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During the Arizona Winter School 2008 (held in Tucson, AZ) we worked on the following problems: a) (Expanding a remark by S. Lang). Define $E_0 = overline{mathbb{Q}}$ Inductively, for $n geq 1$, define $E_n$ as the algebraic closure of the field generated over $E_{n-1}$ by the numbers $exp(x)=e^x$, where $x$ ranges over $E_{n-1}$. Let $E$ be the union of $E_n$, $n geq 0$. Show that Schanuels Conjecture implies that the numbers $pi, log pi, log log pi, log log log pi, ldots $ are algebraically independent over $E$. b) Try to get a (conjectural) generalization involving the field $L$ defined as follows. Define $L_0 = overline{mathbb{Q}}$. Inductively, for $n geq 1$, define $L_n$ as the algebraic closure of the field generated over $L_{n-1}$ by the numbers $y$, where $y$ ranges over the set of complex numbers such that $e^yin L_{n-1}$. Let $L$ be the union of $L_n$, $n geq 0$. We were able to prove that Schanuels Conjecture implies $E$ and $L$ are linearly disjoint over $overline{mathbb{Q}}$.
We prove the Mumford--Tate conjecture for those abelian varieties over number fields whose extensions to C have attached adjoint Shimura varieties that are products of simple, adjoint Shimura varieties of certain Shimura types. In particular, we prov
Let $F$ be a totally real field in which a fixed prime $p$ is inert, and let $E$ be a CM extension of $F$ in which $p$ splits. We fix two positive integers $r,s in mathbb N$. We investigate the Tate conjecture on the special fiber of $G(U(r,s) times
This paper summarizes recent progresses in our theoretical understanding of particle acceleration at relativistic shock waves and it discusses two salient consequences: (1) the maximal energy of accelerated particles; (2) the impact of the shock-gene
Strongly Turing determinacy, or $mathrm{sTD}$, says that for any set $A$ of reals, if $forall xexists ygeq_T x (yin A)$, then there is a pointed set $Psubseteq A$. We prove the following consequences of Turing determinacy ($mathrm{TD}$) and $mathrm{s
The paper compares the asymptotic of the expressions $frac {1} {x} sumlimits_{n leq x} {f(n)}$ and $sumlimits_{n leq x} {frac {f(n)} {n}}$, $frac {1} {x} sumlimits_{p leq x} {f(p)}$ and $sumlimits_{p leq x} {frac {f(p)} {p}}$. The asymptotic of sums