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
 Nalini Anantharaman,
Université ParisSud, Orsay, France
"Spectral Geometry"
 Martin Andler,
Université VersaillesSaintQuentin, France
"A stroll through representation theory for Lie
groups"
 Alexander Bobenko,
Technische Universität Berlin, Germany
"Quadrilateral surfaces"
 John H. Conway,
Princeton University, USA
"Topics decided in consultation with
participants"
 Hans Feichtinger,
Universität Wien, Austria
"Fascination TimeFrequency Analysis:
Between real world applications and abstract harmonic analysis"
(Background
reading)
 Gerhard Frey,
Universität DuisburgEssen, Essen, Germany
"Elliptic Curves in Theory and Practice"
 Matthias Görner,
Pixar Animation Studios, USA
"Surfaces from two sides"
 Olga Holtz, Technische
Universität Berlin, Germany
"Zero localization: from simple algebra to
tantalizing open problems" (Slides)
"Around the RouthHurwitz problem"
 Ilya Itenberg,
Université Pierre et Marie Curie, Paris, France
"Real Algebraic Curves"
 Alexandre Kirillov,
University of Pennsylvania, USA
"A bit of old Geometry seeing from Modern
Mathematics"
 Victor Kleptsyn,
Université de Rennes, France
"Asymptotic problems in combinatorics"
 Yair N. Minsky, Yale
University, USA
"Manifolds, their geometry, topology and
transformation groups"
 Gaiane Panina, St.
Petersburg Institute for Informatics and Automation of the Russian
Academy of Sciences
"Configuration Spaces"
 Dierk Schleicher,
Jacobs University, Germany
"On Newton's rootfinding method and the
Mandelbrot set  or: on Useless and Useful Mathematics"
 Sergei Tabachnikov,
Pennsylvania State University, USA
"Equiareal triangulations of a
square"
"Proofs (not) from The Book"
 Vlad Vicol, Princeton
University, USA
"Mathematical aspects of incompressible fluid
dynamics"
 Don Zagier, Max Planck
Institute Bonn, Germany; Collège de France
"The Dilogarithm"
 Günter M. Ziegler,
Freie Universität Berlin, Germany
"Five Math Girls and their Images"
"New Proofs for THE BOOK"
Talks and Abstracts
The vibration of a membrane can be described by a simple partial
differential equation: the 2dimensional D'Alembert equation. The
acoustic vibrations of a membrane is a superposition of harmonics,
which mathematically correspond to eigenvalues and eigenfunctions
of the laplacian. The relation between the geometry of the membrane
and its spectrum of vibration is still not fully understood, and I
will present a few classical results, but also recent research, on
the subject. The talks will focus on the geometry of nodal lines
and on the question asked by Mark Kac in the 60s: "Can you hear the
shape of a drum?" (that is, can 2 membranes with different shapes
produce the same sound?).
TAs: Thanasin Nampaisarn, Sabyasachi Mukherjee,
Michele Triestino
Using simple examples, and a historical perspective, the talk
will try to explain how this field has evolved from its origins in
the mathematical foundations of quantum mechanics to being central
in several domains of mathematics.
An emerging field of discrete differential geometry aims at the
development of discrete equivalents of notions and methods of
classical differential geometry, in particular of surface theory.
The latter appears as a limit of a refinement of the
discretization. Current interest in discrete differential geometry
derives not only from its importance in pure mathematics but also
from its applications in computer graphics, theoretical physics,
architecture and numerics.
These lectures are about quadrilateral surfaces, i.e. surfaces
built from planar quadrilaterals. They can be seen as discrete
parametrized surfaces. Discrete curvatures as well as special
classes of quadrilateral surfaces, in particular discrete minimal
surfaces are considered.
No knowledge of differential geometry is expected.
TAs: Mikhail Hlushchanka, Alexander Minets, Russell
Lodge
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.
Timefrequency analysis is based on a local variant of the
Fourier transform. In a musical context a spectrogram would tell
the user at which time which "harmonies" are active, very much like
a score. Due to the redundancy of such a continuous
representation/visualization of a signal various discretizations
have been considered, leading to what is called Gabor analysis. An
important part of Gabor analysis is to understand such
nonorthogonal and redundant signal representations and to use them
in order to study the mapping properties of certain
pseudodifferential operators. On the other hand there are nice
applications to digital signal processing (up to MP3).
