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We will start by introducing the phenomenon of the KPZ (Kardar-Parisi-Zhang) universality. KPZ problem was a very active research area in the last 20 years. The area of KPZ is essentially interdisciplinary. It is related to such fields as probability theory, statistical mechanics, mathematical physics, PDE, SPDE, random dynamics, random matrices, and random geometry, to name a few.

In most general form the problem can be formulated in the following way. Consider random geometry on the two-dimensional plane. The main aim is to understand the asymptotic statistical properties of the length of the geodesic connecting two points, which are far away from each other, in the limit as distance between the endpoints tends to zero. One also wants to study the geometry of random geodesics, in particular how much they deviate from a straight line. It turn out that the limiting statistics for both the length and the deviation is universal, that is it does not depend on the details of the random geometry. Moreover, many limiting probability distributions can be found explicitly.

In the second part of the talk we will proceed with discussion of the geometrical approach to the problem of the KPZ universality which provides an even broader point of view on the problem of universal statistical behavior.

No previous knowledge of the subject will be assumed.