Termine Angewandte Mathematik Münster

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Spin models in statistical mechanics have a history of almost one hundred years. They first occurred as models of ferromagnetism and Ernest Ising's PhD thesis. In this minicourse we will start with a brief survey of some results in the Ising model. We will then turn to its mean-field version the Curie-Weiss model and see how phase transition can be proved there. In a final step we will introduce some simple models of so-called spin glasses and discuss some of the results that have been proved in this context.

We start with a short introduction to the concept of Semmes families of curves in metric measure spaces and its role to derive regularity properties of such spaces. Then we indicate how these ideas are used in scalar curvature geometry and relativity to formulate dimensional reduction arguments also known as splitting theorems.

In this seminar we will study advanced numerical methods for the solution of parameterized PDEs in various application contexts. A paricular emphasise will be on model order reduction and recent machine learning approaches. Among other approaches, we will discuss the usage of data driven methods based on deep neural networks (DNNs), as well as on physics informed neural networks (PINNs).

There will be a preliminary discussion for interested students on Wednesday, January 26th 2022 at 1 p.m. in room 120.029 in the second floor of Orléans-Ring 10. Virtual participation is also possible.
Alternatively you my also participate virtually. Please connect via Zoom to https://wwu.zoom.us/j/69240672478?pwd=ZklXcTJtQ09GUkhOZG5PVmswL01HZz09.
See also at HISLSF .

Spin models in statistical mechanics have a history of almost one hundred years. They first occurred as models of ferromagnetism and Ernest Ising's PhD thesis. In this minicourse we will start with a brief survey of some results in the Ising model. We will then turn to its mean-field version the Curie-Weiss model and see how phase transition can be proved there. In a final step we will introduce some simple models of so-called spin glasses and discuss some of the results that have been proved in this context.

By embedding a Markov-modulated random recurrence equation in continuous time, in this talk I will derive the Markov-modulated generalized Ornstein-Uhlenbeck process. This process turns out to be the unique solution of a stochastic differential equation driven by a bivariate Markov-additive process that will be presented and solved explicitely. Afterwards we discuss basic properties of the obtained processes such as stationarity conditions and some moments. If time permits, we may also discuss a possible application and introduce a Markov-modulated risk model with investment that generalizes Paulsen?s risk process to a Markov-switching environment.

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