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07.11.2022 16:30 Stefan Grosskinsky (Universität Augsburg):
Asymptotics of generalized Pólya-urns with non-linear feedback joint work with Thomas GottfriedB 252 (Theresienstr. 39, 80333 München)

Generalized Pólya urns with non-linear feedback are an established model for the competition in markets. Depending on the feedback function, the model can exhibit monopoly, where a single agent achieves full market share. We examine the asymptotic behaviour with diverging initial market size for a large class of feedback functions, and establish a scaling limit for the evolution of market shares, including a functional central limit theorem. In the monopoly case find a criterion to predict the (in general random) monopolist with high probability under generic initial conditions. Our results reveal an interesting difference between exponentially and more realistic polynomially growing feedback.

10.11.2022 13:30 research talk by Tyler Helmuth (Durham), Minicourse by Roland Bauerschmidt, research talk by Rémy Poudevigne (Cambridge):
Mini Workshop on Reinforcement and Statistical MechanicsEI 02.5901.013 (Hans-Piloty-Straße 1, 85748 Garching)

13:30-14:20 Research talk by Tyler Helmuth: The Arboreal Gas

14:30-16:00 Minicourse by Roland Bauerschmidt: Phase transition for the Arboreal Gas in d≥3

16:30-17:20 Research talk by Rémy Poudevigne: Slow phase transition for the VRJP on the tree

more details on the workshop website:

https://www.math.cit.tum.de/math/forschung/gruppen/probability-theory/singleview/article/mini-workshop-on-reinforcement-and-statistical-mechanics/

11.11.2022 08:30 Minicourse by Constanza Rojas-Molina (Cergy-Pontoise), research talk by Xiaolin Zeng (Strasbourg), research talk by Christophe Sabot (Lyons), Minicourse by Sabine Jansen :
Mini Workshop on Reinforcement and Statistical MechanicsEI 02.5901.013 (Hans-Piloty-Straße 1, 85748 Garching)

8:30-10:00 Minicourse by Constanza Rojas-Molina: The Multiscale Analysis Method in the Theory of Random Schrödinger Operators

10:30-11:20 Research talk by Xiaolin Zeng: The random Schrödinger operator related to H2|2 model, integrated density of state

11:30-12:20 Research talk by Christophe Sabot: Stochastic calculus aspects of the Vertex Reinforced Jump Process

14:00-15:30 Minicourse by Sabine Jansen: Witten Laplacian for spin systems vs. Stein's method in probability theory

more details on the workshop website:

https://www.math.cit.tum.de/math/forschung/gruppen/probability-theory/singleview/article/mini-workshop-on-reinforcement-and-statistical-mechanics/

14.11.2022 15:00 Maximilian Schemel TUM:
Mixed-Mode oscillations in the Olsen modelMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

This talk is about the occurrence of mixed-mode oscillations in Olsen model for the peroxidase-oxidase reaction. The model is studied using geometric-singular perturbation theory. It is four dimensional, contains two small parameters and is a three-time scale system, with one fast, two medium and one slow variable. I used the slowest variable as the only slow variable and the other three as fast variables. The small amplitude oscillations in the model arise due to a dynamic Hopf-bifurcation. The oscillations are organised by canard-like orbits (crossings of stable and unstable manifolds of saddle-slow manifolds). Upon parameter change, an arithmetic progression of mixed-mode oscillations is obtained. This can be readily explained by the canard-like orbits. In the second part of the talk, I compare my results, with the results by Kühn and Szmolyan, who analysed the Olsen model using only the fastest variable as the fast-variable and the other three as slow variables. Combining both analysis gives insights about the double limit, where both parameters are small.

14.11.2022 16:30 Alexandra Quitmann (WIAS Berlin):
Macroscopic loops in interacting random walk loop soupsB 252 (Theresienstr. 39, 80333 München)

We consider a general interacting random walk loop soup that is related to several well-known statistical mechanics models, such as the Spin O(N) model, the double dimer model or the interacting Bose gas. We discuss the system in $\mathbb{Z}^d, d>2$, and present some recent results about the occurrence of macroscopic loops whose length is proportional to the volume of the system as the inverse temperature is large enough.

15.11.2022 16:15 Frederik Ran Klausen (Copenhagen):
Single particle open quantum systems in one dimension: dissipation and disorder imply decoherence. MI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

We introduce single particle open quantum systems in one spatial dimension and review how spectral properties of the Lindbladian influence the relaxation towards the steady state in finite volume.

Afterwards, we briefly discuss spectral properties of infinite volume systems and how the spectra of the Lindbladian operators can be calculated using a direct integral of Laurent operators.

