Please find more information about the lectures at the detail pages.
For rooms at the MPI MiS please note: Use the entry doors Kreuzstr. 7a (rooms A3 01, A3 02) and Kreustr. 7c (room G3 10), both in the inner court yard, and go to the 3rd. floor. To reach the Leibniz-Saal (E1 05, 1st. floor) and the Leon-Lichtenstein Room (E2 10, 2nd. floor) use the main entry Inselstr. 22.
Please remember: The doors will be opened 15 minutes before the lecture starts and closed after beginning of the lecture!
About this lectureDue to the rather broad spectrum of topics within the IMPRS, the curriculum consists of a core curriculum to be attended by all students and a variety of more specialized lectures and courses. The heart of our teaching program certainly is the Ringvorlesung. Each semester the Ringvorlesung focuses on one field and is usually delivered by scientific members of the IMPRS who introduce different approaches and visions within this field.
Topics
Lecturer: Fatihcan Atay
Introduction to dynamical systems. Existence, uniqueness, continuous dependence of solutions to ODEs
Flows, invariant sets, alpha and omega limit sets
Concepts of stability: Lyapunov, orbital, global. Lyapunov functions
Periodic solutions, Floquet multipliers, Poincare maps, Hyperbolicity and bifurcations
Lecturer: Felix Otto
Stable, unstable and center manifolds
Poincare normal forms
Topological conjugacy (Hartman-Grobman)
Examples and applications (pattern formation)
Lecturer: Jürgen Jost
Dynamical systems, Morse-Conley theory
Entropy of dynamical systems
Entropy concepts (topological and measure theoretic), entropy and information
Invariant measures
Date and time infoMondays 13:30 - 15:00
In the winter term 2012/2013, I taught the first part of my course on information theory. I introduced basic information-theoretic quantities such as the entropy, the conditional entropy, the mutual information, and the relative entropy. Here, I tried to present a general measure-theoretic perspective, which provides strong tools for the treatment of information sources and information channels. Based on the developed information-theoretic quantities, I presented elementary results on the Kolmogorov-Sinai entropy of dynamical systems and proved the Shannon-McMillan-Breiman theorem. This theorem is the main theorem of the first part of my course and will serve as a prerequisite for the second part, which I am going to present during the summer term 2013. In this part, I will concentrate on information channels. I will introduce the transmission rate and capacity of information channels. The central theorems there will be Shannon's celebrated coding theorems. To this end, I will develop Feinstein's fundamental lemma, which constitutes, together with the Shannon-McMillan-Breiman theorem, the main tool for the proofs of Shannon's coding theorems.ReferencesA. I. Khintchine. Mathematical Foundations of Information Theory. Dover, New York, 1958.
Y. Kakihara. Abstract Methods in Information Theory. World Scientific, Singapore, 1999.
P. Walters. An Introduction to Ergodic Theory. Springer, 1982.Date and time infoWednesday 11.30 - 13.00
This is not a regular lecture, but rather a reading seminar, in which students and postdocs will be asked to give lectures on some selected papers about the Navier-Stokes equations.
The Navier-Stokes equations (suitably coupled with other supplemental equations) model in a quite accurately way the motion of viscous fluids under several conditions. These equations have been studied since a long time, but still the main basic questions, such as the existence of solutions for all times, remain open. The reasons for this lack of understanding have to be found in the structure of the equations, which are nonlinear and involve a certain Lagrange multiplier, the pressure.
The aim of the seminar is to give an introduction to the subject, by presenting and discussing some of the fundamental papers on this topic. In particular, the following results (but not only!) will be discussed:
existence of Leray's weak solutions;
Serrin's regularity conditions;
Caffarelli-Kohn-Niremberg's partial regularity;
Recent results by Constantin, Sverak and others on suitable regularity criteria.Date and time infoWednesday 13.15 - 14.45KeywordsNavier-Stokes equations, existence, regularityPrerequisitesbasic functional analysis, basic PDE's, basic fluid mechanicsLanguageEnglish
Cellular automaton was introduced by von Neumann. It has many important applications in dynamical systems. In this course I will explain the relations between amenability, residually finiteness, soficity, and surjunctivity of groups and cellular automata theory. More precisely, I am planning to teach
The dynamical characterization of residually finiteness.
