Chebyshev Laboratory, St. Petersburg State University
St. Petersburg Department
of Steklov Mathematical Institute
International conference
COMPLEX ANALYSIS
AND RELATED TOPICS
ABSTRACTS
St. Petersburg, April 14–18, 2014
Organizing Committee:
Alexander Aptekarev, Co-chair
Keldysh Institute of Applied Mathematics
Stanislav Smirnov, Co-chair
University of Geneva and St. Petersburg State University
Anton Baranov
St. Petersburg State University
Evgenii Chirka
Steklov Mathematical Institute
Sergey Kislyakov
St. Petersburg Department of Steklov Mathematical Institute
Konstantin Fedorovskiy
Bauman Moscow State Technical University
Local Organizing Committee:
Anton Baranov
St. Petersburg State University
Yurii Belov
St. Petersburg State University
Svetlana Likhova
Chebyshev Laboratory
Nikolai Osipov
St. Petersburg Department of Steklov Mathematical Institute
Sponsors:
Chebyshev Laboratory
under RF Government grant 11.G34.31.0026
JSC «Gazprom Neft»
INVITED PLENARY LECTURES
Kari ASTALA (University of Helsinki)
Multifractal spectra for bi-Lipschitz and quasiconformal mappings
I will describe a recent joint work with T. Iwaniec, I. Prause and E. Saksman
on the theme:
’Given an L-bi-Lipschitz mapping of the plane, how large can be the set E
where the map rotates or spirals with a given rate γ?’
We describe the optimal bounds for the size of such sets in terms of the
dimension; it turns out that always the Hausdorff dimension dim(E) ≤ 2 −
|γ|2L/(L2 − 1), where for some sets E and maps f the equality holds.
The key points of the proof are holomorphic deformations of bi-Lipschitz
mappings and a new kind of interpolation on Lp -spaces, where we can interpolate
with complex exponents p.
∗ ∗ ∗
Farit AVKHADIEV (Kazan Federal University)
Hardy type inequalities
We will present our main results about Hardy inequalities in plane and space
domains including inequalities a) plane domains with uniformly perfect boundary,
b) for convex domains and c) for arbitrary domains. In particular we will present
some new results on a geometrical description of non-convex domains for which
Hardy’s constants are best possible.
∗ ∗ ∗
Bo BERNDTSSON (Chalmers University of Technology)
Convexity of some functionals in K¨
ahler geometry
In the study of extremal metrics on a K¨ahler manifold, certain functionals
defined on the space of all K¨ahler metrics in a fixed cohomology class play an
important role. The most important of these are the Mabuchi K-energy and the
Ding functional whose critical points are metrics of constant scalar curvature and
K¨ahler–Einstein metrics respectively. We will discuss convexity properties of these
functionals and relations to questions of existence and uniqueness of extremal
metrics. This is partly joint work with Robert Berman.
∗ ∗ ∗
Vladimir DUBININ (Far Eastern Federal University, Vladivostok )
Some resent applications of the capacitary approach
symmetrization to the geometric function theory
and
We discuss the basic properties of the generalized condensers on the open
subsets of the Riemann sphere [1]. As simple applications of the monotonicity of
3
the capacity, some theorems for the univalent holomorphic functions are proved.
We obtain distortion theorems, inequalities for the Schwarzian derivative and
propositions describing the boundary behavior of such functions. We also give
new applications of the Steiner symmetrization. A lower bound for the half-plane
capacity of the compact sets and the inequality for the initial coefficient of some
class of univalent functions are established [1] - [3]. Thereafter a new version of
circular symmetrization for condensers on the Riemann surface is considered [4].
As application of such symmetrization, a two-point distortion theorem for complex
polynomials is proved [5]. The corollaries of this theorem are an exact lower bound
for maximal moduli of critical values of polynomials and a new version of Markovtype inequality for an arbitrary compact sets.
[1] V.N. Dubinin, Condenser capacity and symmetrization in the geometric
function theory of a complex variable, Dalnauka, Vladivostok, 2009, 401 p.
[2] V.N. Dubinin, Lower bounds for the half-plane capacity of compact sets and
symmetrization, Sbornik: Mathematics, 201:11 (2010), 1–12.
[3] V.N. Dubinin, Steiner symmetrization and the initial coefficients of univalent
functions, Izvestiya: Mathematics, 74:4 (2010), 735–742.
[4] V.N. Dubinin, A new version of circular symmetrization with applications to
p-valent functions, Sbornik: Mathematics, 203:7 (2012), 996–1011.
[5] V.N. Dubinin, On one extremal problem for complex polynomials with
constrains on critical values, Siberian Mathematical Journal, 55:1 (2014), 63–71.
∗ ∗ ∗
Stephen GARDINER (University College Dublin)
Universal Taylor series and potential theory
In various mathematical contexts it is possible to find a single object which,
when subjected to a countable process, yields approximations to the whole
universe under study. Such an object is termed "universal"and, contrary to
expectations, such objects often turn out to be generic rather than exceptional.
This talk will focus on this phenomenon in respect of the Taylor series of a
holomorphic function, and how the partial sums behave outside the domain of the
function. It will discuss how potential theory reveals much about the boundary
behaviour of such functions, and their relationship with conformal mappings.
∗ ∗ ∗
Dmitry KHAVINSON (University of South Florida)
The fundamental theorem of algebra, complex analysis and astrophysics
The Fundamental Theorem of Algebra first rigorously proved by Gauss states
that each complex polynomial of degree n has precisely n complex roots. In
recent years various extensions of this celebrated result have been considered.
We shall discuss the extension of the FTA to harmonic polynomials of degree n.
4
In particular, the theorem of D. Khavinson and G. Swiatek that shows that the
harmonic polynomial z¯ − p(z), deg p = n > 1 has at most 3n − 2 zeros as was
conjectured in the early 90´s by T. Sheil-Small and A. Wilmshurst. L. Geyer was
able to show that the result is sharp for all n.
G. Neumann and D. Khavinson proved that the maximal number of zeros of
rational harmonic functions z¯ − r(z), deg r = n > 1 is 5n − 5. It turned out
that this result confirmed several consecutive conjectures made by astrophysicists
S. Mao, A. Petters, H. Witt and, in its final form, the conjecture of S. H. Rhie
that were dealing with the estimate of the maximal number of images of a star if
the light from it is deflected by n co-planar masses. The first non-trivial case of
one mass was already investigated by A. Einstein around 1912.
We shall also discuss the problem of gravitational lensing of a point source
of light, e.g., a star, by an elliptic galaxy, more precisely the problem of the
maximal number of images that one can observe. Under some more or less
“natural” assumptions on the mass distribution within the galaxy one can prove
(A. Eremenko and W. Bergweiler – 2010, also, DK - E. Lundberg – 2010) that
the number of visible images can never be more than four in some cases and six
in the other. Interestingly, the former situation can actually occur and has been
observed by astronomers. Still there are much more open questions than there are
answers.
