] M.T. Barlow, A.A. Jarai, T. Kumagai and G. Slade, Random walk on the incipient infinite clusterfor oriented percolation in high dimensions, Commun. Math. Phys. 278 (2008), 385–431.

[2] D. Ben-Avraham and S. Havlin, Diffusion and reactions in fractals and disordered systems, CambridgeUniversity Press, Cambridge, 2000.

[3] G. Kozma and A. Nachmias, The Alexander-Orbach conjecture holds in high dimensions, Preprint2008 (available at arXiv:0806.1442).

Minimal currents in codimension larger than one

Camillo De Lellis

Institut fur Mathematik, Universitat ZurichWinterthurerstrasse 190, CH-8057 Zuerich (CH)

delellis@math.uzh.ch

(Canceled due to health condition)

The famous Plateau problem consists in finding the surface of least area spanning a givencontour in a given Riemannian manifold. In a celebrated work of the sixties (see [2]),Federer and Fleming introduced the theory of currents, probably the most successfulframework for proving the existence of a solution using the direct methods of the calculusof variations.Due to the work of De Giorgi, Almgren, Simons, Federer and Simon (among others),there is a quite satisfactory regularity theory for area-minimizing currents of codimension1, which are in fact regular up to a singular set of codimension 7. In codimension higherthan 1 the situation is dramatically different, because branching (and hence singularitiesof codimension 2) can occur. A deep regularity theory has been developed 30 years agoby Almgren and it is contained in a monograph of 950 pages (see [1]).In two recent works we have found much shorter proofs for a good portion of Almgren’smonograph. These results come as a combination of his ideas with new techniques, theygive some new points of view on Almgren’s theory and provide links to other topics inanalysis.

Joint work with Emanuele Spadaro (Universitat Zurich).

References

[1] F. J. Almgren, Jr., Almgren’s big regularity paper. Q-valued functions minimizing Dirichlet’s inte-gral and the regularity of area-minimizing rectifiable currents up to codimension 2, World ScientificMonograph Series in Mathematics, 1. World Scientific Publishing Co., Inc., River Edge, NJ, 2000.

[2] H. Federer, W. H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458-520.

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Commensurators of infinite volume hyperbolic groups.

D. D. Long

Department of Mathematics,University of California

Santa Barbara, CA 93106, USAlong@math.ucsb.edu

There is a celebrated theorem of Margulis which describes the commensurator of a latticein a semi-simple Lie group. This has motivated much study of the commensurator of vari-ous classes of groups, together with its role in geometry and topology. We will discuss thecase of infinite co-volume Kleinian groups which are not free groups. The commensuratoris shown to be discrete and a lattice in the case of a virtual fibre group.

Joint work with C. J. Leininger (University of Illinois at Urbana-Champaign) and A. W. Reid (Universityof Texas, Austin).

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On the problem of local connectivity of the Mandelbrot set

Misha Lyubich

Mathematics Department, Stony Brook UniversityStony Brook, NY 11794-3651, USA

mlyubich@math.sunysb.edu

The problem of local connectivity of the Mandelbrot set (abbreveated as “MLC”) is thecentral open question about dynamics of quadratic polynomials. In the talk, we willdescribe recent developments in this problem. It is based on a joint work with JeremyKahn.

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Dynamics on character varieties for free groups

Yair Minsky

Department of MathematicsYale University

New Haven, CT 06520-8283yair.minsky@yale.edu

The group Out(Fn) of outer automorphisms of the free group on n letters acts naturally onthe PSL(2,C) character variety for Fn. We discuss some curious features of this action.In particular, the set of Schottky characters is not the domain of discontinuity, and weintroduce a strictly larger invariant open set (denoted the primitive stable characters) onwhich the action is properly discontinuous. In particular this implies that the action is notergodic on the characters of representations with dense image. We do identify a subsetof the dense characters, called the redundant characters, on which the action is ergodicbut is still not weakly mixing (joint with T. Gelander). Similar definitions can be givenin the setting of surface groups and their mapping class groups, but there the situation isnot as well understood.

Nevanlinna theory in higher dimensions and related Diophantine problems

Junjiro Noguchi

Graduate School of Mathematical Sciences, The University of Tokyo3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan

noguchi@ms.u-tokyo.ac.jp

I will begin with abc-Conjecture: a+b = c. For (S-) units in a number field this was studiedby Siegel and by many, later. For units in the ring of entire functions in one variable it isPicard’s Theorem. Its quantitative version is Nevanlinna’s Second Main Theorem (SMT).Nevanlinna theory may be taken in two senses: one is an intersection theory and the otheris an approximation theory. The First Main Theorem (FMT) provides the frame work forthe intersection theory. Here I will give the FMT with respect to coherent ideal sheaveson a compact complex manifold M and for a holomorphic mapping (curve) f : C → M .The obtained order function of f will play the central role.I will survey the development of the Nevanlinna theory in higher dimensions followingafter the history, and then discuss some new topics and applications:(i) Classics due to Nevanlinna, Cartan, Weyls, Ahlfors, Stoll.(ii) Equidimensional SMT due to Griffiths et al.(iii) Log Bloch-Ochiai Theorem and an inequality of SMT type:(iv) Nochka and Corvaja-Zannier-Min Ru’s improvement:(v) Yamanoi’s abc-Theorem;(vi) SMT for a semi-abelian variety due to N.-Winkelmann-Yamanoi:

References

[1] J. Noguchi, J. Winkelmann, and K. Yamanoi, The second main theorem for holomorphic curves intosemi-abelian varieties, Acta Math. 188 No. 1 (2002), 129-161.

[2] J. Noguchi, Value distribution and distribution of rational points, Spectral Analysis in Geometryand Number Theory, Ed. M. Kotani et al, Contemp. Math. 484, pp. 165–176, Amer. Math. Soc.Providence, Rhode Island, 2009.

[3] J. Noguchi, J. Winkelmann, and K. Yamanoi, The second main theorem for holomorphic curves intosemi-abelian varieties II, Forum Math.20 (2008), 469–503.

[4] P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math. 1239,Springer-Verlag, 1987.

[5] K. Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192(2004), 225-294.

On random qc-maps

Eero Saksman

Department of Mathematics and Statistics, University of HelsinkiP.O. Box 68, 00014 University of Helsinki, Finland

eero.saksman@helsinki.fi

We will consider some results on random quasi-conformal maps. These include homog-enization results for random qc-maps (joint with K. Astala, S. Rohde and T. Tao), andSLE-inspired random weldings (joint with Astala, P. Jones and A. Kupiainen).

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Sharp estimates for the Ricci flow

Burkhard Wilking

Westfalische Wilhelms-Universitat Munsterwilking@math.uni-muenster.de

We show that every adjoint orbit of SO(n,C) gives rise to a Ricci flow invariant curva-ture condition. Although the proof is just a few lines it recovers all previously knowninvariant nonnegativity conditions. Moreover a similar statement for the complexified Liealgebra of the isometry group of Rn gives rise to Harnack inequalities, including Bren-dles generalization of Hamiltons Harnack inequality to manifolds of nonnegative complexcurvature.The second part of the talk is a progress report on joint work with Christoph Bohmconcerning manifolds with scal ≥

√2(n− 1)(n− 2)‖Rw‖. We show that this condition

is invariant under Ricci flow in dimensions above n ≥ 12. We show its invariant undersurgery of codimension > n

2+1. Finally we indicate we can deform the condition through

invariant conditions to constant curvature. Namely the conditions scal ≥ 0 combinedwith (n − 2 + x)‖R‖2 ≤ ‖Ric‖ are also invariant under the Ricci flow for n ≥ 12 andx ∈ [0, n]. Finally we speculate why this should help with the singularity analysis.

Mini Courses

Lagrangian Floer theory

Kenji FUKAYA

Department of Math., Faculty of Science, Kyoto University.Kyoto 606-8502, Japan

fukaya@math.kyoto-u.ac.jp

This is a survey of Lagrangian Floer theory together with an application to Mirror sym-metry. Below is a tentative plan of the talk.

