Jens Walter Fischer

Jens Walter Fischer

Mathematician with an interest in all quantitative domains.

Posts by Collection

portfolio

Convergence of Deffuant model with opinions absolutely continuous with respect to Lebesgue measure (ID: CDM230517)

We lift the analysis of the Deffuant model with \(k\) agents exhibiting opinions absolutely continuous with respect to Lebesgue measure to the manifold of the densities implied by the model. We conjecture that there is an embedding into 3-dim. space based on the two model parameters and a fixed initial density for i.i.d. opinions with a unique singularity. This singularity gives the limit law of the model and is probably given by a Dirac delta over the projection of the support of the \(k\) product of the initial density to the digonal of the \(k\) cube.

The log grid and its measurements (ID: LGM230517)

We consider the positive cone of the $d$-dimensional grid and apply to any vertex a logarithm, where the basis is at first irrelevant. We observe in the $2$-dimensional case that along the diagonal the cab driver distance of grid points converges to $0$ but the distance between points of the form $(a,x)$ and $(x,x)$ grow infinitely far appart as $x$ goes to infinity. The project is designed to investigate the implied geometry of the log grid.

preprints

publications

Random population dynamics under catastrophic events

Published in Journal of Applied Probability, 2022

This paper extends the classical birth-death process by a point catastrophe event which eradicates a part of the population immediately. We consider times to catastrophe as well as the long term development of the expected populaton size.

Recommended citation: Patrick Cattiaux, Jens Fischer, Sylvie Roelly and Samuel Sindayigaya. (2022). "Random population dynamics under catastrophic events." Journal of Applied Probability 1.

talks

Consensus Dynamics and Exclusion processes

Published:

I presented a first result linking opinion dynamics with exclusion processes in a random absorbing environment. The lift to token graphs was the central focus of this talk which formed the basis of further discussion regarding the applicability of Markov chain results to complex systems of heterogeneous agents.

Day of the PhD Students

Published:

I gave an introdution to mathematical modelling of population dynamics based on my work on generalized birth-death processes with catastrophes and opinion dynamics in heterogenous populations. The focus lay on the explanation of the basic assumptions and ideas without too many technical details.

Social conflict and exclusion processes - a link

Published:

I presented a negativ result concerning the reversibility of some generalized exclusion processes due to the strong lokal dependencies on the structure of the underlying discrete state space. Using combinatorical methods I showed that for a fixed transition structure, possible state spaces can be separated into two classes based on the reversibility property of the exclusion process. Both classes are completely defined by solutions of a Diphontine equation of order four in four variables, which encodes cycles of length four of the process.

Random Network Dynamics - A reduced Echo Chamber model

Published:

The content of this talk was a summary of my research work of my PhD up to this point with a focus on the necessity to employ algebraic, combinatoric and graph theoretic techniques to answer questions regarding generalized exclusion processes associated to opinions dynamics. I explained the difficulties one might encounter while using functional analytical technqiues based on Eigenvalues of some associated operator and the trade-off I had to accept between the two points of view.

Absorption times for generalized exclusion processes in random absorbing environments

Published:

I presented a theoretical framework for absorption times for generalized exclusion processes in random absorbing environments on a graph motivated by the Echo Chamber Model as an example. I discussed the dependency on the underlying graph structure relativ to the structure of the induced subgraph of the absorbing sites. I gave a general upper bound, combining the quasi-stationary distribution of the processes with an extended notion of Token graphs and quantified its quality for a special case of the Echo Chamber Model.

High-dimensional grid exploration using self-avoiding exclusion processes

Published:

I discussed an approach based on self-avoiding exclusion processes inspired by swarm/genetic algorithms to explore high dimensional grids. Those are considered now as one option in post-quantum cryptography and pose interesting problems for self-avoiding exlcusion processes which form a natural extension of a self-avoiding random walk. The possible particle transitions, I discussed, left the next neighbor structure of graphs behind them and focused on transitions based on a given lattice basis and geometrically distributed jump lengths in one randomly picked direction. This talk used many notions from my thesis but applied in a completely different field.

teaching

Teaching experience 1

Undergraduate course, University 1, Department, 2014

This is a description of a teaching experience. You can use markdown like any other post.

Teaching experience 2

Workshop, University 1, Department, 2015

This is a description of a teaching experience. You can use markdown like any other post.