∗ ∗ ∗
Ari LAPTEV (Imperial College London & Institute Mittag-Leffler )
Spectral inequalities for Partial Differential Equations and their
applications
We shall discuss properties of the discrete and continuous spectrum of different
classes of self-adjoint differential operators including Schr¨odinger operators.
∗ ∗ ∗
Yurii LYUBARSKII (Norwegian University of Science and Technology,
Trondheim)
Composition operators in model spaces
We consider the composition operator acting from a model space KΘ into the
Hardy space. We study when such operators are compact or belong to Schatten
classes. In the case of one component Θ one can obtain complete answer to these
questions. We also give necessary and (separately) sufficient conditions for the
general case.
∗ ∗ ∗
5
Stefan NEMIROVSKI (Steklov Mathematical Institute, Moscow )
Rational approximations and symplectic topology
The talk will survey recent advances in the understanding of the topology and
geometry of rationally convex sets and domains achieved using ideas and methods
from symplectic geometry.
∗ ∗ ∗
Eero SAKSMAN (University of Helsinki)
On the multiplicative chaos
We consider some basic properties of multiplicative chaos, especially at
criticality. Also we try to explain connections to complex analysis and related
topics. The talk is based on joint work with J. Barral, A. Kupiainen, M. Nikula
and C. Webb.
∗ ∗ ∗
Kristian SEIP (Norwegian University of Science and Technology, Trondheim)
Poisson integrals on polydiscs
We consider the problem of estimating the Poisson integral of the square of
the modulus of a holomorphic polynomial at a point in the infinite dimensional
polydisc. The problem can be seen to originate in metric number theory and is
related to Granville and Soundararajan’s resonance method. We obtain optimal or
nearly optimal results in the particular case when the point in the polydisc is the
sequence of a fixed negative power of the primes, thus solving a problem studied
by Dyer and Harman in 1986. The talk is based on joint work with Christoph
Aistleitner and Istvan Berkes and joint work with Andriy Bondarenko.
∗ ∗ ∗
Armen SERGEEV (Steklov Mathematical Institute, Moscow )
Universal Teichm¨
uller space and its quantization
Universal Teichm¨
uller space T is the quotient of the group QS(S 1 ) of
quasisymmetric homeomorphisms of S 1 modulo M¨obius transformations. In
particular, this space contains the quotient S of the group Diff+ (S 1 ) of
diffeomorphisms of S 1 modulo M¨obius transformations. Both groups act naturally
1/2
on Sobolev space H := H0 (S 1 , R). Quantization problem for T and S arises
in string theory where these spaces are considered as phase manifolds. To
solve the problem for a given phase manifold means to fix a Lie algebra of
functions (observables) on it and construct its irreducible representation in a
Hilbert (quantization) space. For S the algebra of observables is given by the
Lie algebra Vect(S 1 ) of Diff+ (S 1 ). For quantization space we take the Fock space
1/2
F (H), associated with Sobolev space H = H0 (S 1 , R). Infinitesimal version
of Diff+ (S 1 )-action on H generates an irreducible representation of Vect(S 1 ) in
6
F (H), yielding quantization of S. For T the situation is more subtle since QS(S 1 )action on T is not smooth. So there is no classical Lie algebra, associated to
QS(S 1 ). However, we can define a quantum Lie algebra of observables Derq (QS),
generated by quantum differentials, acting on F (H). These differentials are given
by integral operators dq h on H with kernels, given essentially by finite-difference
derivatives of h ∈ QS(S 1 ).
∗ ∗ ∗
Yum-Tong SIU (Harvard University)
Application of d-bar estimates to effective very ampleness and
abundance
We will discuss some analytic methods of construction of sections of vector
bundles motivated by algebraic and complex geometric problems. We will
especially focus on the very ampleness part of the Fujita conjecture and the
abundance conjecture.
∗ ∗ ∗
Xavier TOLSA (ICREA & Universitat Autonoma de Barcelona)
Square functions, Riesz transforms, and rectifiability
In this talk I will survey recent results about the characterization of uniform
rectifiability in terms of square functions involving densities and in terms of L2
boundedness of Riesz transforms.
∗ ∗ ∗
Avgust TSIKH (Siberian Federal University, Krasnoyarsk )
Singular strata of cuspidal type for the classical discriminant
We consider an algebraic equation with variable complex coefficients. For
the reduced discriminant set of this equation we get a parametrization of
singular strata accountable for existence of roots with prescribed multiplicity.
This parametrization is the restriction to a flag of linear subspaces of the Horn–
Kapranov uniformization for the whole reduced discriminant set. We prove that
the mentioned strata are birationally isomorphic to some A-discriminant sets, and
therefore they are of maximum likelihood degree one. This is a joint submitted
paper with E. N. Mikhalkin.
∗ ∗ ∗
7
TALKS AND POSTERS
Irina ANTIPOVA (Siberian Federal University, Krasnoyarsk )
On the structure of the residue current of the Bochner–Martinelli type
Let X be a complex analytic n-dimensional manifold. Consider a holomorphic
mapping f = (f1 , . . . , fp ) on X (p is less or equal n). We consider the residue
integral I() over the tube defined by the mapping f with vector radius for the
compactly supported test form of the bidegree (n, n − p). For the case p = 1
the limit of the residue integral as tends to zero defines the Herrera–Lieberman
current. The local structure of this current was studied by P. Dolbeault. If p > 1
then the limit of the function I() does not exist in general. So it is reasonable
to consider some kinds of mean values of the residue integral. The mean value
on the base of Bochner-Martinelli kernel was realized by M. Passare, A. Tsikh
and A. Yger. We present the description of the local structure for the currents of
the Bochner–Martinelli type in the case of monomial mapping f . We prove the
following assertion.
Theorem. The residue current of the Bochner–Martinelli type is the finite
sum, where every summand is a product of q Herrera–Lieberman residue currents
and (n − q)-fold principal value, multiplied by certain hypergeometric function.
∗ ∗ ∗
Laurent BARATCHART (INRIA)
Pseudo-holomorphic functions with L2 coefficients and conjugate
Beltrami equation with tanh-W 1,2 coefficients
For D the unit disk and α ∈ L2 (D), we consider the equation
¯ = αw.
∂w
¯
(1)
This equation defines pseudo-holomorphic functions w and it has been extensively
studied by I. N. Vekua, L. Bers and many others, when α lies in Lr (D) for some
r > 2, see e.g. [1–5]. The main novelty here is that we consider r = 2, a case
where solutions need no longer be continuous nor even locally bounded.
