Pseudo-ovals in even characteristic and ovoidal Laguerre planes

Pseudo-arcs are the higher dimensional analogues of arcs in a projective plane: a pseudo-arc is a set $\mathcal{A}$ of $(n-1)$-spaces in $\mathrm{PG}(3n-1,q)$ such that any three span the whole space. Pseudo-arcs of size $q^n+1$ are called pseudo-ovals, while pseudo-arcs of size $q^n+2$ are called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from applying field reduction to an arc in $\mathrm{PG}(2,q^n)$. We explain the connection between dual pseudo-ovals and elation Laguerre planes and show that an elation Laguerre plane is ovoidal if and only if it arises from an elementary dual pseudo-oval. The main theorem of this paper shows that a pseudo-(hyper)oval in $\mathrm{PG}(3n-1,q)$, where $q$ is even and $n$ is prime, such that every element induces a Desarguesian spread, is elementary. As a corollary, we give a characterisation of certain ovoidal Laguerre planes in terms of the derived affine planes

Veech groups of Loch Ness monsters

We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of $\mathbf{GL}_+(2,\R)$ avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surface is either countable, or one of three specific types.Comment: 15 page

Direction problems in affine spaces

This paper is a survey paper on old and recent results on direction problems in finite dimensional affine spaces over a finite field.Comment: Academy Contact Forum "Galois geometries and applications", October 5, 2012, Brussels, Belgiu

The classification of inherited hyperconics in Hall planes of even order

AbstractIn this note we complete the classification of inherited hyperconics in Hall planes of even order that was started by O’Keefe and Pascasio by proving that in the cases left open in [C.M. O’Keefe, A.A. Pascasio, Images of conics under derivation, Discrete Math. 151 (1996) 189–199] there are no inherited hyperconics

Hyperovals in Hall planes

AbstractIn this paper we construct two classes of translation hyperovals in any Hall plane of even orderq2 ≥ 16. Two hyperovals constructed in the same Hall plane are equivalent under the action of the automorphism group of that Hall plane iff they are in the same class

Brief introduction to tropical geometry

The paper consists of lecture notes for a mini-course given by the authors at the G\"okova Geometry \& Topology conference in May 2014. We start the exposition with tropical curves in the plane and their applications to problems in classical enumerative geometry, and continue with a look at more general tropical varieties and their homology theories.Comment: 75 pages, 37 figures, many examples and exercise