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!
Geometric group theory studies finitely presented groups using geometric properties of associated graphs. We discuss new developments, such as Kleiner's new proof of Gromov's theorem on polynomial growth and Bridson's lower bounds for the Andrews- Curtis conjecture.Date and time infoMonday 11:00 - 13:00KeywordsGeoemtric group theory, growth of groups, topologial group theoryAudienceMSc students, PhD students, PostdocsLanguageEnglish
This is a continuation of my Information Theory I course, which I offered in the winter term 2016/2017. I concluded with elementary results on the Kolmogorov-Sinai entropy of dynamical systems and proved the Shannon-McMillan-Breiman theorem for information sources. This theorem will serve as a prerequisite for the second part of my course, Information Theory II, in which I will concentrate on information channels. I will introduce the transmission rate and the capacity of information channels. The central theorems of this part will be Shannon's celebrated coding theorems. 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.References
A. 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.
T. M. Cover, J. A. Thomas. Elements of Information Theory. Wiley, 2006.
Date and time infoTuesday, 11:00 - 12:30KeywordsPartial Differential Equations, Applications of PDEs in sciencePrerequisitesBasic knowledge in probability and measure theory is required.AudienceMSc students, PhD students, PostdocsLanguageEnglishRemarks and notesThis course consists of six lectures, which will take place on April 4, 25, and May 2, 9, 16, 23. In the first lecture on April 4, I will provide a brief summary of the basic results of Information Theory I that will be required for the course.
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.
Schedule
Tanja Eisner: Ergodic theory (5 lectures, 4.4. - 25.4.)
Hans-Bert Rademacher: Geodesic flow on Riemannian manifolds (9.5. - 30.5.)
Matthias Schwarz: Hamiltonian systems and symplectic geometry (6.6. - 4.7.)Date and time infoTuesday 13:15 - 14:45KeywordsErgodicity, Recurrence, ergodic theorems, Geodesic ow, Riemannian metrics, curvature, Hamiltonian systems, stability, symplectic formAudienceMSc students, PhD students, Postdocs
In the last decade ergodic theorems for actions of non-amenable groups on probability spaces have been actively researched by A. Nevo and collaborators. An interesting feature is that for a broad class of groups, such as semisimple Lie groups with Kazhdan's property (T), the exponential convergence rate in the ergodic theorem arises naturally without any spectral assumptions, which is not the case, for instance, for actions of amenable groups.
The aim of this seminar is to understand in depth various techniques behind proofs of ergodic theorems for amenable and non-amenable groups. After reviewing ergodic theorems in the case of amenable groups, we proceed with that of non-amenable ones, following in parts the book "The ergodic theory of lattice subgroups" by A. Gorodnik and A. Nevo. Necessary tools will be introduced and discussed.References A. Paterson, Amenability. American Mathematical Society, Providence, RI, 1988.T. Tao, Some notes on amenability, blog. A. Garrido, An introduction to amenable groups, lecture notes. D. Ornstein, B. Weiss, "The Shannon-McMillan-Breiman theorem for a class of amenable groups", Israel J. Math. 44 (1983), 53–60.E. Lindenstrauss, "Pointwise theorems for amenable groups", Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 82-90. (link) A. Gorodnik, A. Nevo, The ergodic theory of lattice subgroups. Princeton University Press, Princeton, NJ, 2010. B. Bekka, P. de la Harpe, A. Valette, Kazhdan's property (T). Cambridge University Press, Cambridge, 2008. (link to pdf) G. A. Margulis, Discrete subgroups of semisimple Lie groups. Springer-Verlag, Berlin, 1991. G. A. Margulis, A. Nevo, E. M. Stein, "Analogs of Wiener's ergodic theorems for semisimple Lie groups, II", Duke Math. J. 103 (2000), 233–259.A. Nevo, E. M. Stein, "A generalization of Birkhoff's pointwise ergodic theorem", Acta Math. 173 (1994), 135–154.R. Howe, E.-Ch. Tan, Nonabelian harmonic analysis. Applications of SL(2,R). Springer-Verlag, New York, 1992.Date and time infoTue, 09:15-11:45LanguageEnglish
In this course we will consider the variational approach to stochastic partial differential equations with monotone drift, going back to N. V. Krylov, B. L. Rozovskii and E. Pardoux. A main benefit of this approach is that it allows to deal with degenerate quasilinear equations such as the (stochastic) porous medium equation $$du = \Delta (|u|^m u) dt + B(u_t)dW_t$$ and the (stochastic) p-Laplace equation $$du = \textrm{div} (|\nabla u|^p \nabla u) dt + B(u_t)dW_t.$$ After having established the well-posedness of solutions to this class of equations we will investigate qualitative questions on the long-time behavior, ergodicity and random dynamics.Date and time infoWednesday, 11:00 - 12:30KeywordsStochastic Partial Differential Equations, Stochastic AnalysisPrerequisitesbasic measure theory, functional analysis, probability theoryAudienceMSc students, PhD students, PostdocsLanguageEnglish
This course introduces students with basic knowledge of algebraic geometry to the theory of moduli spaces of sheaves. While curves and surfaces may often be studied via explicit equations it becomes more and more difficult in higher dimensions to detect interesting geometric properties from equations. Moduli spaces of sheaves (or stable objects) are an important class of examples of higher dimensional algebraic varieties and their geometric properties are studied through the geometry of the objects they parametrize. During the course we will encounter the following topics(semi-)stability of sheaves
