Involutive automorphisms of N∘∘N_\circ^\circ groups of finite Morley rank

We classify a large class of small groups of finite Morley rank: $N_\circ^\circ$-groups which are the infinite analogues of Thompson's $N$-groups. More precisely, we constrain the $2$-structure of groups of finite Morley rank containing a definable, normal, non-soluble, $N_\circ^\circ$-subgroup

On function field Mordell-Lang: the semiabelian case and the socle theorem

We here aim to complete our model-theoretic account of the function field Mordell-Lang conjecture, avoiding appeal to dichotomy theorems for Zariski geometries, where we now consider the general case of semiabelian varieties. The main result is a reduction, using model-theoretic tools, of the semiabelian case to the abelian case.Comment: 43 pages. Some minor corrections and clarifications were made following a referee's repor

Remarks concerning the Lyapunov exponents of linear cocycles

We impose a condition of pointwise convergence on the Lyapunov exponents of a d-dimensional cocycle over a compact metric minimal flow. This condition turns out to have significant consequences for the dynamics of the cocycle. We make use of such classical ODE techniques as the Lyapunov-Perron triangularization method, and the ergodic-theoretical techniques of Krylov and Bogoliubov

Periodic Higgs bundles over curves

In this article, we study periodic Higgs bundles and their applications. We obtain the following results: i). an elliptic curve has infinitely many primes of supersingular reduction if and only if any periodic Higgs bundle over it is a direct sum of torsion line bundles; ii). the uniformizing de Rham bundle attached to a generic projective hyperbolic curve is not one-periodic, and it is motivic iff it admits a modular embedding (e.g. Shimura curves, triangle curves); iii). there is an explicit Deuring-Eichler mass formula for the Newton jumping locus a Shimura curve of Hodge type. We propose the periodic Higgs conjecture, which would imply an arithmetic Simpson correspondence. The conjecture holds in rank one case.Comment: 36 page