Third meeting, Weizmann, 30.3.16, Ziskind 1
Linear non-split sharply two transitive groups.
A permutation group is called sharply two transitive if it is acts transitively and freely on pairs of distinct unordered points. Such is the action of the group of affine transformations {x-->ax+b} with its natural action on the field. Such an affine action over a field or even a near field is called split.
In the 1930's Zassenhaus classified the finite sharply two transitive groups, and in particular showed that they are all split. And ever since it was established in other special settings that all sharply two transitive groups are split. Only very recently Rips Segev and Tent gave the first examples of non-split sharply two transitive groups.
I will discuss a joint work with Dennis Gulko in which we solve the problem of exactly when linear sharply two transitive groups are split.
Diophantine Estimates in Positive Characteristic.
We study the Diophantine Approximation problem for function fields of positive characteristic.We begin with a brief overview of developments in the theory of Diophantine approximation. Using techniques developed by Athreya, Parrish and Tseng we prove a weaker version of Schmidt's theorem, on distribution of solutions for the Diophantine problem, for function fields of positive characteristic.
Counting points on varieties for dummies.
The Garside groups and some of their properties.
Garside groups have been first introduced by P.Dehornoy and L.Paris in 1990. In many aspects, Garside groups extend braid groups and more generally finite-type Artin groups. These are torsion-free groups with a word and conjugacy problems solvable, and they are groups of fractions of monoids with a structure of lattice with respect to left and right divisibilities. It is natural to ask if there are additional properties Garside groups share in common with the intensively investigated braid groups and finite-type Artin groups. In this talk, I will introduce the Garside groups in general, and a particular class of Garside groups, that arise from certain solutions of the Quantum Yang-Baxter equation. I will describe the connection between these theories arising from different domains of research, present some of the questions raised for the Garside groups and give some partial answers to these questions.
You are cordially invited.