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
Markov chains, martingales, stochastic processes
Lecturer
Benjamin Gess, Max von Renesse, Artem SapozhnikovDate and time infoTuesday 11.15 - 12.45KeywordsStochastic processesAudiencePhD studentsLanguageEnglish
In this course, I shall discuss various dynamical systems in neurobiology, from the dynamics of individual neurons to activity patterns of populations. Both deterministic and stochastic phenomena will be treated. This course can therefore also serve as an introduction to the mathematical theory of dynamical systems, because I shall use the examples to explain general concepts and results from the theory of dynamical systems or, in the terminology of physicists, nonlinear dynamics.ReferencesJ. Jost, Mathematical methods in biology and neurobiology, Springer, 2014Date and time infoFriday 13:30 - 15:00AudienceMSc students, PhD students, PostdocsLanguageEnglish
This will be a self-organizing seminar. Any participant could request any topic to be discussed or offer to share his/her knowledge with others. Lectures are supposed to be accessible to any non-specialist mathematician.Date and time infoFriday: 15:00 - 16:30KeywordsAny topic by request of one of the participants or an offer from lecturers.PrerequisitesCuriosity and appreciation of beauty of mathematicsAudienceMSc students, PhD students, Postdocs, Group leaders, Directors of the MPI MISLanguageEnglishRemarks and notesParticipants need only attend lectures, that are of interest to them.
We are interested in linear elliptic differential equations (including the case of systems) with random coefficient fields. Provided the underlying probability measure on the space of coefficient fields is shift invariant and ergodic, the solution operator acts on large scales like the solution operator of an elliptic equation with deterministic and spatially homogeneous coefficients; a phenomenon called homogenization. We are interested in quantitative aspects of this phenomenon. It turns out that this is intimately linked to elliptic regularity theory; in fact, the above type of randomness generates a large-scale regularity theory as for instance encoded in Liouville properties.Date and time infoTuesday 09:15 - 11:00Keywordslinear elliptic differential equations with random coefficient fields, elliptic regularity theoryPrerequisitesSome knowledge in elliptic PDE is helpful, practically no knowledge in Probability theoryAudienceMSc students, PhD students, PostdocsLanguageEnglish
Results about existence and stability of closed geodesics on compact manifolds with a Riemannian or Finsler metric can be derived using Morse theory for the energy functional. This functional is defined on the free loop space of the underlying manifold.Date and time infoWednesday 09:15 - 10:45Keywordsenergy functional, free loop space, critical point theory, index of iteratesPrerequisitesBasic knowledge of differential geometry and algebraic topologyAudienceMSc students, PhD students, PostdocsLanguageEnglish
G-Manifolds, principal bundles and connections
Classical Yang-Mills theory
Classification of principal bundles and Chern classes
Introduction to the theory of instantons
Associated bundles, matter fields and the Higgs-mechanism
Clifford algebras and spinor fields
The standard model
Linear connections, Riemannian geometry and the Einstein equations
Date and time infoThursday 15:30 - 17:00, Friday 11:00 - 12:30AudienceMSc students, PhD students, PostdocsLanguageEnglishRemarks and notesThere will be tutorials. Matching the lecture, the advanced seminar "Quantum Field Theory and Gravitation" centers around magnetic monopoles.