We then look at Lindblad operators with a random potential in the Hamiltonian and discuss what localization could mean in this context. In particular, we describe how a gap in the effective Non-Hermitian Hamiltonian implies exponential decay in the off-diagonal elements of the steady state. We discuss how it might more generally be true that dissipation and disorder imply decoherence in our setup.

17.11.2022 15:00 Marco Bresciani (Friedrich-Alexander-Universität Erlangen-Nürnberg):
Existence results in magnetoelasticityRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

We study a variational model of magnetoelasticity at large strains both in the static and in the quasistatic setting. The model features a mixed Eulerian-Lagrangian formulation, as magnetizations are defined on the deformed configuration in the actual space. The magnetic saturation constraint is formulated in the reference configuration and involves the Jacobian determinant of deformations. These belong to the class of possibly discontinuous deformations excluding cavitation introduced by Barchiesi, Henao and Mora-Corral. We establish a compactness result which, in particular, yields the convergence of the composition of magnetizations with deformations. In the static setting, this result provides the existence of minimizers by means of classical lower semicontinuity methods. Our compactness result also allows us to address the analysis in the quasistatic setting, where we examine rate-independent evolution driven by applied loads and boundary conditions in presence of dissipative effects. In this case, we prove the existence of energetic solutions.

17.11.2022 16:30 Sophie Thery (Université Grenoble Alpes):
Study of an iterative method on ocean-atmosphere coupling algorithms including boundary layer parametrisationsRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

The interactions between atmosphere and ocean play a major role in many geophysical situations (climate modelling, cyclone, weather forecast). Therefore, geophysical scale numerical simulation requires coupled atmospheric and oceanic models, which properly represent the behaviour of the boundary layers encompassing the air-sea interface and their two-way interactions. Nowadays, the currently used coupling methods implement a one way coupling through the interface condition. We propose to couple these models through a Schwarz iterative algorithm to represent better the interaction between atmosphere and ocean. In this talk, I will present Schwarz algorithm and how it can be used to implement better the two-way coupling between ocean and atmosphere dynamics. We study in particular the convergence of the iterative algorithm on a one dimensional two-way coupled diffusion equation with Ekman boundary layers. The particularity of this study is to take into account the turbulent phenomena near the ocean-atmosphere interface, considering a spatially variable viscosity coefficient. We first focus on a linear version of the diffusion equation. We will see how the Coriolis effect and the vertical parametrization of the turbulence influence the convergence of the algorithm. We will then consider the full non linear equation that includes non linear interface conditions and non linear diffusion coefficient. I will eventually discuss future research directions for this work.

17.11.2022 16:30 Lars Hesselholt:
Dirac geometryA027 (Theresienstraße 39, 80333 Mathematisches Institut, LMU)

The discovery of solitons and completely integrable partial differential equations (PDEs) provides a paradigm in mathematics and modern physics. Its impact on the development of PDEs in physics (both classical and quantum), pure analysis, and differential geometry can hardly be overrated. In this colloquium talk, I will give an introduction to a class of newly discovered completely integrable PDEs, which exhibit turbulent behavior, i.e., the degree of smoothness of solutions cannot be generally controlled by an infinite hierarchy of conservation laws and thus singularities can form. Indeed, these systems can be seen as infinite-dimensional continuum versions of classical so-called Calogero-Moser systems introduced by Francesco Calogero and Jürgen Moser in 1970s. Part of my talk is based on joint work with Patrick Gérard (Paris-Sud).

21.11.2022 16:30 Tamas Makai (LMU):
Degree sequences of random uniform hypergraphsB 252 (Theresienstr. 39, 80333 München)

Consider the probability that a random graph selected uniformly from the set of r-uniform hypergraphs with n vertices and m edges, has a given degree sequence. Previously the value of this probability has been investigated by Kamčev, Liebenau and Wormald, where they examined degree sequences from very sparse to moderately dense hypergraphs when r=o(n^{1/4}) and the variation of the degrees is small, but exceeds the typical degree variation in random hypergraphs. We extend their results, by establishing this result for dense hypergraphs, which hold for any value of r and allow for a greater variation on the degrees.This is joint work with Catherine Greenhill, Mikhail Isaev and Brendan McKay.

23.11.2022 16:00 Mario Ayala (TUM):
A short introduction to duality for interacting particle systems and its applications to macroscopic fieldsMI 03.04.011 (Boltzmannstr. 3, 85748 Garching)

In this talk we will introduce the concept of duality for a class of interacting particle systems that includes the well-known symmetric inclusion and exclusion processes. By means of examples we will show how to derive self-duality relations from the knowledge of reversible measures, using the so-called algebraic approach. We will also show how to derive orthogonal dualities by means of the well known three terms recurrence relations satisfied by the classical (discrete) families of orthogonal polynomials.