Below we put T to mean the unit circle and, for any function f defined on D,
we let fρ (z) := f (ρz), z ∈ D. We prove a M. Riesz-type theorem as follows:
Theorem 1. For 1 < p < ∞, λ a real number, and ψ ∈ Lp (T), there uniquely
exists w meeting (1) such that
sup rkwr kLp (T) < c(kψkLp (T) + λ),
0<r<1
Z
p
lim <(wr ) = ψ in L (T),
r→1−
8
=(wr ) = λ.
lim
r→1−
T
(2)
In (2), the constant c depends only on p and α.
Theorem 1 has the following consequences. If we let σ be a nonnegative
function on D such that log σ ∈ W 1,2 (D) with W 1,2 (D) the familiar Sobolev
space on D, then the Dirichlet problem for the conductivity equation
div(σ∇u) = 0
is well posed in D with boundary value φ in the weighted space Lp (T, σ 1/2 dm)
where m is Lebesgue measure on T. Note that this conductivity equation may
not be strictly elliptic for σ could vanish on a set of 2-Riesz capacity zero, and
that coefficients may not even be bounded for σ could be +∞ on another set of
2-Riesz capacity zero. In fact, u solving the above Dirichlet problem is the real
part of a function f = u + iv, unique up to addition of a pure imaginary constant,
satisfying the conjugate Beltrami equation
¯ = ν ∂¯f¯,
∂f
ν = (1 − σ)/(1 + σ),
tanh−1 ν ∈ W 1,2 (D),
f being moreover such that
sup rkσ 1/2 ur + iσ −1/2 vr kLp (T) < +∞.
0<r<1
The results extend to Dini-smooth domains. We shall discuss extensions to
complex valued dilation coefficient ν. This is joint work with A. Borichev and S.
Chaabi.
[1] L. Baratchart, Y. Fischer, J. Leblond, Dirichlet/Neumann problems and Hardy
classes for the planar conductivity equation, to appear in Complex variables and
elliptic equations.
[2] L. Baratchart, J. Leblond, S. Rigat, E. Russ, Hardy spaces for the conjugate
Beltrami equation in smooth domains of the complex plane, J. Funct. Anal. 259
(2010), 384–427.
[3] L. Bers, Theory of pseudo-analytic functions, New York University, 1953.
[4] I. N. Vekua, Generalized Analytic Functions, Addison–Wesley, 1962.
[5] S. B. Klimentov, Hardy classes of generalized analytic functions, Izvestia Vuzov
Sev.-Kav. Reg., Natural Science 3 (2003) 6–10 (in Russian).
∗ ∗ ∗
Aleksandr BEKNAZARYAN (Kazan State Power Engineering University)
Bohr–Riemann surfaces
The talk is about the Bohr–Riemann surfaces over the generalized plane ∆.
The group structures of the covering spaces will be presented which allow to show
that each n-fold covering of the punctured generalized plane ∆0 is an algebraic
Galois covering. Also, the notion of equivalent points on the Bohr–Riemann
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surfaces will be given and it will be shown that the points of a Bohr–Riemann
surface locally have the same number of equivalent points.
∗ ∗ ∗
Yurii BELOV (St. Petersburg State University)
Mixed completeness problems in the spaces of entire functions
Let {kλ }λ∈Λ be a system of reproducing kernels in some space of entire
functions and {gλ }λ∈Λ be a biorthogonal system. We consider the completeness
problem for the system {kλ }λ∈Λ1 ∪ {gλ }λ∈Λ2 , Λ = Λ1 ∪ Λ2 . This property
corresponds to the spectral synthesis property for some linear operator. We give a
survey of results of this type in various situations. For example using this technique
we answer a question posed by A. Aleman and B. Korenblum about differentiation
invariant subspaces of C ∞ (a, b).
The talk is based on joint works with Anton Baranov, Alexander Borichev and
Alexandru Aleman.
∗ ∗ ∗
Sergey BEZRODNYKH, Vladimir VLASOV (Dorodnicyn Computing
Centre of RAS, Moscow )
Riemann–Hilbert problem in a singularly deforming domain
The Riemann–Hilbert problem with piecewise constant coefficients and a
growth condition of its solution is considered in a domain G, which represents
a decagon being an exterior of system of cuts. The considered problem gives the
Somov model of magnetic reconnection emerging in Solar flares. In the work an
analytic solution of the considered Riemann–Hilbert problem has been obtained.
With the help of the conformal mapping theory of singularly deforming domains,
the asymptotics of the obtained solution has been derived corresponding to special
cases of singular deformation of the domain G. These asymptotics show that
magnetic fields of Syrovatskii’s and Petscheck’s models follow from Somov’s model
as special limit cases.
∗ ∗ ∗
Markus BORN (University of Trier )
A note on lacunary approximation of Mergelyan type
Let K ⊂ C \ (−∞, 0) be compact and star-like with respect to the origin. For
a given subset Λ of the natural numbers we consider the set P
PΛ of polynomials
with exponents in Λ. We show that if 0 ∈ Λ, the condition n∈Λc n−1 < ∞ is
sufficient for PΛ to be dense in A(K), the space of all functions continous on K
and holomorphic in the interior, endowed with it’s usual norm. This is related
to a result of Anderson which states that for a given α ∈ [0, π), a M¨
untz-type
condition is necessary and sufficient for PΛ to be dense in every A(K), where K
is contained in the sector Kα = {z ∈ C : arg z < α}.
∗ ∗ ∗
10
Petr BORODIN (M. V. Lomonosov Moscow State University)
Approximation by simplest partial fractions
The talk contains results on the density of the set of simplest partial fractions
(logarithmic derivatives of polynomials) and their generalizations in different
spaces of functions defined on subsets of the complex plane.
∗ ∗ ∗
Victor BUSLAEV (Steklov Mathematical Institute, Moscow )
Convergence of two-point Pade approximants
Assume that algebraic functions f0 and f∞ are holomorphic at the points
z = 0 and z = ∞ respectively and have nonempty finite sets of branch points.
Let πn be a two-point Pade approximant of f0 and f∞ , i.e. πn is a rational function
Pn /Qn such that deg Pn ≤ n, deg Qn ≤ n, Qn 6≡ 0 and
(Qn f0 − Pn )(z) = O(z n ), z → 0,
(Qn f∞ − Pn )(z) = O 1/z , z → ∞.
Using potential theory methods we will show that the sequence of Pade
approximants {πn }∞
n=1 converges in capacity to the function f0 in some domain
D0 3 0 and to the function f∞ in some domain D∞ 3 ∞. The geometrical
properties of the complement C \ (D0 ∪ D∞ ) will be described.
∗ ∗ ∗
Andrey DNESTRYAN (Moscow Institute of Physics and Technology)
On the lower bound of the output entropy of the tensor product of two
Markov maps
A lower bound of von Neumann entropy of an operator is obtained, which is
the modus operandi of the tensor product of two Markov maps on the algebra of
bounded operators on a Hilbert space.
