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Department of Mathematical Sciences, University of Liverpool
Mathematical Sciences Building, Liverpool L69 7ZL, United Kingdom
+44 (0)151 794 4001
Selected Topics in Mathematics
Speakers
7 March (Thursday, 5pm)
Poj Lertchoosakul, University of Liverpool

On the complexity of the Liouville numbers in positive characteristic

Abstract. Badly approximable numbers are in one of the central research in Diophantine approximation. In the classical real case, Davenport and Schmidt gave two characterizations of these numbers, based on their continued fraction expansions and the criteria of an improvement on Dirichlet's theorem. Kit Nair asks whether it is also true in the field of formal Laurent series. In this talk, we try to answer this question. Indeed, we shall prove that an irrational number in positive characteristic is badly approximable if and only if the elements in its continued fraction expansion are bounded. However, we shall see that the criteria of an improvement on Dirichlet's theorem does not hold in this setting.
Note. This is a small joint work with Kit. We assume no prior knowledge, and the talk should be accessible by the general maths audience. We shall give a brief introduction to the field of formal Laurent series and the continued fraction algorithm in this setting, as well as the definitions relating to the results.
21 March (Thursday, 4pm)
Dr. Elisabetta Candellero, University of Birmingham

Clustering in random geometric graphs on hyperbolic spaces

Abstract. In this talk we introduce the concept of random geometric graphs on hyperbolic spaces and discuss its applicability as a model for social networks. In particular, we will discuss issues that are related to clustering, which is a phenomenon that often occurs in social networks: two individuals that have a common friend are somewhat more likely to be friends of each other. We give a mathematical expression of this phenomenon and explore how this depends on the parameters of our model. This is a joint work with Nikolaos Fountoulakis
18 April (Thursday, 4pm)
Dijana Kreso, TU Graz

Polynomial Decomposition and Applications to Diophantine Equations

Abstract. In the 1920's J. F. Ritt studied the question of non-uniqueness of the 'prime factorization' of polynomials over complex numbers under the operation of functional composition. Ritt's results have been applied to a variety of topics. One such topic is a classification of polynomials $f$ and $g$ with rational coefficients for which the Diophantine equation $f(x)=g(y)$ has infinitely many integer solutions. In 2000 Bilu and Tichy succeeded in fully joining polynomial decomposition theory with the classical theorem of Siegel on finiteness of integral points on curves of genus greater than $0$, to give a complete ineffective criterion on the finiteness of the number of integer solutions $x$, $y$ of Diophantine equations of the type $f(x)=g(y)$. Ritt's methods are further applied to determine polynomial decomposition invariants. In this talk I will present some recent results on polynomial decomposition invariants, as well as some recent applications of the theorem of Bilu and Tichy.
2 May (Thursday, 4pm)
Florian Lehner, TU Graz

Symmetries of random graph colourings

Abstract. Let $G =(V,E)$ be a graph and let $\mathcal C$ be the set of all $C$-colourings of $G$, that is, maps $c \colon V \to C$. Then there is a natural right action of $\operatorname{Aut} G$ on $\mathcal C$ defined by $(c \varphi) (v) = c (\varphi v)$. A colouring $c$ is said to be distinguishing if its stabiliser in $\operatorname{Aut} G$ is trivial. It has been conjectured that, if $G$ is an infinite, locally finite graph and every automorphism of $G$ moves infinitely many vertices, then there is a distinguishing $2$-colouring of $G$. We investigate properties of the stabiliser of a random colouring and point out some special cases, where a random colouring is almost surely distinguishing.
9 May (Thursday, 4pm)
Alena Jassova, University of Liverpool

p-adic continued fraction map

Abstract. I would like to talk about a $p$-adic analogue of the regular continued fraction expansion which was introduced by T. Schneider. Let $p$ be a prime. We will consider the continued fraction expansion of a $p$-adic integer $x$ in $pZ_p$. In this talk I will discuss some metric and ergodic properties of its map. This is a joint work with J.Hancl, R. Nair and P. Lertchoosakul.
6 June (Thursday, 2pm)
Fawaz Alharbi, Mecca

Quasi-Projections of graphs of mappings

Abstract. We classify all simple quasi-projections of graphs of map germs from a plane to a plane, and simple quasi-projections of parametrized planar curves. We also discuss the classification of the graphs of map germs from R^2 to R^3. The classifications are rougher than the standard one via the group of the diffeomorphisms preserving a given fibration. In fact, the idea is similar to the one introduced by V.Zakalyukin in [1].
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[1] Zakalyukin V.M., Quasi projections, Proceedings of Steklov Mathematical Institute, Vol. 259, 2007, 279-290.
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