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!
Category theory is a language which formalizes the commonalities among different kinds of mathematics. Modern algebraic topology and geometry crucially rely on categorical concepts, and there are indications that categorical thinking may also be relevant to probability theory and network science. The topics of the second half of this course will be chosen according to the audience's requests. Possible choices include the following:
Homological algebra: abelian categories, derived functors, triangulated categories, applications in geometry
Sheaves and stacks: gluing and locality, toposes, Grothendieck fibrations, applications in geometry
Higher categories: 2-categories, categorical homotopy theory, model categories
Categorical geometry: Grothendieck bifibrations, cohesion, generalized smooth spaces
Date and time infoMonday 11:00 - 13:00KeywordsHomological algebra, Sheaves and stacks, Higher categories, Categorical geometryAudienceMSc students, PhD students, PostdocsLanguageEnglish
This course is an attempt to give a reasonably self-contained presentation of the basic theory of stochastic partial differential equations, taking for granted basic measure theory, functional analysis and probability theory, but nothing else. In view of the available time we shall focus on semilinear parabolic problems driven by additive noise, such as stochastic reaction diffusion equations \[du = \Delta u\ dt + f(u)dt + dW_t\] and stochastic Navier-Stokes equations \[du = \Delta u\ dt - (u\cdot\nabla u)u\ dt-\nabla p\ dt + dW_t,\quad \textrm{div}\ u =0,\] where \(W\) is an infinite-dimensional Wiener process. Such SPDE can be treated as stochastic evolution equations in some infinite-dimensional Banach space and they already form a rich class of problems with many interesting properties. Furthermore, this class of problems has the advantage of allowing to completely pass under silence many subtle problems arising from stochastic integration in infinite-dimensional spaces. The reader who is interested in a more detailed exposition of these more technically subtle parts of the theory might be advised to read the works
Da Prato, Zabczyk; Stochastic equations in infinite dimensions, 1992
Da Prato, Zabczyk; Ergodicity for infinite-dimensional systems, 1996
Prevot, Röckner; A concise course on stochastic partial differential equations, 2007.
Date and time infoThursday, 16:00 - 18:00KeywordsStochastic Partial Differential Equations, Stochastic AnalysisPrerequisitesbasic measure theory, functional analysis and probability theoryAudienceMSc students, PhD students, PostdocsLanguageEnglish
The lecture will start with Quantum Field Theory (QFT) on Minkowski spacetime and discuss the relevant concepts and results. The second half will give an introduction to QFT on curved spacetimes and applications.Topics include:
quantum fields and particles
the mathematical framework of QFT
spin and statistics, PCT
models of quantum fields
concepts of locality and covariance
Hawking effect
semi-classical gravity
QFT and cosmological inflation
This lecture is one half of the "Quantum/Gravity Double Feature", the other half being the Lecture on General Relativity by Stefan Hollands. The lectures are planned as a double feature, but are each self-contained and can hence also be attended separately.Date and time infoWednesday 11:15 - 12:45 and Thursday 13:30 - 15:00KeywordsQuantum Field Theory on Minkowski Spacetime, Quantum Field Theory on Curved Spacetimes and ApplicationsAudienceMSc students, PhD students, PostdocsLanguageEnglish
These lectures will cover the basic concepts and methods of special and general relativity theory, followed by introductions to all the most important physical applications of Einstein's theory of gravitation.Topics include:
time and space in special relativity
curvature of spacetime and Einstein's equations
general covariance and the equivalence principle
geodesics
gravitational waves
cosmological models and expansion of the universe
singularities of spacetime and black holes
general relativity and the GPS system
This lecture is one half of the "Quantum/Gravity Double Feature", the other half being the Lecture on Quantum Field Theory on Curved Spacetimes by Rainer Verch and Thomas-Paul Hack. The lectures are planned as a double feature, but are each self-contained and can hence also be attended separately.Date and time infotbaKeywordsSpecial Relativity, General Relativity and ApplicationsPrerequisitesClassical MechanicsAudienceMSc students, PhD students, PostdocsLanguageEnglish
Information theoretic concepts, information geometry (mathematical approach to parametric statistics), information decompositions, processing of information in neurons and neuronal networks.
The topics may somewhat change in response to the interests of the participants.Date and time infoFriday 13:30 - 15:00KeywordsModern methods of information theory, some background from the neurosciences, conceptual discussionAudienceMSc students, PhD students, PostdocsLanguageEnglish
The length of the largest leaf of a one-dimensional foliation of a two-dimensional disk is not less than the diameter of the disk. The generalization of this fact to higher (co-)dimension requires beautiful and deep geometric insights. The relevant min-max techniques have been crucial in the recent proof of the Willmore conjecture and in the solutions of many other open problems.
