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!
In this course we introduce the core concepts of stochastic calculus of finance: Brownian motion, Itô integral, Itô formula, etc. We apply them to derive the pricing formula and the replicating portfolio of vanilla options within the Black-Scholes model. Furthermore, risk-neutral probability, its relation to the fundamental theorems of asset pricing, and connections with partial differential equations are presented. All these theoretical concepts are guided by and applied to the analysis of financial market data.Date and time infoFriday 11.00 - 12.30Keywordsstochastic calculus of finance, financial data analysisPrerequisitesBasic Probability and PDE Theory might be helpfulAudienceMSc students, PhD students, PostdocsLanguageEnglish
Self-organization is a process by which systems, usually composed of many individual parts, spontaneously develop structure or function without specific guidance. Self-organization occurs in a variety of physical, chemical, biological, social and cognitive systems. Self-organized processes are in addition to environmental and genetic factors crucial for the development of the brain. We will discuss the phenomenon of self-organization in the realm of computational neuroscience. It is a highly inter-disciplinary topic that uses analysis tools from statistical physics, mathematics and computer science. Some of the discussed topics will be:
Self-organization by means of synaptic plasticity
Organization of visual cortex
Self-organized criticality
Information-theoretic aspects of self-organization
Date and time infoWednesday 11.30-13.00Keywordscomputational neuroscience, self-organization, simulation and analysisPrerequisitesfundamental mathematical knowledge (ODEs, probability theory, ...)AudienceMSc students, PhD students, PostdocsLanguageEnglish/German (on demand)
Two-dimensional variational problems are of particular interest in geometry (minimal surfaces, harmonic and conformal mappings, J-holomorphic curves,. . . ) and theoretical physics (nonlinear sigma model, string theory,. . . ) and they also present some of the guiding problems of geometric analysis. There are many important phenomena, like conformal invariance, selfduality, supersymmetry,. . . that, while not necessarily restricted to two dimensions, acquire a special significance in two dimensional problems. This is on one hand fundamental for their applications in geometry and physics, and on the other typically provides the keys for their successful analytical treatment.References
J.Jost, Riemannian geometry and geometric analysis, 6th edition, 2011
J.Jost, Two-dimensional geometric variational problems, 1991
J.Jost, Geometry and physics, 2009
Date and time infoFriday 13.30 - 15.00Keywordsvariational problems, geometry, applications to theoretical physicsAudienceMSc students, PhD students, PostdocsLanguageEnglish
Tensor numerical approximation provides the efficient separable representation of multivariate functions and operators on large n⊗d-grids, that allows the solution of d-dimensional PDEs with linear complexity scaling in the dimension, O(dn). Modern methods of separable approximation combine the canonical, Tucker, as well as the matrix product state (MPS) formats (also known as tensor train (TT) decomposition).The recent quantized-TT (QTT) approximation is proven to provide the logarithmic data-compression on a wide class of functions and operators. It makes possible to solve high-dimensional steady-state and dynamical problems in quantized tensor spaces, with the log-volume complexity scaling in the full-grid size, O(dlog n), instead of O(nd).In this lecture we will discuss how the grid-based tensor approximation applies to hard problems arising in electronic structure calculations, such as many-electron integrals and solution of the Hartree-Fock equation.We present the algorithms for the nonlinear Tucker decomposition of function related tensors represented in full grid size and in the canonical formats, the basic rank-structured operations with tensors, and the rank reduction algorithms. The numerical results are given for 3D tensors corresponding to functions (also with strong cusps) 1/r, e−r, ∑c ke−αrk2 , r ∈ R3, etc.We present on-line Matlab simulations for computing the ab initio ground state energy of compact molecules including glycine and alanine amino acids.KeywordsTensor numerical methods, canonical tensor decomposition, multilinear algebra, quantics tensor approximation, Hartree-Fock-equation, many-electron integrals, molecular dynamics, chemical master equations, Coulomb potential, lattice sums, periodic systemsPrerequisitesPDE's, ODE's, introduction to numerics and (multi) linear algebraAudienceMSc students, PhD students, Postdocs, ResearchersLanguageEnglish
In the last decades there has been quite some interest in the derivation of continuous models from discrete ones. Examples are in nonlinear elasticity, dislocation energies in crystals, surface energies in spin models, phase transitions, fracture mechanics, stochastic homogenization and many more. The mathematical tools used are formal asymptotics, homogenization techniques, Γ-convergence, geometric measure theory and large deviations. Depending on the interest of the participants, we would like to give an overview of some of the relevant results in the vast literature. There will be weekly presentations on selected papers.
