The admissibility of M11 over number fields

A group G is $\mathbb{Q}$-admissible if there exists a G-crossed product division algebra over $\mathbb{Q}$. The $\mathbb{Q}$-admissibility conjecture asserts that every group with metacyclic Sylow subgroups is $\mathbb{Q}$-admissible. We prove that the Mathieu group $M_{11}$ is $\mathbb{Q}$-admissible, in contrast to any other sporadic group.     

Valuations of semirings

We develop notions of valuations on a semiring, with a view toward extending the classical theory of abstract nonsingular curves and discrete valuation rings to this general algebraic setting; the novelty of our approach lies in the implementation of hyperrings to yield a new definition (hyperfield valuation). In particular, we classify valuations on the semifield ${\mathbb{Q}}_{max}$ (the max-plus semifield of rational numbers) and also valuations on the ‘function field’ ${\mathbb{Q}}_{max}(T)$ (the semifield of rational functions over ${\mathbb{Q}}_{max}$) which are trivial on ${\mathbb{Q}}_{max}$. We construct and study the abstract curve associated to ${\mathbb{Q}}_{max}(T)$ in relation to the projective line ${\mathbb{P}}_{{\mathbb{F}}_1}^1$ over the field with one element ${\mathbb{F}}_1$ and the tropical projective line. Finally, we discuss possible connections to tropical curves and Berkovich's theory of analytic spaces.     

Stable Ulrich bundles on Fano threefolds with Picard number 2

In this paper, we consider the existence problem of rank one and two stable Ulrich bundles on imprimitive Fano 3-folds obtained by blowing-up one of ${\mathbb{P}}^3$, Q (smooth quadric in ${\mathbb{P}}^4$), $V_3$ (smooth cubic in ${\mathbb{P}}^4$) or $V_4$ (complete intersection of two quadrics in ${\mathbb{P}}^5$) along a smooth irreducible curve. We prove that the only class which admits Ulrich line bundles is the one obtained by blowing up a genus 3, degree 6 curve in ${\mathbb{P}}^3$. Also, we prove that there exist stable rank two Ulrich bundles with $c_1=3H$ on a generic member of this deformation class.     

Uniqueness of Fq-quadratic perfect nonlinear maps from Fq3 to Fq2

Let q be a power of an odd prime. We prove that all ${\mathbb{F}}_q$-quadratic perfect nonlinear maps from ${\mathbb{F}}_{q^3}$ to ${\mathbb{F}}_q^2$ are equivalent. We also give a geometric method to find the corresponding equivalence explicitly.