Colored versions of Tverberg's theorem

Günter M. Ziegler (Berlijn)

Donderdag 22 april, 9:55-10:45, Megaron

In the winter 1964, the young Norwegian mathematician Helge Tverberg was sitting in a hotel room in Manchester, freezing, and proved a d-dimensional version of the following result by Bryan Birch (1958): Given 3r-2 points in the plane, one can always divide them into r groups of at most three points, whose convex hulls intersect. One point less is not enough. For the d-dimensional version of this result, the minimal number of points is (d+1)(r-1)+1, according to "Tverberg's theorem", published 1966.

In 1989, Bárány, Füredi, and Lovász found that they needed a "colored version" of Tverberg's theorem. Such a result was achieved by Vrecica and Zivaljevic (1992). Their proof introduced elegant topological results, as well as fascinating combinatorial structures ("chessboard complexes"), but the result was not sharp -- they required more than the conjectured number of points.

Now we can present a surprising new, sharp "colored" version of the original Tverberg theorems, and new tools are used for the proofs. So there is progress to report about...

(Joint work with Pavle V. Blagojevic and Benjamin Matschke.)

Testing local monotonicity of a hazard rate

Geurt Jongbloed (Delft)

Donderdag 22 april, 11:15-12:00, Blauwe zaal

The hazard rate of a distribution is a function that makes precise the idea of ageing of products or people. A hazard rate that is increasing reflects deterioration in time, whereas a decreasing hazard rate means that a product actually gets more reliable having survived longer. In this talk, I will introduce the problem of testing for local monotonicity of a hazard rate. Various test statistics and approximations to their null distributions have appeared in the literature. These approaches will be discussed and compared to a new procedure, introduced in joint work with Piet Groeneboom.

Calabi-Yau periods

Duco van Straten (Mainz)

Donderdag 22 april, 11:15-12:00, Rode zaal

The remarkable properties of periods of families of elliptic curves are well-know. The talk will be about extensions of these properties to families of Calabi-Yau varieties.

Braids and chaos

Jan Bouwe van den Berg (VU)

Donderdag 22 april, 11:15-12:00, Megaron

Pieces of string or curves in three dimensional space may be knotted or braided. This physical idea can be used as a topological tool to study certain types of dynamical systems. In particular, such an approach leads to forcing theorems in the spirit of the famous "period three implies chaos" for interval maps. We discuss an application to a differential equation from the field of pattern formation. This involves several illuminating topological arguments, which are complemented by an illustrative computer-assisted approach.

Minisymposium Analyse en Dynamische Systemen

Donderdag 22 april, Blauwe zaal

Minisymposium Meetkunde en Topologie

Donderdag 22 april, Megaron

Minisymposium Stochastiek

Donderdag 22 april, Rode zaal

Probing the Planckian Structure of Spacetime

Renate Loll (Utrecht)

Donderdag 22 april, 16:00-16:45, Megaron

Already Riemann contemplated the need for modifying our conventional notions of the metric properties of space on scales which are "immeasurably small", a need that should be driven empirically by new insights gained in physics. Great strides have been made since in understanding the theoretical foundations of the physical world, in the form of special and general relativity, quantum theory and quantum field theory. Taken together they strongly suggest the existence of a theory of quantum gravity, which should provide a consistent and quantitative description of the nature of "quantum spacetime" on ultrashort, Planckian length scales. After decades of research, the problem of finding this theory is still outstanding.



I will report on recent, unprecedented progress in a new formulation of quantum gravity, called Causal Dynamical Triangulation. It is based on performing a "sum over histories" by using an intrinsically geometric way of regularizing this quantum superposition in terms of triangulated, piecewise flat spacetimes. In two dimensions, evaluating the sum takes the form of a combinatorial problem, which can be solved explicitly. In the physically relevant case of four spacetime dimensions, nontrivial properties of the sum over spacetimes can be extracted with the help of numerical simulations, yielding some intriguing results which confirm the highly nonclassical nature of spacetime geometry at the Planck scale, and the emergence from it of classical geometry on large scales.

