The colloquium of the Centre for Mathematical Sciences, Lund University, normally runs once a month, Wednesdays from 14.00 until 15.00 in the Hörmander, Riesz or Gårding lecture halls. It is aimed at the entire Centre for Mathematical Sciences with overview talks by renowned experts about exciting mathematical topics. The purpose of our colloquium is twofold: firstly, it is to provide an inspiring overview of a specific field of mathematics, secondly, it is to bring together students and staff from the entire department and to serve as the proverbial waterhole where contacts are made and maintained. For more information, see the guidelines for colloquium speakers.

The colloquium is organized by Dragi Anevski, Magnus Goffeng, Magnus Oskarsson, Tony Stillfjord and Anitha Thillaisundaram. Feel free to contact any one of us for questions or suggestions for colloquia speakers. See also the information for suggesting colloquium speakers.

January 22 at MH:Riesz

Speaker

Christian Bär (Potsdam)

Title

Counterintuitive approximations

Abstract

One of the oldest questions in differential geometry is whether every curved space arises as a subspace of Euclidean space ℝⁿ. More precisely, does every Riemannian manifold admit an isometric (i.e. length preserving) smooth embedding into ℝⁿ? Nash showed that the answer is yes but the dimension n of the Euclidean space may be much larger than that of the manifold. Often there are non-isometric embeddings of much lower codimension.

The Nash-Kuiper embedding theorem is a prototypical example of a counterintuitive approximation result in geometry: any short (but highly non-isometric) embedding of a Riemannian manifold into Euclidean space can be approximated by isometric embeddings. They are generally not smooth but of regularity C¹. This implies that any surface with a given geometry can be isometrically C¹-embedded into an arbitrarily small ball in ℝ³. For C²-embeddings this completely false due to curvature restrictions.

After explaining this historical and conceptual context, I will present a general result which ensures approximations by maps satisfying strongly overdetermined equations (such as being isometric) on open dense subsets. This will be illustrated by three examples: Lipschitz functions with surprising derivative, surfaces in 3-space with unexpected curvature properties, and a similar statement for abstract Riemannian metrics on manifolds. Our method is based on “cut-off homotopy”, a concept introduced by Gromov in 1986.

This is based on joint work with Bernhard Hanke.

February 19

Speaker

Tomas Persson (Lund)

Title

Linear response

Abstract

TBA

March 19

Speaker

Carl Olsson (Lund)

Title

TBA

April 9 at MH:Hörmander

Speaker

Timo Reis (Ilmenau)

Title

Circuits and Maxwell Equations from a Systems Theoretical Point of View

Abstract

We consider the modelling of electrical circuits interacting with fields. The former is described by differential-algebraic equations, while the latter is governed by partial differential equations. Together, they form a coupled system known as a "partial differential-algebraic equation." In addition to discussing the fundamental modelling aspects, we will also examine energy considerations and present some solvability results.

May 21 at MH:Hörmander

Speaker

Richard Samworth (Cambridge)

Title

Nonparametric inference under shape constraints: past, present and future

Abstract

Traditionally, we think of statistical methods as being divided into parametric approaches, which can be restrictive, but where estimation is typically straightforward (e.g. using maximum likelihood), and nonparametric methods, which are more flexible but often require careful choices of tuning parameters. The area of nonparametric inference under shape constraints sits somewhere in the middle, seeking in some ways the best of both worlds. I will give an introduction to this active area, providing some history, recent developments and a future outlook.

June 11 at MH:Hörmander

Speaker

Colva Roney-Dougal (St Andrews)

Title

The number and nature of subgroups of the symmetric group

Abstract

The symmetries of any object are described by a group, so it is natural to ask: What does a random group look like? This talk will start with a brief survey of how we might go about counting various algebraic structures. We'll then go on to see what a random group might be, in various different contexts.

A symmetric group on some set Omega is the group of all permutations of Omega, under composition of functions. Every group arises as a subgroup of some symmetric group, so fully understanding the symmetric group means understanding all groups. An elementary argument shows that there are at least 2^{n^2/16} subgroups of a symmetric group on n points, and it was conjectured by Pyber in 1993 that up to lower order error terms this is also an upper bound. The same year, Kantor conjectured that a random subgroup of the symmetric group is nilpotent. This talk will present a proof of one of these conjectures, and a disproof of the other.

The new results in this talk are joint work with Gareth Tracey (Warwick).