algebraic groups and geometric invariant theory derived and triangulated categories, Bridgeland stability
birational geometry and wall-crossing
Many of the notions and techniques relevant to the theory of moduli spaces have already been introduced by Mumford in the late sixties. In the last twenty years moduli spaces have played an important role in mathematical theories coming from theoretical physics and in the light of the derived approach they have recently reentered the focus through the fascinating connection to birational geometry.Date and time infoFriday 10:00 - 12:00KeywordsModuli spaces, stable sheaves, geometric invariant theory, stability conditions, birational geometry, wall-crossingPrerequisitesUndergraduate course in algebraic geometry, e.g. scheme theory, divisors, line bundles, coherent sheaves. Also basic knowledge of (algebraic) topology would be helpful. Remarks: If necessary a small repetition of basics from algebraic geometry at the beginning is possible.AudienceMSc students, PhD students, PostdocsLanguageEnglish
I will discuss the notion of Tropical Probability Spaces and their commutative diagrams. This new theory streamlines many ad hoc arguments used prior in Information Theory, AI, Causal Inference, and provides new techniques to deal with questions arising in the forementioned areas of research as well as in such areas as non-equilibrium thermodynamics and others.Date and time infoThursday, 11:00 - 12:30KeywordsTropical probability spaces, higher order relations between RV's beyond entropy, new exciting theoryPrerequisites10 x 10 -multiplication table. Other background topics will be provided on demand.AudienceMSc students, PhD students, Postdocs, Group Leaders, Directors of the InstituteLanguageEnglish
This lecture is an introduction to tropical geometry with focus on its techniques and its applications. It is split in two parts. We begin with studying tropical varieties, how they arise and what structure they have. Along the way, we highlight important concepts and techniques that are useful beyond tropical geometry, like Gröbner bases and triangular decompositions. This serves as a theoretical foundation for the next part. The second part is dedicated to applications of tropical geometry. It consists of a series of expository talks. Possible topics include but are not limited to:
algebraic geometry: enumerative geometry.
biology: phylogenetic trees
economics: product-mix auctions, Ricardian economics.
optimization: mean payoff games, multiobjective integer linear programming.
physics: central configurations in the n-body problem.
Near the end of the first part, there will be a short overview of possible topics, for which suggestions are welcome. The topics will be selected based on the interests of the audience.Date and time infoMonday, 09:00 - 10:30KeywordsTropical Geometry, Algebraic Geometry, Computer AlgebraPrerequisitesBasic knowledge about algebra and algebraic geometry is helpful, but not necessary.AudienceMSc students, PhD students, PostdocsLanguageEnglishRemarks and notesThis lecture will feature some computer presentations. Bringing your laptop along is highly recommended.
Which space lies halfway between \(C([0,1])\) and \(C^1([0,1])\)? Is it the Hölder space \(C^{1/2}([0,1])\)? Does \(C^1([0,1])\) lie halfway between \(C([0,1])\) and \(C^2([0,1])\)? These questions lead to the theory of interpolation spaces (and it is fair to say the answers are, somewhat surprisingly, "yes" for the first question but "no" for the second). Also, many trace theorems, domains for fractional powers of operators and maximal regularity spaces for abstract Cauchy problems are obtained from interpolation between suitable Banach spaces.The plan of this lecture is to
show that the \(L^p\)-scale of \(p\)-integrable functions fits in this picture (Theorems of Riesz-Thorin and Marcinkiewicz), introduce the real, trace and complex interpolation methods,establish duality and reiteration theorems,give concrete examples for interpolation spaces (Höolder spaces, Sobolev-Slobodeckii spaces, Besov spaces), and thereby extract elegant proofs for Young's inequality, Sobolev embeddings and trace theorems, investigate interpolation spaces of domains of closed operators and how they enter the study of abstract Cauchy problems.KeywordsReal interpolation, complex interpolation, reiteration theorem, domains of operatorsPrerequisitesCalculus, functional analysisAudienceMSc students, PhD students, PostdocsLanguageEnglish
The Navier-Stokes equations, governing the motion of an incompressible viscous fluid, are derived based on certain physical assumptions concerning the motion of the fluid. In the words of J.Leray in 1934, "these hypothesis need to be justified a posteriori by establishing the following existence theorem: there is a solution which corresponds to a state of velocity given arbitrarily at an initial instant" This problem, although still without answer 85 years later, inspired a wealth of mathematical development, the introduction of many different notions of solutions (classical, weak, mild, suitable,...) and different approaches (fixed point methods, energy methods, blow-up arguments,...). In the lectures we will give an overview of these different directions, mainly focussing on the theory of weak solutions introduced by Leray and Hopf, including such topics as (i) existence, (ii) partial regularity, and (iii) criteria for uniqueness and regularity.Date and time infoThursday and Friday 09:00 - 11:00KeywordsGeoemtric group theory, growth of groups, topologial group theoryPrerequisitesThe course does not require any previous knowledge of fluid mechanics or familiarity with the Navier-Stokes equations, but some previous experience with PDE is strongly recommended.AudienceMSc students, Diploma students, PhD students, PostdocsLanguageGerman or English