∗ ∗ ∗
Evgueni DOUBTSOV (St. Petersburg Department of Steklov Mathematical
Institute)
Reverse estimates in spaces of holomorphic functions
Given a radial weight w on the unit disk D, the growth space Aw consists
of those f ∈ Hol(D) for which |f (z)| ≤ Cw(z), z ∈ D. A natural problem
motivated by applications is to find test functions f1 , . . . , fk ∈ Aw such that the
following reverse estimate holds: |f1 | + · · · + |fk | ≥ Cw(z), z ∈ D. First, we
characterize, up to equivalence, those w for which the reverse estimate problem is
solvable. Second, we consider the weighted Bloch space B w defined by the estimate
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|f 0 (z)|(1 − |z|) ≤ Cw(z), z ∈ D. We show that the sharp reverse estimates in B w
depend on the quadratic integral
Z 1 2
w (t)
dt.
t
x
Similar results hold for the complex unit ball and for circular strictly convex
domains. Also, we discuss applications related to Carleson measures and
composition operators; in particular, we give a solution of the Bloch-to-BMOA
problem in several variables.
∗ ∗ ∗
Konstantin DYAKONOV (ICREA & Universitat de Barcelona)
Gauss–Lucas type theorems on the disk
The classical Gauss–Lucas theorem describes the location of the critical points
of a polynomial. There is also a hyperbolic version, due to Walsh, in which the role
of polynomials is played by finite Blaschke products on the unit disk. We study
similar phenomena for generic inner functions, as well as for certain "locally inner"
self-maps of the disk.
∗ ∗ ∗
Yurii DYMCHENKO (Far Eastern Federal University, Vladivostok )
The condition of small girth on Carnot groups
We show that if a compact set is removable for capacity on the Carnot group,
then it satisfies the condition for small girth in terms of horizontal curves, that
is, for any positive Borel function and almost all curves of a certain family of
horizontal curves with a finite module the following condition holds: this curve
can be deformed to a finite number of sections so that the resulting curve does
not pass through a removable set and the integral of this function on the modified
sections of the curve is less than any given positive number.
∗ ∗ ∗
Anastasia FROLOVA (University of Bergen)
Cowen–Pommerenke type inequalities for univalent functions
We present a new estimate for angular derivatives of univalent maps at the
fixed points. The method we use is based on the properties of the reduced modulus
of a digon and the problem of extremal partition of a Riemann surface. Joint work
with M. Levenshtein, D. Shoikhet and A. Vasil’ev.
∗ ∗ ∗
12
Anna GLADKAYA, Oleg VINOGRADOV (St. Petersburg State
University)
Entire functions of exponential type deviating least from zero with a
weight
Classical results of P.L. Chebyshev and S.N. Bernstein on polynomials of least
deviation from zero with a weight are extended to entire functions of exponential
type. Let ρm be a function from the Carthwright class C of type m which is
positive on the real axis and let σ > m. Put
fσ (z) =
1 ∗
(G (z) + G(z)) ,
2
Fσ (z) =
1
(G∗ (z) − G(z)) ,
2iz
2
(z), ρm (x) = |gm (x)|2 , G∗ (z) = G(z). For the functions fσ
where G(z) = e−iσz gm
and Fσ the following theorems are proved.
Theorem 1. For any nonzero entire function q from the class C of type less
than σ we have the inequality
fσ − q > sup fσ .
sup ρm ρm R
R
Theorem 2. For any nonzero entire function Q of type less than σ summable
on the real axis with the weight | · |/ρm we have the inequality
Z∞
−∞
|Fσ (x) − Q(x)| − |Fσ (x)|
|x| dx > 0.
ρm (x)
Thus, the functions fσ and Fσ are strictly deviating least from the zero in
the uniform and integral metrics with the weights 1/ρm and | · |/ρm , respectively.
Analogous statements hold true for fσ in the integral metric with the weight 1/ρm
and for Fσ in the uniform metric with the weight | · |/ρm . Functions fσ and Fσ
generalize Chebyshev polynomials of first and second kind. We note that they are
not summable with respect to the above-mentioned weights.
∗ ∗ ∗
Victor GORYAINOV (Moscow Institute of Physics and Technology)
Loewner–Kufarev equation for conformal mappings of strips
The infinitesimal description of semigroups of conformal mappings of strips is
obtained. Some approximative property of the semigroup is given.
∗ ∗ ∗
13
Antti HAIMI (Norwegian University of Science and Technology, Trondheim)
Polyanalytic Bergman kernels
We discuss Hilbert spaces of polyanalytic functions on the complex plane. The
norm on these spaces is given by integration against a weight e−mQ(z) where Q
is a strictly subharmonic function and m a large positive scaling parameter. We
obtain a near-diagonal asymptotic expansion for the reproducing kernels as m
tends to infinity. In the setting of one complex variable, this generalizes the work
of Tian–Yau–Catlin–Zelditch from analytic functions to polyanalytic functions.
We also study reproducing kernels of corresponding polynomials spaces. These
are spanned by functions z¯r z j where 0 ≤ r ≤ q − 1 and 0 ≤ j ≤ n − 1. The inner
product is induced by the same weight as before. Keeping q fixed and letting n and
m go to infinity, we obtain scaling limits for the kernels in so called bulk regime.
In the model case Q(z) = |z|2 , this investigation has applications in statistical
quantum mechanics. Some of the results are joint work with Haakan Hedenmalm.
∗ ∗ ∗
Olena HLADKA (National University of Water Management and Nature
Resources, Rivne)
The complex analysis method of numerical identification of parameters
of nonlinear quasiideal processes
The method for solving nonlinear model problems of complex potential
theory in doubly-layered curvilinear domain with identification of parameters
(conductivity coefficient, characteristic values of the potential on the equipotential
lines of section layers, and the flows across the border parts bounded by the
flow lines that separate layers) was developed on the basis of numerical methods
of quasiconformal mappings and summary representations in conjunction with
domain decomposition method by Schwartz.
∗ ∗ ∗
Andreas JUNG (University of Trier )
Universality of composition operators for Fatou components of an
entire function
Let f be an entire function and let U be an invariant component of the Fatou
set of f . For a domain Ω ⊃ U in the complex plane, we investigate which level
of ‘universality’ a sequence of composition operators Tf n : H(Ω) → H(U ), g 7→
g ◦ f n can have (here f n denotes the n-th iterate of f ). In this situation, we are
looking for g ∈ H(Ω) having the property, that the set {Tf n (g) : n ∈ N} is
dense in H(U ). Such functions are called universal for (Tf n ). Starting with local
universality results in open neighborhoods
W of (super-)attracting or neutral fixed
points of f , for which the restriction f W is injective, we try to expand those local
results via conjugation to finite Blaschke products into more global results, which
14
are valid in an entire invariant Fatou component U of f where in general f U is
not injective.
