Modern Mathematics - International Summer School for Students


The instructors of the School include some of the leading international mathematicians who are able to give exciting presentations to talented students from high schools and universities and who enjoy discussing with them.

Speakers (tentative list)

Martin Andler, Université Versailles-Saint-Quentin, France
"Representation of Groups: a Scenic Road "

Kai-Uwe Bux, University of Bielefeld, Germany
"Graph Theory"

John H. Conway, Princeton University, USA
"Topics Decided in Consultation with Participants"

Gábor Domokos, Budapest University of Technology and Economics, Hungary
"The Gömböc, the Sphere and the Pebbles of Mars"

Theodore P. Hill, Georgia Institute of Technology, USA
"Benford's Law: The Significant-Digit Phenomenon"
"Knowing When to Stop"
"Fair-Division Problems"

Mikhail Hluschanka, Jacobs University Bremen, Germany
"From Rational Maps to Tilings and Groups"

Marc-Thorsten Hütt, Jacobs University Bremen, Germany
"Simple Models of Interacting Biological Neurons"

Ivan Izmestiev, Université de Fribourg, Switzerland

Victor Kleptsyn, Institut de Recherche Mathématique de Rennes, France
"Sphere Packings in Higher Dimensions"

Anke Pohl, Friedrich-Schiller-Universität Jena, Germany
"Dynamical Methods in Number Theory"

Dierk Schleicher, Jacobs University Bremen, Germany
"On Dynamics, Geometry, Groups, Numerics, and a Lot of Money"

Bernd Sturmfels, University of California Berkeley, USA and Max Planck Institute Leipzig, Germany
"Invitation to Tropical Mathematics"
"Sixty-Four Curves of Degree Six"

Yuri B. Suris, Technische Universität Berlin, Germany
"What Is Discrete Differential Geometry?"

Sergei Tabachnikov, Pennsylvania State University, USA
"Variations on the Tait-Kneser Theorem and the Sylvester Problem"

Rebecca Waldecker, Martin-Luther-Universität Halle-Wittenberg, Germany
"Permutation Groups"

Don Zagier, Max Planck Institute Bonn, Germany

Günter M. Ziegler, Freie Universität Berlin, Germany
"Semi-Algebraic Sets of Integer Points"

Anton Zorich, Institut Mathématiques de Jussieu, France
"Butterflies, Cats, and Billiards in Polygons"

Talks and Abstracts

Representation of Groups: A Scenic Road

Martin Andler

In the lecture, I will try to address an existential issue : "Why can't I answer the following question: 'are you an algebraist, an analyst, a geometer, ..., or perhaps a combinatorist, ..., a mathematical physicist?'" I will give a number of examples of representation theoretic situations in different contexts.

Graph Theory

Kai-Uwe Bux

I would talk about the graph-minor theorem of Robertson and Seymour and its variations, in particular Krushkal's tree theorem and Wagner's characterization of planar graphs. I would also say how the graph-minor theorem leads to the strange situation that sometimes we know that a problem has an (even fast) algorithmic solution, yet we do not know the algorithm.

TAs: Bernhard Reinke, Steffen Maaß, Bayani Hazemach.

Topics Decided in Consultation with Participants

John H. Conway

John Conway prefers to decide the topics of his presentations on short notice. He enjoys discussing with the participants of the summer school (which is the main reason why he is coming all the way), and likes to choose the topics depending on interests and requests of participants. Some of his most popular talks are on "games and numbers", on "sphere packings", on "lattices and (lexi-)codes", on "knot theory", on "fractran: universal computing with fractions", or on "the free will theorem" (a relatively new piece of work). Some of his more recent work is on generalizations of Fibonacci sequences; one may also ask him to speak on Euclidean geometry and triangles, or on a relation between his various works on computability and provability, including the question on whether the famous \(3n+1\) problem is solvable.

