Mathematical Research Seminar - Archive

The model theory of valued fields has been extensively studied after, in 1965, the work of Ax-Kochen and Ershov had found remarkable applications in number theory.
To a valued field the value group (an ordered abelian group) and the residue field are naturally associated. The theorem of Ax-Kochen and Ershov can be stated as follows: the theory of henselian valued fields of residue characteristic 0 is complete relative to the value group and the residue field.

Model theorists have later asked what model theoretical properties of henselian valued fields can, as completeness, be understood at the level of the value group and the residue field. I will focus in particular on the problem of quantifier elimination.

It turns out that the statement “the theory of henselian valued fields of residue characteristic 0 admits quantifier elimination relative to the value group and the residue field” is in general false. In this setting, the value group and the residue field do not always carry enough information about the original valued field. As a consequence, other structures associated to valued fields have been considered. In 2011 Flenner obtains quantifier elimination for henselian valued fields of residue characteristic 0 relative to the RV-structures which he introduced with this name,
while developing some less recent ideas of other authors such as Basarab and F.-V. Kuhlmann.

During the talk, while discussing with more details the problem of quantifier elimination for henselian valued fields, I will also argue that RV-structures are, in essence, the same structures that M. Krasner studied in 1957, not in relation to the model theory of valued fields, and that led him to the definition of his hyperfields.

Everyone is welcome and encourage to attend.

Given a finite simple graph on n vertices, its complementary prism is a graph on 2n vertices, which is obtained from the disjoint union of the graph and its complement, if we add n edges that join identical vertices in the graph and in its complement. Complementary prisms generalize the Petersen graph. In the talk I will describe few properties of these graphs.

Everyone is welcome and encouraged to attend.

While domination in (undirected) graphs is one of the most investigated topics in graph theory, domination in digraphs has been studied much less extensively. In this talk, we present some new results on (total) domination in digraphs with an emphasis on some digraph products. We present a generalization of the classical result of Meir and Moon from graphs to digraphs by proving that in an arbitrary ditree (a directed tree of which underlying graph is a tree) the domination number coincides with the packing number. In addition, a similar result is proved for the total domination number of a ditree. Then we focus on the total domination number of direct products of digraphs and the domination number of Cartesian products of digraphs. While the Vizing-type inequality is not true in all Cartesian products of digraphs, we present a different lower bound on the domination number of the Cartesian product of two digraphs expressed in terms of the domination numbers of factor digraphs, and demonstrate its sharpness.

Everyone is welcome and encouraged to attend.

The classical No-Three-In-Line asks for the largest set of lattice points in the n by n grid that lie in general position; that is, so that no three points are on a common line.  It is well known that for any n, the largest such set is between n and 2n.  Erde asked for an infinite set of points in general position, having many points on an n by n grid (for every n).  In recent work joint with Dáni Nagy and Zoli Nagy, we have produced an infinite set with Theta(n/log^(1+ε)) points when intersected with each n by n grid, and suggested a construction that might give exactly n/2 points on such grids.

Everyone is welcome and encouraged to attend.