Mathematics, Magic and the Electric Guitar

David Acheson (Oxford)

Donderdag 22 april, 20:30, Academiegebouw

Why are so many people frightened of mathematics? Even at its simplest level, the subject is full of surprises that anyone can enjoy. I will start with a simple number trick that always gives the answer 1089 and then move on to surprises in geometry, chaos theory and electric guitar dynamics. I will even examine if mathematics can explain the magical Indian Rope Trick. Most importantly of all, however, I will suggest ways in which anyone can see how mathematics has a certain magic of its own.

Mathematical Awareness 

Rainer Kaenders (Keulen)

Vrijdag 23 april, 9:30-10:15, Blauwe zaal

What does it mean to have learned mathematics? It certainly involves knowledge and proficiencies. But how can we make the objectives of mathematics teaching precise? The usual way (like in most national curricula, or assessments like PISA or TIMSS) is to demand certain competences. However, competences can more and more be taken over by new media like computer algebra etc. Asa consequence aiming for such competences often modifies traditional mathematics courses into a superficial treatment of machines. Mathematical awareness is an attempt of Ladislav Kvasz en myself to formulate objectives of mathematics teaching which allows to distinguish different qualities of insight and knowledge -- and in particular deep from superficial ones. It follows from this approach that the role of new media is recognized but does not put long-established mathematics teaching in jeopardy. As illustration, we will broaden the usual awareness for the solution of polynomial equations by the surprising method of Lill.

(Lezing in het Nederlands)

φ and σ, from Euler to Erdös

Florian Luca (Morelia)

Vrijdag 23 april, 9:30-10:15, Megaron

Beeger lecture

In the first part of this talk, we will survey various old and new results related to the distribution of the values of the Euler function φ(n) and the sum of divisors function σ(n) of a positive integer n, their popular values, their champions, as well as to the distribution of those positive integers satisfying certain equations involving such functions, like the perfect numbers and the amicable numbers. In the second part of the talk, we will give some of the ideas involved in a proof of a recent result obtained jointly with Kevin Ford and Carl Pomerance which says that there are infinitely common values in the ranges of these two functions. This settles a 50 year old question of Paul Erdös.

Modeling the Immune System

Rob de Boer (Utrecht)

Vrijdag 23 april, 9:30-10:15, Rode zaal

The immune system is a fascinating complex system taking decisions on how to respond to a wide variety of stimuli, varying from lethal pathogens to harmless proteins in the food. Decisions are remembered for life in the form of immunological memory. By mathematical modeling, computer simulation, and bioinformatics we aim to better understand how this complex system is functioning. This requires a quantitative approach of estimating various population sizes, the turnover rates of the cells within each population, their migration rates, and the rates at which cells form contacts with other cells. Devolpment of the proper formal models for this can be challenging.

Minisymposium Biowiskunde

Vrijdag 23 april, Blauwe zaal

Minisymposium Geschiedenis van de wiskunde

Vrijdag 23 april, Megaron

Thema: Verspreiding van wiskundekennis rond 1600 (lezingen in het Nederlands)

Minisymposium Impact nieuwe ICT-mogelijkheden

Vrijdag 23 april, Rode zaal

Philips wiskundeprijs

Vrijdag 23 april, Zaal A

Minisymposium Getaltheorie

Vrijdag 23 april, Blauwe zaal

Minisymposium Leraar en wiskunde

Vrijdag 23 april, Megaron  (lezingen in het Nederlands)

Minisymposium Scientific Computing

Vrijdag 23 april, Rode zaal

Minisymposium Freudenthal's Selecta

        een terugblik op Freudenthal en zijn werk

Vrijdag 23 april, zaal A

At the occasion of the publication of a volume containing selected articles from Freudenthal's work, in the series  Heritage of European Mathematics of the European Mathematical Society.

Historical Reflections on Teaching Calculus/Analysis

David Bressoud (St. Paul)

Vrijdag 23 april, 16:05-16:50, Megaron

The history of mathematics can and should play three important roles in
support of the teaching of mathematics: It helps students to understand
the process of discovering mathematical truths, it explains the
motivation behind the creation of definitions and assumptions and thus
enriches their meaning, and it informs pedagogy by displaying the
historical difficulties even the best mathematicians of their era
encountered. This talk will illustrate all three of these roles as they
inform the teaching of calculus/analysis, with emphasis on the
conceptual difficulties encountered in the 19th century and how they are
reflected in the work of Cauchy, Riemann, and Borel.