1st talk: I want to illustrate definition and application in the simplest case. I plan toexplain a calculation in the case of plumbing two cotangent bundle and taking Lagrangiansurgery. Besides the definition, two points will be explained. Relation of the LagrangianFloer homology to usual homology. Relation of Lagrangian surgery to the moduli spaceof pseudo-holomorphic disc. ([3].)

2nd talk: In the first talk, we considered the case of exact Lagrangian submanifold,which was the case originally studied by Floer. If one goes beyond exact case, we needto introduce Novikov ring, deformation theory, algebraic structure based on A infinityalgebra. In this talk I want to illustrate those points by taking the case of P 2 and itsblow up as an example. ([1, 2].)

3rd talk: Starting around 1998, there had been a project to prove homological mirrorsymmetry conjecture by using family of Floer homologies. In this talk I want to surveythis project. Especially I will explain how to realize it in the 2 dimensional case, using‘rigid analytic family of Floer homologies’ (which is related to the material of the 2ndtalk) and ‘compactification of this family to the singular fiber by studying Lagrangiansurgery’, (which is explained of the 1st talk).Joint work with Y.G.-Oh, H.Ohta and K. Ono.

References

[1] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer theory – Anomaly andobstructions to appear in AMS/IP studies in advanced Math.

[2] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds I, toappear in Duke. Math. J., arXiv:0802.1703.

[3] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono Lagrangian surgery and metamorphosis of pseudo-holomorphic polygons, in preparation. A preliminary version is available at http:// www.math.kyoto-u.ac.jp/∼fukaya/fukaya.html as ‘Chapter 10’ of [1].

Nevanlinna-Galois Theory for Pseudo-Algebraic Minimal Surfaces

Ryoichi Kobayashi

Graduate School of Mathematics, Nagoya UniversityNagoya 464-8602, Japan

ryoichi@math.nagoya-u.ac.jp

This lecture is based on the preprint [2]. We have introduced the concept of pseudo-algebraic minimal surfaces in [1] and studied them from algebro-geometric view point.In this lecture, we introduce the Nevanlinna analogue of [1]. We hope that this attemptopens a new possibility of the application of Nevanlinna Theory to problems in geometry.We propose Nevanlinna theoretic interpretations of two basic aspects in the theory of(pseudo-)algebraic minimal surfaces. One is the Cohn-Vossen inequality and the other isthe period condition. The Weierstrass data lifted to the universal cover D (the unit disk)consists of the lifted Gauss map g : D → P1 and the lifted 1-form partner h] : D → TP1

(h] := (g, h)).The Nevanlinna analogue of the Cohn-Vossen inequality is the estimate for the small-est number κ satisfying

0exp(κTg(t))dt = ∞. This is equivalent to the collective

Cohn-Vossen inequality for all pieces of fundamental domains cut out by concentric disksD(r)0<r<1. The Nevanlinna analogue of the Cohn-Vossen inequality holds for pseudo-algebraic minimal surfaces.The period condition (i.e., the condition for the minimal surface to close up) [P ] holdonly for algebraic minimal surfaces. The condition [P ] is equivalent to the invarianceof the modulus of a certain holomorphic function on D which is constructed from theWeierstrass data (g, h). The condition [P ] is then translated into a certain system ofinequalities arising from applying Nevanlinna’s Lemma on Logarithmic Derivative (LLDin short) to the above mentioned holomorphic function. The point is that the algebraicityof our minimal surface allows us to define two kinds of proximity functions and LLDencodes the relationship between them.

References

[1] Y. Kawakami, R. Kobayashi and R. Miyaoka, The Gauss map of pseudo-algebraic minimal surfaces,Forum Math. 20, No. 6 (2008), 1055-1069.

[2] R. Kobayashi and R. Miyaoka, Nevanlinna-Galois theory for pseudo-algebraic minimal surfaces andvalue distribution of the Gauss map, preprint, 2009.

Invited Talks

Scl, sails and surgery

Danny Calegari

California Institute of TechnologyPasadena, CA 91125 USAdannyc@its.caltech.edu

Given a group G and an element g ∈ [G,G], the commutator length of g, denoted cl(g), isthe smallest number of commutators in G whose product is g, and the stable commutatorlength of g is the limit scl(g) := limn→∞ cl(gn)/n. Commutator length in a group extendsin a natural way to a pseudo-norm on the real vector space of 1-boundaries (in grouphomology), and should be thought of as a kind of relative Gromov-Thurston norm. Weshow that the problem of computing stable commutator length in free products of abeliangroups reduces to a (finite dimensional) integer programming problem. Moreover, certainfamilies of elements in such groups (i.e. those obtained by surgery on some element in abigger group) give rise to families of integer programming problems that are related inexplicit ways. In particular one can use this to establish the existence of limit points inthe range of scl in such groups, and produce elements whose stable commutator lengthis congruent to any rational number modulo the integers. This technology relates stablecommutator length to the theory of multi-dimensional continued fractions, and Kleinpolyhedra, and suggests an interesting conjectural picture of scl in free groups. See [1, 2]for background and more details.

References

[1] D. Calegari, scl, MSJ Memoirs, 20. Mathematical Society of Japan, Tokyo, 2009

[2] D. Calegari, Scl, sails and surgery, preprint arXiv:0907.3541

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Moduli spaces of hyperbolic 3-manifolds

Richard D. CanaryDepartment of Mathematics

University of Michigan at Ann Arbor

In the study of 2-dimensional hyperbolic manifolds, one often studies the Teichmullerspace of marked hyperbolic surfaces homeomorphic to a fixed surface. The mapping classgroup of the surface acts properly discontinuously on Teichmuller space and its quotientmoduli space is an orbifold.

In 3-dimensional hyperbolic geometry, one studies the space AH(M) of marked hyperbolic3-manifolds homotopy equivalent to a fixed compact 3-manifold M. The mapping classgroup of M often fails to act properly discontinuously on AH(M) and its quotient is oftena rather bad topological space. However, the topological properties of this quotient arerelated to the topological properties of M. This talk will describe joint work with PeterStorm.

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The space of flat metrics

Moon Duchin

University of MichiganAnn Arbor, MI

mduchin@umich.edu

We define Flat(S), the space of flat metrics on a surface of finite type. We obtainresults about the length spectrum of flat surfaces, and about a compactification which isanalogous to the Thurston compactification of Teichmuller space.

Joint work with Christopher Leininger and Kasra Rafi.

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Integrability of rational maps.

Charles Favre

CNRS-Institut de Mathematiques de Jussieu (Paris)Rue du Chevaleret, Paris, France

favre@math.jussieu.fr

In dynamical systems, the notion of integrability refers to a system for which one can insome way compute all its iterates explicitly. This notion is however very hard to formalizeand most of the time, one is looking for sufficient conditions ensuring integrability.We shall discuss classification results of rational maps of the complex projective planepreserving algebraic structures. In particular we shall focus on those maps preserving aholomorphic foliation or a web. We shall see that these maps all exhibit some kind ofintegrability property.

Joint work with Jorge Vitorio Pereira (IMPA-Rio).

References

[1] C. Favre and J. V. Pereira, Foliations invariant by rational maps, arxiv.

Dynamics of holomorphic self-embeddings of Teichmuller spaces

Ege Fujikawa

Department of Mathematics and Informatics, Chiba University1-33 Yayoi-cho, Inage, Chiba 263-8522, Japan

fujikawa@math.s.chiba-u.ac.jp

We consider a Riemann surface with a non-injective holomorphic self-cover. Coveringis always meant to be unbranched and unlimited. Non-trivial cases appear only whenRiemann surfaces are of topologically infinite type. Construction of self-covering hasbeen given in [2]. Also self-covering naturally appears in the theory of complex dynamics,see [3]. We first give the structure theorem of non-injective holomorphic self-covers ofRiemann surfaces, which is a basic tool to clarify the nature of topologically infiniteRiemann surfaces.The Teichmuller space is a deformation space of a complex structure of a Riemann surface.Every non-injective holomorphic self-cover of a Riemann surface induces a non-surjectiveholomorphic self-embedding of its infinite-dimensional Teichmuller space. We investigatethe dynamics of such self-embeddings by applying our structure theorem and examine thedistribution of its isometric vectors on the tangent bundle over the Teichmuller space. Wealso extend our observation to quasiregular self-covers of Riemann surfaces and give ananswer to a certain problem on quasiconformal equivalence to a holomorphic self-cover.