Both surjectivity and pre-injectivity of the cellular automata over an amenable groups are equivalent to the fact that the image of the configuration space has maximal entropy. Then one can get the Garden of Eden theorem for the case of amenable groups.
A characterization of amenability of groups in terms of cellular automata (Ceccherini Silberstein,Machi,Scarabotti and Bartholdi).
Gromov-Weiss's proof of Gottschalk conjecture for sofic groups.
Using cellular automata to give another proof of the Kaplansky's Direct Finiteness conjecture for sofic groups (Elek-Szabo, Ceccherini Silberstein-Coornaert).
Zero divisor conjectures of group rings and their reformulations in linear cellular automata.ReferencesCeccherini-Silberstein, Coornaert, Cellular Automata and Groups. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010.Date and time infoThursday 9.30 - 11.00KeywordsCellular automata, Amenable groups, Residually finite groups, Sofic groups, Entropy, The Garden of Eden theoremLanguageEnglish
In this lecture, we investigate the function theory on Riemannian manifolds. It is divided into three parts. The first part is devoted to the local properties of harmonic functions and eigenfunctions on Euclidean spaces and Riemannian manifolds. This involves detailed analysis of the local behaviors of solutions to elliptic PDEs. Our main interests lie on the study of nodal sets of harmonic functions and eigenfunctions on manifolds [4-7]. The second part of the lecture is about the global properties of harmonic functions on Riemannian manifolds, e.g. the asymptotical behaviors of harmonic functions on noncompact manifolds such as Liouville theorem of harmonic functions by Cheng-Yau gradient estimate. We will introduce the solution of Yau's conjecture on polynomial growth harmonic functions on manifolds with nonnegative Ricci curvature by Colding-Minicozzi [2-3]. The third part consists of the heat kernel estimates, the wave kernel estimates and their applications. The heat kernel is widely used in the study of spectral theory on Riemannian manifolds, traced back to the classical works by Li-Yau and Grigor'yan. The classical wave kernel method by Cheeger-Gromov-Taylor [1] and its geometric applications will be explained in detail.References Cheeger-Gromov-Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom. (1982).
Colding-Minicozzi, Harmonic functions with polynomial growth, J. Diff. Geom. (1997).
Colding-Minicozzi, Harmonic functions on manifolds, Ann. of Math. (2) (1997).
Colding-Minicozzi, Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys. (2011).
Han-Lin, Nodal sets of solutions of elliptic differential equations, preprint.
Hardt-Hoffmann-Ostenhop-Hoffmann-Ostenhop-Nadirashvili, Critical sets of solutions to elliptic equations, J. Diff. Geom. (1999).
Hardt-Simon, Nodal sets for solutions of elliptic equations, J. Diff. Geom. (1989).
Date and time infoThursday 13.15 - 14.45KeywordsNodal set theory, harmonic functions, polynomial growth harmonic functions, heat kernel, wave kernelLanguageEnglish
This lecture will discuss methods commonly used for the simulation and mathematical analysis of networks of spiking neurons, the basic building block of most nervous systems. No previous knowledge in neuroscience will be assumed, so this lecture is well-suited for the neuroscientifically interested but not yet well-adept.
We will take a look at the mathematical properties of single cell models and networks thereof as well as software suited for the simulation of single cells and neuronal networks in silico (NEST, NEURON, BRIAN).
Furthermore, we investigate upon learning in such networks (by means of dynamic synapses) and develop mathematical tools for the qualitative and quantitative analysis of information coding and transfer in such networks.
