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!
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-delnin-w25-join@mis.mpg.deIn this course I will introduce the theory of currents in Euclidean spaces, which is nowadays a standard setting to study the Plateau problem in the context of minimal surfaces. Currents represent a notion of generalized oriented manifold, which might have corners and singularities, and enjoy good compactness properties making them useful to study variational problems.
The first part of the course is devoted to the foundations of the theory, and by the end of the course the goal is to introduce the transport equation for currents, which describes their motion through a given velocity field. We will discuss the well-posedness of this PDE in the class of Lipschitz vector fields, and its relation to fluid dynamics and the motion of dislocations in materials.
The lectures will tentatively touch upon the following topics:
- Existence of minimizers for the Plateau problem in the class of integral currents;
- Flat distance;
- Decomposition theorems (Smirnov's theorem; decomposition of integral currents);
- Polyhedral deformation theorem;
- Isoperimetric inequality;
- Transport equation for currents.
The contents can also be adapted or expanded according to the audience.
Several tools and notions from geometric measure theory will be introduced along the way, such as rectifiable sets, Rademacher's theorem, the area formula, and the decomposability bundle.
I intend to keep the course rather informal, giving proofs but sometimes skipping the more technical details, with the idea of giving an overview of the theory. Useful prerequisites are the theory of distributions and basic measure theory. For some background on rectifiable sets you can have a look at section 5 (plus possibly 1,3,4) of my lecture notes for the course "Selected topics in Geometric Measure Theory" that I gave in 2023/24 and that you can find in my homepage in the teaching section or with the direct link.KeywordsCurrents, Plateau problem, minimal surfaces, transport equation, geometric measure theoryPrerequisitesBasic measure theory. Knowledge of the theory of distributions, differential forms and BV functions is also helpful but not strictly necessary
The program of the seminar will be flexibly adjusted to the preliminary knowledge of the participants. There will be talks available for everybody with an honest interest in Fourier analysis. Suggested topics:
Fourier series
Fourier transform for functions and distributions
Payley-Wiener theorems
Hardy spaces
Applications to PDEs
Optional: basic harmonic analysis
Date and time infoWednesday 09:15 - 10:45PrerequisitesAnalysis I-II, Linear Algebra I-II, Measure and Integration Theory. Ideally: Lie Groups and Lie Algebras
Test functions, Schwartz functions, distributions: tempered distributions, regular and singular distributions, compactly supported distributions
Distributional derivatives and further operations with distributions
Wavefront set and singular support, products of distributions
Oscillatory integrals, symbol classes, pseudodifferential operators
(Semiclassical) quantisation and symbol calculus
Ellipticity, parametrix construction
Sobolev spaces, Fredholm theory
Schwartz kernel theorem, operator wavefront set, microlocalisation
Date and time infoTuesday 17:15 - 18:45, Wednesday 11:15 - 12:45PrerequisitesAnalysis I-II, Linear Algebra I-II, Measure and Integration Theory. Ideally: Functional Analysis, Differential Geometry
In this combined course we will introduce some of the most important notions for discrete groups, such as e.g. amenability, Kazhdan's property (T), residual finiteness and sofic approximations, as well as related properties of their probability measure preserving actions. The aim of the lecture is to lay the foundations for these topics and present some recent research advances. The seminar, in which students will give the talks, will focus on additional topics accompanying the lecture.
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-farre-w25-join@mis.mpg.deWe are interested in fine structural and statistical features of certain continuous dynamical systems of a geometric origin. Our motivating examples are the geodesic and horocyclic flows on the unit tangent bundle of a surface with a complete, negatively curved metric. The simplest case to consider is when the surface is compact or finite volume, and the metric has curvature everywhere equal to -1, i.e., the surface is hyperbolic.
We will prove that the geodesic flow on such a closed hyperbolic surface is exponentially mixing, has positive entropy, is uniformly hyperbolic (in fact Anosov), and thus exhibits both structural stability and chaos. It can also be coded using a Markov partition and studied in terms of symbolic dynamics, and we will use this coding to compute the growth rate of periodic orbits of the flow, among other applications.