Joint work with Katsuhiko Matsuzaki (Okayama University) and Masahiko Taniguchi (Nara Women’sUniversity).

References

[1] E. Fujikawa, K. Matsuzaki and M. Taniguchi, Dynamics on Teichmuller spaces and self-covering ofRiemann surfaces, Math. Z. 260 (2008), 865–888.

[2] T. Jørgensen, A. Marden and C. Pommerenke, Two examples of covering surfaces, In: Riemannsurfaces and related topics, Ann. Math. Studies 97 (1978), 305–319.

[3] C. McMullen and D. Sullivan, Quasiconformal homeomorphisms and dynamics III. The Teichmullerspace of a holomorphic dynamical system, Adv. Math. 135 (1998), 351–395.

Homeomorphisms of bounded variation and homeomorphisms in the Sobolevspace

Stanislav Hencl

Department of Mathematical Analysis, Charles UniversitySokolovska 83, 186 00 Prague 8, Czech Republic

hencl@karlin.mff.cuni.cz

In this talk we address the following problem. Let Ω ⊂ Rn be an open set, f : Ω → Rn

be a homeomorphism in the Sobolev space f ∈ W 1,p(Ω, Rn) for some p ≥ 1. Underwhich assumptions on f can we conclude that the inverse is weakly differentiable, i.e.f−1 ∈ W 1,1

loc (f(Ω), Ω)? (or at least f−1 ∈ BV )?Our main two theorems are the following.

Theorem 1. Let Ω ⊂ Rn be an open set. Suppose that f ∈ W 1,n−1(Ω, Rn) is ahomeomorphism. Then f−1 ∈ BVloc(f(Ω), Rn).

Moreover f−1 ∈ W 1,1 if we assume that f has finite distortion, i.e. the jacobian Jf isstrictly positive almost everywhere on the set where the derivative |Df | does not vanish.

Theorem 2. Let Ω ⊂ Rn be an open set. Suppose that f ∈ W 1,n−1(Ω, Rn) is a homeo-morphism of finite distortion. Then f−1 ∈ W 1,1

loc (f(Ω), Rn) and has finite distortion.

We will also discuss various counterexamples showing the sharpness of the above results.Some results about the regularity of the composition u f−1 and about the regularity ofthe second weak derivatives will be also mentioned.

Joint work with Marianna Csornyei (University College London) and Jan Maly (Charles University). Itis also based on some previous work with Pekka Koskela (University of Jyvaskyla) and Jani Onninen(University of Jyvaskyla).

References

[1] Marianna Csornyei, Stanislav Hencl, Jan Maly: Homeomorphisms in the Sobolev space W 1,n−1, toappear in J. Reine Angew. Math.

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Boundary behavior of superharmonic functions satisfying nonlinearinequalities

Kentaro Hirata

Faculty of Education and Human Studies, Akita UniversityAkita 010-8502, Japan

hirata@math.akita-u.ac.jp

In this talk, we discuss the class of positive superharmonic functions satisfying a nonlinearinequality, which includes positive solutions of semilinear elliptic equations like as −∆u =V up. Many properties may depend on the exponent of nonlinearity. We give a boundarygrowth estimate, which is applicable to obtain a Harnack type inequality and a refinementof the Naım-Doob limit theorem. Also, a Fatou type theorem and a Littlewood typetheorem concerning the boundary behavior are presented.

References

[1] K. Hirata, Boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniformdomains, to appear in Trans. Amer. Math. Soc.

[2] K. Hirata, Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth do-mains, preprint.

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Noncoherence of rank 1 lattices

Michael Kapovich

Department of Mathematics, University of California, Davis1 Shields Ave., Davis, CA 95616, USA

kapovich@math.ucdavis.edu

A group Γ is called coherent if every finitely-generated subgroup of Γ is also finitelypresentable. Among lattices in semisimple Lie groups, lattices in SL(2,R) and SL(2,C)are known to be coherent. In this talk I will discuss the following

Conjecture. If G is a semisimple Lie group which is not locally isomorphic to SL(2,R)and SL(2,C), then every lattice Γ < G is non-coherent.

This conjecture is wide-open for Lie groups of rank ≥ 2, e.g., even for SL(3,Z). I willsketch a proof of this conjecture for some classes of lattices in rank 1 Lie groups, i.e.,isometry groups of the real-hyperbolic spaces, complex-hyperbolic spaces, etc. For in-stance, assuming the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds,every arithmetic lattice Γ in SO(n, 1), n ≥ 4, n 6= 7, is non-coherent. If Γ < SU(2, 1) is auniform lattice with infinite abelianization, then Γ is non-coherent. In particular, everytype I uniform arithmetic lattice in SU(n, 1), n ≥ 2, is non-coherent.

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Measure equivalence rigidity of amalgamated free products

Yoshikata Kida

kida@math.kyoto-u.ac.jp

Two discrete countable groups Γ and Λ are said to be measure equivalent (ME) if one hasa standard Borel space (Σ,m) with a σ-finite positive measure and a measure-preservingaction of Γ×Λ on (Σ,m) such that there exist Borel subsets X,Y ⊂ Σ satisfying the equa-tion Σ =

⊔γ∈Γ γY =

⊔λ∈Λ λX up to m-null sets. It is known that ME is an equivalence

relation between discrete countable groups and that lattices in a locally compact secondcountable group are ME to each other. The aspect of rigidity is one of the major focusesof recent advance in the study of ME. Among other things, fascinating rigidity is discov-ered for lattices in SL(n,R) with n ≥ 3, and for mapping class groups of non-exceptionalcompact orientable surfaces. The latter groups are known to be ME rigid, where a groupΓ is said to be ME rigid if any group which is ME to Γ is virtually isomorphic to Γ. Thistalk presents a construction of ME rigid groups given by amalgamated free products oftwo rigid groups in the sense of ME. A class of amalgamated free products is introduced,and groups which are ME to a group in the class are described.

References

[1] Y. Kida, Rigidity of amalgamated free products in measure equivalence theory, preprint,arXiv:0902.2888.

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Rigidity and Flexibility of surface group representations in semisimple Liegroups

Inkang KIM

School of Mathematics, KIASHoegiro 87, Seoul 130-722, Korea

inkang@kias.re.kr

We will survey some recent results about rigidity vs flexibility phenomena of surface groupand complex hyperbolic group representations in semisimple Lie groups. See [1] and [2].

Joint work with Pierre Pansu (Orsay).

References

[1] I. Kim, B. Klingler and P. Pansu, Local quaternionic rigidity for complex hyperbolic lattices, preprint.

[2] I. Kim and P. Pansu, Density of Zariski density for surface groups, preprint.

Thurston’s Pullback Map

Sarah C. Koch

Department of Mathematics, Harvard UniversityOne Oxford Street, Cambridge MA 02138

kochs@math.cornell.edu

Let f : P1 → P1 be a rational map with finite postcritical set Pf . Thurston showed thatf induces a holomorphic map σf : Teich(P1, Pf ) → Teich(P1, Pf ) of the Teichmuller spaceto itself. The map σf fixes the basepoint corresponding to the identity map id : (P, Pf ) →(P1, Pf ). We give examples of such maps f showing that the following cases may occur:

1. σf has an attracting fixed point, the image of σf is open and dense in Teich(P1, Pf )and σf : Teich(P1, Pf ) → σf

(Teich(P1, Pf )

)is a covering map,

2. σf has a superattracting fixed point, the image of σf is Teich(P1, Pf ) and σf :Teich(P1, Pf ) → Teich(P1, Pf ) is a ramified Galois covering map, or

3. the map σf is constant.

Joint work with Xavier Buff (Universite Paul Sabatier), Adam Epstein (University of Warwick), KevinPilgrim (Indiana University).

References

[1] X. Buff, A. Epstein, S. Koch, & K. Pilgrim, On Thurston’s Pullback Map, Complex Dynamics,Families and Friends, edited by D. Schleicher, in press. Adv. Math. 158: (2001) 154-168.