The Gömböc, the Sphere and the Pebbles of Mars

Gábor Domokos

In 2012 NASA's Curiosity sent home stunning images of rounded pebbles on Mars [1]. It seemed to be a plausible hypothesis that these pebbles have been rounded while they were carried by an ancient river. However, more evidence was called for. The mathematical theory that finally led to the required geophyiscal argument is based on a curious idea of one of 20th century's most prominent mathematicians, V.I. Arnol'd, who in 1995 conjectured that convex, homogeneous solids with just two static balance points may exist. Ten years later, based on a constructive proof [2] by P.L. Várkonyi and the author, the first such object (dubbed Gömböc) was built. In everyday terms, such a body (or pebble) would behave like a weeble toy, and it would always right itself on a flat surface under gravity to one and the same rest position. The Gömböc is quite sensitive and almost never found in Nature, so it appeared to be a mathematical curiosity.
It came as a surprise as it turned out that the Gömböc plays and important role in the theory explaining the evolution of shapes in the non-living Nature, such as sand grains, pebbles or asteroids. Although at first sight the connection appeared to be far-fetched, laboratory and field data showed a convincing match with theory-based predictions and this turned out to be the key to understanding the provenance of Martian pebbles [3].

[1] R.M.E. William et al.: Martian fluvial conglomerates at Gale crater. Science 340 (6136) 1068-1072,
[2] P.L. Varkonyi and G. Domokos: Mono-monostatic bodies: The answer to Arnold's question. The Mathematical Intelligencer, 28(4) pp 34-38. (2006),
[3] T. Szabó, G. Domokos, J.P. Grotzinger, and D.J. Jerolmack: Reconstructing the transport history of pebbles on Mars.Nature Communications 6, Article number: 8366 (2015) doi:10.1038/ncomms9366

Benford's Law: The Significant-Digit Phenomenon

Theodore (Ted) P. Hill

A century-old empirical observation called Benford's Law states that the significant digits of many datasets are logarithmically distributed, rather than uniformly distributed as might be expected. For example, more than \(30\%\) of the leading decimal digits are \(1\), and fewer than \(5\%\) are \(9\). This talk will briefly survey some of the colorful history and empirical evidence of Benford's law, and then discuss re cent mathematical discoveries that help explain the ubiquity of Benford datasets. For example, it has now been shown that iterations of many common functions (including compositions of polynomial, power, exponential, and trigonometric functions), large classes of ordinary differential equations, random mixtures of data from different sources, and many numerical algorithms like Newton's method, all produce Benford distributions. Applications of these theoretical results to practical problems of fraud detection, analysis of round-off errors in scientific computations, and diagnostic tests for mathematical models will be mentioned, as well as several open problems in dynamical systems, probability, number theory, and differential equations.

TAs: Alexander Minets, Anton Shemyakov, and Dimitry Rumyansev.

Knowing When to Stop

Theodore (Ted) P. Hill

In many basic processes in science (and the rest of life) there is an element of chance involved, and a crucial problem is deciding when to stop. The process could be waiting to buy or sell Google stocks, proofreading a paper or debugging a large software program, deciding when to switch to a new medication, updating eBay auction bids, or interviewing for a new secretary (or spouse). At some point you need to stop, and your objective is to do it in a way that optimizes your reward (e.g. maximum profit or satisfaction, minimum cost or errors). The mathematical theory of optimal stopping, including Secretary Problems (also known in the literature as Marriage, Dowry, or Best-Choice Problems), has a long and colorful history, complete with excellent rules of thumb, counterintuitive surprises, colorful paradoxes, and famous unsolved problems. The elegant and unexpected solution to the classical "no-information" Secretary Problem will be reviewed, along with several game-theoretic extensions, analogs for "full-information" and "partial-information" stopping, and several basic open problems.

TAs: Alexander Minets, Anton Shemyakov, and Dimitry Rumyansev.

Fair-Division Problems

Theodore (Ted) P. Hill

The general subject of this talk is the question of whether an object (such as an irregular cake or piece of land) can be divided among \(N\) people in such a way that each person receives what they consider a fair portion, even though they may have different values. Formally, there are \(N\) values (measures) on the same object and a typical goal is to partition the object into N pieces, and assign each person a piece that they themselves value at least \(\frac{1}{N}\) of the total. Classical fair-division includes Steinhaus's "Ham Sandwich Problem," Dubins and Spanier's "Sliding Knife Algorithm," Neyman and Pearson's "Bisection Problem," and Fisher's "Problem of the Nile." Generalizations of these, in both continuous and discrete settings, use tools such as Lyapounov's Convexity Theorem. The talk will include several open problems, and applications to disarmament, dividing inheritances, and selection of leaders.

