Modern Mathematics  International Summer School
for Students
Instructors
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 VersaillesSaintQuentin, France
"Representation of Groups: a Scenic Road
"
 KaiUwe 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 SignificantDigit
Phenomenon"
"Knowing When to Stop"
"FairDivision Problems"
 Mikhail Hluschanka,
Jacobs University Bremen, Germany
"From Rational Maps to Tilings and
Groups"
 MarcThorsten Hütt,
Jacobs University Bremen, Germany
"Simple Models of Interacting Biological
Neurons"
 Ivan Izmestiev,
Université de Fribourg, Switzerland
"Porisms"
 Victor Kleptsyn,
Institut de Recherche MathÃ©matique de Rennes, France
"Sphere Packings in Higher Dimensions"
 Anke Pohl,
FriedrichSchillerUniversität 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"
"SixtyFour 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 TaitKneser Theorem and
the Sylvester Problem"
 Rebecca Waldecker,
MartinLutherUniversität HalleWittenberg, Germany
"Permutation Groups"
 Don Zagier, Max Planck
Institute Bonn, Germany
"TBA"
 Günter M. Ziegler,
Freie Universität Berlin, Germany
"SemiAlgebraic Sets of Integer Points"
 Anton Zorich, Institut
Mathématiques de Jussieu, France
"Butterflies, Cats, and Billiards in
Polygons"
Talks and Abstracts
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.
I would talk about the graphminor 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 graphminor 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.
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.
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 nonliving Nature, such as sand grains, pebbles or asteroids.
Although at first sight the connection appeared to be farfetched,
laboratory and field data showed a convincing match with
theorybased predictions and this turned out to be the key to
understanding the provenance of Martian pebbles [3].
References
[1] R.M.E. William et al.: Martian fluvial conglomerates at Gale
crater. Science 340 (6136) 10681072,
[2] P.L. Varkonyi and G. Domokos: Monomonostatic bodies: The
answer to Arnold's question. The Mathematical Intelligencer, 28(4)
pp 3438. (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
A centuryold 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 roundoff 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.
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 BestChoice 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
"noinformation" Secretary Problem will be reviewed, along with
several gametheoretic extensions, analogs for "fullinformation"
and "partialinformation" stopping, and several basic open
problems.
TAs: Alexander Minets, Anton Shemyakov, and Dimitry
Rumyansev.
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 fairdivision 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.
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.
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.
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.
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.
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.
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.
The core of the organizing committee of our summer school are
members of my "European Research Council Advanced Grant" called
"HOLOGRAM: holomorphic dynamics, geometry, rootfinding, 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 âModern
Mathematicsâ 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
Newtonâs 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.
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
RokhlinNikulin classification.
TAs: Thanasin Nampaisarn, Konstantin Bogdanov, Sergei
Shemyakov, and Maik Sowinski.
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.
The TaitKneser 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.
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.
TBA
TAs: TBA.
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.
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.
