|C: Models and Approximations|
C1: Evolution and asymptotics
C3: Interacting particle systems and phase transitions
|Current Publications||• Gusakova A, Kabluchko Z, Thäle C The $β$-Delaunay tessellation II: The Gaussian limit tessellation. arxiv.org/abs/2101.11316 Vol. 2021, 2021 online|
• Gusakova A, Kabluchko Z, Thäle C The $β$-Delaunay tessellation III: Kendall's problem and limit theorems in high dimensions. arxiv.org/abs/2104.07348 Vol. 2021, 2021 online
• Gusakova A, Kabluchko Z, Thäle C The $β$-Delaunay tessellation IV: Mixing properties and central limit theorems. arxiv.org/abs/2108.09472 Vol. 2021, 2021 online
• Götze F, Gusakova A, Kabluchko Z, Zaporozhets D Distribution of complex algebraic numbers on the unit circle. Journal of Mathematical Sciences Vol. 251 (1), 2020 online
• Gusakova A, Kabluchko Z, Thäle C The $β$-Delaunay tessellation I: Description of the model and geometry of typical cells. arxiv.org/abs/2005.13875 Vol. 2020, 2020 online
|Current Projects||• SPP 2265: Random Geometric Systems - Subproject: Random polytopes The aim of the project is to investigate random polytopes. online|
• EXC 2044 - C1: Evolution and asymptotics In this unit, we will use generalisations of optimal transport metrics to develop gradient flow descriptions of (cross)-diffusion-reaction systems, rigorously analyse their pattern forming properties, and develop corresponding efficient numerical schemes. Related transport-type- and hyperbolic systems will be compared with respect to their pattern-forming behaviour, especially when mass is conserved. Bifurcations and the effects of noise perturbations will be explored.
Moreover, we aim to understand defect structures, their stability and their interactions. Examples are the evolution of fractures in brittle materials and of vortices in fluids. Our analysis will explore the underlying geometry of defect dynamics such as gradient descents or Hamiltonian structures. Also, we will further develop continuum mechanics and asymptotic descriptions for multiple bodies which deform, divide, move, and dynamically attach to each other in order to better describe the bio-mechanics of growing and dividing soft tissues.
Finally, we are interested in the asymptotic analysis of various random structures as the size or the dimension of the structure goes to infinity. More specifically, we shall consider random polytopes and random trees.For random polytopes we would like to compute the expected number of faces in all dimensions, the expected (intrinsic) volume, and absorption probabilities, as well as higher moments and limit distributions for these quantities. online
• EXC 2044 - C3: Interacting particle systems and phase transitions The question of whether a system undergoes phase transitions and what the critical parameters are is intrinsically related to the structure and geometry of the underlying space. We will study such phase transitions for variational models, for processes in random environments, for interacting particle systems, and for complex networks. Of special interest are the combined effects of fine-scalerandomly distributed heterogeneities and small gradient perturbations.
We aim to connect different existing variational formulations for transportation networks, image segmentation, and fracture mechanics and explore the resulting implications on modelling, analysis, and numerical simulation of such processes. We will study various aspects of complex networks, i.e. sequences of random graphs (Gn)n∈N, asking for limit theorems as n tends to infinity. A main task will be to broaden the class of networks that can be investigated, in particular, models which include geometry and evolve in time. We will study Ising models on random networks or with random interactions, i.e. spin glasses. Fluctuations of order parameters and free energies will be analysed, especially at the critical values where the system undergoes a phase transition. We will also investigate whether a new class of interacting quantum fields connected with random matrices and non-commutative geometry satisfies the Osterwalder-Schrader axioms. Further, we will study condensation phenomena, where complex network models combine the preferential attachment paradigm with the concept of fitness. In the condensation regime, a certain fraction of the total mass dynamically accumulates at one point, the condensate. The aim is a qualitative and quantitative analysis of the condensation. We willalso explore connections to structured population models. Further, we will study interacting particle systems on graphs that describe social interaction or information exchange. Examples are the averaging process or the Deffuant model.
We will also analyse asymmetric exclusion processes (ASEP) on arbitrary network structures. An interesting aspect will be how these processes are influenced by different distribution mechanisms of the particles at networks nodes. If the graph is given by a lattice, we aim to derive hydrodynamic limits for the ASEP with jumps of different ranges for multiple species, and for stochastic interactingmany-particle models of reinforced random walks. Formally, local cross-diffusion syste ms are obtained as limits of the classical multi-species ASEP and of the many-particle random walk. We will compare the newly resulting limiting equations and are interested in fluctuations, pattern formation, and the long-time behaviour of these models on the microscopic and the macroscopic scale. Further, we will analyse properties of the continuous directed polymer in a random environment. online
|zakhar dot kabluchko at uni-muenster dot de|
|Phone||+49 251 83-33773|
|FAX||+49 251 83-32712|
Frau Anita Kollwitz
Telefon +49 251 83-33770
Fax +49 251 83-32712
|Address||Prof. Dr. Zakhar Kabluchko |
Institut für Mathematische Stochastik
Fachbereich Mathematik und Informatik der Universität Münster
|Diese Seite editieren|