Walter Bergweiler
Christian-Albrechts-Universität Kiel, Germany
e-mail: bergweiler@math.uni-kiel.de
Fatou-Julia Theory for non-uniformly quasiregular maps
Montel’s theorem plays a central role in the Fatou-Julia theory of iteration of rational
functions. Montel’s theorem has been extended to quasiregular maps in higher dimensions
by Rickman and Miniowitz. This has been used by Hinkkanen, Martin, Mayer and others
to develop a Fatou-Julia theory for uniformly quasiregular maps. These are, by definition,
quasiregular maps for which the dilatation of the iterates is uniformly bounded. We show
that some results of the Fatou-Julia theory also hold for non-uniformly quasiregular maps,
provided the degree exceeds the inner dilatation. In dimension two such results were obtained
by Sun and Yang.
William Cherry
University of North Texas, USA
e-mail: wcherry@unt.edu
An effective Schottky-Landau Theorem
for holomorphic curves in projective space
In 1944, Dufresnoy published a generalization of Landau’s theorem: if a holomorphic
map from the unit disc to complex projective n-space omits 2n+1 hyperplanes in general
position, then the Fubini-Study derivative of the map at the origin is bounded above by
a constant. Dufresnoy’s argument, making use of a normal families argument, does not
effectively estimate the constant, and Dufresnoy commented that from his argument, the
constant depends in an “unkown way” on the omitted hyperplanes. I will discuss joint work
with Alex Eremenko about how the potential theoretic method of Eremenko and Sodin can be
used to give an effective estimate for Dufresnoy’s constant which, although non-sharp, gives
a good sense of how Dufresnoy’s constant depends on the geometry of the configuration
of the omitted hyperplanes and has almost the best possible asymptotic behavior as the
hyperplanes degenerate away from general position.
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Demetrios Christodoulou
ETH Zurich, Switzerland
e-mail: demetri.christodoulou@math.ethz.ch
Acoustical geometry and shock formation
This talk will be based on my monograph on the formation of shocks in 3-dimensional
fluids. My approach to the problem is in the spirit of general relativity because I introduce
a metric on the spacetime manifold which I call “acoustical metric”, the null cones of which
are the sound cones of the fluid. I then study how this “acoustical geometry” degenerates
when shocks begin to form. In fact, I give a complete description of the singular boundary.
I found that there is another differential structure relative to which the boundary is smooth.
It is this “de-singularization” which allows me to give a complete description. Perhaps the
method will in the future find some use in general relativity although surely, entirely new
ideas will also be needed.
Vladimir Dubinin
Far-Eastern Branch of RAS, Russia
e-mail: dubinin@iam.dvo.ru
Some inequalities for the Schwarzian derivative
of holomorphic functions via capacity and symmetrization
We discuss capacity and symmetrization methods for proving distortion theorems that
involves the Schwarzian derivatives of holomorphic functions. First, some basic properties
of generalized condensers, an asymptotic formula for capacity, and examples of reduced
moduli are demonstrated [1]. Applications include distortion theorems in various classes
of univalent functions, inequalities for coefficients, and extremal decomposition problems.
Some proofs based on simple properties of generalized condensers and symmetrization are
also presented [1], [2]. Second, under certain constraints on the holomorphic function f ,
new inequalities for the real part of the Schwarzian derivative Sf are established [3]. These
inequalities supplement and refine some known boundary versions of the Schwarz lemma.
Our results come near the Burns-Krantz rigidity theorem for the regular self-mappings of
the unit disk and its generalizations obtained by R. Tauraso, F. Vlacci and D. Shoikhet.
Finally, we consider the Schwarzian derivative via the half-plane capacity [4, p.69, 139].
New lower bounds for this capacity are established in terms of capacities of sets obtained by
an application of a series of geometric transformations to the set E, including Steiner and
circular symmetrizations. Its behavior under linear and radial averaging transformations of
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families of compact sets Ek is examined [5]. In this way, we supplement some recent results
of S. P. Lalley, G. F. Lawler and H. Narayanan.
