Moduli of families of curves and extremal problems for conformal maps (2-3 students)
Abstract. The project is aimed at thorough study of moduli of free families of homotopy classes of curves and their application to concrete extremal problems for conformal maps.
Description. The project includes topics from Differential Geometry, Real and Complex Analysis, Differential Equations. Students are supposed to learn Riemann surfaces and conformal invariants on them. Free families of homotopy classes of curves are to be defined. Modulus of a family of curves is a quantitative description of so-called “length-area” principle and possesses some properties of measure. Extremal families of curves is of particular importance. They are geodesics in certain Riemannian metric on the underlying Riemann surface generated by a quadratic differential. The method of moduli goes back to Ahlfors’ extremal length method and represents a powerful tool to solve extremal problems for conformal maps being combined with symmetrization and polarization. Students will be offered some concrete extremal problems:
References:
Dynamical systems in the unit disk and boundary behavior of conformal maps (1-2 students)
Abstract. The project is devoted to the Denjoy-Wolff theory of the dynamical systems in the unit disk. Students will study it and apply to fine problems of boundary behavior of univalent dynamical systems.
Description. The project includes topics from Dynamical Systems, Real and Complex Analysis, Ordinary Differential Equations. Conformal maps defined on some canonical domain (the unit disk in most of the cases) being analytic inside this domain show quite irregular behavior near the boundary. Many properties of boundary limits of conformal maps are known, in particular, angular limits have been studied deeply starting from the famous Beurling-Ahlfors paper in 1950. It is known that the angular limit exists at almost all points of the unit circle. However generally, very little can be said about the existence of the boundary derivatives. Moreover, one may observe the twisting effect at the boundary. For self-maps of the disk Julia-Wolff’s lemma implies that the angular derivative exists at almost all points where the angular limit exists and is equal to 1. Boundary derivatives are closely connected with reduced moduli of triangles and digons. The project is aimed to the study of mutual behavior of angular and usual internal derivatives applying the moduli technique as well as other extremal problems are expected to be solved.
References:
Classical actions for evolution dynamics of conformal maps (1-2 students)
Abstract: The project is devoted to applications of String Theory to some classical problems of growth evolution in the complex plane.
Description: The project includes topics from Mathematical Physics, Differential Geometry, Real and Complex Analysis, Hamiltonian Systems. Classical actions have been proved to be one of the principal subjects of the research in Field Theory in general, and in Gauge String Theory in particular, where the Liouville action plays an important role in the definition of the two-dimensional gravity. This leads to so called Quantum Geometry of Riemann Surfaces. Takhtajan and Zograf made a rigorous mathematical treatment of Polyakov’s discovery that first-quantized bosonic string propagation can be described as a theory of free bosons coupled with the two-dimensional quantum gravity. It is not surprising that several classical mechanical processes possess some quantum features (KdV and the soliton theory, e.g.). Some two-dimensional evolution processes in Fluid Mechanics can be described by means of conformal maps. The project is aimed at study of Liouville and similar classical actions first of all for the Laplacian Growth and then, for general evolution families of conformal maps described by the Löwner-Kufarev equation and its modifications for quasiconformal and quasiconformally extendable maps. This study is closely connected with the Kirillov definition of infinite-dimensional Lie algebras (Virasoro, in particular), and with their links to the Teichmüller theory. It is expected to obtain also variations of the actions and these variations are expected to clear up that the actions represent potentials of the Weil-Petersson metric on the Teichmüller spaces. In the project we expect to discover other connections between evolution equations for conformal maps and problems of Quantum Field Theory.
References:
Exceptional sets in first-year Calculus
Abstract: The project adresses the problem of construction of a function by its derivative.
Description: A well-known theorem in the first-year Calculus states that a continuous function in the interval [a,b] differentiable in this interval is constant if and only if its derivative is zero. The result remains true if we assume differentiability in [a,b]\E, where E is a finite number of points in [a,b]. How big can the set E be? The project includes some bibliography work.
