## Events

18.07.2024 Oberseminar Darmstadt

Jakob Björnberg (Universität Göteborg)

### Dimerisation in mirror models and quantum spin chains

weiterlesen12.07.2024 Rhein-Main-Kolloquium

Nicolas Champagnat (Nancy)

Anita Winter (Duisburg-Essen)

### RMK Frankfurt

The grapheme-valued Wright-Fisher Diffusion with mutation --- In Athreya, den Hollander and Röllin (2021) models from population genetics were used to de fine stochastic dynamics in the space of graphons that arise as continuum limits of dense graph sequencess. In this talk we extend this framework to a model with mutation. In particular, we define a finite graph valued Markov chain that can be associated with the infinite many alleles model, and establish a diffusion limit as the number of vertices goes to infinity. For that we encode finite graphs as graphemes. Graphems are those graphons that can be represented as a triple consisting of a topological vertex space, an adjacency matrix and a sampling measure. The space of graphons is equipped with convergence of sample subgraph densities. (joint work with Andreas Greven, Frank den Hollander and Anton Klimovsky) -------------------------------------------------------------------- Scaling limits of individual-based models in adaptive dynamics and local extinction of populations (N. Champagnat) --- Starting from an individual-based birth-death-mutation-selection model of adaptive dynamics with three scaling parameters (population size, mutation rate, mutation steps size), we will describe several scaling limits that can be applied to this model to obtain macroscopic models of different natures (PDE, Hamilton-Jacobi equation, stochastic adaptive walks, canonical equation of adaptive dynamics), which allow to characterize the long-term evolution of the population. Motivated by biological criticisms on the time-scale of evolution and the absence of local extinctions in the obtained macroscopic models, we propose new parameter scalings under which we can characterize the evolution of population sizes of the order of $K^\beta$, where $K$ is the order of magnitude of the total population size, and which allows for local extinction of subpopulations. This presentation will gather results obtained with several collaborators: Régis Ferrière, Sylvie Méléard, Amaury Lambert, Viet Chi Tran, Sepideh Mirrahimi, Vincent Hass. weiterlesen27.06.2024 Oberseminar Darmstadt

David Heson (Mississippi State University)

### Computational Methods for Analyzing 1-Dimensional Heisenberg Spin Rings and Similar Systems

weiterlesen21.06.2024 Rhein-Main-Kolloquium

Jiří Černý (Universität Basel)

### RMK Mainz

In the presentation, I will consider two related models: the one-dimensional branching Brownian motion in random environment (BBMRE), and the randomized F-KPP equation (rFKPP). I will discuss the following natural questions: (1) Given a typical realisation of the environment, are the distributions of the maximal particle of the BBMRE (re-centred around their medians) tight? (2) For the same environment, is the width of the front of the "travelling-wave" solution to rFKPP uniformly bounded in time? Surprisingly, it turns out that the answers to these questions can be different. This highlights that, when compared to the settings of homogeneous branching Brownian motion and the F-KPP equation in a homogeneous environment, the introduction of a random environment leads to a much more intricate behaviour. The presentation is based on joint works with A. Drewitz, L. Schmitz, and P. Oswald. weiterlesen21.06.2024 Rhein-Main-Kolloquium

Stein Andreas Bethuelsen (University of Bergen)

### RMK Mainz

Assign to each lattice point of Z^d a Poissonian number of particles and let each of them evolve independently as discrete-time simple random walks. On top of this dynamically evolving environment we consider an additional random walk whose jump transition depends on whether there are particles present at its location or not. This is the so-called random walk on random walks model. Previous studies have concluded that this model on Z^d, d\geq1, has a diffusive scaling when the density of particles is sufficiently low or sufficiently high. We will argue that this holds for all densities for the model on Z^d with d\geq 5. Our proof of this rely on a novel domination result for the dynamic environment that, when combined with coupling arguments and standard random walk estimates, yield uniform mixing bounds for the so-called local environment process. Based on joint work, partly in progress, with Florian Völlering (University of Leipzig) weiterlesen20.06.2024 Oberseminar Darmstadt

Tejas Iyer (WIAS Berlin)

