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References (59)
-
(1916)
Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln -
A. Kirillov, D. Yuriev (1988)
Representations of the Virasoro algebra by the orbit methodJournal of Geometry and Physics, 5
-
(1979)
On the rational solutions of the Zaharov–Shabat equations and completely integrable systems of N particles on the line -
M. Mineev-Weinstein, P. Wiegmann, A. Zabrodin (2000)
Integrable structure of interface dynamicsPhysical review letters, 84 22
-
J. Gervais (1985)
Infinite family of polynomial functions of the Virasoro generators with vanishing Poisson bracketsPhysics Letters B, 160
-
B. Janssens (2016)
Loop groups -
M. Bauer, D. Bernard (2002)
Conformal Field Theories of Stochastic Loewner EvolutionsCommunications in Mathematical Physics, 239
-
I. Markina, A. Vasil’ev (2010)
Virasoro algebra and dynamics in the space of univalent functions -
G. Segal, G. Wilson (1985)
Loop groups and equations of KdV typePublications Mathématiques de l'Institut des Hautes Études Scientifiques, 61
-
(1975)
Univalent Functions, with a Chapter on Quadratic Differentials by G -
(1978)
Semigroups of holomorphic functions and composition operators -
H Airault, Yu Neretin (2008)
On the action of Virasoro algebra on the space of univalent functionsBull. Sci. Math., 132
-
P Raboin (1979)
Le problème du $$\bar{\partial }$$ ∂ ¯ sur en espace de HilbertBull. Soc. Math. Fr., 107
-
E. Witten (1988)
Quantum field theory, Grassmannians, and algebraic curvesCommunications in Mathematical Physics, 113
-
L. Lempert (1995)
The Virasoro group as a complex manifoldMathematical Research Letters, 2
-
V. Kac (1968)
SIMPLE IRREDUCIBLE GRADED LIE ALGEBRAS OF FINITE GROWTHMathematics of The Ussr-izvestiya, 2
-
A. Kirillov (1998)
Geometric approach to discrete series of unirreps for Vir.Journal de Mathématiques Pures et Appliquées, 77
-
D. Prokhorov, A. Vasil’ev (2004)
Univalent Functions and Integrable SystemsCommunications in Mathematical Physics, 262
-
(2008)
Differentiable Manifolds, 2nd edn -
(1984)
Remarks on infinite-dimensional Lie groups -
(1979)
Le problème du ∂̄ sur en espace de Hilbert -
E. Date, M. Kashiwara, T. Miwa (1981)
Vertex operators and $\tau$ functions transformation groups for soliton equations, II, 57
-
V. Matveev (1979)
Some comments on the rational solutions of the Zakharov-Schabat equationsLetters in Mathematical Physics, 3
-
(2005)
Discretized Virasoro Algebra -
(1972)
Optimal controls and univalent functions -
G. Wilson (1998)
Collisions of Calogero-Moser particles and an adelic Grassmannian (With an Appendix by I.G. Macdonald)Inventiones mathematicae, 133
-
Mikio Sato (1983)
Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann ManifoldNorth-holland Mathematics Studies, 81
-
G. Wilson (1993)
Bispectral commutative ordinary differential operators.Journal für die reine und angewandte Mathematik (Crelles Journal), 1993
-
R. Friedrich, W. Werner (2003)
Conformal Restriction, Highest-Weight Representations and SLECommunications in Mathematical Physics, 243
-
R. Friedrich (2009)
The Global Geometry of Stochastic L{\oe}wner EvolutionsarXiv: Mathematical Physics
-
R. Douglas (1972)
Banach Algebra Techniques in Operator Theory -
G. Goodman (1968)
Univalent functions and optimal control -
L Conlon (2008)
Differentiable Manifolds -
G Segal, G Wilson (1985)
Loop groups and equations of KdV typePubl. Math. IHES, 61
-
Karl Löwner (1923)
Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. IMathematische Annalen, 89
-
DV Prokhorov (1990)
Value sets for systems of functionals in the classes of univalent functionsMath. Sb., 181
-
I. Markina, D. Prokhorov, A. Vasil’ev (2006)