TAs: Palina Salanevich, Sabyasachi Mukherjee
While timefrequency analysis starts to play an increasingly
important role in abstract analysis, e.g. in the context of
pseudodifferential operators, or the modelling of mobile
communication channels, it has also become a versatile tool for
signal processing applications.
The problem concerns the (linear) representation of signals
(also images) as superpositions of well structured, but
nonorthogonal collections of building blocks. Such an approach
requires considerations of numerical stability and use of memory as
well as the use of efficient numerical linear algebra methods.
On the other hand it allows to produce intuitively appealing
images supporting intuition very much, and to run systematic
experiments, which was not possible in such a way just 25 years
ago. Meanwhile extensive collections of MATLAB files are available
from NuHAG resp. within the LTFAT toolbox of Peter
Soendergaard.
TAs: Palina Salanevich, Sabyasachi Mukherjee
Elliptic curves E over fields K are fascinating
objects for mathematical research. On the one hand, they are very
simple objects, namely cubic curves in the projective plane and so
easily accessible for elementary approaches. On the other side,
they are abelian varieties of dimension one and so the deep theory
of these objects can be used, too. A particular, the
Krational points on elliptic curves form an abelian group
E(K). This gives rise to a very interesting aspect:
The operation of the Galois group on K on algebraic points
of E yields representations that carry a lot of information
both about the field K and the curve E. A spectacular
consequence is Wiles proof of Fermat's Last Theorem. But this is
only part of the reason for the high interest in elliptic curves
nowadays. Based on the theory of elliptic curves there is a strong
algorithmic aspect that has surprisingly far going applications in
data security and is one of the backbones of public key
cryptography.
TAs: Thanasin Nampaisarn, Khudoyor Mamayusupov,
Sabyasachi Mukherjee
In this talk, surfaces will be illuminated from two sides. First
from the point of view of abstract math considering questions such
as: What are invariants of surfaces? How does one classify
surfaces? What are higher structures on surfaces and how do they
generalize to higher dimensions? Then from the point of view of
computer graphics using surfaces for character animation and
considering questions such as: How do we find a good subdivision
algorithm?
TAs: Brennan Bell, Palina Salanevich, Constantin
Bogdanov
This talk will be devoted to (precise) zero localization of
polynomials, entire and meromorphic functions. This subject goes
back at least to Rene Descartes and was developed further by
Hermite, Laguerre, Jacobi, Hurwitz, Polya, Szego, Schoenberg, Krein
and other giants of our field. I will discuss its historic
developments and its connections with other fields, most notably
linear algebra.
In my second talk, I will focus on the many aspects of the
RouthHurwitz problem about polynomial stability: algebraic,
analytic and matrixtheoretic.
The lecture is devoted to the objects that can be described by
an algebraic equation f(x, y)=0 in the real
plane with Cartesian coordinates x and y (here
f is a polynomial in two variables with real coefficients).
Examples of such objects are provided by a line (it can be
described by a polynomial of degree 1) and an ellipse (it can be
described by a polynomial of degree 2). What can be said about
curves described by polynomials of higher degrees? In the lecture,
we will be mainly interested in topological properties of real
algebraic curves.
TAs: Palina Salanevich, Alexander Minets, Michele
Triestino
Tropical geometry is a new mathematical domain which has deep
and important relations with many branches of mathematics. It has
undergone a spectacular progress during the last ten years. In
tropical geometry, algebrogeometric objects are replaced with
piecewiselinear ones. For example, tropical curves in the plane
are certain rectilinear graphs. We will present basic tropical
notions and first results in tropical geometry, as well as
applications of tropical geometry in enumerative problems.
TAs: Palina Salanevich, Alexander Minets, Michele
Triestino
The most intriguing feature of mathematics is the unexpected
relations between its different parts. I discuss the triple
intersections of lines, counting of integral points in plane
figures, tiling of a sphere by tangent discs and geometry of
special relativity.
TAs: Mikhail Hlushchanka, Aleksander Minets, Russell
Lodge
There are many interesting problems in the combinatorics that
are stated as "how does a random big object look like" or as "how
many are there of these objects". For instance, in a randomly
chosen sequence of zeros and ones of big length n, the
proportions of zeros and ones are most probably close to one half;
in a randomly chosen permutation of n elements most probably
there is a "large" cycle (of length comparable to n).
