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!

This course will review core mathematical concepts and results that play an important role within the ﬁeld of machine learning. Various models and architectures of neural networks will be presented, together with their corresponding universal approximation properties. Aspects of learning and generalisation will be addressed within the framework of statistical learning theory. The generality of this theory will be exempliﬁed in the context of neural networks and support vector machines. Further theoretical approaches to learning, in particular gradient-based approaches, will be reviewed. Here, the information-geometric perspective of the natural gradient method will be highlighted.Date and time infoThursday, 11:15 - 12:45, first lecture: Nov. 15, 2018AudienceMSc 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
Stefan Hollands: The Einstein Equation Date: Monday 15.10., 29.10., 05.11., Time: 09:15 - 10:45, Location: MPI MiS G3 10André Uschmajew: Introduction to Compressed Sensing Date: Tuesday 13.11., 20.11., 27.11., Time: 10:15 - 11:45, Location: G3 10 Leibniz-SaalNihat Ay: Information Geometry Date: Tuesday 04.12., 11.12., 18.12., Time: 10:15 - 11:45, Location: G3 10 Leibniz-SaalGuido Montúfar: Topics from Deep Learning Date: Wednesday 09.01., Tuesday 15.01., 22.01., Time: 10:15 - 11:45, Location: G3 10 Leibniz-SaalOrlando Marigliano: Graphical Statistical Models Date: 29.01. (MPI MiS G3 10), Time: 10:15 - 11:45AudienceMSc students, PhD students, PostdocsLanguageEnglish

Numerical data rarely is given exact, but usually comes with errors due to noise, measurement errors or errors caused by prior calculations. Consequently, algorithms which take such data as input can't produce correct answers.
In this course we will learn about the concept of condition numbers and how it helps to understand how much the solution of a computational problem is changed, if the input data is perturbed. We will also discuss how condition numbers are related to the complexity of numerical algorithms. The concepts of average and smoothed analysis will be introduced.
We will consider the condition numbers of the following problems: linear equation solving, polynomial equation solving and computing tensor decompositions. If time permits, we will also cover condition numbers of problems in linear programming.ReferencesCondition: the geometry of numerical algorithms by Bürgisser and Cucker (Springer, 2013)The condition number of join decompositions by Breiding and Vannieuwenhoven (SIAM J. Matrix Anal. and Appl., 39(1), 287–309)Date and time infoTuesday, 09:00 - 10:00KeywordsCondition numbers, complexity of numerical algorithms, loss of precision, average analysis, smoothed analysisPrerequisitesParticipants should have a good understanding of linear algebra and a basic knowledge about probability theory and differential geometry. Expertise in linear programming, algebraic geometry or multilinear algebra is helpful but not required.AudienceMSc students, PhD students, PostdocsLanguageEnglish

Groups are a fundamental construct to understand “symme- tries”. They are best understood by studying their action on linear vector spaces, that is via linear representations i.e. their concrete realization as matrices. This for example appears inharmonic analysis (think: the Fourier transform of periodic functions)tensor decomposition (a matrix (a 2-tensor) can be split into a symmetric part and an antisymmetric part; what about higher order tensors?)the study of certain Markov chainsThis is a reading group of
Fulton, William, and Joe Harris. Representation theory: a ﬁrst course. Vol. 129. Springer Science & Business Media, 2013.
We will go through Chapter 1-15; at the speed required by the group. Starting essentially from scratch in Chapter 1, at the end we will have a solid understanding of representations of the symmetric group (permutations) and the general linear group (invertible matrices). Each week one designated “leader” will guide the session, but all attendants are asked to read the relevant chapter before. We will work with examples (by hand and by code) as much as possible.Date and time infofirst lecture: Tuesday 16.10.2018 at 11:00hKeywordsrepresentation theory; symmetric group; GLAudienceMSc students, PhD students, Postdocs

The problem of the generation of random dynamical systems by stochastic partial diﬀerential equations (SPDE) is one of the open problems in the application of dynamical systems theory to SPDE. Despite its fundamental nature, most results are restricted to ”simple” random perturbations of aﬃne-linear structure. However, as we will see in this course, applications like scaling limits of particle systems with interaction and branching and non-equilibrium statistical mechanics, lead to porous media equations perturbed by nonlinear multiplicative or nonlinear conservative noise. We will ﬁrst convince ourselves that established methods such as the variational approach to SPDE cannot be applied to these equations, let alone prove the generation of random dynamical systems. Then, based on entropy and kinetic theory we will prove their well-posedness. This will lead to a strong notion of uniqueness, so-called path-by-path uniqueness, based on rough path theory which in turn proves the generation of a corresponding random dynamical system and opens the way to a qualitative analysis of the (stochastic) ﬂow of the solutions.Date and time infoMonday, 16:15 - 17:45KeywordsPartial Diﬀerential Equations, Applications of PDEs in sciencePrerequisitesBasic PDE courses, functional analysisAudienceMSc students, PhD students, PostdocsLanguageEnglish