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Teichmuller theory of large surfaces and hyperbolic three manifolds

Vladimir MarkovicMathematics Department, University of Warwick

CV4 7AL, Coventry UKv.markovic@warwick.ac.uk

I will talk about my recent work with J. Kahn on the Ehrenpreis conjecture and immersingnearly geodesic surfaces in hyperbolic three manifolds.

Structure of nonnegative solutions for parabolic equations and perturbationtheory for elliptic operators

Minoru Murata

Department of Mathematics, Tokyo Institute of TechnologyTokyo 152-8551, Japan

minoru3@math.titech.ac.jp

This talk is concerned with structure of nonnegative solutions for parabolic equations,perturbation theory for elliptic operators, and their relations. We consider a locallyuniformly elliptic operator L on a noncompact domain D of a Riemannian manifold M .Let 0 < T ≤ ∞. Our main problem is:Determine all nonnegative solutions of a parabolic equation

(∂t + L)u = 0 in D × (0, T ). (1)

This problem is closely related to [UP] (i.e., uniqueness of nonnegative solutions of thecorresponding Cauchy problem) and ”smallness” of the constant function 1 on D in com-parison with L (here 1 is regarded as a perturbation of L). It is known that if [UP] holds,any nonnegative solution of (1) is represented by an integral of the fundamental solutionwith respect to a Borel measure on D. In the case [NUP] (i.e., [UP] does NOT hold),however, no explicit and general answer to our problem has been given until recently (see[3]). The aim of this talk is to give a complete answer under the condition [SSP] (i.e.,1 is a semismall perturbation of L), which is known to imply [NUP]. We also show byseveral concrete examples that [SSP] is a rather optimal sufficient condition for [NUP],by exploiting sharp and general sufficient conditions for [SSP] and [NUP] (see [1, 2]).

References

[1] A. Ancona, First eigenvalues and comparison of Green’s functions for elliptic operators on manifoldsor domains, J. Analyse Math. 72, (1997), 45-92.

[2] K. Ishige and M. Murata, Uniqueness of nonnegative solutions of the Cauchy problem for parabolicequations on manifolds or domains, Ann. Scuola Norm. Sup. Pisa, 30, (2001), 171-223.

[3] P. J. Mendez-Hernandez and M. Murata, Semismall perturbations, semi-intrinsic ultracontractivity,and integral representations of nonnegative solutions for parabolic equations, J. Funct. Anal. to appear.

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Classification of geometric limits and its applications

Ken’ichi Ohshika

Department of Mathematics, Graduate School of Science, Osaka UniversityToyonaka, Osaka 560-0043, Japanohshika@math.sci.osaka-u.ac.jp

I shall show that a geometric limit of any (algebraically convergent) sequence in thedeformation space of a finitely generated Kleinian groups can be embedded topologicallyin a compact 3-manifold, and its isometry type is determined by the topological type andthe end invariants. (The assumption of algebraic convergence is redundant in the case ofsurface Kleinian groups.) As for applications of this result, I shall talk about a sufficientcondition for a sequence of Kleinian groups diverges, and the growth of volumes for acertain family of surface bundles.

Partially joint work with Teruhiko Soma (Tokyo Metropolitan University).

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Rational dynamics and horizontal conformal structures in the Heisenberggroup

Kirsi Peltonen

Department of Mathematics, Helsinki University of TehnologyFIN-02015 Espoo, Finlandkirsi.peltonen@tkk.fi

We report on recent developments of a higher dimensional real counterpart of the iterationtheory of rational functions in the extended complex plane. A subclass of quasiregularmappings, called uniformly quasiregular mappings (UQR) have been studied first in [1]acting on the Riemann sphere and further for example in [2] acting on a compact Rie-mannian manifold so that all the iterates of the mapping are K-quasiregular for fixeddistortion K, independently of the number of iterates.We describe some basic constructions that can be extended further to the sub-Riemanniansetting in the Heisenberg group. Moreover, we construct a non-injective uniformly quasireg-ular mapping g acting on the one point compactification of the Heisenberg group equippedwith a sub-Riemannian metric. We further show that there exits a measurable horizontalconformal structure which is equivariant under the semigroup Γ generated by g. This isequivalent to the existence of a equivariant CR structure. This fact is interesting also fromthe point of view of several complex variables since it was known already for Poincare [3]that the only semigroup of CR maps with respect to the standard CR structure must berestrictions to the sphere of a subgroup of the conformal automorphisms of the unit ballin the Euclidean space of two complex variables.

Joint work with Zoltan Balogh and Katrin Fassler (Bern University).

References

[1] T. Iwaniec and G. Martin Quasiregular semigroups, Ann. Acad. Sci. Fenn. 21, No. 2 (1996), 241-254.

[2] K. Peltonen, Examples of uniformly quasiregular mappings, Conform. Geom. Dyn. 3, (1999), 158-163.

[3] H. Poincare, Les fonctions analytiques de deux variables et la representation conforme, Rend. Circ.Math. Palermo II No. 23 (1907), 185-220.

Spectral properties of Laplacians on the trihexagonal tiling

Norbert Peyerimhoff

Department of Mathematical Sciences, Durham UniversitySouth Road, Durham, DH1 3LE, United Kingdom

norbert.peyerimhoff@durham.ac.uk

The trihexagonal tiling (also called “Kagome lattice”) is a particular periodic tessellationof the plane. We are particularly interested in the spectral distribution function (alsocalled “Integrated Density of States”) of different Laplacians on this graph. In the firstpart of the talk we consider the jumps and their heights of this function of the combina-torial Laplacian on the trihexagonal tiling and of the Kirchhoff Laplacian on the inducedequilateral metric graph. We also explain their connection with finitely/compactly sup-ported eigenfunctions. In the second part of the talk we ”randomize” this graph byallowing the lengths of the edges of the metric graph to vary in a particular way. It turnsout that the almost sure spectral distribution function of the Kirchhoff Laplacian on thisrandom model no longer exhibits jumps and is even Lipschitz continuous. This is aninstance where randomness improves the regularity of this function. The results hold truein much more generality and the trihexagonal tiling serves only as a model of illustration.

Joint work with Daniel Lenz (Universitat Jena, Germany), Olaf Post (Humboldt-Universitat zu Berlin,Germany) and Ivan Veselic (TU Chemnitz, Germany).

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Canonical triangulations of Dehn fillings

Saul SchleimerUniversity of Warwick

CV4 7AL, Coventry UKs.schleimer@warwick.ac.uk

The canonical triangulations of a cusped hyperbolic three-manifold are delicate combina-torial invariants. They have been computed for a few special families, namely puncturedtorus bundles over the circle and two-bridge knot complements. We will discuss how thecanonical triangulation of a twice cusped manifold M determines, in many cases, thecanonical triangulation of the Dehn fillings of M.

This is joint work with Francois Gueritaud.

On the coarse geometry of Weil-Petersson’s metric on Teichmuller space

Ken Shackleton

IPMU, University of TokyoKashiwa, Chiba, Japank314j159s@gmail.com

We discuss the synthetic geometry of the pants graph in comparison with the Weil-Petersson metric, whose geometry the pants graph coarsely models following work ofBrock’s. We also restrict our attention to the 5-holed sphere and the 2-holed torus,studying the Gromov bordification and the dynamics of pseudo-Anosov mapping classes.

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An unstable free boundary problem

Henrik Shahgholian

Dept. of Math. KTHStockholm, Sweden

henriksh@math.kth.se

In this talk I will present some recent developments for free boundary problems of theobstacle type, where the equation is unstable

∆u = −χu>0.

For this problem, classical well-known methods fail, and we need to introduce new methodsand tools for analyzing the so called singular points u = ∇u = 0.In particular, we show that in two space dimensions these points are isolated, and threespace dimension they are either isolated or can be embedded into a C1 manifold.

