Loewner theory and non-commutative probability theory

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This project focuses on exploring connections between geometric function theory, in particular Loewner theory, and non-commutative probability theory.In 1923, C. Loewner has introduced certain evolution equations for conformal mappings, which soon became an important tool in geometric function theory. They are used to model two-dimensional growth processes or to tackle extremal problems for univalent functions. Since the introduction of the Schramm-Loewner Evolution (SLE) by O. Schramm in 2000, Loewner theory has become an active research field with interdisciplinary topics from complex analysis, probability theory, statistical mechanics, and conformal field theory.Non-commutative probability theory provides frameworks for abstract probability spaces consisting of random variables which do not commute in general. This is motivated by quantum mechanics, where observables can be regarded as non-commutative random variables. Many goals in this field ask to transfer notions and theorems from classical probability theory to this non-commutative setting. For instance, there is a theory of quantum stochastic processes and quantum stochastic differential equations, which in turn is useful for providing mathematical models for certain quantum systems. The notion of independence plays a central role in classical probability theory and it has been shown that, in a certain sense, there are five ways of defining it in non-commutative probability theory. This leads to tensor, free, Boolean, monotone and anti-monotone probability theory.All five notions lead to certain convolutions of holomorphic mappings, and at this point, complex analysis enters the theory. So far, however, the methods used for this purpose are rather elementary. I noticed that there is a deeper relation between complex analysis and monotone probability theory: both, most studied Loewner equations (the ``radial'' and the "chordal" equation) can be regarded as the Lévy-Khintchine representation of quantum stochastic processes with monotonically independent increments. (And a time reversion of these equations corresponds to the anti-monotone analogues.)This interpretation of Loewner's differential equation leads to several interesting questions and in this project, I plan to investigate the connection between the two theories systematically. Solutions to the problems I describe would be of interest in non-commutative probability theory on the one hand, but they would also enrich complex analysis on the other hand, as they would add a new, probabilistic perspective to Loewner theory.For instance, Loewner theory and univalent mappings have been studied also in higher dimensions. So far, applications to other disciplines have not been found yet, in contrast to the one-dimensional case. However, a multivariate generalization of monotone independence naturally leads to an evolution equation for univalent mappings in several variables.