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!
Representation theory is about understanding and exploiting symmetry using linear algebra. The central objects of study are linear actions of groups on vector spaces. This gives rise to a very structured and beautiful theory. The aim of this course dealing with finite groups and complex vector spaces is to introduce this theory.
Representation theory plays a major role in mathematics and physics. For example, it provides a framework for understanding finite groups, special functions, and Lie groups and algebras. In number theory, Galois groups are studied via their representations; this is closely related to modular forms. In physics, representation theory is the mathematical basis for the theory of elementary particles.
After introducing the concept of a representation of a group, we will study decompositions of representations into irreducible constituents. A finite group only has finitely many distinct irreducible representations; these are encoded in a matrix called the character table of the group. One of the goals of this course is to use representation theory to prove Burnside's theorem on solvability of groups whose order is divisible by at most two prime numbers. Another goal is to construct all irreducible representations of the symmetric group.References
We will follow the book Representation Theory of Finite Groups by Benjamin Steinberg.
Another reference is Representation Theory: A First Course by William Fulton and Joe Harris.
Date and time infoThursdays 7:30-9:00 and Fridays 9:15-10:45KeywordsRepresentation theory, Group actions, CharactersPrerequisitesBasic knowledge about groups and vector spaces
In this online course we aim to get acquainted with some of the recent progress in the mathematical understanding and theory of machine learning. Particular emphasis will be laid upon overparametrization, implicit bias and aspects of unsupervised learning. The aim of the course is to get an overview of a series of recent articles and the methods developed therein. The course is directed to non experts with a solid background in mathematics aiming to get an idea of recent progress in the mathematics of machine learning. Active contribution in form of a presentation of a research paper (to be chosen) is required for each participant.Date and time infoWednesdays 16:15-17:45
Over the years, the mathematical approach to biology and neurobiology has shifted from more model oriented to more data oriented methods.
I shall describe this development.
Dynamical system models aim at describing the physical dynamics in biological systems in detail, Nonlinear and stochastic mechanisms can generate very rich dynamical behavior.
Network models instead focus on the underlying interaction structure, and methods should robustly extract qualitative information even if the data are large, heterogeneous and noisy or perturbed.
In contrast, information theoretical approaches are concerned with the function rather than the structure of a system.
Methods from topological or geometric data analysis only utilize distance relations between data points.Date and time infoFriday, 14.00
There are several lines of research studying entropy-like quantities for a collection of random variables. Many constructions of such quantities are proposed, that satisfy certain conditions or have desired properties.
Tropical probability proposes to study entropy-like quantities in bulk. The closest analogy will be the relation between calculus and functional analysis.
Very roughly, entropy-like quantity associates a value to the collection of random variables. This association is required to be non-negative, additive (with respect to taking iid copies) and continuous with respect to some natural topology.
The word "tropical" refers to the certain "tropicalization" procedure, in spirit very similar to the "tropicalization" as used in algebraic geometry. Tropicalization is used for the construction of the "tropical cone" of diagrams of probability spaces -- a subcone in a certain Banach space and topology thereupon. The dual of the tropical cone is the cone of entropy-like quantities.
The ultimate goal would be to understand the space of all entropy-like quantities, from which one could then pick those needed for particular applications, such as causal inference, info decomposition, etc.
This area of research is little understood, open questions are behind every corner, and conjectures are abundant.Date and time infoWednesday, 11.00-12.30
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.Date and time infoThursdays, 9.15 - 10.45KeywordsTopological Phases, Combinatorics, PDE, Bayesian InferenceAudienceIMPRS studentsLanguageEnglish
In this course we introduce non-backtracking graphs, non-backtracking random walks and two non-backtracking operators. We investigate the main spectral properties of such operators, and we discuss computational aspects.Date and time infoMondays at 3pm, starting May 2PrerequisitesBasic knowledge in linear algebra and graph theory
In this lecture, we give an overview of recent developments concerning discrete Ricci curvature. Discrete Ricci curvature has proven to be a useful tool in network analysis. It has been applied for detecting local clusters within a network and for finding most important connections between two nodes.
The course focuses on the theoretical background of discrete Ricci curvature. Particularly, we study relations between curvature bounds, random walks and the heat equation. As applications, we can estimate the spectral gap of the graph Laplacian and derive geometric properties as diameter bounds, Gaussian measure concentration and volume growth in terms of the curvature.Date and time infoWednesday, 15.00Keywordsgraphs, curvature, heat equationPrerequisitesBasic analysis and linear algebra
In this class we will study first-order optimization methods for constrained and unconstrained optimization methods. In addition, a major part of the lecture will be devoted to aspects of online convex optimization, which is a combination of convex optimization, statistical learning, and game theory. Online optimization is motivated from practical applications in which the environment is so complex that it is difficult to design robust optimization models and apply classic algorithmic theory. In the online optimization framework, the optimization is instead considered as a process that learns from experience as one goes along and more aspects of the problem are observed. In the exercise class (on demand) a practical application to recommender systems will be considered.Date and time infoLectures: Tuesdays 11:00-12:30, Exercises (biweekly): Tuesdays 14:00-15:00Keywordsonline convex optimization, optimization on manifolds, multi-armed bandit, games and saddle point problemsPrerequisitesBasics of linear algebra, analysis, and probability
In this course we will study disordered (classical) spin systems on the lattice. After a short introduction to the statistical mechanics of lattice spin systems, in particular the DLR formalism (Gibbs states), we will move on to systems with quenched disorder. The first goal will be to discuss random Gibbs measures and metastates. In the context of the random field Ising model, we will consider uniqueness and non-uniqueness questions (Aizenman-Wehr, Bricmont-Kupiainen). The final part will be about mean-field models of spin glasses, where we will have a look at the (generalised) random energy and Sherrington-Kirkpatrick models.
The course will follow A. Bovier's book "Statistical Mechanics of disordered systems. A mathematical perspective" (Cambridge series in statistical and probabilistic mathematics, CUP 2006)Date and time infotbaKeywordsspin glasses, quenched disorder, phase transitions, random Gibbs measures, metastates