Joint work with J. Andersson, and G.S. Weiss

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Trace fields, Margulis numbers and exponential growth

Peter Shalen

Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at Chicago

Chicago, IL, USAshalen@math.uic.edu

I will describe interactions between the number-theoretic properties of a closed, orientablehyperbolic 3-manifold M and its quantitative geometric properties. The number-theoreticproperties in question are captured by the trace field K of M , while the geometric prop-erties in question are expressed through such quantities as the Margulis number andestimates for tube radii about closed geodesics. The link between the two involves thealgebra of finite simple groups, deep number-theoretic results due to Mahler and Siegel,and results from [3] and [1].The following result, which improves on a preliminary version established in the preprint[2], illustrates the method:Theorem. Let K be any number field. Then for all but finitely many closed, orientablehyperbolic 3-manifolds M with trace field K such that H1(M ; Zp) is trivial for p = 2, 3,the number 0.183 is a Margulis number for M .I am working on a partial improvement of this result in which the homological restrictionis removed at the expense of replacing 0.183 by a somewhat smaller number. I will discussthis.The method used to prove the theorem stated above can also be used to prove an analogousresult in which the Margulis number of M is replaced by the uniform exponential growthrate of π1(M).

References

[1] J. W. Anderson, R. D. Canary, M. Culler and P. B. Shalen, Free Kleinian groups and volumes ofhyperbolic 3-manifolds, J. Differential Geom. 43 No. 4 (1996), 738–782.

[2] P. B. Shalen, Margulis numbers and number fields, arXiv:0902.1011.

[3] P. B. Shalen and P. Wagreich, Growth rates, Zp-homology, and volumes of hyperbolic 3-manifolds,Trans. Amer. Math. Soc. 331, No. 2 (1992), 895–917.

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Holonomies and the slope inequalities of Lefschetz fibrations

Hiroshige Shiga

Tokyo Institute of TechnologyTokyo 152-8551, Japan

shiga@math.titech.ac.jp

Let f : M → B be a Lefschetz fibration over a closed orientable surface B whose generalfiber is a surface of genus g ≥ 2. Two necessary conditions are known for the fibration toadmit a complex structure.One is so-called the slope inequality, which comes from a complex geometric point of view.It is known that the slope of of the fibration of relatively minimal holomorphic fibrationof genus g on non-singular algebraic surface is at least 4−4/g. (cf. [7]). Second conditionis irreducibility of the holonomy group, which comes from a topological point of view. Itis known that the image of the holonomy of a surface bundle f : X → B is irreducible ifthe bundle is non-trivial and holomorphic (cf. [6]).

In this talk, we discuss interactions between those two conditions.

Joint work with Hideki Miyachi (Osaka University).

References

[1] H. Endo, M. Korkmaz, D. Kotschick, B. Ozbagci, and A. Stipsicz, Commutators, Lefschetz fibra-tions and the signatures of surface bundles, Topology 41 (2002), 961–977.

[2] H. Endo and S. Nagami, Signature of relations in mapping class groups and non-holomorphicLefschetz fibrations, Trans. Amer. Math. Soc. 357 (2004), 3179–3199.

[3] Y. Matsumoto, Lefschetz fibrations of genus two—a topological approach, Topology and Teichmullerspaces, World Sci. Publ. (1995), 123–148.

[4] W. Meyer, Die Signatur von Flachenbunbeln, Math. Ann. 201 (1973), 239–264.

[5] R. Penner, Construction of Pseudo-Anosov homeomorphism, Trans. Amer. Math. Soc. 301 (1988),179–197.

[6] H. Shiga, On monodromies of holomorphic families of Riemann surfaces and modular transforma-tions, Math. Proc. Cambridge Philos. Soc. 122 (1997), 541–549.

[7] G. Xiao, Fibered algebraic surfaces with low slope. Math. Ann. 276 (1987), 449–466.

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Hyperbolic transformations and Algebraic transformations on moduli spaces

Robert Silhol

Universite Montpellier IIPlace E. Bataillon 34095 Montpellier cedex 5

rs@math.univ-montp2.fr

Generically actions of fractional Dehn twists induce globally real analytic (non holomor-phic) transformations on moduli. However, by restricting to certain subspaces of modulispace one can define hyperbolic transformations, of the above form, acting holomorphi-cally, and hence algebraically, on these subspaces. We will explore some consequences ofthe existence of these transformations.

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(Non)-actions of the mapping class group

Juan Souto

Department of Mathematics, University of MichiganAnn Arbor, MI 48109-1043, USA

jsouto@umich.edu

Let Σ be a closed surface of sufficiently large genus. Morita proved that the mappingclass group Map(Σ) does not lift to the group of diffeomorphisms of Σ. This resulthas been extended by Markovic, who proved that it also does not lift to the group ofhomeomorphisms. On the other hand, it is well-known that the mapping class groupacts in a natural way on the unit tangent bundle T 1Σ and it follows from the workof Sullivan and Deroin-Kleptsyn-Navas that this action is conjugated to an action byLipschitz homeomorphisms. In this talk I will prove that the action of Map(Σ) on T 1Σis not homotopic to a smooth action.

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Instanton approximation, periodic ASD connections, and mean dimension

Masaki Tsukamoto

tukamoto@math.kyoto-u.ac.jp

One of the most surprising feature of the Nevanlinna theory is that it gives several non-trivial results about infinite energy holomorphic curves, i.e, holomorphic maps f : C→ Xwith

∫C f ∗ω = +∞. (X is a projective manifold with a Kahler form ω.) In this talk, I

plan to talk about our attempt to develop infinite energy Yang-Mills gauge theory.Let X = S3×R, and E = X × SU(2) (the product principal SU(2)-bundle over X). Fixa point θ0 ∈ S3. For d ≥ 0, we define the “periodically framed ASD moduli space” Md

as the space of the gauge equivalence classes of (A, p) where A is an ASD connection onE satisfying

||FA||L∞ ≤ d, (2)

and p is a map from Z to E with p(n) ∈ E(θ0,n) for every n ∈ Z. The condition (2) isa “L∞-condition”, and hence the L2-energy

∫X|FA|2dvol is infinite in general. Md is an

infinite dimensional space. We study this infinite dimensional moduli space.

Joint work with Shinichiroh Matsuo (University of Tokyo).

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Dimension comparison in sub-Riemannian geometry

Jeremy Tyson

Department of Mathematics, University of Illinois1409 West Green Street, Urbana, IL 61801 USA

tyson@math.uiuc.edu

Gromov posed the problem of finding sharp estimates relating the topological and Haus-dorff dimensions of submanifolds or subsets of a general sub-Riemannian manifold. We

find sharp comparison theorems relating Euclidean and sub-Riemannian Hausdorff mea-sures and dimensions on an arbitrary Carnot (nilpotent stratified Lie) group G. To showsharpness we construct sets of minimal sub-Riemannian dimension for fixed Euclideandimension. Such sets are “horizontal”: they follow the lowest possible layers in the sub-Riemannian decomposition of the tangent bundle of G. Typically these sets are fractalfrom the perspective of both Euclidean and sub-Riemannian geometry. As a consequence,we obtain exact dimension formulas for a class of invariant sets of nonlinear, nonconfor-mal Euclidean iterated function systems of polynomial type. Inspired by Falconer’s workon almost sure dimensions of Euclidean self-affine fractals we show that sub-Riemannianself-similar fractals are almost surely horizontal. By way of contrast, smooth submani-folds of sub-Riemannian spaces are generically maximally non-horizontal. Understandingthe precise relationship between regularity and horizontality remains an interesting openproblem.

Joint work with Zoltan Balogh (Universitat Bern, Switzerland) and Ben Warhurst (Banach Center–IMPAN, Poland).

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Some topics in Nevanlinna theory concerning derivatives of meromorphicfunctions

Katsutoshi Yamanoi

Graduate School of Science and Technology, Kumamoto UniversityKurokami, Kumamoto 860-8555, Japan

yamanoi@kumamoto-u.ac.jp

In this talk, I will discuss about the following two (related) problems in Nevanlinnatheory, concerning derivatives of meromorphic functions in the plane. (1) The sharpdefect relation for the derivatives of meromorphic functions (called Mues conjecture). (2)The higher derivatives of meromorphic functions have asymptotically more zeros thanpoles (called Gol’dberg conjecture).