TAs: Alexander Minets, Anton Shemyakov, and Dimitry Rumyansev.

From Rational Maps to Tilings and Groups

Mikhail Hluschanka

Given a rational map, one can naturally associate to it an algebraic object, called the iterated monodromy group (IMG). IMG's not only help to answer questions in holomorphic dynamics, but also provide examples of groups with complicated structure and exotic properties which are hard to find among groups defined by more "classical" methods. In the talk I will discuss how one can study properties of the IMG's using certain tilings of the complex plane which are associated to the maps. At the end I will outline some open questions in the field in the context of the ERC project "HOLOGRAM" of Prof. Dierk Schleicher.

Simple Models of Interacting Biological Neurons

Marc-Thorsten Hütt

In my lecture I will discuss simple mathematical models of excitable elements. Such models can be thought of as minimal representations of neurons. I will show some properties of these models and discuss the relevance of these properties for understanding brain dynamics. One focus will be on collective excitation patterns. If the excitable elements form a chain or a lattice we observe spatiotemporal patterns. If the elements form a (less regular) network we can explore, how network architecture shapes excitation patterns.


Ivan Izmestiev

Porisms are geometric theorems of the following form: "If a statement holds for some initial data, then it holds for any other initial data." In this series of lectures several porisms will be presented, some will be proved, and a general perspective will be given on all of them. The methods involve geometric transformations, parameterizations of conics, and (if time permits) elliptic functions.

TAs: Roman Prosanov, Khudoyor Mamayusupov, and Sergei Shemyakov.

Sphere Packings in Higher Dimensions

The classical idea of a Farey sequence is to list in increasing order all the irreducible fractions from the interval \([0,1]\), whose denominator does not exceed a given number \(N\). Studying how such sequences change, we will pass to the continued fractions, to the action of \(\text{SL}(2,\mathbb{Z})\) and its properties. Continued fractions are also related with the circle rotations, and the latter with the Sturmian words. An example of a Sturmian word, corresponding to the golden ratio, can be constructed using an iterated substitution \(0\mapsto 01\) and \(1\mapsto0\); more complicated substitutions lead to the concept of the Rauzy fractal.

TAs: Roman Prosanov, Thanasin Nampaisarn, and Sergei Shemyakov.

Dynamical Methods in Number Theory

Anke Pohl

Some questions in number theory can be formulated and investigated with the help of dynamical systems. A combination of number theoretical and dynamical methods may lead to new insights or even solutions. In the last few decades we have seen several such examples. We will focus on exploring one such example, namely continued fractions, starting with some rather classical observations and then study more recent results.

TAs: Bayani Hazemach, Alexander Minets, and Sabyasachi Mukherjee.

Invitation to Tropical Mathematics

Bernd Sturmfels

In tropical arithmetic, the sum of two numbers is their maximum and the product of two numbers is their usual sum. Many results familiar from algebra and geometry, including the Quadratic Formula, the Fundamental Theorem of Algebra, and Bezout's Theorem, continue to hold in the tropical world. Here we learn how to draw tropical curves and why evolutionary biologists might care about this.

TAs: Thanasin Nampaisarn, Konstantin Bogdanov, Sergei Shemyakov, and Maik Sowinski.

On Dynamics, Geometry, Groups, Numerics, and a Lot of Money

Dierk Schleicher

The core of the organizing committee of our summer school are members of my "European Research Council Advanced Grant" called "HOLOGRAM: holomorphic dynamics, geometry, root-finding, algebra, and the Mandelbrot set." I will try to give an overview on the research agenda of this research program and about the topics, and some of its success. This is research that includes a number of young students, some of them current or former “Modern Mathematics” participants.
The Mandelbrot set is a famous icon of "holomorphic dynamics" (also known as "complex dynamics") and includes Fatou and Julia sets, "fractals," and many exciting questions and surprising results. Root finding is done by various iteration methods such as Newton’s method that is closely related to complex dynamics in different versions. It is less obvious how geometry and algebra are related, and I will try to outline some connections.
PS.: The best I will be able to accomplish in 50 minutes about "algebra" is to give a forward reference to the talk by Dr. Hlushchanka.

