The theory of billiards in rational polygons leads to the theory of Riemann surfaces and quadratic differentials. We will present, in an elementary way, the relations between these objects. Then we will give recent results on these moduli spaces.
We shall recall the remarkable properties of the O(1) Temperley-Lieb loop model and how they led Razumov and Stroganov to formulate a conjecture relating it to Fully Packed Loops and Alternating Sign Matrices. We shall then discuss recent attempts at proving this conjecture; in particular we shall try to generalize it by introducing inhomogeneities and explain the connection with the Izergin-Korepin determinant formula for the six-vertex model.
A generalization of particular soliton cellular automata, described by the q to 0 limit of R-matrix, to the of general q is given.
We give a new definition for randomness of a subset A of natural numbers as follows: A is weak-mixing (any other ergodic type property) if the point 1_A of {0,1}^{infinity} is a generic point for the weak-mixing (any other ergodic type property) system (cl{T^n 1_A},B,\mu,T) (T is the usual shift to the left) and additionally A has a positive density in natural numbers. We list some Ramsey-type results for weak-mixing (totally ergodic) sequences, for example, if A is weak-mixing, then there exist x,y in A such that x+y=square. We will remind some Ramsey-type results for partitions of N, in particular, the Rado theorem for regularity of a linear system of equations and will explain a connection between our results and some open questions of the Ramsey theory of partitions of natural numbers. Finally, we will present the following result with a proof (sketch): There exists a normal subset A of natural numbers for which the equation xy=z is not solvable inside A; this means that for any x,y,z in A xy \not= z.
We give conditions when a cocycle $\Phi$ does not trivially satisfy the equation $\gamma\Phi =\lambda\frac{g\circ T}{g}$, where $\lambda\in S^1$, $g$ is an $S^1$-valued function, $\gamma$ is a character, and $T$ is a measure-preserving automorphism.