Mathematical and Computational Logic

Differential geometry and global analysis

Special structures on Riemannian manifolds Spinor field equations in mathematical physics

biogeochemistry, isotope geochemistry, soil science

soil erosion in Alpine areas radiogenic isotopes to track soil erosion stable isotopes to determine soil degradation wetland processes and degradation

Bioinformatics, computational genomics and modeling molecular evolution

Together with my students I work on theoretical aspects of modeling molecular evolution as well as data-driven applications of our new methodologies. We are broadly interested in applying computational methods to study the process of gene and genome evolution and in particular the process of adaptive genetic change. Growing genome-scale sequences and information from complimentary disciplines of transcriptomics, proteomics and metabolomics have begun to seriously impact genomics. But the challenge of understanding the dynamics of such large-system data can only be met through an integration of organism, molecular, and mathematical disciplines. New techniques need to be developed for discovering complex patterns from multi-faced systems biology data.

Evolutionary biodemography

Development of new concepts, models, and theories for understanding the evolution of aging in light of biodiversity

Algebra, Representation Theory

Cluster algebras, cluster categories, surface combinatorics, surface categories

Mathematics education

Topology and Categorification

Link homology theories, quantum invariants of links and 3-manifolds, topological quantum field theories and their categorification

Analysis

Harmonic analysis, wavelets, the Radon transform, inverse problems, Clifford analysis

Algebra, discrete mathematics

Representation theory of finite groups Representation theory of algebras Theory of symmetric functions Enumerative and algebraic combinatorics Additive number theory

Epidemiology and statistics

Epidemiology of cancer and radiation illnesses Research concentrations: biometric methods in epidemiology and epidemiological studies of radiation-induced tumours

Wissenschaftsgeschichte, Sinologie

History of Mathematics, particularly in China Relations between the history of Mathematics and Linguistics Mathematics and Culture

Non-classical logics in computer science

Foundations (proof theory, algebraic semantics,...) and applications in various fields

Algebra, Combinatorics, Theoretical Computer Science

Group Theory, Discrete Mathematics, Languages and Automata

Robust design, networks (gas, circuits, water, etc.), scientific computing, computational engineering

Network simulation and graph mining Multicriteria Parameter Optimization in scientific and technical applications Metamodelling (Response Surfaces) Sensitivity and robustness studies for high resolution simulation results Correlation analysis for high resolution simulation results Hierarchical linear solvers, in particular, algebraic multigrid methods

Harmonic Analysis and Probability

Random walks, random recursions, stationary measures, Lipschitz iterative models, branching processes in random environment

Solar Physics

Models and observations of dynamics in the solar atmosphere; Coronal heating; Computational modelling of oscillations in the Sun's atmosphere

Algebraic Geometry and its applications

Statistics

Causal inference, graphical Models, event history analysis, epidemiology

Cognitive psychology, mathematical psychology

Perception (intersensory interaction) Decision theory (stochastic dynamic models) Prioritisation in medicine (empirical research, qualitative and quantitative methods)

Nonlinear optimisation

Quadratic models Optimisation over convex cones Copositivity Matrix theory Blackbox optimisation

Functional analysis and operator theory; dynamical systems and ergodic theory

Ergodic theory: ergodic theorems Operator theory: asymptotics of operators and operator semigroups, stability

Arithmetic algebraic geometry

Mathematics Education

teaching and learning teacher professional education and development (education of teacher educators) assessment of learning textbook analysis

Pure Mathematics

The interplay of algebra, geometry, topology and combinatorics, more specifically, in arrangements of hyperplanes, toric varieties and tropical geometry

Applied mathematics: approximation theory and applied harmonic analysis.

Harmonic analysis: Wavelet analysis, complex wavelets, phase information, complex splines Integral transforms for signal and image analysis Applications in signal and imaging processing Approximation theory, sampling and interpolation Function theory: Nonharmonic Fourier and Dirichlet series, growth estimates for entire functions of special type

Population economics and applied macroeconomics

Economic effects of an aging population

Statistical modelling, robust statistical methods, quality engineering and industrial statistics, online monitoring methods

Interdisciplinary collaborative work on the modelling of data from the life sciences and engineering sciences

Coding Theory, Commutative Algebra, Cryptography, Algebraic Geometry

Liaison theory, determinantal ideals, Groebner bases, network coding, elliptic curve cryptography, torus-based cryptography

Applied Mathematics: Numerical analysis, differential equations, parameter estimation, control theory, dynamical systems and computational neuroscience

Mathematical modeling of complex biological systems

Analysis

Theory of function spaces Fractal analysis Approximation theory

Algebraic topology and its applications, particularly in the life sciences (especially neuroscience), but also in material science

Quantified formal specification, i.e. formal methods enhanced with quantitative information such as probabilities and stochastic time.