The geodesic flow on a complete hyperbolic surface is an example from the world of homogeneous dynamics. The basic objects are: a (non-compact) Lie group G, a discrete subgroup Γ ≤ G, and a 1-parameter subgroup A ≤ G acting continuously on Γ \G preserving a finite invariant (Haar) measure. Sometimes the dynamics of this action can be used to count orbits of Γ on G/A. We will explore some examples of homogeneous dynamical systems and counting problems. Some of the motivation comes from number theory. Homogeneous dynamics is a very active field of current research - especially the dynamics of unipotent flow (like the horocycle flow). This course will serve as an introduction to some of the basic principles with an emphasis on (hyperbolic) geometry.Date and time infoOctober 14, 2025 - February 3 2026. Tuesdays 15:15 - 16:45
The plan of the lecture is to give overviews and proofs of some special topics in regularity theory of (non-linear) PDE and the Calculus of Variations.In the first part of the lecture we will discuss the De Giorgi-Nash-Moser scheme including the very weak Harnack and weak Harnack inequalities. We will continue with degenerate equations. Thereafter, possible topics are Evan's proof of the $C^{1,\alpha}$-regularity for the $p$-Laplacian or Savin's proof of the continuity of solutions to degenerate equations in 2D.I am also open to any other suggestions.Date and time infoWednesday 9:15 - 10:45 amKeywordsPDE, Calculus of Variations, Regularity theoryPrerequisitesPDE 1+2, Functional analysis, Measure theoryAudiencegraduate students, PhD students, PostDocsLanguageenglish
Algebraic curves in projective space have a long history, blending the classical geometry of the nineteenth century with modern tools from scheme theory and moduli spaces. This lecture series will examine practical aspects of curves in projective space, emphasizing examples in low genus. We will follow excerpts from Eisenbud and Harris's recent textbook, The Practice of Algebraic Curves. See here for errata.
The lectures will be given by Nathan Pflueger, supported by Bernd Sturmfels.
SCHEDULE
07.01.2026, 14:00-16:00 Linear series §1 & Rational functions and plane curves §2 (Nathan Pflueger)
07.01.2026, 16:00-18:00 Exercises (led by Bernd Sturmfels)
08.01.2026, 10:00-12:00 Rational curves (Lakshmi Ramesh) §3 Newton polygons (Vincenzo Galgano and Jessica Alessandri)
08.01.2026, 14:00-16:00 Canonical curves §2 & Low-genus curves§4,6,9 (Nathan Pflueger)
12.01.2026, 10:00-11:30 Catching Up (led by Smita Rajan & Bernd Sturmfels)
12.01.2026, 14:00-16:00 Hilbert schemes §7 & Hurwitz spaces and Severi varieties §8.5-6 (Nathan Pflueger)
12.01.2026, 16:00-18:00 Exercises (led by Bernd Sturmfels)
13.01.2026, 10:00-12:00 The moduli space (Dmitrii Pavlov) & Intersection theory on moduli spaces (Smita Rajan)
13.01.2026, 14:00-16:00 Curves on scrolls §17 & Castelnuovo's theorem §10 (Nathan Pflueger)
14.01.2026, 14:00-16:00 Brill--Noether theory §12 & Brill--Noether varieties (Nathan Pflueger)
14.01.2026, 16:00-18:00 Exercises (led by Bernd Sturmfels)
19.01.2026, 9:00-11:00 Moduli of stable maps (Felix Lotter and Carl Waller) & Curves in Macaulay2 (Alejandro Ovalle and Svala Sverrisdottir)
19.01.2026, 11:00-12:00 Discussions (led by Bernd Sturmfels)
19.01.2026, 14:00-16:00 Inflection points §13 & The interpolation theorem (Theorem 12.11) (Nathan Pflueger)Date and time info07.01.2026: 14:00-18:00, 08.01.2026: 10:00-12:00 + 14:00-16:00, 12.01.2026: 10:00-11:30, 14:00-18:00, 13.01.2026: 10:12:00 + 14:00-16:00, 14.01.2026: 14:00-18:00, 19.01.2026: 9:00-12:00 + 14:00-16:00