[1] Dubinin V. N. Condenser capacity and symmetrization in the geometric function theory
of a complex variable [in Russian], Dalnauka. Vladivostok. 2009.
[2] Dubinin V. N. Steiner symmetrization and the initial coefficients of univalent functions,
Izvestia: Mathematics 74:4 (2010), 735-742.
[3] Dubinin V. N. On the boundary Schwarzian derivative of regular function, Sbornik:
Mathematics (2011), to appear.
[4] Lawler G. F. Conformally invariant processes in the plane, Math. Surveys. Monogr.
Vol.114, Amer. Math. Soc., Providence, RI 2005.
[5] Dubinin V. N. Lower bounds for the half-plane capacity of compact sets and symmetrization, Sbornik: Mathematics 201:11 (2010), 1-12.
Alexandre Eremenko
Purdue University, USA
e-mail: eremenko@math.purdue.edu
Spectral loci of one-dimensional Schrödinger operators
We consider eigenvalue problems
−y ′′ + P (a, z)y = λy,
y = y(z),
with boundary conditions at ∞ in the complex z-plane, and polynomial potentials P depending on the parameter a. The dependence of the spectrum on a is described by the
equation F (a, λ) = 0, where F is an entire function. We study topological properties of the
spectral locus, which is the zero-set of F .
Based on joint work with A. Gabrielov.
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Peter Fenton
University of Otago, New Zealand
e-mail: pfenton@maths.otago.ac.nz
Subharmonic functions that are convex between two curves
This talk concerns joint work with John Rossi on functions that are subharmonic in the
plane and equal to a convex function of log |z| outside the region D contained between two
simple, non-intersecting curves that join 0 to ∞. If the angular separation of the curves is
no more than 2α, where 0 < α < π, and
AD (r) = inf{u(z) : z ∈ D and |z| = r}
is bounded, what can be said about the growth of u? This question is connected with an
unresolved conjecture of Kövari on gap power series.
Jürgen Grahl
Universität Würzburg, Germany
e-mail: grahl@mathematik.uni-wuerzburg.de
Some results on normal families,
shared values and differential polynomials
We show how the Zalcman-Pang rescaling method can be applied to deduce a connection
between results on exceptional values of differential polynomials on the one hand and Picard
type and Montel type results for differential polynomials sharing a value with their derivative
on the other hand. Furthermore, we discuss a modification of the Nevanlinna theory and its
application to normality problems for generalized differential polynomials containing rotations and dilations in the argument. Finally, we present a counterpart to Marty’s Theorem
for families of meromorphic functions whose spherical derivative is uniformly bounded away
from zero.
Several of the results in this talk are joint work with Shahar Nevo.
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Lawrence Harris
University of Kentucky, USA
e-mail: larry@ms.uky.edu
Markov inequalities for multivariate and Banach space polynomials
We discuss a proof of the following:
Theorem. Let X and Y be real normed linear spaces and let P : X → Y be a polynomial
of degree at most m satisfying ∥P (x)∥ ≤ 1 for all x ∈ X with ∥x∥ ≤ 1. Then
∥D̂k P (x)∥ ≤ Tm(k) (1) for all x ∈ X with ∥x∥ ≤ 1 and
∥D̂k P (x)∥ ≤ Tm(k) (∥x∥) for all x ∈ X with ∥x∥ ≥ 1.
Here D̂k P (x) denotes the kth order Gâteaux derivative of P at x and Tm denotes the
Chebyshev polynomial of degree m.
The proof of this theorem reduces to establishing just the second inequality in the case
where X = R2 with the max norm, Y = R and x = (r, r) with r real. We establish this
by using bivariate Lagrange interpolation at a set of nodes in R2 whose coordinates are
Chebyshev points and a corresponding Christoffel-Darboux formula to determine the sign of
the derivatives of the Lagrange polynomials.