Performance modelling Systems biology Collective adaptive systems

Numerical mathematics

Numerical methods for time-dependent partial differential equations Iterative methods for solving large linear systems of equations and eigenvalue problems Approximation of matrix functions Computational physics

Arithmetic geometry

Motives and special values of L-functions

Combinatorics, Discrete Probability, Algorithms

Probabilistic Combinatorics, Enumerative Combinatorics, Combinatorics in Statistical Physics Random Graphs and Hypergraphs, Random Graphs on Surfaces, Random Simplicial Complexes, Random Discrete Structures Randomised Algorithms, Analysis of Algorithms

Nonlinear control

Differential algebraic and differential geometric methods, systems on time scales, relations between different system descriptions, output feedback solutions

Applied analysis

Applied harmonic analysis and geometric multiscale analysis, in particular time–frequency analysis, wavelet theory and shearlet theory Approximation theory, in particular sparse approximation and compressed sensing Frame theory, in particular theory of redundant systems Mathematic data analysis, in particular signal processing and image processing Inverse Probleme Mathematical Data Science Numerical analysis of partial differential equations

Scientific Computing

Approximation theory, radial basis function approximation, numerical solution of partial differential equations, task based parallel computing, computational finance, computational biomechanics

Mathematical optimisation, algorithmics

My research area is in the field of mathematical optimisation. The (further) development of effective algorithms for both NP-hard problems and polynomial solvable problems is of particular interest to me. It is my aim to develop practically efficient but, above all, exact algorithms, to implement them in terms of algorithm engineering and to evaluate them for use in relevant applications. I enjoy studying problems that have applications in the natural sciences, for instance in physics, but also in operations research. My work focuses on combinatorial optimisation, polyhedral combinatorics and graph algorithms. An additional focus of my research is the further development of general methods in nonlinear optimisation, for example in binary constrained quadratic optimisation. I am the head of a young researcher group in the Department of Computer Science at the University of Cologne within the framework of the German Research Foundation Emmy Noether Programme. The research topic for this project is the development of exact optimisation algorithms for problems in theoretical physics. Frequently, these algorithms can be applied to solving problems in other fields.

Mathematics

Noncommutative Algebraic geometry

Convex and discrete geometry, geometric analysis

Valuations on convex bodies Affine isoperimetric inequalities Affine surface areas Valuations on function spaces

Mathematical modelling in life sciences, applied analysis, multiscale analysis, differential equations, cell biology

Mechanisms of pattern formation and cell differentiation and their mathematical description Analysis of differential equations

Bioengineering. Molecular and Micro- Biology. Computational and Systems Biology. Stochastic modelling and simulation. Bioinformatics.

(1) Host-microbiota interactions in complex human diseases, (2) microbial multi-drug resistance, (3) gene expression and chromatin organization, and (4) the role and effect of mechanical forces ('mechanicanobiology') and cell compartmentalization in pathogenesis .

Number theory

Additive combinatorics, combinatorial and analytic number theory, applications to Diophantine geometry

combinatorial optimization, efficient algorithms

Approximation algorithms Scheduling, resource management, network design and routing Coping with uncertainty: online, stochastic, and robust optimization

Mathematical physics, quantum gravitation, algebra, representation theory, differential geometry

Statistics, Environmental Statistics

Statistics, spatiotemporal data, environment, water quality

Statistics for the social and behavioral sciences.

Optimal experimental design Statistical power analysis Multilevel analysis Survival analysis

Applied statistics and econometrics

Credit scoring and credit rating Nonparametric and semiparametric regression Computer-based statistics Analysis of financial market data

Integral representations of finite groups, discrete mathematics, algebraic number theory, lattices, codes, modular forms

Explicit and concrete constructions of discrete structures; experimental mathematics

Applied analysis

Non-linear partial differential equations, kinetic equations, coagulation processes, multiscale problems, free boundary value problems Applications in materials science, physics and biology

Harmonic analysis

Biometry

Graphical models Genetic epidemiology Biometry Use of secondary data in drug safety research Primary prevention and its evaluation (especially of childhood obesity)

Computational Neuroscience, Computational Biology, Bioinformatics

Modelling of neural cells and circuits, focusing on unraveling the role of dendrites and dendritic computations in learning and memory. Development of computational methods and tools for bioinformatics analysis of biological data, focusing on miRNAs and cancer diagnostics. Systems biology approaches for understanding neural and gene functions.

History of mathematics in the 16th and 19th/20th centuries History of astronomy from the 16th to the 19th centuries Current area of research: Gauss and Russia, a ca. 800-page comprehensive work written with Dr Elena Roussanova

Mathematics education

Teaching and learning Educational monitoring

Mathematical Analysis

Nonlinear and partial differential equations Well-posedness of problems arising from phase transition-separation phenomena Memory effects in phase transitions phoenomena Long time behaviour and existence of the attractor for nonlinear/singular phase-field systems Inverse problems: identification of memory kernels in PDE systems

Stochastic, Probability Theory, Random Processes

Interacting systems Branching processes Random diffusions Mathematical statistical mechanics Stochastic Differential Equations Reziprocal processes

Mathematical modelling of biological systems. Especially modelling of plant-soil interaction, blood and lymphatic system function in mammalian body, cancer modelling.

Imaging and modelling in biology. Especially imaging using multi scale X-ray CT and image based model development.

