In this paper we study the variety Mnil of nilpotent elements of a reductive monoid M . In general this variety has a completely different structure than the variety Guni of unipotent elements of the unit group G of M . When M has a unique non-trivial minimal or maximal G×G-orbit, we find a precise description of the irreducible components of Mnil via the combinatorics of the Renner monoid of M...

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindžuk in 1964. We also prove several related hypotheses of Baker and Sprindžuk formulated in 1970...

In these notes we construct a quantization functor, associating a Hilbert space H(V ) to a finite dimensional symplectic vector space V over a finite field Fq. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp (V ). The main technical result is a proof of a stronger form of the Stonevon Neumann theorem for the Heisenberg group over Fq. Our result an...

In these notes we construct a quantization functor, associating an Hilbert space H(V ) to a finite dimensional symplectic vector space V over a finite field Fq. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp (V ). The main technical result is a proof of a stronger form of the Stone-von Neumann theorem for the Heisenberg group over Fq. Our result ...

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindžuk in 1964. We also prove several related hypotheses of Baker and Sprindžuk formulated in 1970...

In this paper, we construct a quantization functor, associating a complex vector space H(V ) to a finite dimensional symplectic vector space V over a finite field of odd characteristic. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp (V ). The main new technical result is a proof of a stronger form of the Stone-von Neumann property for the Heisenb...

We explore connected affine algebraic groups G, which enjoy the following finiteness property (F): for every action of closure G-orbit contains only finitely many G-orbits. obtain two main results. First, we classify such groups. Namely, prove that a group G enjoys (F) if and is either torus or product one-dimensional unipotent group. Secondly, characterization in terms modality sense V. Arnol’...

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain ows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprind zuk in 1964. We also prove several related hypotheses of Baker and Sprind zuk formulated in 1970...

The earthquake flow and the Teichmüller horocycle flow are flows on bundles over the Riemann moduli space of a surface, and are similar in many respects to unipotent flows on homogeneous spaces of Lie groups. In analogy with results of Margulis, Dani and others in the homogeneous space setting, we prove strong nondivergence results for these flows. This extends previous work of Veech. As coroll...

Given a toric affine algebraic variety $X$ and collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that group $G$ generated verifies following alternative Tits' type: either is group, or it contains non-abelian free subgroup. We deduce if $2$-transitive $G$-orbit in then subgroup, so, exponential growth.