## Colloquia Abstracts 2013/14

Multiplicative Problems in Diophantine Approximation

Speaker: Professor Simon Kristensen

Diophantine approximation is concerned with the approximation of real numbers by rationals, where the quality of the approximation is measured in terms of the size of the denominator of the approximating rationals. In simultaneous Diophantine approximation, one approximates several numbers by rationals with the same denominator. Multiplicative Diophantine approximation is an instance of simultaneous approximation. The most famous problem in the field is the celebrated Littlewood conjecture, which has remained open for more than 80 years. In my talk, I will give an overview of known results about the Littlewood conjecture and related problems. In addition, I will present some new results on a family of Littlewood type problems whose proofs involve techniques from Fourier analysis, measure theory and uniform distribution theory.

Interplay between spectrally bounded operators and Complex Analysis

Speaker: Dr. Martin Mathieu

We give an overview on recent advances in the understanding of spectrally bounded and spectrally isometric operators between Banach algebras with an emphasis on methods coming from Complex Analysis.

Representations of Finite Groups Through Geometry

Speaker: Dr. Jay Taylor

For over a century we have studied the representations of finite groups with the vain hope of gaining a deeper insight into the internal structure of such groups. Amongst all finite groups the class of finite reductive groups plays a prominent role. This is because almost all finite simple groups (the building blocks of all finite groups) arise as central quotients of finite reductive groups. Finite reductive groups are subgroups of connected reductive algebraic groups, which are highly geometric objects. In 1976 Deligne and Lusztig gave a powerful method for obtaining representations of finite reductive groups, which uses heavily the geometry of the ambient algebraic group. During the last 35 years Lusztig has built on this work and developed a deep geometric theory for the representations of these groups. In this talk I will give a short survey of this work.

Density of orbits in compact spaces, chromatic number, and Diophantine approximation

Speaker: Dr. Alan Haynes

Suppose we are given a compact topological space and a collection of maps from the space to itself. A natural problem is to determine when the collection of images of a point is dense in the whole space. This problem has led to some beautiful mathematics in the last 100 years, and cases in which we are able to establish (and possibly even quantify) density or non-density of orbits often have applications to other problems. In this talk we will discuss recent work along these lines, focusing on endomorphisms of compact Abelian groups and isometries of compact metric spaces, with an emphasis on connections to graph theory (bounding the chromatic number of infinite Cayley graphs) and number theory (Diophantine approximation).

Means, Braids and Triangle Equations

Speaker: Dr. Bernd Kreussler

The Triangle Equation, also known as the Yang-Baxter equation, was discovered around 1970 by Yang (in quantum field theory) and Baxter (in statistical mechanics). It now plays an important role for quantum groups and was used in the construction of invariants of knots. In this talk I shall start with two elementary topics: the arithmetic-geometric mean and braid groups. The arithmetic-geometric mean is closely related to elliptic integrals (Lagrange and Gauss). From there it is only one step to elliptic functions. Representations of braid groups will lead us to the Triangle Equation. Its classical limit is known as the classical Yang-Baxter equation. I shall explain how vector bundles on elliptic curves give rise to elliptic solutions of this equation. The degeneration of such solutions into trigonometric or rational solutions can be explained geometrically and is linked to the degeneration of an elliptic curve into a nodal or cuspidal rational curve. This is joint work with Igor Burban (Cologne).

Nilpotent completely positive maps and majorization.

Speaker: Professor B V Rajarama Bhat

The theory of majorization provides a way of comparing real vectors. This notion appears in a wide variety of fields. Jordan block sizes of nilpotent linear maps obey a bunch of inequalities coming from Littlewood-Richardson rules, including majorization inequalities. In the context of nilpotent completely positive maps, we prove a new type of majorization. This is a joint work with Nirupama Mallick.

Fusion and bisets for finite groups.

Speaker: Dr Sejong Park

Fusion systems are categories modelling conjugation relations among $p$-subgroups of a finite group. They appear naturally in the study of the $p$-local structure of finite groups and p-blocks of finite groups, and have applications in representation theory and homotopy theory. I'll give an overview of the theory of fusion systems and show why bisets (sets with compatible group actions on the left and on the right) appear naturally in the study of fusion systems. Finally, I'll present some results which showcase the interplay between fusion and bisets.

Threshold concepts and student learning – the example of function

A threshold concept can be seen as a portal to a new and previously unreachable understanding. The notion was introduced by Meyer and Land in 2003. Threshold concepts are initially troublesome to learn and students’ development of understanding involves a potential to transform the understanding of the whole subject area where the concept is included. In mathematics there are several threshold concepts. In the seminar I will present research on student learning of threshold concepts and how focus on these concepts may improve study results. As an example I will present results from my recent study on university students’ transformation of their understanding of the threshold concept of function.

'On the intersection of $(m,p)$-isometric and $(μ,∞)$-isometric operator tuples'.

Speaker: Dr. Philipp Hoffmann

So-called $(m,p)$-isometric operator tuples are tuples $T=(T_1,...,T_d)$ of commuting, bounded linear operators on a normed space $X$, which pointwise satisfy a certain multi-nomial equation. Closely related to these tuples are so-called $(m, ∞)$-isometric tuples, whose definition is derived by letting the $p$ in the definition of $(m,p)$-isometric tuples tend to infinity. The question arises what (necessary and sufficient) conditions a commuting operator tuple has to satisfy to be simultaneously $(m,p)$-isometric and $(m, ∞)$-isometric. We present some recent (partial) results for this problem.

Inhomogeneous multiplicative Littlewood conjecture and logarithmic savings.

Speaker: Dr. Pankaj Vishe

We use dynamics on SL(3,R)/SL(3,Z) to get logarithmic savings in the inhomogeneous multiplicative Littlewood setting. This is a joint work with Alex Gorodnik.

Responding to the Mathematics Problems: 2000 - 2013

Speaker: Mr. Michael Grove

In June 2000, the Engineering Council published its resort entitled 'Measuring the Mathematics Problem'. The report was produced following a seminar held the year before and noted compelling evidence "of a serious decline in students mastery of basic mathematical skills and level of preparation for mathematics-based degree courses. This decline is well established and affects students at all levels. As a result, acute problems now confront those teaching mathematics and mathematics-based modules across the full range of universities." One of the reasons for this decline in the mathematical skills and preparedness of students as they commenced programmes in mathematics, physics and engineering was attributed to "insufficient candidates with satisfactory A-level Mathematics grades for the number of degree places available" and represented a wider problem of an increasing decline in the number of students choosing to study the Science, Technology, Engineering and Mathematics (STEM) disciplines at University level. Since 2000, there have been an increasing number of projects and initiatives that have sought to address the issue of declining student numbers within the STEM disciplines and ensure students begin their university studies by having access to a range of mathematics support interventions. This talk will discuss some of the interventions that have sought to tackle these issues and will highlight a range of practices and approaches that have been successfully implemented by universities to increase and widen participation in mathematics (for example, there has been a 28% increase in participation in the mathematical sciences at undergraduate level within the UK between 2007 and 2012). It will also discuss the efforts of universities to ensure students are supported do develop their mathematical skills upon entry which is another known barrier to progression and retention within the STEM disciplines.