∗ ∗ ∗
Georgy IVANOV (University of Bergen)
Slit holomorphic stochastic flows
We consider the general case of holomorphic stochastic flows that have slit
geometry, as well as their deterministic counterparts. We compare the random
families of curves generated by these flows with those generated by SchrammLoewner evolution, which is a particular example of a slit holomorphic stochastic
flow. This is joint work with Alexey Tochin and Alexander Vasil’ev.
∗ ∗ ∗
Valery KALYAGIN (Higher School of Economics, Nizhny Novgorod )
Asymptotics of sharp constants in Markov–Bernstein inequalities
We apply a new methods of matching asymptotics of solutions of difference
equations with boundary value problems in the complex plane to the investigation
of sharp constants in Markov–Bernstein inequality in a weighted functional spaces.
As a result we obtain asymptotics in terms of the smallest zero of appropriate
Bessel functions.
∗ ∗ ∗
Ilgiz KAYUMOV (Kazan Federal University)
An efficient algorithm for computation the integral means spectrum for
conformal mappings generated by lacunary power series with Hadamard
gaps
A computation of the integral means spectrum for certain domains with fractal
type boundaries is a very nontrivial problem. Such problems were investigated
by J. Littlewood, Ch. Pommerenke, N. G. Makarov, L. Carleson, P. W. Jones,
S. K. Smirnov, D. Beliaev and by many other mathematicians. Lacunary series
with Hadamard gaps was firstly used by Makarov to disprove one famous
conjecture about the behavior of the integral means spectrum near the point
t=1/3. In the report we (jointly with D. V. Maklakov) present an efficient
algorithm that allows us to compute such the spectrum with a very high precision
for conformal mappings generated by lacunary power series with Hadamard gaps
of the type 2n . In particular, we will show that the integral means spectrum is
less than t2 /4 for t < 1 in this case.
∗ ∗ ∗
15
Vladimir KESELMAN (Moscow State Industrial University)
Conformal capacity and conformal classification of non-compact
Riemannian manifolds
Non-compact connected Riemannian manifolds can be divided into two
conformally invariant classes (types): manifolds of conformally parabolic type (as
Euclidean space) and manifolds of conformally hyperbolic type (as Lobachevski
space) according to the conformal capacity of the manifold at infinity which is
equal to zero or is positive, respectively.
We give some geometric conditions and prove criteria for conformal type of a
Riemannian manifold. In particular, we prove that the isoperimetric inequality (or
isoperimetric function) of the manifold can be reduced, by the conformal change
of the initial Riemannian metric of the manifold, to the normal form of a manifold
of Euclidian or Lobachevski type, depending of the conformal type of the manifold
under consideration.
∗ ∗ ∗
Vitaly KIM (Institute of Mathematics, Ufa)
Hypercyclic operators on spaces of holomorphic functions
An operator (linear and continuous) on a topological vector space is
hypercyclic if there exists a vector whose orbit under the operator is dense.
In this talk I will discuss various aspects of hypercyclic operators on spaces of
holomorphic functions and their connection with complete systems of analytic
functions.
∗ ∗ ∗
Bulat KHABIBULLIN (Bashkir State University, Ufa)
Zero subsequences of entire functions of exponential type with
restrictions on their growth along the real line
Denote by R and C the sets of real and complex numbers. Let M : C → R
be a continuous function such that M is harmonic on C \ R, M (z) ≤ O |z| ,
R
(x)|
z → ∞, sup M (w) ≤ M (z) + const, z ∈ C, R |M
1+x2 dx < +∞, and
|z−w|≤1
1
M (x) ≤ 2
π
t + r
1
dt,
M (x + t) log t
t
−
r
R
Z
0 < r < r(x),
x ∈ R.
Let Λ = {λk }∞
k=1 ⊂ C be a point sequence without limit points in C. We give a
complete description of Λ for which there is an entire function f 6= 0 vanishing
on Λ and such that f (z) ≤ M (z) for all z ∈ C. This description differs from
the description given in our paper [1] for special weight function M (z) = σ|Im z|,
z ∈ C, 0 < σ ∈ R.
[1] B.N. Khabibullin, G.R. Talipova, F.B. Khabibullin, Zero subsequences for
16
Bernstein’s spaces and the completeness of exponential systems in spaces of
functions on an interval, Algebra i Analiz (St. Petersburg Math. Journal), 26:2
(2014), 193–223.
∗ ∗ ∗
Alexander KOMLOV (Steklov Mathematical Institute, Moscow )
Second order differential equation and strong asymptotics for a twopoint analogue of Jacobi polynomials
First of all we consider two-point Pade approximants for function from some
class of multivalued analytic functions with p > 1 branch points. We obtain a
differential equation of the second order with accessory parameters satisfied by
numerators of two-point Pade approximants of this function and the corresponding
remainder functions. After that we consider the case p = 2, then the numerators
of two-point Pade approximants are two-point analogue of Jacobi polynomials.
Applying the classical Liouville–Steklov method to obtained differential equation
we get strong asymptotic formula for this analogue.
∗ ∗ ∗
Evgeny KOROTYAEV (St. Petersburg State University)
Global properties of resonances for Dirac and Stark operators
We consider the Dirac and Stark operators perturbed by a compactly
supported potential (of a certain class) on the real line. We determine upper
and lower bounds of the number of resonances in complex discs at large radius
and express the trace formula in terms of resonances only.
Moreover, we estimate the sum of the negative power of all resonances and
eigenvalues in terms of the norm of the potential and the diameter of its support.
The proof is based on harmonic analysis and Carleson measures arguments.
∗ ∗ ∗
Olesya KRIVOSHEYEVA (Bashkir State University, Ufa)
A basis in an invariant subspace of entire function
The existence problem for a basis in a differentiation-invariant subspace of
entire function is investigated. It is proved that in any such subspace a basis
composed of linear combinations of eigenfunctions and associated functions exists.
The linear combinations is generated in exponents clusters of an arbitrarily small
relative diameter.
∗ ∗ ∗
17
Nikolay KRUZHILIN (Steklov Mathematical Institute, Moscow )
Isolated singularities, finite maps, and function algebras
Relations of isolated singularities of complex hypersurfaces to relevant
functiona algebras are discussed. The problem of the reconsruction of a
homogeneous singularity from its Milnor algebra is solved.
∗ ∗ ∗
Galina KUZ’MINA (St. Petersburg Department of Steklov Mathematical
Institute)
The extremal metric method in the geometric function theory
The method of extremal metric is a general method of the function theory. The
determining role in the development of this method is due to J. A. Jenkins. One
form of this method in the geometric function theory is the method of modules of
curve families. This method obtained the development in the recent works of the
Leningrad mathematical school and found many applications. The characteristic
feature of the modules method is the immediately use of the properties of
quadratic differentials.
