Twisted loop groups and their affine flag varieties

We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a "twisted case"; a consequence of our results is that our construction also includes the flag varieties for Kac-Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t)) and some applications to local models of Shimura varieties.Comment: LaTex, 73 page

Local models of Shimura varieties, I. Geometry and combinatorics

We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties such as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians and wonderful completions of symmetric spaces.Comment: 86 pages, small corrections and improvements, to appear in the "Handbook of Moduli

Non-ergodicity of the motion in three dimensional steep repelling dispersing potentials

It is demonstrated numerically that smooth three degrees of freedom Hamiltonian systems which are arbitrarily close to three dimensional strictly dispersing billiards (Sinai billiards) have islands of effective stability, and hence are non-ergodic. The mechanism for creating the islands are corners of the billiard domain.Comment: 6 pages, 8 figures, submitted to Chao