Resumo: Nesta palestra, baseada em trabalho conjunto com C. Tresser, examinaremos o chamado problema da persistência formulado por N. Sloane em 1973 sob o ponto de vista dinâmico, relacionando-o com a dinâmica de transformações do círculo e ações ergódicas de Z^d.
Título: Trying to generalize Pick's theorem
Resumo: We will review Pick's theorem in dimension 2 and see some ideas for an extension. Based on understanding the Todd class of toric varieties, a variety of local formulas have been proposed. Each of these local formulas can be seen as a higher Pick's theorem. I will present a conjecture and evidence for one particular formula to become "the" natural extension.
Título: Integrality relations among areas of plane polygons
Resumo: In 1970, Paul Monsky proved that a square cannot be dissected into an odd number of triangles of equal area. This theorem raises the more general question of what other restrictions there might be on the areas of the triangles in a dissection of a square. It turns out that if one fixes the combinatorics of the dissection, then there is a single irreducible polynomial that must be satisfied by the areas of the triangles in the dissection. Further, the area of any triangle of the dissection turns out to be integral over the ring generated by the areas of the other triangles in the dissection. Similar relations exist for dissections of trapezoids and general 4-gons. These integrality relations and the mysteries surrounding them will be the focus of this talk.
Resumo: In recent years, more and more discrete variants of classical inequalities from convex geometry, such as Brunn-Minkowski, have been
investigated. The goal is, roughly speaking, to replace the volume functional in these classical inequalities by the discrete volume, i.e., the number of lattice points inside the convex body.
In the lecture we will present some of these discrete variants, with particular emphasis on slicing inequalities and Minkowski's successive minima
inequalities.
(Based on ongoing joint works with Ansgar Freyer)