Matter fields as sections in associated bundles (1 talk)
The Yang-Mills-Higgs system (2 talks)
The Dirac magnetic monopole (1 talk)
Status of the experimental search for magnetic monopoles (1 talk)
The ’t Hooft-Polyakov monopole (2 talks)
The Prasad-Sommerfield limit (2 talks)
The Seiberg-Witten model* (1 talk)
Date and time infoMonday 16:00 - 17:00AudienceMSc students, PhD students, PostdocsLanguageEnglish
Metric curvatures represents a minimalistic, foundational approach to the development of a Discrete Differential Geometry on a very general call of geometric objects (spaces).We survey the essential types of metric curvatures, namely Menger and Haantjes curvatures of curves, and Wald curvature for surfaces, special accent being placed upon the last one. We investigate the relationship between these notions, as well as those with the far better known notion of Alexandrov curvature. Special accent is placed upon the various applications, both intra-mathematical (Complex Analysis, fractals, construction of Lipschitz functions, metric Ricci curvature for PL cell complexes and a metric Ricci flow for PL surfaces) as well as in "real life" (DNA Microarray Analysis, Protein Folding, Networks, Imaging and Graphics, Wavelets, Pattern Recognition).Date and time infoThursday: 11:15 - 13:00KeywordsMetric Geometry and its ApplicationsPrerequisitesBasic Mathematical Analyis, rudiments of Differential Geometry (preffered), curiosity and a quest for "real life" applications of "Pure" Mathematics.AudienceMSc students, PhD students, PostdocsLanguageEnglishRemarks and notesThese lectures are envisioned, at this stage, as an overview. However, given specific interests of the attendance and/or requests, more technical details can be added, as well as special emphasis/focus on specific areas/subjects. Lecture series will only be offered till December!
This course is meant to give a compact introduction to symplectic geometry with an overview over essential main phenomena, methods and results. Major emphasis will lie on the phenomenon of symplectic rigidity. The method of pseudoholomorphic curves will be introduced and explained along examples, but not thoroughly be deduced. A large part of the course will concern the field of Hamiltonian dynamical systems.Date and time infoFriday 9:15 - 10:45KeywordsSymplectic structures, Symplectic Topology, Symplectic Rigidity, Hamiltonian systemsPrerequisitesBasic knowledge of differential geometry and algebraic topologyAudienceMSc students, PhD students, PostdocsLanguageEnglish
This is a reading seminar (not a regular lecture) on geometric measure theory: the participants will be asked to contribute a talk related to the topic of the seminar.As for the content, the aim of this seminar is to provide an introduction to the striking recent results on the rectifiability of the singular set of minimal hypersurfaces proven by Naber and Valtorta (see arXiv:1504.02043 and arXiv:1505.03428). A preliminary list of topics which will be covered is the following:Monotonicity formula;Stratification of singularities;Minimality of Simons' cone;Introduction to the rectifiability of the singular set.Date and time infoThursday 09:00 - 11:00KeywordsGeometric Measure Theory, Minimal Surfaces, Singularities, RectifiabilityPrerequisitesBasic knowledge on measure theory and mathematical analysisAudiencePhD students, PostdocsLanguageEnglish
The aim of the lectures is to present the basic theory of distributions and linear partial differential operators. Topics to be discussed are:Distributions and Fourier transformLocal solvability of linear partial differential equationsPseudodifferential operatorsPlease note: There will also be a Fachseminar (jointly with Bernd Kirchheim) mainly for university students about "convex geometry". PhD students willing to learn something about concentration of measure, Brunn-Minkowski and Dvoretzky’s Theorem are welcome on Wednesdays from 13.00 to 15.00 at Leipzig University, SG A-314.Date and time infoWednesday 09:15 - 10:45PrerequisitesSolid background in functional analysis and knowledge of partial differential equations. Sobolev spaces are useful but not required.AudienceMSc students, PhD students, PostdocsLanguageEnglish
Malliavin calculus is an abstract infinite dimensional calculus on abstract Gaussian probability spaces. Combined with corresponding probabilistic representations it can be an extremely useful tool in the analysis of PDE problems of which the Hörmander theorem for hypoelliptic operators is a celebrated example. In the second part of the lecture we will extend this to the case of Riemannian manifolds where we discuss probabilistic methods for the derivation of heat kernel bounds and the prominent role of Ricci curvature.Date and time infoMonday 15:00 - 17:00KeywordsProbability Theory & Elements of Ito Stochastic Calculus, Elements of Differential GeometryAudienceMSc students, PhD students, PostdocsLanguageEnglish