Selected topics:
Validity and failure of Cauchy-Born rule for derivation of elastic energies.
Non-zero temperature case and free energy derivation.
Higher order limits and derivation of surface energies in multiphase problems.
Limit of dilute dislocations in crystals.
Discrete models for fracture and image processing.
Stochastic homogenization
Motion and depinning of random interfaces.
Numerical schemes for atomistic to continuum coupling.
Date and time infoMonday 11:15 - 12:45, MPI MIS, A 01KeywordsHomogenization, Phase Transitions, Discrete InterfacesPrerequisitesAnalysis I-III, basic probability theoryAudienceMSc students, PhD students, PostdocsLanguageEnglish
Although polynomial equations are much more complicated than linear equations, they still have a lot of structure, and there are many algorithms to bound the number of solutions and to describe their solution sets. Due to their simple structure, polynomials are often used in mathematical modelling, and this is just one reason why they abound in applications. The lecture will present different ideas from algebra and analysis to study and solve polynomial equations.
Some topics I plan to cover are:
Polynomials in a single variable: How to count the roots (over C and over R), and how to find them.
Polynomial systems with finitely many solutions; that is, zero-dimensional polynomial ideals.
Systems of sparse polynomials (fewnomials).
Primary decomposition of polynomial ideals.
Real polynomial equations and sums of squares.
The main reference for the course will be Bernd Sturmfels' book that carries the same name as the lecture.Date and time infoThursday 11.00 - 12.30Keywordspolynomials, Gröbner bases, fewnomials, commutative algebraPrerequisitesKnowledge of elementary commutative algebra will be helpful, but not essentialAudienceMSc 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.
Topics
The aim of this course is to give an overview of some basic concepts and results in probability.
Artem SapozhnikovProbability space, random variable, expectation, independence, Borel-Cantelli lemma
Types of convergence of random variables, weak and strong laws of large numbers, Cramer's large deviations theorem
Characteristic functions, central limit theoremInfinitely divisible and stable distributions
Felix Otto
Conditional expectation and probability
Martingales
Markov chains, random walk
Max von Renesse
Continuous time Markov processes: Markov Kernels, Kolmogorov Consistency, Generators
Brownian Motion: Levy Construction, Strong Markov Property, Reflection Principle
Path regularity of Brownian Motion: Kolmogorov-Chentsov and Non-Differentiability (Paley-Wiener-Zygmund), Law of the Iterated Logarithm
Connection between Brownian Motion and PDE: Heat and Poisson Equation in regular Domains. Recurrence-Transience of BM.
Date and time infoMonday 13:30 - 15:00PrerequisitesMeasure theory
The mathematical theory of information has been initiated by the mathematician and engineer Claude E. Shannon to answer the purposes of communication technologies - the seminal paper (Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379--423, 623--656 (1948)) has been published in 1948. Nowadays this branch of probability theory plays an important role in various areas of the natural as well as social sciences. In response, it appears as an interdisciplinary research field. Technically, it has a significant overlap with mathematical statistics, ergodic theory or combinatorics.We want to discuss selected articles from the volume 44 (6) of IEEE Transactions on Information Theory (1998) which has been published on the occasion of 50 years of Shannon theory. It gives an excellent overview of the major issues of information theory which link to topics such as lossless data compression, universal coding, reliable communication through noisy channels, universal prediction, statistical pattern recognition or pattern matching.Date and time infoWednesday 13.30 - 15.00LanguageEnglsih
