You are invited to a:

** Distinguished Lecture Series by**

**Prof. Sergei Tabachnikov**

**Pennsylvania State University **

**Title: ***Frieze patterns. Cross-ratio dynamics on ideal polygons.*

**Abstract: **In the first lecture I shall describe basic properties of frieze patterns. These are are beautiful combinatorial objects, introduced by Coxeter in the early 1970s. He was about 30 years ahead of time: only in this century, frieze patterns have become a popular object of study, in particular, due to their relation with the emerging theory of cluster algebras and to the theory of completely integrable systems. I shall prove the theorem of Conway and Coxeter that relates arithmetical frieze patterns with triangulations of polygons. There is an intimate, and somewhat unexpected, relation between three objects: frieze patterns, 2nd order linear difference equations, and polygons in the projective line (or ideal polygons in the hyperbolic plane).

In the next lectures I shall outline some recent work on frieze patterns, including their relation with the Virasoro algebra. Then I shall present cross-ratio dynamics on ideal polygons. Define a relation between labeled ideal polygons in the hyperbolic 3-space by requiring that the complex distances (a combination of the distance and the angle) between their respective sides equal a constant; the constant is a parameter of the relation. This gives a 1-parameter family of maps on the moduli space of ideal polygons in the hyperbolic space (or, in its real version, in the hyperbolic plane). I shall discuss complete integrability of this family of maps and related topics, including a continuous version of this relation that is intimately related with the Korteweg-de Vries equation.

**1st lecture**: Monday, January 6, 2020 at 15:30

**2nd lecture**: Wednesday, January 8, 2020 at 15:30

**3rd lecture**: Thursday, January 9, 2020 at 15:30

All lectures will be in Amado 232.

Light refreshments will be given in the faculty lounge on the 8th floor.