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!
We will cover with details all the basic and most important results in the regularity theory of the lecturers' favourite PDE's. If time alouds us, after January we may decide together with the audience which direction to go. Suggestions are: GMT, or harmonic map heat flow , or further topics in parabolic equations (the most probable one!).Date and time infoWednesdays, 15-17KeywordsSchauder theory, Krylov-Safonov Theorem, Allard's Regularity Theorem, Elliptic and Parabolic PDE'sPrerequisitesMaster level analysisAudiencePost-docs, PhD.LanguageEnglish
The aim of this seminar is to learn basics of a fundamental tool in harmonic analysis, namely the Littlewood-Paley decomposition, keeping in mind its applications to nonlinear PDEs.The Littlewood-Paley decomposition boils down to representing a function \(f\) by a sum of functions having only certain frequencies, i.e. \[ f = \sum_{k \in \mathbb{Z}} P_kf \] where \(P_kf\) have frequencies localised to \(\sim 2^k\). In other words, it is 'localisation on the Fourier side' or 'a pointwise discrete approximation of the Plancharel theorem'. Despite simplicity of this classical idea (having its roots in 1930s) it is widely applicable to PDEs, since it allows a careful control of derivatives (via Bernstein inequalities) and nonlinearities (via Bony's paraproduct). Therefore, after a short introduction of harmonic analysis preliminaries, we intend to focus on Littlewood-Paley methods in PDEs, including
multiplier theorems
useful definition of Besov spaces and the related solvability of nonlinear evolutionary problems for scaling invariant initial data
commutator estimates with applications to turbulence
Date and time infoThursdays, 15:15- 16:45LanguageEnglish
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
Part A by Angkana Rüland (25.10., 01.11., 08.11.,15.11.):
The Calder'on Problem -- A Prototypical Inverse Problem
Part B by Daniela Cadamuro (22.11., 29.11., 13.12. 20.12.):
Scattering Theory
This course offers an introduction to the mathematical and foundational aspects of scattering theory in quantum mechanics. We consider the one-dimensional Schrödinger operator on the line with a multiplicative potential, and present some facts about both the forward and inverse scattering problems. In the first case one studies the properties of the scattering matrix (and of the Møller operators) depending on those of the potential, while in the second case one determines the potential from scattering data, such as the reflection coefficient.
As an application, we consider the stability of "quantum inequalities" under scattering, which is relevant in physics, for example for the phenomenon of "quantum backflow".
IMPRS student lecture (06.12., 10.30 o'clock)
Sharwin Rezagholi: Symbolic dynamics and cellular automata
Symbolic dynamics is a subfield of dynamical systems and theoretical computer science where elementary parts of topology, algebra, and discrete mathematics meet in a beautiful way. This minicourse will cover the central definitions and results, such as shifts, subshifts, shifts of finite type, cellular automata, and entropy/word complexity. There are no prerequisites.
IMPRS student lecture (10.01. , 09.30)
Felix Gaisbauer: A brief introduction to opinion dynamics and a model of opinion expression
I will give a very short introduction to opinion dynamics, including the Ising model and the voter model. I will then turn to a model our group is currently developing, which does not investigate opinion change, but public opinion expression. I will present a game-theoretic as well as a dynamical systems approach to opinion expression. There are no prerequesites.
Part C by Matteo Smerlak (17.01., 24.01., 31.01., 07.02., 9.15 ):
Evolutionary dynamicsDate and time infoFriday 09.15-11.00 (Rüland) 09.30-11.00 (Cadamuro) 09.15-11.00 (Smerlak) Exercice classes: Friday 11.00-12.00KeywordsScattering theory, Calder'on Problem, Evolutionary dynamicsAudienceIMPRS students, Phd students, postdocs
The goal of the Seven Lectures in Algebraic Statistics is to give a practical introduction to several topics in Algebraic Statistics. No previous knowledge of statistics will be assumed. Familiarity with algebra at the level of Cox, Little, O’Shea’s book “Ideals, varieties, and algorithms” is a plus. The lecture will be 60 minutes long and be followed by a 30 minute practical session in R or Macaulay2.Date and time infoMonday 13:00 to 14:30, starting October 21KeywordsAlgebra, Statistics, Causality, conditional independence, Fisher's exact test, Markov basisPrerequisitesSome knowledge in Algebraic Geometry at the level of Cox, Little, Oshea's book "Ideals varieties and algorithms"LanguageEnglishRemarks and notesThis course consists of seven lectures. The last three lectures will serve as preparatory lectures for those interested in attending the Spring School in Mathematical Statistics taking place March 30-April 3 at MPI MIS Leipzig.
In this lecture we consider the stochastic thin-film equation. The stochastic thin-film equation is a forth-order, degenerate stochastic PDE with nonlinear, conservative noise. This renders the existence of solutions a challenging, largely open problem (since 2006). Due to the forth order nature of the equation, comparison arguments do not apply and the analysis has to rely on integral estimates.