∗ ∗ ∗
Andrei LISHANSKII (St. Petersburg State University)
A class of Toeplitz operators with hypercyclic subspaces
We use a theorem by Gonzalez, Leon-Saavedra and Montes-Rodriguez to
construct a class of coanalytic Toeplitz operators which have an infinitedimensional closed subspace, where any non-zero vector is hypercyclic.
∗ ∗ ∗
Alexander LOBODA (Voronezh State Academy of Building and Architecture)
Homogeneous real hypersurfaces of three-dimensional complex spaces
The classification problems for the affine homogeneous and holomorphically
homogeneous real hypersurfaces will be discussed. The scheme will be presented
of complete description of strictly pseudo convex homogeneous surfaces in affine
case and full solution of the question under consideration in some subcases.
Some necessary conditions for the Taylor coefficients will be obtained for the
holomorphic homogeneous hypersurfaces with 5-dimensional transitive group
actions.
∗ ∗ ∗
Lev MAERGOIZ (Siberian Federal University, Krasnoyarsk )
Extensions of a class of entire functions of many variables
The object of investigation is the class H(Tn ) of holomorphic functions in the
multidimensional torus space Tn . We study the subclass of functions equivalent
18
to a class of entire functions in the following sense: a function g ∈ H(Tn )
belongs to the mentioned subclass if there exists a monomial map F such that
f = g ◦ F is an entire function. Description of functions in these subclasses in
terms of geometrical properties of supports of their Laurent series is obtained. A
multidimensional analog of holomorphic function expansion in Laurent series and
structure of functions in H(Tn ) are presented. Characteristics of growth of these
functions are studied. New results and problems are discussed.
[1] L.S. Maergoiz, Multidimensional analogue of the Laurent series expansion of
a holomorphic function and related issues, Doklady Mathematics, 88:2 (2013),
569–572.
∗ ∗ ∗
Eugenia MALINNIKOVA (Norwegian University of Science and Technology,
Trondheim)
Coefficients of harmonic functions in growth spaces and multipliers
Let h∞
g be the space of harmonic functions in the unit ball that are bounded
by some increasing radial function g(r) with limr→1 g(r) = +∞; these spaces
are called growth spaces. We describe functions in growth spaces by the Ces`aro
means of their expansions in harmonic polynomials and apply this characterization
to study coefficient multipliers between growth spaces. A series of examples of
multipliers will be given and the law of the iterated logarithm for the radial
oscillation will be explained. The talk is based on joint works with Yu. Lyubarskii,
K. S. Eikrem, and P. Mozolyako.
∗ ∗ ∗
Alexey MARKOVSKIY (Kuban State University)
Equilibrium constant of compact plane set
We consider the variational problem of finding the function which is least
deviating from zero on a given compact set. This function is from the class of
complex products with a given number of zeros concentrated on this compact
set and having non-algebraic singularities with sum of multiplicities equal to one.
Properties of solutions of this problem are investigated. For plane compact set the
equilibrium constant is determined as the limit values of extreme products; it is
proved that this constant is equal to the capacity of the compact set.
∗ ∗ ∗
Maxim MAZALOV (Moscow Power Engineering Institute, Smolensk Branch)
Conditions of C m approximability by harmonic and bianalytic functions
Some recent results of C m approximability of functions on compact subsets of
Rd by harmonic and bianalytic functions will be discussed.
∗ ∗ ∗
19
Mark MELNIKOV (Universitat Aut´onoma de Barcelona)
On one conjecture of the ‘domain expansion principle’ type
Let G be a domain in the complex plane, a ∈ G, and γ(G, a) = max |f 0 (a)|,
where the maximum is taken over all holomorphic in G functions f such that
|f | ≤ 1 in G. One conjecture concerning the behavior of the quantity γ(G, a)
when the domain G expands is discussed. We consider variation of the quantity
γ(G, a) under special variations of the boundary of G.
∗ ∗ ∗
Pavel MOZOLYAKO (Norwegian University of Science and Technology,
Trondheim)
On the minimum of harmonic functions in the n-dimensional ball
A well-known theorem by M. Cartwright states that if a harmonic in the unit
disk function u satisfies the one-sided growth condition
u(z) ≤ w(|z|),
where w(r) =
1
(1−r)p
z∈D
for some p > 1, then the reverse inequality holds
u(z) ≥ −Cw(|z|),
z ∈ D,
where C depends only on p. This result was extended in many ways to different
types of weights, the works by N. Nikolskii and A. Borichev should be mentioned in
particular. Some of the techniques they used however were limited to the complex
plane, so it was not obvious how one should obtain these kinds of results in higher
dimensions. We are planning to prove the extension of Cartwright’s theorem for
the unit ball in Rn for sufficiently regular weights w.
∗ ∗ ∗
¨
J¨
urgen MULLER
(University of Trier )
Limit functions on small parts of the Julia set
In the theory of dynamical systems the notion of ω-limit sets of points is
classical. In this talk, the existence of limit functions on subsets of the underlying
space is treated. It is shown that in the case of topologically (weak-)mixing
systems, plenty of limits functions on small subsets of X exist. On the other
hand, such sets necessarily have to be small in various respects. The results for
general discrete systems are applied in the case of Julia sets of rational and entire
functions. Further results are formulated in the case of the existence of Siegel
disks.
∗ ∗ ∗
20
Ildar MUSIN (Institute of Mathematics, Ufa)
Weighted spaces of entire functions
New results on relationship between a global growth of entire functions of
several complex variables of certain classes and behavior of these functions
on sets of multidimensional real space will be reported. For the considered
spaces theorems of Paley-Wiener type (or Paley–Wiener–Schwartz type) will be
established.
∗ ∗ ∗
Valentin NAPALKOV (Institute of Mathematics, Ufa)
Generalized Bargmann-Fock spaces and spectral synthesis
Applying the solution of the problem about the description of the operator
adjoint to the operator of multiplication in the Bargmann–Fock spaces we solve
the problems of spectral synthesis.
∗ ∗ ∗
Valerii NAPALKOV (Institute of Mathematics, Ufa)
Ortosimilar expansion systems in Hilbert spaces of analytic function
We consider ortosimilar expansion systems (expansion systems similar to
orthogonal ones) in Hilbert spaces of analytic function. We study the problem
of describing the adjoint operator to the operator of multiplication by the
independent variable in the Bargmann–Fock space. Newman–Shapiro’s problem
is considered. In this talk we will show how to apply the theory of orthosimilar
expansion systems to solve this problem. Also we will discuss one problem
associated with Ahlfors’ quasiconformal reflection in the complex plane.
