Mathematical Research Seminar - Archive

We briefly present the theory of slice-regular functions of one quaternionic variable and compare this regularity properties with Fueter regular functions (the regularity used physics).

We give a possible extension for shears and overshears in the case of two non-commutative (quaternionic) variables in relation with the associated vector fields and flows. We present a possible definition of volume preserving automorphisms, even though there is no quaternionic volume form on \mathbb{H}^2.

Using this, we determine a class of quaternionic automorphisms for which  the Andersen-Lempert theory applies. Finally, we exhibit an example of a quaternionic  automorphism, which is not in the closure of the set of finite compositions of volume preserving quaternionic shears, though its restriction to complex subspace is in the closure of the set of finite compositions of volume preserving complex shears.

Majorana theory was introduced by A. A. Ivanov as an axiomatic framework in which to study objects related to the Monster group and its 196884 dimensional representation, the Griess algebra. The objects at the centre of the theory are known as Majorana algebras and can be studied either in their own right or as Majorana representations of certain groups. I will give a brief introduction to the theory before presenting the methods and results of my recent work developing an algorithm to construct the Majorana representations of a given group. 

This work is based on a paper by Á. Seress and is joint with M. Pfeiffer.

The concept of k-regular maps was introduced into topology by Karol Borsuk in 1950'. Long before it was investigeted in interpolation and approximation theory by (among others) Chebyshev and Kolmogorov, and had distinctly applied mathematics flavor.

A continuous map f: X\to V, from a topological space to a vector space is k-regular of for any k distinct points in X their images are linearly independent.

Example: 
If f:X\to V is an embedding, then the map F: X\to R\oplus V, defned by F(x)=(1,f(x))
is 2-regular.

In analogy to theory of embeddings, one of central problems in the study of k-regular maps is (given k, X)  to construct a k-regular map of X into space V of minimal possible dimension. Unlike for embeddings, this is interesting already for X=R^d I will discuss  lower bounds (obstructions) to the existence to the existence of k-regular maps (this involves some algebraic topology of configuration spaces, following work of many people, most recent being Blagojevic, Cohen, Lueck and Ziegler) and upper bounds (constructions) (this involves some algebraic geometry of secant varieties and  Hilbert schemes, following the work of Buczynski, Januszkiewicz, Jelisiejew and Michalek).