As a first application, we will use duality to show the propagation of positive correlations for one of our processes. Later we will also provide applications of (self-)duality in the study of macroscopic fields of interacting particle systems, in particular in the derivation of hydrodynamic limits and fluctuations from the hydrodynamic limit. Finally, if time permits, we will try to make the case (via examples) that interacting particle systems that enjoy the property of self-duality can be used as a sort of laboratory to test the validity of general claims about interacting particle systems.

Disclaimer: This material is mostly based on the work of others (to mention a few: C. Franceschini, G. Carinci, C. Giardina, and F. Redig). However, some of the applications of orthogonal duality are based on the speaker's own work together with Gioia Carinci and Frank Redig.

24.11.2022 15:45 Tobias Ried (MPI für Mathematik in den Naturwissenschaften, Leipzig):
A variational approach to the regularity of optimal transportationMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

In this talk I want to present a purely variational approach to the regularity theory for the Monge-Ampère equation, or rather optimal transportation, introduced by Goldman—Otto. Following De Giorgi’s strategy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, which leads to a one-step improvement lemma, and feeds into a Campanato iteration on the C^{1,\alpha}-level for the displacement. The variational approach is flexible enough to cover general cost functions by importing the concept of almost-minimality: if the cost is quantitatively close to the Euclidean cost function |x-y|^2, a minimiser for the optimal transport problem with general cost is an almost-minimiser for the one with quadratic cost. This allows us to reprove the C^{1,\alpha}-regularity result of De Philippis—Figalli, while bypassing Caffarelli’s celebrated theory. (This is joint work with F. Otto and M. Prod’homme)

24.11.2022 17:15 Michiel Renger (TUM):
Macroscopic Fluctuation Theory on discrete spacesMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

Often thermodynamical phenomena are described microscopically by a randomly evolving particle system, or macroscopically by an evolution equation, and the two levels of descrip- tion are connected by sending the number of particles to infinity. Onsager and Machlup pos- tulated that microscopic systems in detailed balance (reversible Markov process) behave as a gradient flow on the macroscopic level. This principle is now well-understood and can be made precise via the theory of large deviations. In order to understand the behaviour of non-equilibrium systems (not in detailed balance/nonreversible), one commonly studies par- ticle densities as well as particle fluxes; this is the topic of Macroscopic Fluctuation Theory. Classically the large deviations yield a natural Hilbert structure that allows to decompose the dynamics into a gradient flow component (dissipating free energy) and a Hamiltonian com- ponent (conserving energy). For systems where the particles hop between discrete sites, and the macroscopic equation is a system of ODEs, such natural Hilbert space is not available. We present a generalised Macroscopic Fluctuation Theory, sufficiently general to apply to, for example zero-range processes and chemical reactions. This work lies on the boundary between probability, analysis and physics, but I will mostly fo- cus on the analysis and physics part.

29.11.2022 16:00 Mirjeta Palloshi:
Kombinatorische Matching-Algorithmen für die KurszuweisungenMI 02.04.011 (Boltzmannstr. 3, 85748 Garching)

Eine bestmögliche Einteilung von Studierenden zu Kursen stellt für viele Algorithmen eine große Herausforderung dar. Dabei sind Eigenschaften wie Strategiesicherheit, Pareto-Effizienz und Fairness schwer einzuhalten. Es haben sich in diesem Zusammenhang zwei bekannte Algorithmen bewährt und sollten bezüglich ihrer Vor- und Nachteile verglichen werden.

30.11.2022 12:15 Elizabeth Gross (University of Honolulu, USA):
Phylogenetic network inference with invariantsBC1 2.01.10 (Parkring 11, 85748 Garching)

Phylogenetic networks provide a means of describing the evolutionary history of sets of species believed to have undergone hybridization or horizontal gene flow during the course of their evolution. The mutation process for a set of such species can be modeled as a Markov process on a phylogenetic network. Previous work has shown that a site-pattern probability distributions from a Jukes-Cantor phylogenetic network model must satisfy certain algebraic invariants, i.e. polynomial relationships. As a corollary, aspects of the phylogenetic network are theoretically identifiable from site-pattern frequencies. In practice, because of the probabilistic nature of sequence evolution, the phylogenetic network invariants will rarely be satisfied, even for data generated under the model. Thus, using network invariants for inferring phylogenetic networks requires some means of interpreting the residuals when observed site-pattern frequencies are substituted into the invariants. In this work, we propose an approach that combines statistical learning and phylogenetic invariants to infer small, level-one phylogenetic networks, and we discuss how the approach can be extended to infer larger networks. This is joint work with Travis Barton, Colby Long, and Joseph Rusinko.