This summer school combines various current research topics in geometry and mathematical physics, the unifying theme being the role of discrete models and of intelligent discretization methods. It is organized by the Berlin Mathematical School and the SFB/TRR 109, Alexander Bobenko and Yuri Suris.
Discrete Models in Geometry and Mathematical Physics
11-21 September 2017
(arrival 10 September, departure 21 September)
Venue: Institute of Mathematics at TU Berlin
This summer school is organized by the Berlin Mathematical School and the SFB/TRR 109, Alexander Bobenko and Yuri Suris. It combines various current research topics in geometry and mathematical physics, the unifying theme being the role of discrete models and of intelligent discretization methods.
A diverse program of introductory lecture series will treat subjects ranging from integrability of discrete models, discrete differential geometry and its application in computer graphics to conformal dynamics, cluster algebras and discrete models in quantum field theory.
The summer school is aimed at graduate students in mathematics. Postdocs are also encouraged to attend.
All lectures will take place in the lecture hall MA 041 on the ground floor of the Institute of Mathematics, TU Berlin, Strasse des 17. Juni 136, 10623 Berlin.
Alexander I. Bobenko (TU Berlin) - Discrete confocal Quadrics
Abstract: Quadrics in general and confocal quadrics in particular play a prominent role in classical mathematics due to their beautiful geometric properties and numerous relations and applications to various branches of mathematics. In this lectures we describe a discretization of confocal quadrics (and of the corresponding confocal coordinate systems). It leads to factorizable discrete nets with a novel discrete analog of the orthogonality property and to an integrable discretization of the Euler-Poisson-Darboux equation. We demonstrate that special discrete confocal conics lead to recently introduced incircular nets, which are systems of lines and touching circles. Classical Ivory, Chasles and Poncelet theorems play a central role there.
David Cimasoni (Université de Genève, Suisse) - The geometry of dimer models
Abstract: The aim of this minicourse is to present an introduction to the dimer model to a geometrically minded audience. We will show how several geometrical tools that we know and love (e.g. (co)homology, spin structures, real algebraic curves) can be applied to very natural problems in combinatorics and statistical physics, such as the enumeration of perfect matchings in arbitrary finite weighted graphs.
Anna Felikson (Durham University, UK) - Cluster algebras and Coxter groups
Abstract: Coxeter groups are classical objects appearing in connection with symmetry groups of regular polytopes and various tessellations. Cluster algebras were introduced by Fomin and Zelevinsky in 2002 and gained a growing wave of interest due to numerous relations to other fields in mathematics and theoretical physics. We will discuss a number of connections between cluster algebras and Coxeter groups.
Vladimir Kazakov (Ecole Normale Supérieure and UPMC Paris, France) - Random matrices, random graphs and integrability
Abstract: 1. Matrix models and counting of planar Feynman graphs. Applications to 2D quantum gravity and solvable statistical mechanics on dynamical planar graphs. Examples of one- and two-matrix models. Solution of one-matrix model by saddle point method and its algebraic curve. 2. Solution of one-matrix model via orthogonal polynomials. Kadomtsev-Petviashvili (KP) equations. Reduction of KP equation for one-matrix model. Double scaling limit, Painleve-I equation and number of large graphs with fixed genus. Non-perturbative corrections. 3. Multi-matrix models and their applications. Two-matrix model and solution of the Ising magnetic on dinamical planar graphs. Matrix quantum mechanics and Berezinsky-Kosterlitz-Thouless spins on dynamical planar graphs.
Franz Pedit (University of Massachusetts, USA) - The geometry of the self-duality equations over a Riemann surface
Abstract: The self-duality equations over a Riemann surface are a 2-dimensional reduction of the Yang-Mills equations over a Riemannian 4-manifold. Depending on the signature of the metric of the 4-manifold the reduced equations describe equivariant minimal surfaces in hyperbolic space (equivalently Higgs bundles or surface group representations), or minimal surfaces in the 3-sphere (and CMC surfaces in R^3). Whereas heat flow techniques allow the construction of minimal surfaces in negatively curved targets, the construction of minimal surfaces in positively curved target spaces (such as the 3-sphere) is much more subtle. Already the genus 1 case leads to a very rich theory of an algebro-geometric integrable system, the sinh-Gordon hierarchy. The construction of compact higher genus >1 minimal surfaces in the 3-sphere from an integrable systems perspective has only recently begun. Here the theory of (families of) Higgs bundles plays a central role in their construction. In the course of the lectures we will cover some basic facts about the self-duality equations and then go into some details of the construction of compact higher genus minimal surfaces. Prerequisites needed are a basic understanding of Riemann surfaces, line (vector) bundles, connections, monodromy, and holomorphic structures.
Dierk Schleicher (Jacobs University Bremen) - On Thurston’s vision in geometry, topology, and dynamics — and aspects of current research
Abstract: Since the 1980’s, Bill Thurston has done fundamental work in apparently quite different areas of mathematics: in particular, on the geometry of 3-manifolds, on automorphisms of surfaces, and on holomorphic dynamics. In all three areas, he proved deep and fundamental theorems that turn out to be surprisingly closely connected both in statements and in proofs. In all three areas, the statements can be expressed that either a topological object has a geometric structure (the manifold is geometric, the surface automorphism has Pseudo-Anosov structure, a branched cover of the sphere respects the complex structure), or there is a well defined topological-combinatorial obstruction consisting of a finite collection of disjoint simple closed curves with specific properties. Moreover, all three theorems are proved by an iteration process in a finite dimensional Teichmüller space (this is a complex space that parametrizes Riemann surfaces of finite type). I will try to relate these different topics and at least explain the statements and their context. I will also try to outline current work on extensions of this research.