We also discuss a theorem of this type at a related set of nodes where the optimal polynomials are the Chebyshev polynomials of the second kind.
W.K. Hayman
Imperial College London, UK
e-mail: k.taylor@imperial.ac.uk
A conjecture of Blumenthal
Let w = f (z) be an entire function and
M (r) = M (r, f ) = sup |f (z)|,
|z|=r
0<r<∞
its maximum modulus. Blumenthal’s Conjecture states that M (r) determines f apart from
rotations and reflections in the z, w planes. We report on joint work with Tom Tyler and
David White on the case in which f is a polynomial.
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Aimo Hinkkanen
University of Illinois, USA
e-mail: aimo@illinois.edu
Growth of second Painlevé transcendents
All solutions of Painlevé’s second equation are meromorphic in the complex plane. It has
been known that the order of any transcendental solution is at least 3/2 and at most 3. In
joint work with Ilpo Laine, we have proved that the order of such a solution is either 3/2 or
3. We provide criteria for determining which case occurs. The proof is based on estimates
on the behavior of the solutions. For example, an upper bound is obtained for the distance
to the nearest pole from a point where the solution is not too large.
Gerhard Huisken
Max-Planck-Institut für Gravitationsphysik, Germany
e-mail: Gerhard.Huisken@aei.mpg.de
Mean curvature flow with surgeries
The lecture discusses joint work with Carlo Sinestrari on mean curvature flow of hypersurfaces that can be extended beyond singularities by a surgery procedure. New a priori
estimates for the flow and their application to singularities and ancient solutions are discussed.
Dmitry Khavinson
University of South Florida, USA
e-mail: dkhavins@usf.edu
Linear extremal problems in Bergman spaces of analytic functions
I shall survey the major problems in the theory of extremal problems in Bergman spaces
and compare them with their well known counterparts in Hardy spaces.
Then I will survey the new advances that occurred in the last few years and discuss the
main open problems.
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Ilpo Laine
University of Eastern Finland, Finland
e-mail: ilpo.laine@uef.fi
Tropical Nevanlinna Theory
In this talk, we briefly describe the background and present key results in the field of
tropical Nevanlinna theory. This version of value distribution theory is devoted to considering
tropical meromorphic functions, i.e., continuous piecewise linear functions f : R → R. Our
main emphasis in this talk will be the tropical counterpart to the Second Main Theorem;
the main ideas in its proof will be presented. Time permitting, we propose to give a short
overview of possible applications, as well as of some ideas for future research problems.
Irina Markina
University of Bergen, Norway
e-mail: irina.markina@uib.no
On value distribution theory on polarizable Carnot groups
This work extends one of Seppo Rickman’s deepest results on the value distribution theory
for quasimeromorphic maps in Euclidean space [3]. We change the abelian group structure
of the vector space into a nonabelian group structure which nonetheless remains close to
the Euclidean setting in the sense that it is a nilpotent graded group. The assumption of
“polarizability” leads in this case to the nontriviality of the conformal capacity of a spherical
condenser; cf. [1]. We define meromorphic maps on these groups and discuss generalizations
[2] of Rickman’s results.
References
[1] Z. M. Balogh and J. T. Tyson, Polar coordinates in Carnot groups, Math. Z. 241 (2002), 697-730.
[2] I. Markina and S. Vodopyanov, On value distributions for quasimeromorphic mappings on H-type Carnot
groups, Bull. Sci. Math. 130 (2006), 467-523.
[3] S. Rickman, Quasiregular mappings, Springer-Verlag, Berlin, 1993, p. 213.
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Shahar Nevo
Bar-Ilan University, Israel
e-mail: shahar.nevo@gmail.com
Qα -normality and entire functions
We introduce the notion of Qα -normality, which is an extension of the notion of Q1 normality (quasinormality) to an arbitrary (possibly infinite) ordinal. Then we explain why
for countable alpha there exists an entire function f = fα such that {f (nz) : n = 1, 2, 3, . . . }
is an “exact” Qα -family.
