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 is a continuation of the course that I taught during the winter term 2018/2019. My general aim is to review core mathematical concepts and results that play an important role within the fields of neural networks and machine learning. I already presented various models and architectures of neural networks, together with their corresponding universal approximation properties. I will continue the course by addressing aspects of learning and generalisation 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 - 12:45KeywordsNeural networks, universal approximation, statistical learning theory, support vector machines, deep learning, information geometryAudienceMSc students, PhD students, PostdocsLanguageEnglish

The aim of this lecture is to discuss well-known and important embedding theorems in complex geometry. The starting point is Kodaira's embedding theorem which asserts that a compact complex manifold admitting a positive line bundle admits a holomorphic embedding into some complex projective space. I will present a proof relying on vanishing theorems for \(\overline\partial\)-cohomology groups. I will then discuss generalizations leading to characterizations of Moishezon manifolds.
Using similar methods, I will discuss holomorphic embeddings of Stein manifolds into Euclidean spaces. I will also review historical and recent results on the minimal embedding and immersion dimensions.Date and time infoTuesday 11:15 - 12:45KeywordsComplex geometry, complex analysisPrerequisitesAnalysis, basic knowledge of differential geometry and functional analysisAudienceMSc students, PhD students, PostdocsLanguageEnglish

Continuum mechanics models behaviour of continuous materials: fluids (water, air) and solids, when forces (or displacements) act upon them. This course presents basics of mathematical theory of mechanics of continua, with a slight bias towards fluid mechanics in the second part of the semester. Depending on the lecture's pace and interests of the audience, we may take a few excursions into continuum thermodynamics or turbulence theory (marked by (*) in the outline below).
The course will be provided as 3 lectures per 1 tutorial (3+1) approximately.
Outline
Part I: General Theory
Preliminaries: Vector and tensor algebra and analysis.
Kinematics: Continuous body, reference con guration, deformation, displacement, motion, trajectories. Strain and stress. Rate quantities. Lagrange criterion and Reynolds theorem.
First mechanical and thermodynamical principles: balance laws, frame indifference, stresses and (*) entropy. Nonsmooth case.
Case-specification: constitutive relations.
Part II: Examples of specific theories
(*) Rigid heat conductors: Coleman-Noll Procedure. Fourier's Law.
Elastic solids: Linear elasticity, Hooke's Law.
Compressible and incompressible ows: Navier-Stokes and Euler equations.
Part III: Turbulence in fluidsDate and time infoWednesday 9:15 - 10:45 and Friday 9:15 - 10:45PrerequisitesOnly general acquaintance with basic notions of linear algebra and calculus will be needed to follow the lecture, and they will be recalled and systemised at the beginning. The course is in English. The requirements to pass will not go much beyond solvingAudienceMSc students, PhD students, PostdocsLanguageEnglish

“Modern”, is this not just a buzzword, a word known from political campaigns? After all, who wants to be old fashioned?
No, the attribute “modern” in “Modern Algebraic Geometry” is not just for advertisement; Modern Algebraic Geometry is a field for itself, distinct from Classical Algebraic Geometry.
And it is not so modern after all. In fact, it was developed mostly by Alexander Grothendieck in the 1960s. So, it is actually already over 50 years old.
In the center of the theory is the notion of scheme. This is a vast generalization of the concept of variety. To any ring, one can associate it “geometric realization”, the corresponding affine scheme, and then one can “glue such objects together”. One can – and one does – consider schemes over arbitrary rings and even over schemes themselves.
The theory of schemes is not just very general, to use it is also like driving a car in between cities rather than walking. One just reaches the desired goals much more quickly and conveniently. Also, it is very safe. For example, if one considers questions where multiplicities are of importance, one usually obtains the correct result “by free”.
Of course, before one can benefit from the theory, one has to learn it, and many mathematicians shy away from it because of some initial mental barrier. Well, they really miss something ...
In the lecture, I will start right away with schemes. I will assume that the audience is familiar with classical algebraic geometry and also with manifolds and with basic category theory.Date and time infoMonday 11:15 - 12:45 and Friday 13:15 - 14:45AudienceMSc students, PhD students, PostdocsLanguageEnglish or German

The lecture gives an introduction to basic concepts of Computer algebra. In more detail, we will discuss the following topics:
Integer and polynomial arithmetics, fast multiplication, gcd computations, complexity of algorithms.
Systems of equations, Hermite- and Smith-normal form, p-adic approximation, lattices, LLL algorithm.
Resultants and extended gcd computations.
Multivariate polynomials, Gröbner bases for ideals and modules, modular methods, Faugère's F4 and F5 algorithm, syzygies, free resolutions.
(Squarefree) Factorization, Hensel lifting, factorization with LLL, primality tests, factorization over algebraic number fields.ReferencesCohen: A Course in Computational Algebraic Number Theory
Cox, Little, O'Shea: Ideals, Varieties, and Algorithms
von zur Gathen: Modern Computer Algebra
Greuel, Pfister: A SINGULAR introduction to Commutative Algebra
Date and time infoWednesday 13:15 - 14:45, Thursday 9:15 - 10:45KeywordsArithmetics, Systems of (non-)linear equations, lattices, resultants, Gröbner bases, factorizationPrerequisitesBasic knowledge about algebra and the usage of computer algebra systemsAudienceMSc students, PhD students, PostdocsLanguageEnglish