### Persistent hubs in generalised preferential attachment trees

weiterlesen06.06.2024 Stochastik-Kolloquium Frankfurt

### B cell phylodynamics and mean-field multi-type birth and death processes

No knowledge of biology will be necessary to understand this talk. Germinal centers (GC) are micro-anatomical structures that transiently form in lymph nodes during an adaptive immune response. In a GC, B cells—the cells that make antibodies—diversify and compete based on the ability of the antibodies they express to recognize a foreign antigen molecule. As GC B cells proliferate, they undergo targeted mutations in the genomic locus encoding the antibody protein that can modify its antigen binding affinity. Via signaling from other GC cell types, the GC can monitor the binding phenotype of the B cell population it contains and provide survival signals to B cells with the highest-affinity antibodies (i.e., birth and death rates depend on type). Motivated by this mechanism, we develop a mean-field model that couples the birth and death rates in a focal multi-type birth and death process (MTBDP) with D types to the empirical distribution of states—i.e., the mean-field over an exchangeable system of N replica MTBDPs. The empirical distribution process of the N replicas converges to a deterministic probability measure-valued flow as N goes to infinity. In the limit, the focal process evolves as a multi-type birth and death process with rates governed by the probability measure-valued flow which is in turn the flow of one-dimensional marginal distributions of the focal process. Individual focal processes become independent in the limit and this holds out the hope of inference being feasible for this model. This is joint work with William S. DeWitt, Ella Hiesmayr, and Sebastian Hummel. weiterlesen04.06.2024 Oberseminar Mainz

Samuel Modee (University of Bergen)

### Some rigorous results on the zombie infection model

The so-called zombie infection model is a one-parameter family of interacting particle systems introduced in the physics literature in the 2010’s. This model can be seen as a natural hybrid between a SIR infection model and the bias-voter model. After a brief survey on the basics of this model, we present in this talk some rigorous results, focusing on monotonicity properties and bounds on the probability of a zombie invasion. This is based on joint work, in progress, with Erik Broman and Stein Andreas Bethuelsen. weiterlesen28.05.2024 Oberseminar Mainz

Stein Andreas Bethuelsen (University of Bergen)

### Mixing for Poisson representable processes and the contact process

In this talk I will present some new insights on so-called Poisson representable processes, a general class of {0,1}-valued processes recently introduced by Forsström, Gantert and Steif. Particularly, I will discuss a new characteristic of these in terms of certain mixing properties. As an application thereof, I will argue that the upper invariant measure of the contact process on Z^d is not Poisson representable, thereby answering a question raised in the above mentioned work. This relies on the upper invariant measure satisfying certain directional mixing properties, but not their spatial equivalent. Moreover, the general approach extends to other processes having similar properties, such as the plus phase of the Ising model on Z^2 in the phase transition regime. weiterlesen16.05.2024 Oberseminar Darmstadt

Mark Sellke (Harvard University)

### Confinement of Unimodal Probability Distributions and an FKG-Gaussian Correlation Inequality

weiterlesen07.05.2024 Oberseminar Mainz

Prof. Dr. Steffen Dereich (Universität Münster)

### New error bounds for the Adam optimisation algorithm

In this talk, I will give an introduction concerning the training of artifcial neural networks based on stochastic gradient descent algorithms. Whereas there are various approaches for carrying out an error analysis for classical stochastic gradient descent, significantly less is known for more elaborate algorithms such as the Adam algorithm. Recently, we established a new approach which allows us to provide error estimates and to reduce this gap. The talk is based on joint work with Arnulf Jentzen. weiterlesen25.04.2024 Oberseminar Darmstadt

Ralph Neininger (Goethe-Universität Frankfurt)

### Recursive distributional equations in applications

weiterlesen24.04.2024 Andere Veranstaltungen

Christoph Thäle (Ruhr-Universität Bochum)

### Mathematisches Kolloquium TU Darmstadt

weiterlesen19.–23.02.2024 Sommer-/Winterschule

Eero Saksmann (University of Helsinki)

Julien Barral (Université Paris 13)

### Spring School 2024

weiterlesen02.02.2024 Rhein-Main-Kolloquium

Peter Mörters (Universität Köln)

Alessandra Bianchi (Università degli Studi di Padova)

### RMKS Frankfurt/Main 02.02.2024

weiterlesen25.01.2024 Oberseminar Darmstadt

Matthias Birkner (Johannes Gutenberg-Universität Mainz)

### Survival and complete convergence for a branching annihilating random walk

weiterlesen11.01.2024 Oberseminar Darmstadt

Christian Mönch (JGU Mainz)

### Inhomogenous long-range percolation in the weak decay regime

weiterlesen29.11.2023 Andere Veranstaltungen

Benjamin Gess (Universität Bielefeld & MPI Leipzig)

### Mathematisches Kolloquium TU Darmstadt

weiterlesen24.11.2023 Rhein-Main-Kolloquium

Véronique Gayrard (HCM Marseille)

Jean-Christophe Mourrat (ENS Lyon)

### RMKS 24.11.2023 Mainz

weiterlesen09.11.2023 Oberseminar Darmstadt

Alexander Glazman (Universität Innsbruck)