Posters

Elementary Multi-scale Riemann-Finsler Geometry and Applications toTensor Valued Medical Image Analysis

Laura Astola

Department of Mathematics and Computer Science, Eindhoven University of Technology,MB5600 Eindhoven, The Netherlands

l.j.astola@tue.nl

We use tools from Riemann-Finsler geometry to analyze medical images. Our main appli-cation is in High Angular Resolution Diffusion Imaging (HARDI), a Magnetic ResonanceImaging modality to infer muscular/neural tissue structure from the diffusion of watermolecules they comprise. The goal is to extract axons connecting neurons at differentareas of the brain from a (multimodal) set of images. We define a HARDI image asfollows:

u : T3 × S2 → R : u(x,y) 7→ r . (3)

We estimate this u with a homogeneous polynomial restricted to sphere using a singletensor representation u(x,y) = Di1···ik(x)yi1 · · ·yik . Some results in our investigation:

1. A strong convexity criterion for a Finsler norm in dimension three [1]. Let u(θ, ϕ)

satisfy F (u) = 1, uiθ := ∂ui

∂θ, ui

θθ := ∂2ui

∂θ2 etc. and put

u1 u2 u3

u1θ u2

θ u3θ

u1ϕ u2

ϕ u3ϕ

,mθ =

u1θθ u2

θθ u3θθ

u1θ u2

θ u3θ

u1ϕ u2

ϕ u3ϕ

,mϕ =

u1ϕϕ u2

ϕϕ u3ϕϕ

u1θ u2

θ u3θ

u1ϕ u2

ϕ u3ϕ

Then F is strongly convex if det(mθ)det(m)

< 0 and det(mϕ)

det(m)< − (gijui

θujϕ)2

gijuiθuj

2. A simple scheme to compute Laplace-Beltrami regularized orientation distributionfunction from a raw HARDI signal using single tensor representation and Clebschprojection [2]. Only one pseudo-inverse and one matrix multiplication needed.

References

[1] Astola, L. and Florack, L., Finsler geometry on Higher Order Tensor Fields and Applications to HighAngular Resolution Diffusion Imaging, Proc. Scale Space and Variational Methods, 2nd InternationalConference (2009)

[2] Muller, C., Analysis of Spherical Symmetries in Euclidean Spaces, Springer (1998)

Stability of Calderon’s inverse conductivity problem in the plane

Albert Clop

Universitat Autonoma de BarcelonaDepartament de Matematiques, Facultat de CienciesCampus de la U.A.B., 08193-Bellaterra (Barcelona)

albertcp@mat.uab.cat

We study stability of Calderon’s inverse conductivity problem in the plane. The unique-ness question for measurable conductivities was completely solved by Astala and Paivarinta[1]. In [2], uniform stability is obtained for conductivities having Holder continuous deriva-tives. This result is improved in [3], where just Holder continuity of the conductivitiesis needed. In the present work, we deal with conductivities in a fractional Sobolev spaceWα,p, α > 0, p > 1. With this smoothness we can only attain integral stability, althoughour results include discontinuous conductivities.

Joint work with Daniel Faraco and Alberto Ruiz (Universidad Autonoma de Madrid)

References

[1] K. Astala, L. Paivarinta, Calderon’s inverse conductivity problem in the plane. Ann. of Math. (2) 163(2006), no. 1, 265–299.

[2] J.A. Barcelo, T. Barcelo, A. Ruiz, Stability of the inverse conductivity problem in the plane for lessregular conductivities, J. Differential Equations 173 (2001), no. 2, 231–270.

[3] T. Barcelo, D. Faraco, A. Ruiz, Stability of Calderon inverse conductivity problem in the plane, J.Math. Pures Appl. (9) 88 (2007), no. 6, 522–556.

[4] A. Clop, D. Faraco, A. Ruiz, Stability of Calderon’s inverse conductivity problem in the plane fordiscontinuous conductivities, Inv. Probl. Imag. to appear

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On semi-symmetric Riemann spaces

Iulia Elena Hirica

University of Bucharest, Faculty of Mathematics and Informatics14 Academiei Str., RO-010014, Bucharest 1, Romania

ihirica@fmi.unibuc.ro

Semi-symmetric manifolds constitute a generalization of spaces of constant sectional cur-vature, along the line of locally symmetric spaces.Conditions of pseudo-symmetric and semi-symmetric type on geodesic and subgeodesicrelated Riemann spaces are studied. Properties of conharmonic and concircular transfor-mations of metrics are also characterized.

References

[1] E. Boeckx, G. Calvaruso, When is the tangent sphere bundle semi-symmetric, Tohoku Math. J., (2)56 (2004), 3, 357-366.

[2] F. Defever, R. Deszcz, A note on geodesic mappings of pseudo-symmetric Riemann manifolds, Coll.Math., LXII, 2 (1991), 313-319.

[3] I.E. Hirica, L. Nicolescu, On Weyl structures, Rend. Circolo Mat. Palermo, II, LIII (2004), 390-400.

[4] I. Kim, H. Park, H. Song, Ricci pseudo-symmetric real hypersurfaces in complex space forms, NihonkaiMath. J., 18, 1-2, (2007), 1-9.

[5] Z.I. Szabo, Structure theorems on Riemannian spaces satisfying R(X, Y )·R = 0. II. The local version.J. Diff. Geom., 17 (1982), 531-582; II. Global version. Geom. Dedicata, 19 (1985), 1, 65-108.

[6] M. Yawata, H. Hideko, On relations between certain tensors and semi-symmetric spaces, J. PureMath., 22 (2005), 25-31.

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Entire functions of small order of growth

Katsuya Ishizaki

Department of Mathematics Nippon Institute of Technology4-1 Gakuendai Miyashiro, Minamisaitama Saitama 345-8501, Japan

ishi@nit.ac.jp

Let f and F be transcendental entire functions. We are concerned with the growthestimate of F f when F satisfies the condition

log M(r, F ) = K(log r)p(1 + o(1)),

where K is a positive constant and p ≥ 2 is an integer. It is shown

log log M(r, F (f)) = p log log M(r, f)(1 + o(1)),

which is an analogue of the Valiron–Mohon’ko theorem [4], [3] for a transcendental entirefunction F . The tools of the proof are estimates of minimum modulus, see e.g. [1], andWiman–Valiron theory for composite functions, see e.g. [2].

References

[1] Boas R. P., Entire Functions, Academic Press 1954.

[2] Clunie, J., The maximum modulus of an integral function of an integral function, Quart. J. Math. 6(1955), 176–178.

[3] Laine, I., Nevanlinna Theory and Complex Differential Equations, W. Gruyter, Berlin–New York,1992.

[4] Mohon’ko, A. Z., The Nevanlinna characteristics of certain meromorphic functions, Teor. FunktsiiFunktsional. Anal. i Prilozhen 14, 1971, 83–87, (Russian).

Lattes-type uniformly quasiregular mappings and their Julia sets

Riikka Kangaslampi

Department of Mathematics and Systems Analysis, Helsinki University of Technology02015 TKK, Finland

riikka.kangaslampi@tkk.fi

A uniformly quasiregular mapping acting on a compact Riemannian manifold distortsthe metric by a bounded amount, independently of the number of iterates. There is aFatou-Julia type theory associated with the dynamical system obtained by iterating thesemappings.We study a rich subclass of uniformly quasiregular mappings that can be produced usingan analogy of classical Lattes’ construction of chaotic rational functions acting on theextended complex plane (see [3]). We construct several essentially different examples ofLattes-type uniformly quasiregular mappings on three and higher dimensional compactRiemannian manifolds with a wide variety of Julia sets of codimensions 0, 1, and 2.Detailed discussion and constructions can be found in [1] and [2].

Joint work with Kirsi Peltonen (Helsinki University of Technology).

References

[1] L. Astola, R. Kangaslampi, K. Peltonen, Lattes-type mappings on compact manifolds, (Preprint).

[2] R. Kangaslampi, Uniformly quasiregular mappings on elliptic Riemannian manifolds, Ann. Acad. Sci.Fenn. Ser. A I Math. Diss. 150 (2008).