Periodic or closed geodesics on a compact Riemannian manifold are of importance for the study of the geometry of the manifold. The existence of closed geodesic can be studied using critical point theory for the energy functional on the free loop space of the manifold. The topology of the free loop space allows to present existence results. In case of positive curvature also geometric methods come into play.Date and time infoMonday 9:15-10:45, Tuesday 9:15-10:45KeywordsClosed geodesics, Riemannian geometry, Critical point theory, Free loop space, Morse theoryPrerequisitesdifferential geometry, algebraic topologyLanguageEnglish
Date and time infoTuesday, 11 to 13, and Thursday 9 to 11Keywordsbasics in algebraic topology, Poincaré duality, cohomology theories (in particular de Rham and Cech)Prerequisitesabstract algebra
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-rvl-w25-join@mis.mpg.dePart I: Bernd Sturmfels
Convex Polytopes
The sources are the two books:
Rekha Thomas: Lectures in Geometric Combinatorics, American Mathematical Society, 2006
Guenter Ziegler: Lectures on Polytopes, Springer Verlag, 1995
The lectures are mostly based on [Thomas]. For additional reading, [Ziegler] is recommended. The library provides both e-books and hard copies.
Each lecture is broken into two parts, with a short break in between. Here is a detailed syllabus:
October 20: 9:30 The Main Theorem [Thomas, Chapter 2]; 10:15 Faces of Polytopes [Thomas, Chapter 3]
October 27: 9:30 Schlegel Diagrams [Thomas, Chapter 4]; 10:15 Steinitz' Theorem [Ziegler, Chapter 4]
November 10: 9:30 Gale Diagrams [Thomas, Chapters 5 and 6]; 10:15 The Upper Bound Theorem [Ziegler, Section 8.4]
November 17: 9:30 Triangulations [Thomas, Chapter 7]; 10:15 The Secondary Polytope [Thomas, Chapter 8]
Exercise session will be organized spontaneously. Computations are strongly encouraged. Try OSCAR.
Part II: Melchior Wirth
Title: Quantum Optimal Transport
Following the success of optimal transport across applied and pure mathematics, in recent years a theory of quantum optimal transport has started to evolve. After a brief overview of classical optimal transport theory from an algebraic point of view, we will introduce some concepts of quantum measure theory and quantum information. We will then discuss various quantum Wasserstein metrics, in particular coupling-based and dynamical distances, and see some applications in the analysis of open quantum systems.
Date and time: November 24, December 1, 8, 15. tbc
Part III: Bian Wu
Date and time: tbaDate and time infoMonday 9:30-11:00KeywordsConvex Polytopes
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-zizza-w25-join@mis.mpg.deIn 1966 Arnold interpreted Euler Equations as a geodesic Equation on the group of volume-preserving diffeomorphisms SDiff (or fluid configurations). Solutions of Euler Equations arise as critical points of the actional functional (i.e. kinetic energy). Unfortunately, the group SDiff is an object that is not much well understood. In dimension $d\geq 3$ it is possible to find $f\in SDiff$ for which a shortest path in SDiff does not exist, and in dimension $d=2$ there exists fluid configurations $f\in SDiff$ that are not attainable. To minimize the action functional, Brenier considered the class of Generalized Incompressible Flows: fluid particles do not follow a specified trajectory, but they can split in a continuum of trajectories. Still, it is not clear how to match the notion of Generalized Flows with the one of weak solutions of Euler.Date and time infoFriday afternoon, possibly starting in NovemberKeywordsGroup of volume-preserving diffeomorphisms, Euler Equations, Generalized FlowsPrerequisitessome differential geometry may help, but it is not needed