Sixty-Four Curves of Degree Six

Bernd Sturmfels

This lecture is an invitation to real algebraic geometry, along with computational aspects, ranging inflection points of curves to eigenvectors and ranks of tensors. We present an experimental study - with many pictures - of smooth curves of degree six in the real plane. The number 64 refers to the rigid isotopy types in the Rokhlin-Nikulin classification.

TAs: Thanasin Nampaisarn, Konstantin Bogdanov, Sergei Shemyakov, and Maik Sowinski.

What Is Discrete Differential Geometry?

Yuri B. Suris

Differential geometry studies objects like curves, surfaces, coordinate systems with the methods of mathematical analysis. Thus, even the very basic notions and results of differential geometry are hardly accessible for high school students. This is still more true for more advanced results, like those related to transformations of surfaces, which were developed in a comprehensive manner by Darboux, Bianchi, Eisenhardt, and other geometers of the late 19th - early 20th century. However, the new discipline of discrete differential geometry (DDG) which emerged in the recent years, changes this situation dramatically. This new discipline develops discrete analogues and equivalents of notions and methods of differential geometry of smooth curves, surfaces etc. The smooth theory appears in a limit of the refinement of discretizations. The basic feature of DDG is that it is based on elementary geometry, in particular, on classical incidence theorems like the Desargues theorem, the M\"obius theorem about pairs of tetrahedra which are mutually inscribed and circumscribed, the Miquel theorem from the geometry of triangles, and so on. For instance, it would be no exaggeration (or only a slight exaggeration) to say the contents of the monumentally voluminous and difficult treatise by Darboux about orthogonal coordinate systems and their transformations can be reduced to the elementary Miquel theorem. In these lectures, I will give a very brief introduction to DDG suitable for high school students

TAs: David Pfrang, Steffen Maaß, and Bernhard Reinke.

Variations on the Tait-Kneser Theorem and the Sylvester Problem

Sergei Tabachnikov

The Tait-Kneser theorem states that the osculating circles of a plane arc with monotone curvature are pairwise nested. I shall explain this theorem and discuss its variations (for example, the osculating circles can be replaced by the osculating conics). I shall outline an analogy with various versions of the Sylvester problem: given a finite set of points in the plane, if the line through every pair contains another point from this collection, must all he points be collinear? I shall describe the curious history of the Sylvester problem and present some of its ramifications.

TAs: Konstantin Bogdanov, Maik Sowinski, Sabyasachi Mukherjee, and Anton Shemyakov.

Permutation Groups

Rebecca Waldecker

Symmetry plays a role in many areas of science and in the arts. Mathematically, symmetry is studied in the language of groups, and a particularly nice way to enter group theory is via permutation groups. Together we will explore the basic concepts, prove some results and of course discuss many examples. In the discussion groups we can also look at specific research questions on permutation groups and try to understand why they are interesting (and sometimes difficult).

TAs: Bernhard Reinke, David Pfrang, Steffen Maaß.


Don Zagier



Semi-Algebraic Sets of Integer Points

Günter M. Ziegler

We look at sets of integer points in the plane, and discuss possible definitions of when such a set is "complicated"--this might be the case if it is not the set of inter solutions to some polynomial equations and inequalities. Let's together work out lots of examples, and on the way let's try to develop criteria and proof techniques.

TAs: Roman Prosanov, Sergei Shemyakov, and Khudoyar Mamayusupov.

Butterflies, Cats, and Billiards in Polygons

Anton Zorich

How can we instantly count the number of digits in the 10,000th term of the Fibonacci sequence without using a computer? Why does the butterfly effect cause a computer to give a completely wrong answer when counting the 100th term of the analogous sequence? Why does chaos behave very regularly over a long period of time? What is the connection between wrapping a bicycle tube over itself, playing billiards, and the Boltzmann gas? We will discuss these and other questions and, hopefully, will arrive at some of the results of the 2014 Fields Medalist, Maryam Mirzakhani.

TAs: Sabyasachi Mukherjee, Khudoyar Mamayusupov, and Bayani Hazemach.