A preliminary table of contents is the following:
Elements of neuronal networks: neurons and synapses
Single cell models: from Hodgkin-Huxley to integrate and fire
Population models
Neuronal dynamics & coding
Synaptic plasticity
Information theoretic analysis of spiking neural networksDate and time infoTuesday 13.30 - 15.00Keywordsspiking neuron models, simulation software, neural networks, synaptic plasticity, neural coding, information theoryLanguageEnglish
In this course, on one hand, I shall use the topic to present a wide spectra of mathematical methods and tools, from dynamical systems and fixed point theorems to aspects of epistemic logic. On the other hand, I also want to discuss the conceptual foundations and problems of game theory.Date and time infoFriday 13.30 - 15.00LanguageEnglish
The topic of this lecture are basic programming techniques needed to exploit parallel computing systems (as can also be found at the MPI MIS or at the "Rechenzentrum Garching'' of the MPG ). Starting with simple vectorisation, e.g. SSE, AVX or MIC, the lecture will carry on with techniques for shared memory systems, e.g. OpenMP and task based parallelisation. Finally, distributed computing using message passing will be discussed, needed to use hundreds of computing cores.
The lecture will concentrate on practical aspects of parallel programming, mostly presented in the form of examples, common to scientific computing. Furthermore, basic theoretical aspects, e.g. algorithm complexity, will also be an important topic.
The lecture may be the basis for future (block) courses for people at the MPI MIS using our parallel computing facilities.Date and time infoMonday 10.15 - 11.45KeywordsOpenMP, thread building blocks, message passing, vectorisationLanguageEnglish
The goal of this course is to give an overview of some classical and recent developments in discrete probability. Possible topics include random walks and electric networks, cover and mixing times for Markov chains on finite graphs, self-avoiding walk, uniform spanning tree, percolation, random planar maps, etc.Date and time infoThursday 11.00 - 12.30LanguageEnglsih
In this class I will present the theory of rectifiable currents. These are a measure theoretic analog of the notion of smooth submanifolds of Riemannian manifolds, and emerge in many applications in geometric analysis and mathematical physics. After an introduction concerning the general aspects of the topic and the basic definitions, the course will cover:
the Federer and Fleming's theory, reviewed in the "metric space'' approach by Ambrosio and Kirchheim;
the regularity of codimension 1 mass minimizing currents;
the analysis of singular cones, and consequent dimension reduction arguments for the singular set;
(time permitting) the boundary regularity of codimension 1 mass minimizing currents, after Hardt and Simon.
The class will be complemented by an exercise section and a reading seminar. The participants will be asked to actively take part to the course by solving some exercise sheets (which will integrate the lectures) and giving a seminar on a previously assigned paper on related topics.References
L. Simon, Lecture notes on Geometric Measure Theory.
F. Lin and X. Yang, Geometric Measure Theory. An introduction.
S. Krantz and H. Parks, Geometric Integration Theory.
Date and time infoTuesday 10.00 - 12.00KeywordsMinimal Surfaces, Geometric Measure Theory, Regularity Theory of nonlinear PDEsPrerequisitesThis is an advanced class in Geometric Measure Theory. It is assumed a good knowledge about differential geometry, measure theory, functional analysis and partial differential equations. The basic results and the techniques borrowed by these fields will be every time suitably introduced and discussed, but no proof will be provided.LanguageEnglish
We will present and discuss in detail basic problems of quantum statistics. These include, in analogy to mathematical statistics, quantum hypothesis testing and state estimation. In the first part of the course it will be necessary to carefully introduce the C*-algebraic formalizm, which allows to describe physical systems, classical or quantum, in an unified way. Within this abstract setting the basic concept of random variable, for example, is generalized by the notion of (quantum) state.Date and time infoFriday 10.00 - 11.30Keywordsquantum theory, probability theory, mathematical statistics, ergodic theory, operator algebras (matrix, C*-, von Neumann), state space, statistical model, quantum tests, decision problems, state estimation, (asymptotic) hypothesis testing, stochastic processesPrerequisitesSome background in probability theory, quantum mechanics, linear algebra, functional analysis, ergodic theory would be helpful.LanguageEnglish