Many data can be organized as networks, or more generally, as simplicial complexes or hypercomplexes, possibly weighted and/or directed.
We develop mathematical tools to analyze such structures systematically.
This is a continuation of the course from the last term, but newcomers are welcome, and the material will be organized accordingly.ReferencesJ.Jost, Mathematical concepts, Springer, 2015Further and more detailed references will be given during the lecture.Date and time infoFriday 13:30 - 15:00, starting October 26AudienceMSc students, PhD students, PostdocsLanguageEnglish

This course will deal with the basic problem of understanding the structure (e.g. the geometry and topology) of the set of solutions of real polynomial equations with random coeﬃcients. The simplest case of interest is the count of the number of real zeroes of a random univariate polynomial $$p(x) = a_{0} +· · ·+a_{d}x^{d}\: , where\: a_{0}, . . . , a_{d}$$ are random Gaussian variables – this problem is “classical” and was pioneered by Kac in the 40s. More generally algebraic geometers might be interested, for example, in the number of components of a random real plane curve of degree d, or in the expected number of real solutions of more advanced counting problems (e.g. enumerative problems). This is a very fresh and modern approach to real algebraic geometry: when the outcome is highly sensitive to the choice of the parameters (in general, there is no notion of “generic” in the real world), the attention is shifted to the “typical” situation. I will present the basic techniques for attacking this type of questions, trying to emphasize the connections of classical algebraic geometry with convex geometry, random matrix theory and random ﬁelds.Date and time infoTuesday, 09:00 - 10:00, from February 6 to March 27KeywordsReal Algebraic Geometry, Random Matrix Theory, Integral GeometryPrerequisitesBasic knowledge from Diﬀerential Geometry, Probability and Algebraic GeometryAudienceMSc students, PhD students, Postdocs

Geometric measure theorem is a very important tool in the studies of many problems of variational nature. It has helped mathematicians in the past to solve important problems in the calculus of variations and geometry. For instance, the concepts of currents and the solution by Rado and Douglas of the Plateau problem. We plan to cover the following topics very carefully:Basics in GMTRectifiable setsVarifoldsCurrents and the solution of Plateau's problemAllard Regularity of Varifolds (if time permits)In this seminar we are reading the book Introduction to Geometric Measure Theory by Leon Simon. We basically follow the book starting at page 1, therefore it is a course that one just needs basics of measure theory to follow. Everybody is invited to speak, although the organizers will speak whenever is necessary.Date and time infoMonday, 15:00 - 17:00AudienceMSc students, PhD students, PostdocsLanguageEnglish

Toric varieties form a special class of algebraic varieties, in some aspects generalizing aﬃne and projective spaces. Due to connections to combinatorics they also form one of the best understood classes. A lot of algebraic invariants turn out to behave in a particularly nice way for toric varieties. The aim of the lecture is to describe interactions of algebra, combinatorics and geometry; to learn about toric varieties, but at the same time gain experience in algebraic geometry.
We will be closely following the book ’Toric Varieties’ by Cox, Little and Schenck. The course will very actively involve participants - everyone will be expected to read the material in advance and some participants will be asked to present various parts. We hope to keep a fast pace, so that the course not only presents connections between polytopes/cones and toric varieties, but also describes more involved constructions including Cox rings, Picard and class groups, canonical divisor, sheaf co-homology, various types of singularities (including Gorenstein, Cohen-Macaulay, rational, normal) etc.Date and time infoFriday, 11:00 - 12:30KeywordsAlgebraic Varieties, Lattice Polytopes, Complex Torus, Lattice Cones, Divisors, Orbits, SingularitiesPrerequisitesBasic knowledge of algebraic geometry and commutative algebra (say at the level of nonlinear algebra Ringvorlesung) or a lot of motivation and enough time to catch upAudienceMSc students, PhD students, PostdocsLanguageEnglish

I will give an introduction into the notions and tools of regularity structures, which were developed to renormalize the nonlinearity in partial diﬀerential equations driven by white noise. I will discuss these tools when applied to (a proxy for) the dynamical Phi-4-3 model. This model amounts to an Allen Cahn equation driven by space-time white noise for analysts, or to the stochastic quantization in quantum ﬁeld theory for math physicists.ReferencesI will follow Hairer’s lecture notes “Regularity structures and the dynamical Phi-4-3 model”, and expand on my mini-course given at the Newton institute.Date and time info09:15 - 11:00, Wednesday, only at 24.10., 07.11., 5.12., 19.12., 16.01., 30.01.AudienceMSc students, PhD students, PostdocsLanguageEnglish