This is joint work with Shai Gul.
Alon Nishry
Tel Aviv University, Israel
e-mail: alonnish@post.tau.ac.il
The hole probability for Gaussian entire functions
We study the hole probability of Gaussian entire functions. More specifically, we consider
entire functions in Taylor series form with i.i.d complex Gaussian coefficients. A hole is the
event where the function has no zeros in a disc of radius r, centered at 0. We find exact
asymptotics for the rate of decay of the hole probability for large values of r, outside a small
exceptional set (which is deterministic).
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Junjiro Noguchi
University of Tokyo, Japan
e-mail: noguchi@ms.u-tokyo.ac.jp
Second Main Theorem, degeneracy of entire curves,
and applications
We will survey the development of the value distribution theory since Cartan’s S.M.T.
in the line of Bloch-Ochiai-Noguchi/Winkelmann/ Yamanoi. The model case is the S.M.T.
for differentially non-degenerate meromorphic maps, due to Griffiths et al. A newly found
geometric proof of Cartan’s S.M.T. in terms of the Fubini-Study metric connection will be
given; the proof is quite analogous to Griffiths’ case.
We will then talk about the S.M.T. for entire curves into semi-abelian varieties with
counting functions truncated to level one, due to Noguchi-Winkelmann-Yamanoi. It has a
number of applications which will be discussed; in particular, we give some degeneracy and
unicity theorems for entire curves and meromorphic functions. Here there is an interesting
analogy between Nevanlinna theory and Diophantine approximation.
The present research has a natural connection to the theory of Kobayashi hyperbolic
manifolds, which we will discuss if time allows.
Anthony G. O’Farrell
National University of Ireland, Ireland
e-mail: anthonyg.ofarrell@gmail.com
Reversible biholomorphic maps in two variables
An element g of a group G of maps is reversible if it is conjugate in G to its inverse, i.e., if
there is a solution h ∈ G to h−1 gh = id. Reversible maps have attracted a lot of interest in
the past two decades. For biholomorphic maps in one variable, reversibility is understood.
We shall report on recent progress (jointly with D. Zaitsev) on the group of biholomorphic
germs in two variables.
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Min Ru
University of Houston, USA
e-mail: minru@math.uh.edu
Value distribution theory of the Gauss maps
of minimal surfaces in R3 and Rn
In this talk, I will give a brief introduction to the value distribution theory of Gauss maps
of minimal surfaces in R3 and Rn , developed by Osserman, Fujimoto and others. I will
then compare the theory with standard Nevanlinna theory. Finally, some new results in this
direction will be discussed.
Mikhail Sodin
Tel Aviv University, Israel
e-mail: sodin@post.tau.ac.il
Random complex zeroes
In my talk, I plan to survey some classical and recent results on zeroes of random Gaussian
entire functions.
Alexander Solynin
Texas Tech University, USA
e-mail: alex.solynin@ttu.edu
Quadratic differentials and weighted graphs on compact surfaces
We will show that every weighted graph (not necessarily connected) embedded in a compact surface can be realized as a critical weighted subgraph of a quadratic differential. Such
realization is not unique in general. Thus, we will discuss minimal sets of restrictions which
guarantee uniqueness. Some interesting special cases and applications to extremal problems
also will be discussed.
This work generalizes previous results of the author where such realization was established
for connected graphs.
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Norbert Steinmetz
Technical University of Dortmund, Germany
e-mail: stein@math.uni-dortmund.de
Normal Families and Differential Equations
Differential equations in the complex domain, re-scaling methods, and normal family criteria and arguments are intimately connected. In this talk we will discuss some of these
connexions exemplarily as follows:
• Normality criteria and linear differential equations
— arXiv[math.CV] 1102.3104
• The Painlevé transcendents and re-scaling
— Israel J. Math. 128
• Normal families and the Yosida class
— arXiv[math.CV] 1103.2073