It turns out that the problems "count the objects" and "find the
limit properties of a random object" are often related. We will
consider (only with handwaving arguments) some problems where such
a link appears:
 Partitions of a large number n into a sum of
nonincreasing numbers (or, what is the same, Young diagrams of
size n. How does a typical partition look like? What is the
number of such partitions (HardyRamanujan formula)?
 What is the affine length of a curve, and what is the typical
shape of a convex broken line, going from (0,1) to (1,0) inside the
unit square, if its vertices are restricted to stay on a lattice
with step 1/ n? (BárányVershik, Sinai)
If the time permits, we will also discuss the questions related
to the domino tilings, mentioning the wellknown "arctic circle"
theorem.
Manifolds are encountered and constructed in topology in a
variety of flexible ways, but they often exhibit a rigid geometric
structure which is both informative and beautiful. We'll give a
guided tour of some of these phenomena in dimensions 2 and 3. One
focus will be the relationship between topology and the symmetries
of tiling patterns in different geometries. Another will be the
ways in which transformations of 2manifolds are used to construct
and study 3manifolds.
TAs: Mikhail Hlushchanka, Alexander Minets, Russell
Lodge
 What (topologically) is the set of unordered pairs of points
lying on the circle S^{1}?
 Take four bars and join them consecutively by revolving joints
in a closed flexible chain.
What is the set of all possible 3D shapes of the chain? These
are relatively simple questions about configuration spaces. More
detailed, we have the following paradigm: here is some
combinatorial (or combinatorial plus metrical) object A. It
can be just everything: a configuration of points with prescribed
collinearities, a graph with rescribed edgelengths, a face lattice
of a polytope, etc. The object A can be geometrically
realized in many ways. The set of all realizations is by definition
the configuration space of A. To describe the configuration
space is always a kind of art. The aim of my lecture is to present
a diversity of methods used in this subject.
TAs: Mikhail Hlushchanka, Palina Salanevich, Russell
Lodge
We will discuss one area of mathematics that he finds beautiful
and intriguing, and on which he likes working, no matter whether or
not it has any practical applications: these are dynamical systems
that arise by iterating polynomials. Then he discusses another area
that seems unrelated but that seems to have much more practical
relevance: finding zeroes of analytic equations or polynomials. At
the end it turns out that on order to make progress on the "useful"
questions, one needs to know everything about the "useless" (but
interesting and beautiful) questions. Which is one more example why
the distinction of research into "useless" and "useful" is 
"useless"!
If a square is partitioned into triangles of equal areas then
the number of these triangles is necessarily even. This theorem,
which is only about 40 years old, has a surprisingly nontrivial
proof that involves combinatorial topology and number theory. I
shall present the proof, describe the history of the problem and
mention various related results and conjectures.
TAs: Thanasin Nampaisarn, Constantin Bogdanov, Michele
Triestino
According to Erdös, God keeps the most elegant proof of
each mathematical result in "The Book". Many mathematicians have
their own private collections of "book proofs". In fact, such a
collection, and a highly successful one, was published by M. Aigner
and G. Ziegler. I shall describe several proofs not included in the
Aigner and Ziegler book that, in my opinion, are serious contenders
for inclusion in The Book.
TAs: Thanasin Nampaisarn, Constantin Bogdanov, Michele
Triestino
We introduce some of the main models (partial differential
equations) which arise in the mathematical modeling of
incompressible fluids. We overview some of the classical results in
the field and present a plethora of open problems, motivated both
by physics and mathematical analysis.
TAs: Sergiy Vasylkevich, Michele Triestino
TAs: Brennan Bell, Khudoyor Mamayusupov, Constantin
Bogdanov
As we all know, a picture is worth a thousand words. And indeed,
"what is the use of a book without pictures or conversations?" asks
a little girl in a famous children's book written by a
notsofamous mathematician. My plan for this lecture is to show
you at least five pictures, and to accompany this with more than
fivehundred words.
According to a legend told by Paul Erdös, God maintains a
book, called THE BOOK, which contains perfect mathematical proofs.
Martin Aigner and I have, some twenty years ago, started to collect
examples, and then made some modest suggestions of proofs that
might be in THE BOOK. In this lecture, I will try to present some
more.
TAs: Brennan Bell, Khudoyor Mamayusupov, Constantin
Bogdanov