The stochastic thin film equation can be, informally, derived via the lubrication/thin film approximation of the fluctuating Navier-Stokes equations and has been suggested in the physics literature to be an improved mesoscopic model, leading to better predictions for film rupture and expansion.
Besides the specific case of the stochastic thin film equation, this lecture will serve as an introduction to the weak-convergence method to the existence of (weak) solutions to stochastic PDE devised by Flandoli and Gatarek [1995] in the case of stochastic Navier-Stokes equations.Date and time infoMo, 16:15-17:45
This will be an introduction to the basic concepts of game theory, roughly following the book "Essentials of game theory: a concise, multidisciplinary introduction" by Leyton-Brown and Shoham. We will cover normal-form and extensive-form representations of games, and a variety of solution concepts: pure and mixed strategy Nash equilibrium, rationalisability, correlated equilibrium, subgame-perfect equilibrium. After this we will cover two more advanced topics: repeated games and Bayesian games. I will try to provide a balance between mathematical rigour and a more informal style, with a variety of examples from different fields and discussion of when game theory is appropriate as a modelling paradigm.Date and time infoTuesday, 11.15-12.45
Related notionsVariational techniques, multiscale models Notions that will be studiedWe will study the classical elliptic equation in divergence form: $$ -{\rm div}\left( a\left(\frac{x}{\varepsilon}\right) \nabla u^\varepsilon(x)\right) = f(x), $$ where \(\varepsilon \ll 1\) is the small scale, and where \(a\) is a \(\mathbb{Z} ^d\)-periodic coefficient field. This equation is relevant for modeling various physical phenomena (e.g. thermal or mechanical equilibrium, electrostatics) in multiscale materials.Figure 1: Example of coefficient field \(a(x/\varepsilon)\) on (a), right-hand side \(f\) on (b), solution \(u^\varepsilon\) on (c), and its derivative \(\partial_1 u^\varepsilon\) on (d).We show that an averaging process occurs when the small \(\varepsilon\) vanishes and that the solution \(u^\varepsilon\) to (1) can be approximated by the (simpler) solution of the homogenized problem $$ -{\rm div}\left( \overline{a} \nabla \overline{u}(x)\right) = f(x), $$ where \(\overline{a}\) is a constant matrix. The oscillating gradient \(\nabla u^\varepsilon\) is then retrieved by means of the two-scale expansion.In this regard, the following mathematical notions will be under our scope:
the Lax-Milgram theorem and the Fredholm alternative,
the correctors and the two-scale expansion,
the div-curl lemma, Tartar's method and the H-convergence,
the Hashin-Shtrikman bounds.ReferencesG. Allaire.Shape optimization by the homogenization method, volume 146 of Applied Mathematical Sciences. Springer-Verlag, New York, 2002.
H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
V. Jikov, S. Kozlov, and O. Oleinik. Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin, 1994.
L. Tartar. The general theory of homogenization, volume 7 of Lecture Notes of the Unione Matematica Italiana. Springer-Verlag, Berlin; UMI, Bologna, 2009.Date and time infoTuesday, 9h15-10h45KeywordsPDEs, elliptic theory, multiscalePrerequisitesCourses PDE 1 & 2, Basic functional analysis H1 spaces, compactness, weak convergence), basics of Partial Differential Equations (elliptic theory, weak solution)
We will present an invitation to basic topics in commutative algebra and algebraic geometry. The lecture will be based on our new book with Bernd Sturmfels "Invitation to Nonlinear Algebra". We plan to cover 2/3 of a big chapter per week. Apart from first three, the lectures will be mostly independent.Date and time infoMonday, 9.30KeywordsVarieties, Rings, IdealsPrerequisitesVery good knowledge of linear algebra and basic algebra (i.e. no prerequisites)
We will give an introduction into the notions of the Spectral Gap and Logarithmic Sobolev Inequality, and display some of their applications. More precisely, we will discuss some criteria, namely the tensorization property, and the criteria of Bakry-Emery and Holley-Stroock. Applications discussed will involve hyper-contractivity of the overdamped Langevin equation, the construction of singular products in stochastic partial differential equations, and a selected application in stochastic homogenization.
All this, as usual, as time permits.Date and time infoThursday, 09.15-11.00KeywordsSpectral Gap, Logarithmic Sobolev Inequality, overdamped Langevin equation, singular products in stochastic partial differential equations, stochastic homogenizationRemarks and notesLectures only at 14.11.2019, 28.11.2019, 5.12.2019, 19.12.2019, 16.01.2020, 23.01.2020
Basic course on unconstrained and constrained optimization of differentiable functions in \(\mathbb{R}^n\).Date and time infoTue 15:15 - 16:45, Wed 15:15 - 16:45LanguageGerman
