Empirical minimization

We investigate the behavior of the empirical minimization algorithm using various methods. We first analyze it by comparing the empirical, random, structure and the original one on the class, either in an additive sense, via the uniform law of large numbers, or in a multiplicative sense, using isomorphic coordinate projections. We then show that a direct analysis of the empirical minimization algorithm yields a significantly better bound, and that the estimates we obtain are essentially sharp. The method of proof we use is based on Talagrand's concentration inequality for empirical processes. Research partially supported by NSF under award DMS-0434393. Research partially supported by the Australian Research Council Discovery Porject DP0343616.     

The Proalgebraic Completion of Rigid Groups

A finitely generated group is called representation rigid (briefly, rigid) if for every n, has only finitely many classes of simple representations in dimension n. Examples include higher rank S-arithmetic groups. By Margulis super rigidity, the latter have a stronger property: they are representation super rigid; i.e., their proalgebraic completion is finite dimensional. We construct examples of nonlinear rigid groups which are not super rigid, and which exhibit every possible type of infinite dimensionality. Whether linear representation rigid groups are super rigid remains an open question.     

Angular dependence of the magnetic-field driven superconductor-insulator transition in thin films of amorphous indium-oxide

A significant anisotropy of the magnetic-field driven superconductor-insulator transition is observed in thin films of amorphous indium-oxide. The anisotropy is largest for more disordered films which have a lower transition field. At higher magnetic fields the anisotropy reduces and even changes sign beyond a sample specific and temperature independent magnetic field value. The data are consistent with the existence of more than one mechanism affecting transport at high magnetic fields.     

Mixing of all orders of Lie groups actions

Summary We show that for Lie groups whose adjoint representation reflects their topology, mixing implies mixing of all orders. In particular we prove that mixing actions of a semisimple Lie group are mixing of all orders, answering a conjecture of B. Marcus.Oblatum 6-VIII-1990 & 4-IV-1991Sponsored in part by the Edmund Landau Center for research in Mathematical Analysis supported by the Minerva Foundation (Federal Republic of Germany)     

Automata and Square Complexes

We introduce a new geometric tool for analyzing groups of finite automata. To each finite automaton we associate a square complex. The square complex is covered by a product of two trees iff the automaton is bi-reversible. Using this method we give examples of free groups and of Kazhdan groups which are generated by the different states of one finite (bi-reversible) automaton. We also reprove the theorem of Macedońska, Nekrashevych, Sushchansky, on the connection between bi-reversible automata and the commensurator of a regular tree.