Research objectives

At Mathematics Münster (MM), our research aims to implement integrated approaches in order to solve important problems across different mathematical fields. We view mathematical research as an organic whole with countless connections between fields and specifically promote the development of mathematical methods that lead to cross-disciplinary scientific breakthroughs. The integration of research within and across various topics and the dynamic transfer of methods, ideas, and knowledge between mathematical disciplines is central to our strategy (see structural objectives).

O1: Push frontiers in K-theory and attack open questions in topology

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We explore various cohomology theories and their utility across different fields, including algebra, geometric topology and operator algebras. Central among those cohomology theories is K-theory, which is intensively studied from several perspectives. For example, we will obtain new computations of K-groups and prove new cases of the Farrell–Jones conjecture to attack key topological questions, such as the Borel conjecture on topological rigidity. We also use K-theory to obtain curvature bounds on manifolds and to classify C*-algebras.

O2: Study moduli spaces to advance the Langlands programme and other fields

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Our research advances the (local) Langlands Programme through the detailed examination of the arithmetic properties inherent in the associated moduli spaces and categories of representations. More broadly, moduli spaces are integral to numerous mathematical fields. In differential topology, they play a crucial role in studying diffeomorphism groups of high-dimensional manifolds. In differential geometry, they are essential for investigating deformations of positive scalar curvature metrics. In mathematical physics, the moduli spaces of stable curves and of Strebel differentials are of particular interest.

O3: Answer open questions in C*-algebras, build models and study their theories and external properties

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Our research focusses on the structure and classification of C*-algebras, with a specific emphasis on proving existence and classification results for Cartan subalgebras within simple nuclear C*-algebras. This research is fundamentally linked to group theory and group actions, which we approach using tools from functional analysis, probability theory, and combinatorics. Furthermore, we develop and explore models encompassing various strengths and properties to address questions across algebra, geometry, and set theory. Additionally, we analyse the interplay of determinacy hypotheses with strong large cardinal hypotheses.

O4: Approach fundamental open questions in the theory of PDEs and differential geometry through a method transfer between the fields

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We study the interplay between the (local) geometry of Riemannian manifolds (like curvature) and (global) topological and analytic properties of the underlying manifold. Specifically, we aim to complete the Grove programme by generalising our structure results for positively curved Riemannian manifolds and Einstein manifolds with symmetry to Riemannian manifolds with arbitrary one-dimensional isometry groups, and, finally, without any symmetry assumption. We further address singularity and rigidity phenomena for Riemannian and Lorentzian Einstein manifolds and try to prove the stability of the Kerr family of black holes in the context of general relativity.

O5: Exploit stochastic analysis tools in the context of non-commutative field theories

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By combining methods from stochastic analysis, free probability, and topological recursion, we will make decisive progress towards constructing field theories on non-commutative spaces in the critical dimension. We plan to develop a theory of stochastic quantisation for scalar non-commutative quantum field theories in all sub-critical dimensions and, ultimately, in the critical dimension. Specifically, we aim to provide the first rigorous construction of a critical bosonic quantum field theory.

O6: Analyse structures and their limits in stochastic models

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We study random geometric structures and their influence on interacting particle systems, investigating fluctuations and the role of disorder in scaling limits. Key objects include geometric structures such as random tessellations, random graphs, and networks, as well as percolation on groups.

O7: Analyse structures in multiscale processes and derive effective models

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We analyse multiscale phenomena in physical, engineering, and biological systems and develop mathematical models and numerical methods to predict their behaviour. We address optimisation and design problems involving multiple scales, develop approximation algorithms for optimal control problems, study convergence rates in non-linear homogenisation, and investigate mixing and equilibration in fluid dynamics and kinetic equations. Identifying relevant scales to derive scaling limits is a unifying theme.

O8: Develop innovative simulation tools by combining model and data-based approaches

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We will advance the fundamental mathematical understanding of artificial neural networks, for instance, by designing and rigorously analysing stochastic gradient descent methods for their training. By combining data-driven machine learning approaches with model order reduction methods, we will develop fully certified multi-fidelity modelling frameworks for parameterised PDEs, design and study higher-order deep learning-based approximation schemes for parametric SPDEs, and construct cost-optimal multi-fidelity surrogate methods for PDE-constrained optimisation and inverse problems.

Structural objectives

We offer a variety of measures and programmes to successfully implement our research approach.

O9: Implement an integrated approach across mathematical disciplines and increase international visibility.

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Our approach views mathematics as an integrated whole, aiming to strengthen and enhance the interplay between its disciplines to drive innovation. In contrast to the national and international trend towards specialisation, we envision Mathematics Münster as a model for interconnected mathematical research and education. Building on the successful initiatives and developments of the first funding period, we aim to take this programme to the next level, developing a sustainable, internationally recognised research centre in mathematics.

O10: Attract and support outstanding early-career researchers and promote equity and diversity.

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We aim to further enhance our doctoral and postgraduate programmes to attract and nurture the very best international talent in mathematics. By fostering a stimulating research environment with active career development and guidance, we encourage early scientific independence, supporting both excellent mathematical research and successful academic careers.

Embracing diversity, equity, and inclusion is vital to our integrated research approach. Our ambition is to significantly improve gender equity at all levels and cultivate an inclusive research, work, and learning environment that thrives on diverse perspectives and backgrounds.