https://www.math.columbia.edu/~okounkov/AMScolloq.pdf

хочется напонить еще текст «Limit shapes, real and imaginary» Андрея Окунькова

«We are surrounded by random surfaces. In fact, every shape around us is random on a sufficiently small scale. The definite shapes that we perceive are manifestations of the law of large numbers that makes averages much larger than the fluctuations around them. Same law is at work e.g. with a balloon floating through the air. Our eyes trace out a smooth trajectory while air molecules hit the balloon randomly from all directions.

Can mathematical physics link the microscopic dynamics to the resulting macroscopic shape, say, for objects of inorganic natural origin? The endless variety of snowflake shapes produced by commonplace water under ordinary conditions illustrates how challenging this question is.

Yet, at or near equilibrium a successful theory explaining the macroscopic shape from the microscopic laws may be developed. It requires deep and powerful mathematics that we can only begin to explore it the course of 3 lectures. In turn, it impacts areas of mathematics that may seem very distant from equilibrium crystals or liquid droplets.

Mathematics and physics are full of random geometric objects, especially when surveyed with a trained eye. In dealing with them, the experience and intuition acquired in the study of equilibrium crystals can be very valuable.

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The main goal of this lecture is to provide an accessible introduction to an area of mathematical physics that I, personally, find captivating. (…)»