∗ ∗ ∗
Petr PARAMONOV (M. V. Lomonosov Moscow State University)
Smooth subharmonic extensions of Runge and Walsh types on open
Riemann surfaces
Several settings of the C m -subharmonic extension problem on open Riemann
surfaces is planned to discuss. The problem was completely solved (for all
m ∈ [0, +∞)) for the so-called Runge-type extensions. Several (in some sense
sharp) sufficient conditions and counterexamples were found also for the Walshtype extensions. As applications, these results allow to prove the existence of
C m -subharmonic extensions, automorphic with respect to some special groups of
conformal mappings. The results of joint work of the author with A. Boivin and
P. M. Gauthier (Canada).
∗ ∗ ∗
21
Sergey POPENOV (Institute of Mathematics, Ufa)
Interpolation with real nodes by series of exponentials in the spaces of
analytic functions
We obtain the necessary and sufficient conditions for interpolation with real
nodes by series of exponentials in the spaces of analytic functions in unbounded
convex domains. Furthermore necessary conditions are found for interpolation by
elements from invariant subspaces.
∗ ∗ ∗
Aleksandr ROTKEVICH (St. Petersburg State University)
A constructive characterization of Besov spaces on convex domains
In 1981 E. M. Dyn’kin gave the characterization of holomorphic Besov spaces
on domains on complex plane by means of polynomial approximations. He
suggested the method of ‘pseudoanalytical continuation’ as a tool to study spaces
of holomorphic functions. This talk concerns the generalization of this method to
the case of convex domains in the space of several complex variables.
∗ ∗ ∗
Timur SADYKOV (Plekhanov Russian State University of Economics,
Moscow )
Maximally reducible monodromy of bivariate hypergeometric systems
I will present a joint work with S. Tanabe. We investigate branching of
solutions to holonomic bivariate hypergeometric systems of Horn type. Special
attention is paid to the invariant subspace of Puiseux polynomial solutions. We
mainly study (1) Horn systems defined by simplicial configurations, (2) Horn
systems whose Ore–Sato polygon is either a zonotope or a Minkowski sum of
a triangle and segments. We prove a necessary and sufficient condition for the
monodromy representation to be maximally reducible, that is, for the space of
holomorphic solutions to split into the direct sum of one-dimensional invariant
subspaces.
∗ ∗ ∗
Anton SAVIN, Boris STERNIN (Peoples Friendship University, Moscow )
Differential equations on complex manifolds
In this talk we deal with the theory of differential equations on complex
manifolds. Namely, we study linear differential equations in CN and, more
generally, on complex manifolds. This theory is remarkable and interesting. It
turns out that to study equations on complex manifolds, one has to use essentially
new methods, than in the classical (real) theory. These methods will be described
in the talk, in particular, we shall describe Sternin–Shatalov transform, which
enables one to solve equations with constant coefficients.
22
Complex theory has important applications in mathematics and physics. In
particular, we shall show how the methods described in the talk enable one to
solve balayage inwards problem (Poincare).
∗ ∗ ∗
Nikolay SHCHERBINA (University of Wuppertal )
On Liouville type properties of the core of a domain
For a domain G in a complex manifold M we introduce the notion of the core
c(G) of G as the set of all points where every smooth and bounded from above
plurisubharmonic function on G fails to be strictly plurisubharmonic. We study
then the following questions related to the structure of the core:
1) Let L be a connected component of c(G). Does it follow that every bounded
from above plurisubharmonic function on G is constant on L?
2) In which cases one can ensure the existence of analytic structure on c(G)?
∗ ∗ ∗
Vladimir SHLYK (Far Eastern Federal University & Vladivostok Branch of
Russian Customs Academy)
On the problem of decomposition and composition of normal rectangle
open sets
There is an extensive literature devoted to investigations of the properties of
normal domains (normal domains in the Gr¨otzsch sense or minimal domains in
the Koebe sense). Here, using the method of extremal metric in Fuglede’s sense
we give a solution of the problem of decomposition and composition of p-normal
rectangle open sets, 1 < p < ∞, in the n-dimensional Euclidean space Rn , n ≥ 2.
∗ ∗ ∗
Mikhail SKOPENKOV (Institute for Information Transmission Problems,
Moscow )
Discrete complex analysis: convergence results
Various discretizations of complex analysis have been actively studied since
1920s because of applications to numerical analysis, statistical physics, and
inegrable systems. This talk concerns complex analysis on quadrilateral lattices
tracing back to the works of J. Ferrand. We solve a problem of S. K. Smirnov
on convergence of discrete harmonic functions on planar nonrhombic lattices to
their continuous counterparts under lattice refinement. This generalizes the results
of R. Courant–K. Friedrichs–H. Lewy, L. Lusternik, D. S. Chelkak–S. K. Smirnov,
P. G. Ciarlet–P.-A. Raviart. We also prove convergence of discrete period matrices
and discrete Abelian integrals to their continuous counterparts (this is a joint
23
work with A. I. Bobenko). The proofs are based on energy estimates inspired by
electrical network theory.
∗ ∗ ∗
Dmitriy TULYAKOV (Keldysh Institute of Applied Mathematics, Moscow )
Abelian integral of Nuttall on the Riemann surface of the cubic root of
the third degree polynomial
We consider the geometry of Hermite–Pade approximants for two functions
with three common branch points. This problem has an interest with the
connection to the Nuttall’s conjecture, which states that an algebraic function
of the third order appears as the Cauchy transform of limiting measures of poles
of the approximants. We discuss the case of the algebraic function of the genus
one. This is a joint work with A. I. Aptekarev.
∗ ∗ ∗
Askhab YAKUBOV (Chechen State University)
On one classical Chebyshev’s problem on inequalities
To be announced
∗ ∗ ∗
Rinad YULMUKHAMETOV (Institute of Mathematics, Ufa)
Entire functions and unconditional bases in weighted spaces
The talk is devoted to a problem of construction of unconditional basis of
exponentials in weighted Hilbert spaces on segment of real axis. We assume that
a weight function has a slow growth.