Yuri B. Suris (TU Berlin) - Lagrangian theory of integrable systems
Abstract: We will discuss the notion of a pluri-Lagrangian structure, which should be understood as an analog of integrability for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated about a decade ago, however having its more remote roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics and their quasiclassical limit, as well as in the theory of variational symmetries going back to Noether. We will discuss main features of pluri-Lagrangian systems in dimensions 1 and 2, both continuous and discrete, along with the relations of this novel structure to more standard notions of integrability. We will also consider some applications of this structure, including the classical billiards in quadrics and the problem of commutativity of multi-valued maps (correspondences).
Alexander Veselov (Loughborough University, UK) - Multivalued dynamics and Klein's Erlangen programme
Abstract: In his famous Erlangen programme Felix Klein proposed an approach to geometry based on the new (at the time) notion of the group. It is often understood in a narrow premise of projective geometry (which is essentially due to Sophus Lie), but Klein's vision was clearly much wider as it was brilliantly demonstrated in his "Lectures on Icosahedron". The purpose of the lectures is to discuss what could be called dynamical Klein's Erlangen programme when we replace group of integers by the group SL(2,Z). In that case we have the multivalued dynamics on the planar binary tree, which can be considered as a discrete version of the hyperbolic plane. We will discuss the growth problem in serveral examples of the corresponding dynamical systems from geometry and number theory, including the celebrated Markov triples. The lectures are based on the results found jointly with K. Spalding.
Registration and Application:
In order to access the application form, please register first here.
Online application is available from 1 April 2017 to 31 May 2017
Application Deadline: 31 May 2017
Applications should include:
- a short letter of motivation
- one letter of recommendation
Subsidiary financial support may be offered to selected participants for travel costs if requested in the application.
For participants with children of up to 6 years, there will be day care opportunities if requested in advance in the application.
Hotel rooms (twin rooms) have been reserved from 10 September 2017 (arrival day) to 21 September 2017 (departure day) for participants from outside Berlin. All acknowledged participants who have confirmed that they need hotel accommodation will be staying at the Hotel KAISER, Berlin, in shared twin rooms (room share with another participant). For this hotel arrangement from 10-21 September the summer school will cover the overnight costs including breakfast at the hotel.
Please see here for travel directions:
Further links to help with local navigation:
How to get around in Berlin?
Berlin has an excellent public transportation system. There are weekly tickets with which you can travel with the public transportation (U-Bahn, S-Bahn, bus, tram) as much as you wish. Costs for a 7-Day-Ticket (German: 7-Tage VBB-Umweltkarte) for zone AB is € 30,-.
Zone AB is within Berlin City limits, thus ideal for the transportation between the hotel and TU Berlin but also to clubs within Berlin. If you intend to travel to Postdam you need an additional ticket for zone C. See for more information: https://shop.bvg.de/index.php/product/267/show/0/0/0/0/buy
Good places to eat on campus:
Cafeteria on 9th floor
Opening hours: Mo - Fr 11:00 - 16:00.
You can eat warm meals here, for prices ranging from about 2.50 - 5.00 €.
Cafeteria on ground floor
Opening hours: Mo - Th 8:00 - 18:00; Fr 8:00 - 17:00
You can eat warm meals here, for prices ranging from about 2.90 - 5.20 €.
TU Mensa, Hardenbergstrasse 34
Opening hours: Mo - Fr 11:00 – 14:30.
You can eat warm meals here, for prices ranging from about 2.75 to 6.00 €.
The meals are paid using a "Mensa Karte"; you can buy one on the ground floor of the Mensa at the cash point (deposit 1,55 €). The card will immediately be loaded with the desired amount. After this first time, the card can be reloaded at cash machines, also located in the Mensa. Before you leave, you can return the card and receive the remaining value on your card and the deposit.
The lower student prices are only available to students enrolled at universities in Berlin or Potsdam.
Cafeteria TU Mensa, Hardenbergstr. 34
Opening hours: Mo - Fr 11:00 – 15:30.
Cafeteria Wetterleuchten, TU Main Building, Strasse des 17. Juni 135
Opening hours: Mo - Fr 8:00 – 15:00
Facing the main building from the Strasse des 17. Juni, the cafeteria is on the right hand side. It is also possible to sit outside.
Cafeteria TU Ernst-Reuter-Platz, Strasse des 17. Juni 152
Opening hours: Mo - Fr 8:00 – 15:00.
There are warm meals available during lunch time, for prices ranging from about 2.45 to 5.45 €.
It is possible to sit outside.
Cafeteria TU "Skyline", Ernst-Reuter-Platz 7
Opening hours: Mo - Fr 8:30 – 15:00.
You can eat warm meals during lunch time, for prices ranging from about 2.45 to 5.45 €.
Great view over Berlin!
Good places outside the university:
Café Hardenberg, Hardenbergstr. 10
Restaurant Filmbühne am Steinplatz, Hardenbergstr. 12
Pasta & Basta (italian cuisine), Knesebeckstr. 94
Manjurani (indian cuisine), Knesebeckstr. 4
Satyam (vegetarian indian cuisine), Goethestr.5
Café Brel, Savignyplatz 1
Mr Hai + Friends (vietnamese cuisine), Savignyplatz 1
Dicke Wirtin (Berliner Kneipe), Carmerstr. 9