MY RESEARCH



 

My main research interests are basically in the following areas:

1) Complex Geometry and Complex Analysis, including Riemann surfaces, conformal and quasiconformal maps, extremal and variational problems, extremal length methods, parametric methods.

2) Teichmüller Theory, including metric problems, compactification, complex structure.

3) Free Boundary Problems of Fluid Mechanics, including Hele-Shaw flows, melting/freezing models, Muskat problem etc.

4) Mathematical Physics including CFT, integrable systems.

5) Differential Geometry and Geometric Control.

6) Loewner evolution, SLE type stochastic processes in the plane.

I have 2 monographs and about 70 published papers in these six areas. Main results and representative recent works are:



Section 1

A very fruitful tool in geometric function theory is the modulus method: given a family of curves one considers admissible metrics with respect to this family and a certain quantity involving the length and the area of a domain swept out by the curves. The infimum of this quantity over admissible metrics is the modulus of the given family of curves. The important fact is that the modulus is conformally invariant and invariant up to multiplicative bounds under quasiconformal maps. The successful application of this method depends on a judicious choice of the family of curves and on finding a way to evaluate the modulus.
We applied this method to a number of interesting and quite difficult extremal problems for conformal maps. In particular, we considered problems of the boundary behavior of conformal maps and their derivatives as well as two-point distortion under conformal maps.
We extend the modulus method to extremal problems for quasiconformal maps and solved several extremal problems, in particular, two-point distortion.

These results were summarized in the monograph
■A.Vasil'ev: Moduli of families of curves for conformal and quasiconformal mappings. - Lecture Notes in Mathematics, vol. 1788, Springer-Verlag, Berlin- New York, 2002, 212 pp.
Other represetative works in this area are:
■ Ch.Pommerenke, A.Vasil'ev: Angular derivatives of bounded univalent functions and extremal partitions of the unit disk.-  Pacific. J. Math.  206 (2002), no.2, 425-450.
■A.Vasil'ev: On distortion under bounded univalent functions with the angular derivative fixed.- Complex Variables  47 (2002), no. 2, 131-147.

Section 2

Teichmüller theory is a substantial area of mathematics that has interactions with many other subjects, first of all, with mathematical physics. It brings together, at an equally important level, fundamental ideas coming from complex analysis, hyperbolic geometry, theory of discrete groups, algebraic geometry, low- dimensional topology, differential geometry, Lie group theory, symplectic geometry, dynamical systems, number theory, topological quantum field theory, string theory and many others, pursuing a modern trend in science to serve as a universal environment in which new mathematics of the next century is forged.
Together with Ruben Hidalgo we constructed a new partial compactification of the Teichmüller space of finitely generated Kleinian groups and call it noded Teichmüller space. It is one of few existent possible compactifications in some sense close to Abikoff’s Augmented Teichmüller space of Reimann surfaces.
■ R.Hidalgo, A.Vasil'ev: Noded Teichmüller spaces.- J. Anal. Math. 99 (2006), 63-73.

We also considered the modulus as a functional on the Teichmüller spaces and characterized the Teichmüller metric in terms of moduli. We also found sufficient conditions for coincidence of invariant Kobayashi and Carathéodori metrics on Teichmüller spaces.

■ R.Hidalgo, A.Vasil'ev: Harmonic moduli of families of curves on Teichmüller spaces.- Scientia. Ser. A, Math. Sci. 8 (2002), 63-73.
■A.Vasil'ev: Harmonic functionals and invariant metrics in the Teichmüller spaces.- Algebra and Analysis 9 (1997), no. 1, 49-71, (English  transl.:  St.-Petersburg Math. J. 9 (1998), no. 1, 33-48).


Section 3
Free boundary problems in fluid dynamics receive constant interest both from physics and mathematics/applied mathematics community. One of the central models here is the Hele-Show flow, which describes the continual injection/suction of fluid into a subordinating chain of domains. We related these flow with geometric function theory and proved several results on inherited geometry of the evolving in time interfacial curve. One of the main results is that in the difficult suction problem there is a unique possible simply connected evolution both in two- and multi- dimensional cases.

These results were summarized in the monograph

■B.Gustafsson, A.Vasil'ev: Conformal and potential analysis in Hele-Shaw cells. - ISBN 3-7643-7703-8, Birkhäuser Verlag, 2006, 231 pp.

Other representative works in this area are:
■B.Gustafsson,  A.Vasil'ev: Nonbranching weak and starshaped strong solutions for Hele-Shaw dynamics.-Arch. Math. (Basel) 84 (2005), no. 6, 551-558.
■B.Gustafsson, D.Prokhorov, A.Vasil'ev: Infinite lifetime for the starlike dynamics in Hele-Shaw cells.- Proc. Amer. Math. Soc. 132 (2004), no. 9, 2661-2669.
■A.Vasil'ev: Univalent functions in two-dimensional free boundary problems.- Acta Applic. Math. 79 (2003), no. 3, 249-280.
■I.Markina, A.Vasil’ev: On geometry of the Hele-Shaw flows with small surface tension.- Interfeces and Free Boundaries, 5 (2003), no. 2, 183-192
■A.Vasil'ev, K.Kornev: Geometric properties of the solutions of a Hele-Shaw type equation.- Proc. Amer. Math. Soc. 128 (2000), no. 9, 2683-2685.



The following sections are my current and forthcoming research.