∗ ∗ ∗
24
LIST OF PARTICIPANTS
Evgeny Abakumov
University Paris-Est Marne-la-Vallee, [email protected]
Alexei Aleksandrov
St. Petersburg Department of Steklov Mathematical Institute, [email protected]
Irina Antipova
Siberian Federal University, Krasnoyarsk, [email protected]
Alexander Aptekarev
Keldysh Institute of Applied Mathematics, Moscow, [email protected]
Kari Astala
University of Helsinki, [email protected]
Farit Avkhadiev
Kazan Federal University, [email protected]
Anton Baranov
St. Petersburg State University, [email protected]
Laurent Baratchart
INRIA, [email protected]
Mikhail Basok
St. Petersburg State University, [email protected]
Aleksandr Beknazaryan
Kazan State Power Engineering University, [email protected]
Yurii Belov
St. Petersburg State University, [email protected]
Bo Berndtsson
Chalmers University of Technology, [email protected]
Sergey Bezrodnykh
Dorodnicyn Computing Centre of RAS, [email protected]
Markus Born
University of Trier, [email protected]
Petr Borodin
M.V. Lomonosov Moscow State University, [email protected]
Ekaterina Borovik
Bauman Moscow State Technical University, [email protected]
Victor Buslaev
Steklov Mathematical Institute, Moscow, [email protected]
25
Evgenii Chirka
Steklov Mathematical Institute, Moscow, [email protected]
Andrey Dnestryan
Moscow Institute of Physics and Technology, [email protected]
Evgueni Doubtsov
St. Petersburg Department of Steklov Mathematical Institute, [email protected]
Mikhail Dubashinskiy
St. Petersburg State University, [email protected]
Vladimir Dubinin
Far Eastern Federal University, Vladivostok, [email protected]
Yuri Dymchenko
Far Eastern Federal University, Vladivostok, [email protected]
Konstantin Dyakonov
ICREA & Universitat de Barcelona, [email protected]
Konstantin Fedorovskiy
Bauman Moscow State Technical University, [email protected]
Anastasia Frolova
University of Bergen, [email protected]
Stephen Gardiner
University College Dublin, [email protected]
Anna Gladkaya
St. Petersburg State University, [email protected]
Victor Goryainov
Moscow Institute of Physics and Technology, [email protected]
Antti Haimi
Norwegian University of Science and Technology, Trondheim,
[email protected]
Olena Hladka
National University of Water Management and Nature Resources, Rivne,
[email protected]
Georgy Ivanov
University of Bergen, [email protected]
Alexei Ilyin
Keldysh Institute of Applied Mathematics, Moscow, [email protected]
Andreas Jung
University of Trier, [email protected]
Valery Kalyagin
Higher School of Economics, Nizhny Novgorod, [email protected]
26
Vladimir Kapustin
St. Petersburg Department of Steklov Mathematical Institute,
[email protected]
Ilgiz Kayumov
Kazan Federal University, [email protected]
Vladimir Keselman
Moscow State Industrial University, [email protected]
Bulat Khabibullin
Bashkir State University, Ufa, [email protected]
Dmitry Khavinson
University of South Florida, [email protected]
Vitaly Kim
Institute of Mathematics, Ufa, [email protected]
Sergey Kislyakov
St. Petersburg Department of Steklov Mathematical Institute, [email protected]
Alexander Komlov
Steklov Mathematical Institute, Moscow, [email protected]
Anna Kononova
St. Petersburg State University, [email protected]
Evgeny Korotyaev
St. Petersburg State University, [email protected]
Aleksandr Kotochigov
St. Petersburg Elektrotechnical University, [email protected]
Olesya Krivosheeva
Bashkir State University, Ufa, [email protected]
Nikolai Kruzhilin
Steklov Mathematical Institute, Moscow, [email protected]
Galina Kuz’mina
St. Petersburg Department of Steklov Mathematical Institute,
[email protected]
Maria Lapik
Keldysh Institute of Applied Mathematics, Moscow, [email protected]
Ari Laptev
Imperial College London, [email protected]
Elena Lebedeva
St. Petersburg State University, [email protected]
Andrei Lishanskii
St. Petersburg State University, [email protected]
27
Alexander Loboda
Voronezh State Academy of Building and Architecture, [email protected]
Alexander Logunov
St. Petersburg State University, [email protected]
Vladimir Lysov
Keldysh Institute of Applied Mathematics, Moscow, [email protected]
Yurii Lyubarskii
Norwegian University of Science and Technology, Trondheim,
[email protected]
Lev Maergoiz
Siberian Federal University, Krasnoyarsk, [email protected]
Eugenia Malinnikova
Norwegian University of Science and Technology, Trondheim,
[email protected]
Vesna Manojlovic
Mathematical Institute of the Serbian Academy of Sciences, [email protected]
Alexey Markovskiy
Kuban State University, [email protected]
Maxim Mazalov
Moscow Power Engineering Institute (Smolensk Branch),
[email protected]
Mark Melnikov
Universitat Aut´onoma de Barcelona, [email protected]
Pavel Mozolyako
Norwegian University of Science and Technology, Trondheim,
[email protected]
Juergen Mueller
University of Trier, [email protected]
Ildar Musin
Institute of Mathematics, Ufa, [email protected]
Valentin Napalkov
Institute of Mathematics, Ufa, [email protected]
Valerii Napalkov
Institute of Mathematics, Ufa, [email protected]
Stefan Nemirovski
Steklov Mathematical Institute, Moscow, [email protected]
Nikolai Osipov
St. Petersburg Department of Steklov Mathematical Institute, [email protected]
28
Petr Paramonov
M.V. Lomonosov Moscow State University, [email protected]
Sergey Popenov
Institute of Mathematics, Ufa, [email protected]
Aleksandr Rotkevich
Higher School of Economics, St. Petersburg, [email protected]
Marianna Russkikh
St. Petersburg State University, [email protected]
Timur Sadykov
Plekhanov Russian State University of Economics, Moscow, [email protected]
Eero Saksman
University of Helsinki, [email protected]
Anton Savin
Peoples Friendship University, Moscow, [email protected]
Walter Sch¨
afer
Technische Universit¨at Darmstadt, [email protected]
Kristian Seip
Norwegian University of Science and Technology, Trondheim, [email protected]
Armen Sergeev
Steklov Mathematical Institute, Moscow, [email protected]
Nikolay Shcherbina
University of Wuppertal, [email protected]
Nikolai Shirokov
St. Petersburg State University, [email protected]
Vladimir Shlyk
Far Eastern Federal University, Vladivostok, [email protected]
Yum-Tong Siu
Harvard University, [email protected]
Mikhail Skopenkov
Institute for Information Transmission Problems, Moscow, [email protected]
Boris Sternin
Peoples Friendship University, Moscow, [email protected]
Dmitriy Stolyarov
St. Petersburg Department of Steklov Mathematical Institute, [email protected]
Alexandre Sukhov
University Lille-1, [email protected]
29
Xavier Tolsa
ICREA & Universitat Aut´onoma de Barcelona, [email protected]
Avgust Tsikh
Siberian Federal University, Krasnoyarsk, [email protected]
Dmitriy Tulyakov
Keldysh Institute of Applied Mathematics, Moscow, [email protected]
Vasily Vasyunin
St.
Petersburg
Department
[email protected]
of
Steklov
Mathematical
Institute,
Ilya Videnskii
St. Petersburg State University, [email protected]
Oleg Vinogradov
St. Petersburg State University, [email protected]
Vladimir Vlasov
Dorodnicyn Computing Centre of RAS, Moscow, [email protected]
Rinad Yulmukhametov
Institute of Mathematics, Ufa, [email protected]
Askhab Yakubov
Chechen State University, [email protected]
Pavel Zatitsky
St. Petersburg Department of Steklov Mathematical Institute, [email protected]
30
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