Dynamical systems, Partial differential equations

Pattern formation in natural systems, Singularly perturbed systems, Blowup phenomena, Applications to physics and biology

Mathematics

Geometric analysis and mathematical general relativity

Complex Networks

Geometry of complex networks, multilayered networks, function and emergent behavior, adaptation and evolution of networks, economic networks, systems biology

The nature of turbulent flows, in particular, physics of turbulent thermal convection. This includes investigation of natural, forced and mixed convection; coherent flow structures, boundary layer structures, small-scale turbulence in buoyancy-driven flows; the influence of rotation, non-Boussinesq effects, non-monotone fluid properties, wall roughness and domain geometry on turbulent convection. Apart from physics of turbulence itself, the research interests include also all numerical issues and aspects of turbulence simulations. Research interests in natural and applied sciences and engineering include in particular the large-scale oceanic circulation, wind chill effects in hot and cold regions, supergranulation in the solar upper convective zone, heat and mass transfer in nanofluids, surface-tension-driven convection and vibration-induced convection in low gravity, improvement of the efficiency of technological heating and cooling processes, control of ventilation processes in living quarters and in transport.

Applied mathematics, discrete mathematics, mathematical modeling

modeling in systems biology discrete dynamical systems hybrid modeling

Theoretical high energy physics

My research interests include all aspects of string theory, gravitational physics and quantum field theory. In recent years much of my work has been focused on holographic dualities and their implications. Holography relates gravitational theories to theories without gravity in one less dimension and represents a completely new understanding of both gravity and the dual non-gravitational theories. My work on holography encompasses both foundational issues (the holographic dictionary between gravity and gauge theory physics) and applications of holography to black hole physics, phenomenology and condensed matter systems.

Algebra and model theory

Epidemiology, Medical Statistics

Missing Data Longitudinal data analysis Lifecourse epidemiology Causal methods in epidemiology

Applied Mathematics, Numerical Analysis

Wave propagation, Imaging in random media

Mathematical statistics Survival analysis Semiparametric M- and Z-estimation Empirical processes Non- and semiparametric regression Copulas Endogeneity problems in econometrics

Pure Mathematics

Number theory, Diophantine Geometry

Statistics

Income distribution, inequality, poverty Extreme value modeling, copulas (for applications in finance) Generalised linear models, latent variable models (for applications in psychology) Time series modeling, wavelet decomposition (for applications in engineering) Model selection with large datasets (for applications in economics and genetics)

Arithmetic geometry

Shimura varieties, Langlands program

Computational and mathematical approaches to music research

Computational musicology Music information retrieval Music cognition Mathematical music theory Computational music analysis Computational humanities Oral transmission in folksongs Musical similarity Musical performance research Rhythm and meter in music

My research focuses on the dynamics of granular materials, with a special interest in geophysical phenomena occurring in nature and engineering problems in industry. Granular materials span a range of scales, from the micro-scale in grain phenomena to the macro-scale in large systems such as avalanches and dunes. Despite their importance, granular materials remain poorly understood; due to the lack of a comprehensive physical description, mathematical tools and techniques are insufficient to fully predict behavior of granular materials in industrial and natural applications at this time.

In my research program I aim to gain a deeper understanding of the physics of granular materials across all different scales and flow regimes. My interests can be classified into the following areas: 1) wave propagation in a particulate material 2) avalanching of a granular material and 3) near-surface geophysical techniques to characterize soil, sand and geophysical mass flows. Although I am first and foremost an experimentalist, I also employ numerical simulations to develop analytical models and to form a comprehensive picture of open research questions.

Topology

Biostatistics, genomics

As a statistician, I work with biological datasets to understand the mechanisms underlying human disease and identify possible treatments. Such work is only possible with close collaboration with biologists, and I moved to the Cambridge Institute for Medical Research in 2009 to pursue these collaborations. During the GWAS era, we, and others, identified many genetic polymorphisms that alter risk of human diseases such as type 1 diabetes and hypertension. My focus has now mostly shifted towards identifying the molecular function through the pleiotropic effects of these variants on disease and intermediate biological traits such as gene expression. Statistically, my current work is focused on variable selection, Bayesian model averaging, and methods to integrate data from GWAS scans of related traits.

Mathematical physics, geometry

Conformal and topological quantum field theories Kähler geometry Singularities

Arithmetic geometry

Varieties over non-Archimedean fields Berkovich spaces Bruhat–Tits building Vector bundles on p-adic curves

Statistics

Applied statistics Psychometrics Test equating Large-scale assessments Item response theory

Geometry

Geometry and topology Deformation spaces of geometric structures on manifolds Representation varieties of surface groups Symmetric spaces Bounded cohomology Rigidity problems Higher Teichmueller spaces Discrete subgroups of Lie groups

Biostatistics

Biostatistics, clinical trials, systematic reviews, core outcome sets

numerical simulation methods domain decomposition methods mortar finite element methods non-standard discretizations iterative solvers a posteriori error indicators and adaptive refinements contact problems solving engineering problems with uncertainties

Systemsbiology,Theoretical Biophysics

Signal transduction Gene regulation Mathematical modeling of biological processes Quantitative biology

Number theory

Modular forms, p-adic Hodge theory, Iwasawa theory