Sub-Riemannian geometry of the coefficients of univalent functions☆Journal of Functional Analysis, 245
-
K. Kirsch (2016)
Methods Of Modern Mathematical Physics -
(1968)
Cohomology of the Lie algebra of vector fields on the circle -
E Date, M Kashiwara, M Jimbo, T Miwa (1983)
Transformation Groups for Soliton Equations. Nonlinear Integrable Systems—Classical Theory and Quantum Theory (Kyoto, 1981) -
D. Mumford (2002)
Pattern Theory: the Mathematics of PerceptionarXiv: Numerical Analysis
-
L. Dickey (2003)
Soliton Equations and Hamiltonian Systems -
R. Moody (1968)
A new class of Lie algebrasJournal of Algebra, 10
-
(1943)
On one-parameter families of analytic functions -
(1950)
Coefficient Regions for Schlicht Functions (With a Chapter on the Region of the Derivative of a Schlicht Function by Arthur Grad) -
L. Branges (1985)
A proof of the Bieberbach conjectureActa Mathematica, 154
-
Bracci Filippo, Manuel Contreras, S. Díaz-Madrigal (2008)
Evolution families and the Loewner equation I: the unit disc, 2012
-
A. Boggess (1991)
CR Manifolds and the Tangential Cauchy-Riemann Complex -
H. Airault, Y. Neretin
Institute for Mathematical Physics on Action of the Virasoro Algebra on the Space of Univalent Functions -
G Wilson (1998)
Collision of Calogero–Moser particles and an adelic Grassmanian (with an appendix by I. G. Macdonald)Invent. Math., 133
-
V. Goryainov (1993)
FRACTIONAL ITERATES OF FUNCTIONS ANALYTIC IN THE UNIT DISK, WITH GIVEN FIXED POINTSMathematics of The Ussr-sbornik, 74
-
O. Babelon, D. Bernard, M. Talon (2003)
Introduction to Classical Integrable Systems: Introduction -
Björn Gustafsson, A. Vasiliev (2006)
Conformal and Potential Analysis in Hele-Shaw Cells -
C. Pommerenke (1965)
Über die Subordination analytischer Funktionen.Journal für die reine und angewandte Mathematik (Crelles Journal), 1965
-
J. Porras, F. Romo (2008)
Generalized reciprocity lawsTransactions of the American Mathematical Society, 360
-
H. Airault, P. Malliavin (2001)
Unitarizing probability measures for representations of Virasoro algebraJournal de Mathématiques Pures et Appliquées, 80
-
N. Parke, W. Kaplan (1958)
Ordinary Differential Equations.American Mathematical Monthly, 67
-
L. Takhtajan, L. Teo (2005)
Weil-Petersson Geometry of the Universal Teichmüller Space -
(1977)
On the characteristic classes of groups of diffeomorphisms
- Publisher
- Springer Journals
- Copyright
- Copyright © 2015 by Springer-Verlag Berlin Heidelberg
- Subject
- Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
- ISSN
- 1617-9447
- eISSN
- 2195-3724
- DOI
- 10.1007/s40315-015-0133-z
- Publisher site
- See Article on Publisher Site
Abstract
We consider a homotopic evolution in the space of smooth shapes starting from the unit circle. Based on the Löwner–Kufarev equation, we give a Hamiltonian formulation of this evolution and provide conservation laws. The symmetries of the evolution are given by the Virasoro algebra. The ‘positive’ Virasoro generators span the holomorphic part of the complexified vector bundle over the space of conformal embeddings of the unit disk into the complex plane and smooth on the boundary. In the covariant formulation, they are conserved along the Hamiltonian flow. The ‘negative’ Virasoro generators can be recovered by an iterative method making use of the canonical Poisson structure. We study an embedding of the Löwner–Kufarev trajectories into the Segal–Wilson Grassmannian, construct the $$\tau $$ τ -function, and the Baker–Akhiezer function which are related to a class of solutions to the KP hierarchy.
Journal
Computational Methods and Function Theory – Springer Journals
Published: Aug 12, 2015