We consider the Löwner differential equation in ordinary derivatives generating univalent self-maps of the unit disk (or of the upper half-plane). If the solution to this equation represents a one-slit map, then the driving term is a continuous function. The reverse statement is not true in general as a famous Kufarev's example shows. We address the following problem: to find a condition for the Löwner equation to generate one-slit solutions. New examples of non-slit solutions to the Löwner equation are presented and a comparison with the Löwner PDE is given. Properties of singular slit solutions in the half-plane are revealed.2000 Mathematics Subject Classification. Primary 30C35, 30C20; Secondary 30C62.

We study one-parameter expanding evolution families of simply connected domains in the complex plane described by infinite systems of evolution parameters. These evolution parameters in some cases admit Hamiltonian formulation and lead to integrable systems. One example of such parameters is complex moments for the Laplacian growth that form a Whitham-Toda integrable hierarchy. Another example we deal with is related to expanding coefficient bodies for conformal maps given by Löwner subordination chains. The coefficients bodies are proved to form a Liouville partially integrable Hamiltonian system for each fixed index and the first integrals are obtained. We also discuss the contact structure of this system.

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We consider coefficient bodies M n for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then M n are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracketgenerating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system to calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case M 3 .

The problem of integer partitions is addressed using the microcanonical approach which is based on the analogy between this problem in the number theory and the calculation of microstates of a many-boson system. For ordinary (one-dimensional) partitions, the correction to the leading asymptotic is obtained. The estimate for the number of two-dimensional (plane) partitions coincides with known asymptotic results.