Raziskovalni matematični seminar - Arhiv
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.
If f:X\to V is an embedding, then the map F: X\to R\oplus V, defned by F(x)=(1,f(x))
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).
In 2017 we have been hearing a lot about blockchain, in 2018 a bit less. But what is a blockchain, really? How is it connected to Bitcoin? Why is there the "crypto" in "cryptocurrencies"? And should we, as mathematicians, be even interested in it? We will be talking about these and similar questions during the seminar. We will briefly look at what kind of mathematics a usual blockchain is based on, how it started, and in what ways today's world is trying to use it.
In this talk we present an algorithm for computing the Wiener index of a tree from the distance matrix of its leaves, known also as the terminal matrix of a tree. The algorithm performs better than the linear time algorithm for trees that have a large number of vertices of valence 2.