[3] V. Mayer, Uniformly quasiregular mappings of Lattes type, Conform. Geom. Dyn. 1 (1997), 104-111.

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Ramification estimates for the Gauss map of various surfaces

Yu Kawakami

Graduate School of Mathematics, Kyushu University744, Motooka, Nishiku, Fukuoka-city, 819-0395, Japan

kawakami@math.kyushu-u.ac.jp

I will give a survey of my recent work on the value distribution of the hyperbolic Gaussmap of surfaces in hyperbolic three-space. In particular, I will explain “algebraic” classfor constant mean curvature one surfaces and flat flonts and give effective estimates forthe totally ramified value number and the number of exceptional values of the hyperbolicGauss map of them.

References

[1] R. Bryant, Surfaces of mean curvature one in hyperbolic space, Asterisque 154–155 (1987), 321–347.

[2] H. Fujimoto, On the number of exceptional values of the Gauss map of minimal surfaces, J. Math.Soc. Japan, 40 (1988), 235 – 247.

[3] J. A. Galvez, A. Martınez, and F. Milan, Flat surfaces in hyperbolic 3-space, Math. Ann., 316 (2000),419–435.

[4] Y. Kawakami, Ramification estimates for the hyperbolic Gauss map, to appear in Osaka Journal ofMathematics, arXiv:0804.0470.

[5] Y. Kawakami, Value distribution of the hyperbolic Gauss maps for flat fronts in hyperbolic three-space,preprint.

[6] Y. Kawakami, R. Kobayashi and R. Miyaoka, The Gauss map of pseudo-algebraic minimal surfaces,Forum Math., 20 (2008), no. 6, 1055–1069.

[7] M. Kokubu, M. Umehara and K. Yamada, Flat fronts in hyperbolic 3-space, Pacific J. Math., 216(2004), 149–175.

[8] R. Nevanlinna, Analytic Function, Translated from the second German edition by Phillip Emig. DieGrundlehren der mathematischen Wissenschaften, Springer, New York, 1970.

[9] R. Osserman, A survey of minimal surfaces, second edition, Dover Publications Inc., 1986.

[10] M. Umehara and K. Yamada, Complete surfaces of constant mean curvature-1 in the hyperbolic3-space, Ann. of Math. 137 (1993), 611–638.

[11] Z. Yu, Value distribution of hyperbolic Gauss maps, Proc. Amer. Math. Soc., 125 (1997), 2997–3001.

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On holomorphic sections of certain holomorphic family of Riemann surfacesof genus two

Yohei Komori

Department of Mathematics, Osaka City UniversityOsaka 558-8585, Japan

komori@sci.osaka-cu.ac.jp

Joint work with Toshihiro Nogi (Osaka City University Advanced Mathematical Institute).

Let (T , 0) be a closed Riemann surface of genus one with the marked point 0 ∈ T . For

any point t ∈ T − 0, we cut T along a simple curve from 0 to t. Next we take twocopies of the torus T with the cut and call them sheet I and sheet II. The cut on eachsheet has two sides, which are labeled + side and − side. We attach the + side of thecut on I to the − side of the cut on II, and attach the − side of the cut on I to the +side of the cut on II. Now we obtain a closed Riemann surface St of genus two, whichis a two-sheeted branched covering surface St → T branched over 0 and t. From thisgeometric consideration, we have a holomorphic family of Riemann surfaces of genus twostudied in [1] and [2]. Our main result is to determine all holomorphic sections of thisfamily.

References

[1] Y. Imayoshi, Y. Komori and T. Nogi, Holomorphic Sections of a Holomorphic Family of RiemannSurfaces induced by a Certain Kodaira Surface, to appear in Kodai Math. J(2009).

[2] G. Riera, Semi-direct products of fuchsian groups and uniformization, Duke Math. J 44 (1977),291-304.

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The earthquake measure map is a homeomorphism

Hideki Miyachi

Department of Mathematics, Graduate School of Science, Osaka University,Machikaneyama 1-1, Toyonaka, Osaka, 560-0043, Japan

miyachi@math.sci.osaka-u.ac.jp

In [3], W. Thurston defined left and right earthquake maps on the unit disk D and showedthat any orientation preserving homeomorphism is obtained as the boundary value of aunique left earthquake maps. To every earthquake, a measured geodesic lamination on Dis associated. Such measured geodesic lamination is called an earthquake measure. In [1],F. Gardiner, J.Hu and N. Lakic, independently D. Saric in [2], showed that an orientationpreserving homeomorphism on ∂D is quasisymmetric if and only if associated earthquakemeasure is bounded, where a measured geodesic lamination is said to be bounded if thevalues of the transversal measure on geodesic arcs of unit length are uniformly bounded.Therefore, we obtain a canonical bijective mapping EM from the universal Teichmullerspace T (D) to the set MLb(D) of bounded measured geodesic laminations. We call themapping EM the earthquake measure map.It is known that any measured geodesic lamination on D is recognized as a linear functionalon the space of Holder functions with compact support on unoriented geodesics on D.From this identification, we have a canonical topology on MLb(D). We will announcethat the earthquake measure map EM is a homeomorphism.

Joint work with Dragomir Saric (Department of Mathematics, Queens College of CUNY)

References

[1] F.Gardiner, J.Hu, and N.Lakic, Earthquake curves, Complex manifolds and hyperbolic geometry,Contemp. Math., 311 (2002), 141–195.

[2] D. Saric, Bounded Earthquakes, Proc. Amer. Math. Soc 138 (2008), 889–897.

[3] W. Thurston, Earthquakes in two-dimensional hyperbolic geometry, Low-dimensional topology andKleinian groups, LMS. Lecture Note Ser., 112, Cambridge Univ. Press, Cambridge (1986), 91–112.

Examples of the coordinates of genus 2-surfaces with regular fundamentaloctagon

Gou Nakamura

Center for General Education, Aichi Institute of TechnologyYakusa-cho, Toyota 470-0392, Japan

gou@aitech.ac.jp

It is known that the Teichmuller space Tg of closed Riemann surfaces of genus g isparametrized by 6g−5 geodesic length functions. P. Schmutz Schaller presented geodesiclength functions determined by a “canonical” polygon ([2]). Following the ideas in [2],we shall give 4 examples of the coordinates of compact Riemann surfaces of genus twoof which fundamental region is the regular octagon. In order to construct the canonicalpolygons from the regular octagon, we make use of the Weierstrass points found by T.Kuusalo and M. Naatanen ([1]).

References

[1] T. Kuusalo and M. Naatanen, Weierstrass points of genus-2 surfaces with regular fundamental do-mains, Quart. J. Math. 54, (2003), no. 3, 355–365.

[2] P. Schmutz Schaller, Teichmuller space and fundamental domains of Fuchsian groups, Enseign. Math.(2) 45 (1999), no. 1-2, 169–187.

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Qualitative analysis to distribution of types of DNAknots by use of topological invariants

Isamu Ohnishi

Graduate School of Science, Hiroshima UniversityKagamiyama, Higashi-Hiroshima, 739-8526 JAPAN

isamu o@math.sci.hiroshima-u.ac.jp

In this poster presentation, a finer mechanism of transcription process of gene as a cost-saving and resource-saving transportation system of codewords information is investigatedby studying distribution of the DNA knots by use of topological invariants to λ phage’sstructural operation as one of interesting examples. This work is just a finer analysis tothe mechanism, but better understanding makes better using, so that, as a first step, wepresent it here. This is also another important point, because it makes the simulationtime be shorter and will be able to make simulation itself be precise even in the case of bigcrossing number. We use two types of topological invariants: Alexander polynomials andJones polynomials. Generally speaking, Alexander polynomials are easy to use, but havelower separating ability. On the other hand, Jones polynomials are somewhat difficult toutilize, but have finer separating ability. We compare the results gotten by both ways

of simulations to discuss about both advantages and disadvantage to suggest that theyshould be used according to the objectives. Moreover, these results are compared withthe actual biochemical experimental results to make an interpretation to what happensin the cell, and we will state perspectives in the future.Joint work with Mr. Takashi Yoshino (Master course student in the second degree).