We all know that there is exactly 1 conic through 5 general points in the plane, but as soon as we ask how many conics are tangent to 5 given smooth conics the answer is somewhat less obvious. This kind of enumerative questions has been considered since the ancient times.
Enumerative geometry is a great playground and provides plenty of concrete problems to work with, but is sometimes left out of a basic course on algebraic geometry. This reading group aims to fill the gap. We will study how to precisely formulate enumerative questions and how to use intersection theory to find the answers.
While doing that, we will incidentally learn a lot of geometry and bump into key concepts such as Chow rings, Schubert calculus, Grassmanians, Hilbert schemes and Chern classes.Date and time infoWednesday 09:30 - 11:00 and Thursday 13:30 - 15:00KeywordsAlgebraic geometry, 27 lines, 3264 conics, intersection theoryPrerequisitesAn undergraduate course on algebraic geometry. Enthusiasm about doing exercises and about teamwork are very welcome.AudienceMSc students, PhD students, PostdocsLanguageEnglish

In this lecture series we will introduce the concept of stochastic variational inequalities (SVI) as a concept of solutions to SPDE. Our interest in this concept of solutions comes from two directions: First, SVI solutions can be used in certain situations in which the "variational" approach to SPDE fails, e.g. for multi-valued SPDE. We will encounter this in the application to the stochastic total variation flow, with links to self-organized criticality. Second, the concept of SVI solutions offers nice stability properties with respect to perturbations, which will be demonstrated by introducing a stochastic analog of Mosco-convergence. This yields a sufficient condition for the convergence of the corresponding solutions to SPDE. The general theory will be laid out by proving the convergence of non-local approximations to local stochastic p-Laplace equations.Date and time infoMonday 16:15 - 17:45Prerequisitesfunctional analysis, basic convex analysisAudienceMSc students, PhD students, PostdocsLanguageEnglish

ReferencesLiterature: J. Jost, Mathematical concepts, Springer, 2015Date and time infoMonday 13:30 - 15:00, starting April 15AudienceMSc students, PhD students, PostdocsLanguageEnglish

Participants will work through the 13 chapters of the book manuscript with that title. The course will be a mix of lectures by participants and study sessions, with 2 hours devoted to each chapter. A tentative schedule is as follows:
1.
Polynomial Rings
(June 4, 10 - 12)
2.
Varieties
(June 4, 14 - 16)
3.
Solving and Decomposing
(June 5, 10 - 12)
4.
Mapping and Projecting
(June 5, 14 - 16)
5.
Linear Spaces and Grassmannians
(June 6, 10 - 12)
6.
Nullstellensaetz
(June 6, 14 - 16)
7.
Tropical Algebra
(June 11, 10 - 12)
8.
Toric Varieties
(June 11, 14 - 16)
9.
Tensors
(June 12, 14 - 16)
10.
Representation Theory
(June 13, 10 - 12)
11.
Invariant Theory
(June 13, 14 - 16)
12.
Semidefinite Programming
(June 14, 10 - 12)
13.
Combinatorics
(June 14, 14 - 16)
Each chapter will be assigned to one or two leading presenters. Participants from nearby universities (Leipzig, Berlin, Magdeburg,...) are most welcome.Date and time infoBlock course: June 4 - 14AudienceMSc 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
Felix Otto: Gradient FlowDate: 02.04., 09.04., 16.04., 23.04., 30.04Peter Stadler: Discrete Structures -- with an emphasis on models from chemistry and biologyDate: 14.05., 21.05., 04.06., 18.06.Orlando Marigliano: The small linear covariance model: Statistics with only one data pointDate: 25.06Christoph Eikemeier: Primes in arithmetic progressionsDate: 02.07., 04.07.Zach Adams: Drift-diffusion limits in models of chemotaxisDate: 09.07., 11.07.Date and time infoLectures Tuesday, 10:15 - 11:45

This course is meant as an introduction to the concepts and methods of category theory from a rather applied point of view. We will review the basic ideas, such as:
Categories, functors, natural transformations. What it means for a construction to be "functorial", or "natural", and why it is important;
The Yoneda lemma, a.k.a. how objects are determined purely by the way they interact with one another. How to reason in terms of diagrams, and what why it is helpful;
Universal properties, limits, colimits, why they are everywhere, and how they generalize constrained optimization.
Category theory has been central to the development of fields such as algebraic topology, algebraic geometry, and group theory. It has also many applications to general topology, analysis, and geometry. Recently, it has been applied to some subjects of more "applied" flavor, such as:
Networks and complexity;
Graphical models and statistical inference;
Data structures and classes in computer science;
...and many more.
Depending on the interests of the participants, we will choose one or two of these particular applications to study in detail.Date and time infoWednesday 11:00 - 12:30KeywordsCategories, functors, universal properties, applicationsPrerequisitesBasic mathematical knowledge (linear algebra, basic analysis)AudienceMSc students, PhD students, PostdocsLanguageEnglish

