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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 Paris-Sud, Orsay, France
"Spectral Geometry"

Martin Andler, Université Versailles-Saint-Quentin, 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 Time-Frequency Analysis: Between real world applications and abstract harmonic analysis" (Background reading)

Gerhard Frey, Universität Duisburg-Essen, 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 Routh-Hurwitz 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 root-finding 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

Spectral Geometry

Nalini Anantharaman

The vibration of a membrane can be described by a simple partial differential equation: the 2-dimensional 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.

Quadrilateral surfaces

Alexander Bobenko

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.

Time-frequency 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 non-orthogonal and redundant signal representations and to use them in order to study the mapping properties of certain pseudo-differential operators. On the other hand there are nice applications to digital signal processing (up to MP3).

TAs: Palina Salanevich, Sabyasachi Mukherjee

While time-frequency analysis starts to play an increasingly important role in abstract analysis, e.g. in the context of pseudo-differential 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 non-orthogonal 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 K-rational 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

Surfaces from two sides

Matthias Görner

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 Routh-Hurwitz problem about polynomial stability: algebraic, analytic and matrix-theoretic.

Real Algebraic Curves

Ilia Itenberg

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

Ilia Itenberg

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, algebro-geometric objects are replaced with piecewise-linear 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 non-increasing 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 (Hardy-Ramanujan 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? (Bárány-Vershik, Sinai)

If the time permits, we will also discuss the questions related to the domino tilings, mentioning the well-known "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 2-manifolds are used to construct and study 3-manifolds.

TAs: Mikhail Hlushchanka, Alexander Minets, Russell Lodge

Configuration Spaces

Gaiane Panina
  • What (topologically) is the set of unordered pairs of points lying on the circle S1?
  • 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 non-trivial 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

Proofs (not) from The Book

Sergei Tabachnikov

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

The Dilogarithm

Don Zagier

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 not-so-famous mathematician. My plan for this lecture is to show you at least five pictures, and to accompany this with more than five-hundred words.

New Proofs for THE BOOK

Günter M. Ziegler

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



 

 
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