Recent years have shown tremendous progress in our understanding of "singular stochastic PDEs", which are ill posed due to the interplay of very irregular noise and nonlinearities which may lead to resonances that have to be removed by a renormalization procedure. In fact all white-noise driven parabolic SPDEs in dimensions larger than \(d=1\) are singular. To solve these equations it was necessary to come up with new, analytic rather than stochastic, notions of solutions, for example with Hairer's regularity structures or with paracontrolled distributions developed by myself together with Gubinelli and Imkeller. In the lecture I will give an introduction to paracontrolled distributions and applications and I will try to communicate the main ideas without getting lost in technicalities. After introducing the basic tools needed to solve singular SPDEs I will focus on applications likeinvariance principlesconstructing domains for singular operators (Anderson Hamiltonian, infinitesimal generators of diffusions with distributional drift)descriptions of singular SPDEs in terms of singular diffusions ("Feynman-Kac formula")Barashkov-Gubinelli's variational approach to constructing Gibbs measures like \(\Phi^4_3\)aspects of the large scale behavior of some singular SPDEs Date and time infoWednesday, 11:15 - 12:45KeywordsStochastic partial differential equations, paracontrolled distributionsPrerequisitesBasic knowledge of (continuous time) stochastic processes and functional analysis, in particular Schwartz distributionsAudienceMSc students, PhD students, PostdocsLanguageEnglish

Morse Theory of the energy functional on the free loop space Existence results for closed geodesics Geometric estimates for the length of closed geodesics on manifolds of positive curvatureDate and time infoWednesday, 13:15 - 14:45, 1st lecture: October, 24KeywordsClosed geodesics, free loop space, index and nullity, sectional curvaturePrerequisitesBasic knowledge about diﬀerential geometry and algebraic topologyAudiencediploma students, PhD students, PostdocsLanguageEnglish

In this seminar, we are going to study the equation $$- y'' + qy = \lambda y \qquad 0 \le x \le 1$$ subject to the boundary data \(y(0)=y(1)=0\). It is assumed that the function \(q : [0,1] \to \mathbb{R}\) is given. The real number \(\lambda\) is called an eigenvalue, if a non-trivial solution to the above problem exists. In this seminar, we investigate the relation between the potential \(q\) and the set of eigenvalues. For example, typical questions are: For which sets of real numbers does there exist a potential which has this given set as eigenvalues? Which potentials are isospectral, i.e. which potentials give the same eigenvalues? Which additional pieces of information are determined by the potential? The theory is surprisingly complete with rich relations to other fields of mathematics.Date and time infoThursdays 09:15 - 10:45KeywordsPartial Differential Equations, Inverse Spectral TheoryPrerequisitesAnalysis I-III and ODEs is required; basic knowledge of functional analysis and PDEs is helpfulAudienceMSc students, PhD students, PostdocsLanguageEnglish, German if desired

The problem of the generation of random dynamical systems by stochastic partial diﬀerential equations (SPDE) is one of the open problems in the application of dynamical systems theory to SPDE. Despite its fundamental nature, most results are restricted to ”simple” random perturbations of aﬃne-linear structure. However, as we will see in this course, applications like scaling limits of particle systems with interaction and branching and non-equilibrium statistical mechanics, lead to porous media equations perturbed by nonlinear multiplicative or nonlinear conservative noise. We will ﬁrst convince ourselves that established methods such as the variational approach to SPDE cannot be applied to these equations, let alone prove the generation of random dynamical systems. Then, based on entropy and kinetic theory we will prove their well-posedness. This will lead to a strong notion of uniqueness, so-called path-by-path uniqueness, based on rough path theory which in turn proves the generation of a corresponding random dynamical system and opens the way to a qualitative analysis of the (stochastic) ﬂow of the solutions.Date and time infoOctober, 29 – November 2 and November 16PrerequisitesMathematical maturity of an advanced graduate student. Interest in combinatorics.AudiencePhD studentsLanguageEnglish

In this course, I willgive a brief introduction to Information Geometry and how to apply it in the context of Mathematical Population Geneticsgive an overview introduction to mathematical models used in population geneticsdiscuss mathematical approaches used in studying such modelsDate and time infoFriday, 15:15 - 16:45KeywordsInformation Geometry, random genetic drift, mutations, selection, migration, recombination, evolutionary processesPrerequisitesBasic knowledge of calculus, linear algebra, PDE, probability theory and diﬀerential geometryAudienceMSc students, PhD students, PostdocsLanguageEnglish

Statistical Physics, Information Theory and Combinatorial Optimization have many concepts in common: bits, complexities, large deviations or partition functions, are common, albeit for diﬀerent questions. The goal of this course is to understand their intersections: How can an algorithm show a phase transition, what is computational complexity in terms of information. In order to keep it accessible, we will not go into detail in any of those ﬁelds, and we will favour intuitive notions over rigorous results. The course will start with an introduction to the three ﬁelds and some probabilistic notions that will take the ﬁrst four to ﬁve weeks. Then we will go into systems with negligible interactions such as the random energy models, shannon codes, or number partitioning. Then, if time allows it, models on graphs (low parity codes, SAT problems, spin glasses on graphs).ReferencesThe course is based on parts I, II and III the book ”Information, Physics and Computation” by Marc Mezard and Andrea Montanari, see the table of contents and the chapters available at Montanaris’ webpage.Date and time infoWednesday 13:00 - 14:30AudienceMSc students, PhD students, PostdocsLanguageEnglish