This lecture is an introduction to basic ideas and tools of microlocal analysis. Central topics will include
basic distribution theory, distributional Fourier analysis,
the definition of oscillatory integrals and the method of stationary phase,
pseudodifferential operators,
elliptic regularity theory through pseudodifferential calculus.
If time permits, we will also deal with the propagation of singularities.Date and time infoThursday 11:15 - 12:45Keywordsdistribution theory, oscillatory integrals, method of stationary phase, pseudodifferential operators, elliptic regularity theoryPrerequisitesbasic PDE theory, functional analysis and basic Fourier analysis AbstractAudienceMSc students, PhD students, PostdocsLanguageEnglish

In the classical mathematical theory of percolation, the edges (or vertices) of an infinite lattice are deleted independently with probability 1-p, and properties of the remaining components are studied. Despite its simple description, this model captures a variety of phenomena, including structural phase transition, self-similarity, universality. It has been used in studies of materials, social and computer networks, epidemic spreading. This course will provide an introduction to the subject of percolation, focusing on basic results and techniques.Date and time infoThursday 13:15 - 14:45KeywordsPhase transition, correlation inequalities, coarse graining, number of infinite components, Russo-Seymour-Welsh theory, conformal invariance of the scaling limitPrerequisitesBasic knowledge of probabilityAudienceundergraduate students, PhD students, PostdocsLanguageEnglish

Starting from 19th century, elliptic and hyperelliptic integrals captivated the imagination of almost every mathematician. The initial observations provided the intuition for much of the development of modern complex geometry. At our present day, one starts learning the subject from these modern abstractions. However, in this lecture, we will work with concrete examples to see for ourselves what the ancients have seen in order to develop our intuition. This is intended so that we can anticipate what the abstract technical framework must look like, without doing anything technically demanding. We will end the first block by looking ahead and studying projective hypersurfaces, their Hodge decomposition and their periods.Date and time infoBlock course: May 6 - 10, time tbaKeywordsCurves, surfaces, periods, cohomologyPrerequisitesLinear algebra, calculus, visualization skillsAudienceMSc students, PhD students, PostdocsLanguageEnglishRemarks and notesThis is a block course, consisting of two blocks with each block lasting a week. There will be 4 hours of lectures per day with an additional 2 hours for working on exercises in groups. The second block will be late in summer.

This lecture aims at giving a leisure introduction to the field of algebraic analysis, that is, the algebraic study of linear partial differential equations with polynomial coefficients. We will start with basics on differential operators and the Weyl algebra as well as on vector bundles with connections. Next we will discuss the notion of holonomicity and how this gives finiteness restrictions on the solutions of a D-module. Depending on time and audience, we will go into some details of direct and inverse images, give the statement of the Riemann-Hilbert correspondence, explain some facts about filtered D-modules as well as on the V-filtration and Bernstein-Sato polynomials. Finally, we may give a small outline on the theory of mixed Hodge modules.Date and time infoFriday 11:00 - 12:30KeywordsDifferential operators, Weyl algebra, vector bundles with connections, functorial properties, filtered D-modules, Hodge modulesPrerequisitesBasic knowledge of algebraic and complex geometry: algebraic varieties, some commutative algebra (basics of dimension theory), complex manifolds, vector bundles, basics of homological algebraAudienceMSc students, PhD students, PostdocsLanguageEnglish

In der Vorlesung werden verschiedene grundlegende stochastische Modelle der mathematischen Populationsbiologie vorgestellt. Dazu gehören Verzeigungsprozesse, das Wright-Fisher Modell für Genfrequenzen, der Kingman-Koaleszent und Infektionsprozesse. Diese verschiedenen Modelle bilden die mathematische Grundlage, mit der die Evolution von Populationen und Genfrequenzen beschrieben werden. Mathematische beschreiben wir diese Modelle als Markov-Ketten. Im Seminar gibt es die Möglichkeit, komplexere Modelle und feinere Eigenschaften dieser Modelle kennenzulernen. Dazu gehören etwa Modelle mit Mutation und Selektion, Modelle mit räumlicher Komponente oder Konvergenz zu Limit-Prozessen.Date and time infoWednesday and Friday 9:00 - 11:00PrerequisitesGrundlagen der Wahrscheinlichkeitstheorie und Markovketten, etwa aus Wahrscheinlichkeitstheorie I und IIAudienceMSc students, PhD students, PostdocsLanguageEnglish or GermanRemarks and notesLectures in the first part of semester, seminar in the second part of the semester.