Section 4

In recent years, many intersections between the field of complex analysis and the field of mathematical physics have presented themselves. One such crossover is in the study of contour dynamics in the complex plane. It has been discovered that this classical subject in complex analysis is related to the Laplacian growth problem in the complex plane, the Loewner evolution equation and its stochastic counterpart, which in turn are found to be related to integrable systems in mathematical physics, such as the Toda hierarchy.
   Since the advent of string theory, conformal field theory in two dimensions has become an important subject in mathematical physics and theoretical physics. It is closely related to the representation of infinite-dimensional algebras. One of the building blocks of conformal field theory is the Virasoro algebra, which is the central extension of the algebra of vector fields on the unit circle. It is also intrinsically related to integrable hierarchies.

Our recent research has shown that there is a deeper relation between contour dynamics and the Virasoro algebra. We develop the study of the conformal field theory viewpoint on contour dynamics. In particular, we proved that the Virasoro algebra is closely related to the classical smooth Loewner-Kufarev evolution. Moreover, we found solution to the KP hierarchy which remain invariant on the Loewner-Kufarev trajectories embedded into the Sato-Segal-Wilson Grassmannian.

Representative works in this area are:

■I. Markina, A.Vasil'ev: Virasoro algebra and dynamics in the space of univalent functions.- Contemporary Math. 525 (2010), 85-116
■A.Vasil'ev: Energy characteristics of subordination chains.- Arkiv för Matematik 45 (2007), 141-156
■R.Hidalgo, I.Markina, A.Vasil'ev: Finite dimensional grading of the Virasoro algebra.- Georgian Math. J. 14 (2007), no. 3, 419-434.
■D.Prokhorov, A.Vasil'ev: Univalent functions and integrable systems.- Comm. Math. Phys. 262 (2006), no. 2, 393-410.


Section 5

Sub-Riemannian geometry is a generalization of Riemannian geometry. In order, to measure distances in a sub-Riemannian manifold, we are allowed to go only along curves tangent to so-called horizontal subspaces.
Sub-Riemannian manifolds carry a natural intrinsic metric called the metric of Carnot–Carathéodory. The Hausdorff dimension of such metric spaces is always an integer and larger than its topological dimension (unless it is actually a Riemannian manifold).
Sub-Riemannian manifolds often occur in the study of constrained systems in classical mechanics, such as the motion of vehicles on a surface, the motion of robot arms, and the orbital dynamics of satellites. Geometric quantities such as the Berry phase may be understood in the language of sub-Riemannian geometry. The Heisenberg group, important to quantum mechanics, carries a natural sub-Riemannian structure.
We studied geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem was completely solved in the case of 3-dimensional sphere and some partial results are obtained in the general case. The Carnot-Carathéodory distance was calculated. We also presented motivations related to quantum mechanics.
We studied analogous geodesic problem for degenerating metrics introducing so-called sub Lorentzian geometry on anti-de Sitter space.
The main result is that we founded the infinite-dimensional analogue of sub-Riemannian geometry on most general locally compact manifolds. Using concrete example of the group of diffeomorphisms of the unit circle. We deduced several non-linear PDE of KdV type as the Arnold-Euler equations on the cotangent bundle.

Representative works in this area are:

■E.Grong, I. Markina, A.Vasil'ev: Sub-Riemannian geometry on infinite-dimensional manifolds.- arXiv: 1201.2251 [math.DG], 2012, 37 pp.
■D.-Ch. Chang, I. Markina, A.Vasil'ev: Hopf fibration: geodesics and distances.-  J. Geom. Physics 61 (2011), 986-1000.
■D.-Ch. Chang, I. Markina, A.Vasil'ev: Sub-Lorentzian geometry on anti-de Sitter space.-J. Math. Pures Appl. 90 (2008), no.1, 82-110.
■I.Markina, D.Prokhorov, A.Vasil'ev: Sub-Riemannian geometry of the coefficients of univalent functions.- J. Funct. Analysis 245 (2007), no. 2, 475-492.


Section 6

Last decade has been marked by burst of interest to Stochastic (Schramm)-Loewner Evolution (SLE), which has implied an elegant description of several 2D conformally invariant statistical physical systems at criticality by means of a solution to the Cauchy problem for a special differential equation with a random driving term given by 1D Brownian motion. The origin of SLE is underlain in the seminal Schramm's paper where he revisited the notion of scaling limit and conformal invariance for the loop erased random walk and the uniform spanning tree. This approach was developed in many further works and many lattice physical processes were thoroughly proved to converge to SLE with some specific diffusion factor, e.g., percolation, Ising model. Another attempt has been launched in order to construct random interfaces by conformal welding. On the other hand, SLE became an approach to conformal field theory, which emphasizes CFT’s roots in statistical physics.
We considered another setup in which interfaces (shapes) are smooth but they evolve randomly. The study of 2D shapes is one of the central problems in the field of applied sciences. A program of such study and its importance was summarized by Mumford at ICM 2002 in Beijing. We proposed a model, which describes deterministic and stochastic evolution of shapes in the complex plane.
On the other hand we made a progress in the study of the slit geometry of the Loewner evolution.

Representative works in this area are:
■G. Ivanov, D. Prokhorov, A.Vasil'ev: Non-slit and singular solutions to the Loewner equation.- Bull. Sci. Math. 136 (2012), no. 3, 328-341.
■G. Ivanov, A.Vasil'ev: Löwner evolution driven by a stochastic boundary point.- Anal. Math. Phys. 1 (2011), no. 4, 387-412.

■D.Prokhorov, A.Vasil'ev: Singular and tangent slit solutions to the Loewner equation.- Analysis and Mathematical Physics, Trends in Mathematics, Birkhäuser 2009, 455-463.
■F. Bracci, M. D. Contreras, S. Díaz-Madrigal, A.Vasil'ev: Classical and stochastic Löwner–Kufarev equations.- Harmonic and Complex Analysis and its Applications. Trends Math., Birkhäuser, Basel, 2013, 39-134. 
 
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