In this course we will familiarize ourselves with these techniques using growth bounds on min-max volumes of families of cycles in an n-dimensional ball by Gromov and Guth as a guideline.
The lectures will link topology, geometry and metric properties of spaces.Date and time infoThursday 13:30 - 15:00KeywordsAlmgren-Pitts min-max theory, Gromov-Guth families, Willmore conjecturePrerequisitesBasic Geometry and TopologyAudienceMSc 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 have content, but also be accessible to any non-specialist but professional mathematician.Bellow is a list of topics covered last semester:
3-Manifolds
Topological Singularities
Category Theory
Kleinian groups
H-principle
Boltzmann Machines
Min-max in GMT
Sectional Curvature via convexit
Topological combinatorics
ABC Conjecture
Operator Algebras
Overview of AI research at MIS and in general.
K-theory and Atiyah-Singer Index theorem.
Date and time infoFriday 14:15 - 15:45KeywordsAny 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.
Systems of conservation laws are evolutionary nonlinear PDEs with several applications coming from both physics and engineering, in particular from fluid dynamics and traffic models. Despite recent progress, the mathematical understanding of these equations is still incomplete. In particular, general well-posedness results for the Cauchy problem are presently available only for systems of conservation laws in one space dimension, while very little is known for systems in several space dimensions.The course aims to be an introduction to the well-posedness theory for the Cauchy problem associated to a system of conservation laws in one space dimension. The approach I would like to adopt is the following. On one side, I will try to present a comprehensive overview of the main well-posedness results available in the literature. On the other side, I will provide the details of the proofs of such results in a simplified setting (a single 1D equation): here, indeed, the same techniques which have been successfully applied in the case of systems can be understood with much less effort.This is a tentative list of topics, to be adapted according to the wishes of the audience:
Classical solutions: the method of characteristics.
Weak solutions, the Rankine-Hugoniot condition, admissibility criteria.
Existence results: the wavefront tracking algorithm and the Glimm scheme.
Kružkov's entropy theorem.
Some notes on the Cauchy problem for systems.
Date and time infoWednesday 09:15 - 10:45KeywordsEntropy solutions, Kružkov's theorem, Riemann problem, Wavefront tracking, Glimm schemePrerequisitesStandard basic results in mathematical analysis. No previous knowledge in PDEs or Conservation Laws is assumed.AudienceMSc students, PhD students, PostdocsLanguageEnglish
This lecture gives an introduction to neural network learning from a theoretical standpoint. The main focus is on supervised learning problems, covering topics on learnability, generalization, and complexity. The lecture will also touch on current developments on the theory of deep learning.ReferencesM. Anthony and P. Bartlett, Neural Network Learning: Theoretical Foundations, Part one.Date and time infoThursday 11:15 - 12:45Keywordsartificial neural networks, supervised learning, pattern classification, VC-dimension, theory of deep learningPrerequisitesbasic linear algebra and analysisAudienceMSc students, PhD students, PostdocsLanguageEnglish
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
Discrete Optimization:
Lecturer: Peter F. Stadler
Dates: 07.04., 12.04., 19.04., 26.04., 03.05.
Graphs and Simplicial Complexes, Algebraic Topology:
Lecturer: Jürgen Jost
Dates 10.05., 17.05., 24.05., 31.05.
Combinatorial Geometry:
Lecturer: Tobias Fritz
Dates: 07.06., 14.06., 21.06., 28.06., 05.07.Date and time infoTuesday 13:30 - 15:00KeywordsDiscrete Optimization, Graphs, Simplicial Complexes, Algebraic Topology, Combinatorial GeometryAudiencePhD studentsLanguageEnglish
Hydrodynamics is as old and many-faceted subject that has motivated the development of several areas of mathematics, including partial differential equations, harmonic analysis, dynamical systems and statistical mechanics. In this lecture course the aim is to give an introduction to several aspects of this vast subject, mostly focussing on incompressible models (Euler and Navier-Stokes equations).Topics to be discussed are
Hydrodynamic stability and instability
Turbulence
Theory of weak solutions
ReferencesD. Acheson Elementary Fluid DynamicsA. Majda and A. Bertozzi Vorticity and incompressible flowC. Marchioro and M. Pulvirenti Mathematical Theory of Incompressible Nonviscous FluidsU. Frisch TurbulenceDate and time infoTuesday and Wednesday 11:15 - 12:45Keywordspartial differential equations, harmonic analysis, dynamical systems, statistical mechanicsPrerequisitesSolid background in functional analysis and knowledge of partial differential equations (FA1, PDG1). Knowledge of continuum mechanics and some theoretical physics is useful but not required.AudienceMSc students, PhD students, PostdocsLanguageEnglish