References:
N.Bourbaki, Functions of a real variable. Elementary theory. Translated from the 1976 French original Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2004.
Description.
The topic takes its origin at the theory of analytic functions in the plane. A nice generalization of an analytic function of one complex variable to the n-dimentation setting is so called quasiregular mapping. This mapping has coordinate functions belonging to the Sobolev space, maps an infinitesimal ball to an infinitesimal ellipsoid with the controlled ratio between the largest and the smallest semi-exes of the ellipsoid. Like an analytic function, a quasiregular mapping is not necesserely a homeomorphism and can have branching points. The Carnot group is R^n equipped with non-euclidean metric and with a special geometrical structure. The notion of Carnot groups can be found in control theory, quantum mechanics, or differential geometry. In the works given in References, the definition of quasiregular mappings were adopted to the Carnot groups. The project will deal with the following questions. Is the set of suitable capacity zero removable for quasiregular mapping on Carnot group? How can one understand the radial limit of a quasiregular map in the unite ball on the Carnot group? It can happen that the radial curve has infinite length. What can we say about the existence of the radial limit? We shall study also the growth of the quasiregular mappings in unbounded domains in terms of capacity and of module of a family of curves.
The project includes topics from Real Analysis, Partial Differential Equations, Non-Linear Potential Theory, and Differential Geometry.
References:
S.Rickman. Quasiregular mappings. Springer-Verlag, Berlin, 1993, 213 pp.
J.Heinonen, I.Holopainen. Quasiregular maps on Carnot groups. J Geom. Anal. 7 (1997), no.1 109—148
S.
Vodop’yanov. Mapping with bounded distortion and with finite
distortion on
Carnot groups. Siberian Mathematical Journal. 40 (1999), no.4,
644—677.
2. Hamiltonian mechanics on the homogeneous groups.
Description:
Since their introduction by A. Kaplan
in 1980, generalized Heisenberg groups, also known as groups of
Heisenberg type or H-type groups, have provided a framework in which
interesting examples in geometry and analysis can be constructed. An
analysis on homogeneous groups, and in particular, on the H-type
groups, is a good test ground for the study of general sub-elliptic
problems arising from vector fields satisfying the Hörmander
hypoellipticity condition. The knowledge about H-type groups also can
be applied to the study of pseudoconvex domains in complex analysis,
sub-Riemannian geometry, control theory, and semiclassical analysis
of quantum mechanics. Recently the real and complex Hamiltonian
mechanics were involved to the study of sub-Riemannian manifolds. It
is well known, that the elliptic Laplace-Beltrami operator induces
the Riemannian geometry. In the same way the sub-elliptic Laplacian,
the sum of squares of vector fields, that jointly with their
commutators generate the tangent bundle on the groups, responds to
the geometry on sub-Riemannian manifolds. Thus, the study of the
Hamiltonian function associated with the sub-Laplacian plays a
crucial role. The project is aimed at the study of geometric
properties of the sub-Laplacian on homogeneous H-type groups. We will
construct the Hamiltonian function defined by sub-Laplacian and solve
the Hamiltonian system of differential equations. The exact equations
of geodesics allow us to consider the connectivity of points on this
type of groups and the number of geodesics, connecting the origin
with different points on groups. We obtain the modified action
function and find the relation between the homogeneous norm of
element on H-type groups, the Carnot-Carathéodory metric and
the modified action function. The next step is to construct the
volume element that permit to calculate the heat kernel. These
results give a good base on which the Hamiltonian mechanics may be
applied to study the geometry of sub-Riemannian manifolds and related
sub-elliptic equations.
References:
V.I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag New-York, 1989.
A. Kaplan, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), 35-42.
R. Beals, B. Gaveau, and P. C. Greiner, Complex Hamiltonian mechanics and parametrices for subelliptic Laplacians, I, II, III, Bull. Sci. Math., 21 (1997), 1-36, 97-149, 195-259.