References

[1] G. Charvin, T. R. Strick, D. Bensimon and V. Croquette, Topoisomerase IV Bends and OvertwistsDNA upon Binding, Biophysical Journal, Vol.89, (2005), 384–392.

[2] J. Arsuaga, M. Vazquez, P. McGuirk, S. Trigueros, D. W. Sumners and J. Roca, DNA knots reveal achiral organization of DNA in phage capsids, PNAS, Vol.102, (2005), 9165–9169.

[3] M. D. Frank-Kamenetskii and A. V. Vologodskii, Topological aspects of the physics of polymers: Thetheory and its biophysical applications, Sov.Phys.-Usp.(Engl.Transl.), Vol.24, (1981), 679–696.

[4] K. Sekine, H. Imai and K. Imai, Computation of the Jones Polynomial, Transactions of JSIAM,Vol.8, (1998), 341–354.

[5] T. Yoshino and I. Ohnishi, Qualitative analysis to distribution of types of DNA knots by use oftopological invariants, Preprint

[6] J. D. Watson, T. A. Baker, S. P. Bell, A. Gann, M. Levine, and R. Losick, “Molecular biology of thegene” (the sixth edition), Pearson, Benjamin Cummings.

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Nonlinearity of morphisms in non-Archimedean and complex dynamics

Yusuke Okuyama

Department of Comprehensive Sciences, Graduate School of Science and Technology, KyotoInstitute of Technology,Kyoto 606-8585 JAPANokuyama@kit.ac.jp

Let K be a commutative algebraically closed field which is complete with respect to anon-trivial absolute value (or valuation) | · |. This | · | is said to be non-Archimedean if∀z, ∀w ∈ K, |z − w| ≤ max|z|, |w|. Otherwise, | · | is said to be Archimedean and K isthen topologically isomorphic to C (with Hermitian norm). We extend | · | to K` (` ∈ N)as the maximum norm |Z| = |Z|` = maxj=1,...,` |zj| for Z = (z1, . . . , z`). Let π : Kn+1 \O → Pn(K) be the canonical projection and set `(n) ∈ N so that

∧2 Kn+1 ∼= K`(n).The chordal distance [·, ·] on Pn(K) is defined as [z, w] := |Z ∧ W |`(n)/(|Z|n+1|W |n+1),where Z ∈ π−1(z),W ∈ π−1(w). For z0 ∈ Pn(K) and r > 0, we consider the ballB(z0, r) := z ∈ Pn(K); [z, z0] ≤ r.Let f : Pn(K) → Pn(K) be a (finite) morphism, i.e., there is a homogeneous polynomialmap F : Kn+1 → Kn+1 over K, which is called a lift of f , such that F−1(O) = O andsatisfies π F = f π. The degree d = deg f is that of F as homogeneous polynomialmap. As in the case of K = C, the Fatou set F (f) is the largest open set at each pointof which the family fk; k ∈ N is equicontinuous.

The Julia set J(f) is defined by Pn(K) \ F (f). In non-Archimedean case, J(f) may beempty even if d ≥ 2. One of main results is

Theorem. Let f : Pn(K) → Pn(K) be a morphism of degree d ≥ 1. If there are a ballB(z0, r) ⊂ Pn(K) and a morphism g : Pn(K) → Pn(K) such that

lim infk→∞

dklog sup

B(z0,r)

[fk, g] = −∞,

then either f is linear or J(f) = ∅.

We give a few its applications to concrete problems.

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Cooperation Principle in Random Complex Dynamics and SingularFunctions on the Complex Plane

Hiroki Sumi

Department of Mathematics, Graduate School of Science, Osaka University1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

sumi@math.sci.osaka-u.ac.jp http://www.math.sci.osaka-u.ac.jp/∼sumi/We investigate the random complex dynamics and the dynamics of semigroups of rationalmaps on the Riemann sphere C. We see that in the random complex dynamical systems,the chaos easily disappears. This phenomenon occurs since many kinds of maps in thesystem “cooperate together.” This is a new phenomenon. Moreover, even though thechaos disappears, we find a new kind of complexity. More precisely, we investigate theiteration of the transition operator M of a random complex dynamical system, where Macts on the space of continuous functions on C. It turns out that under certain conditions,each non-constant finite linear combination ϕ of unitary eigenvectors of M can be regardedas a complex analogue of the devil’s staircase. In fact, ϕ is a continuous function on Cwhich varies only on the Julia set (thin fractal set) of the associated semigroup. By usingBirkhoff’s ergodic theorem and potential theory, we investigate the non-differentiabilityand the pointwise Holder exponent of ϕ. We find many kinds of new phenomena inrandom complex dynamics which cannot hold in the usual iteration dynamics of a singleholomorphic map. We systematically investigate those phenomena. The contents of thispresentation are included in [1].

References

[1] H. Sumi, Random complex dynamics and semigroups of holomorphic maps, preprint 2008,http://arxiv.org/abs/0812.4483. (The contents of this presentation are included.)

[2] H. Sumi, Rational semigroups, random complex dynamics and singular functions on the complexplane, Sugaku, Vol. 61, No. 2, p133-161. Survey article, written in Japanese. (“Sugaku” is a journalpublished by Mathematical Society of Japan. English version of this article will be published in“Sugaku Expositions” by the AMS.)

[3] H. Sumi, Interaction cohomology of forward or backward self-similar systems, to appear in Advancesin Mathematics. Available from http://arxiv.org/abs/0804.3822.

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Toward a rigorous measure estimate of the stochastic parameter set for thequadratic family

Hiroki Takahasi

takahasi@math.kyoto-u.ac.jp

We develop a constructive version of Jakobson’s inducing argument for the quadraticfamily, with a view to giving a computer-assisted lower bound for the Lebesgue measureof the parameter set corresponding to maps which have absolutely continuous invariantprobability measures.

References

[1] M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensionalmaps. Comm. Math. Phys. 81 (1981), 39–88.

[2] M. Jakobson, Piecewise smooth maps with absolutely continuous invariant measures and uniformlyscaled Markov partitions. Smooth ergodic theory and its applications, 825–881, Proc. Sympos. PureMath., 69, Amer. Math. Soc., Providence, RI, 2001

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Weighted Green functions of polynomial skew products on C2

Kohei Ueno

kueno@math.kyoto-u.ac.jp

We consider the dynamics of polynomial skew products on C2. The Green function of apolynomial map is a useful tool to study the dynamics of the map. However, there aremany polynomial skew products whose Green functions are not well behaved on C2. Weintroduce the weighted Green function of a polynomial skew product f , a generalizationof the Green function of f , and show that it is well behaved on C2. Moreover, we considerthe dynamics of the extension of f to a holomorphic map on the weighted projectivespace.

References

[1] C. Favre, M. Jonsson, Dynamical compactifications of C2, arXiv (2007).

[2] K. Ueno, Symmetries of Julia sets of non-degenerate polynomial skew products on C2, to be appearin Michigan Math. J.

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Quasisymmetric maps on the ideal boundary of some negatively curvedsolvable Lie groups

Xiangdong Xie

Department of Mathematical Sciences, Georgia Southern UniversityStatesboro, GA 30460,USAxxie@georgiasouthern.edu

Let A be an n × n matrix. Let R act on Rn by (t, x) → etAx (t ∈ R, x ∈ Rn). Wedenote the corresponding semi-direct product by GA = Rn oA R. If the eigenvalues of Ahave positive real parts, then GA admits left-invariant Riemannian metrics with negativecurvature.When A is diagonal with positive eigenvalues (but not a multiple of the identity matrix),we show all quasisymmetric maps of the ideal boundary of GA are biLipschitz. When

(1 10 1

we describe all quasisymmetric maps of the ideal boundary.As applications, we show all quasiisometries of GA (in the above two cases) are almostisometries (meaning distance is preserved up to an additive constant). When

(1 10 1

we prove a Liouville type theorem, which says every conformal map of the ideal boundaryis the boundary map of an isometry of GA.

Part of this project is joint work with Nageswari Shanmugalingam.

the xxist rolf nevanlinna colloquiumsugawa/nevanlinna/abstract.pdf · 2015-05-11 · the xxist rolf...

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