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Stability Analysis of Multiplayer Games on Simplicial Complexes in Adaptive NetworksVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

We develop models of multiplayer games based on cooperation, the Snowdrift game and the Prisoner’s Dilemma, on adaptive networks. They contain explicit interactions of multiple players on simplicial complexes. All operations of and on the network are based on game theoretical properties of the respective games. The evolution of the models over time is described by moment equations and is closed by pair approximation. The stability of equilibria is examined when irrational decisions in simplices are added into the models.

A percolation model without positiv correlattionB 252 (Theresienstr. 39, 80333 München)

We introduce a bond-percolation model that is a modification of the corrupted compass model introduced by Christian Hirsch, Mark Holmes and Victor Kleptsyn (2021).On a given graph we start in each vertex independent with probability p a random walk of length L. We make an edge occupied if it was used by a random walk. This model does not exhibit positive correlation.If L is choosen such that there is percolation for p=1, we have a sharp phase transition for p. We discuss the question of percolation on the hypercubic lattice and show that on the square lattice percolation occurs for L=2.

Algebraic Statistics with a View towards PhysicsBC1 2.1.10 (with additional stream using Zoom, see http://go.tum.de/410163 for more details) (Parkring 11, 85748 Garching)

We discuss the algebraic geometry of maximum likelihood estimation from the perspective of scattering amplitudes in particle physics. A guiding example is the moduli space of n-pointed rational curves. The scattering potential plays the role of the log-likelihood function, and its critical points are solutions to rational function equations. Their number is an Euler characteristic. Soft limit degenerations are combined with certified numerical methods for concrete computations.

Periodic behavior of mean-field systemsVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

We will study non-linear mean-field Fokker-Planck equations describing the infinite population limit of interacting excitable particles subject to noise. Taking a slow-fast dynamics approach we will describe the emergence of periodic behaviors induced by the noise and the interaction, considering in particular the case in which each unit evolves according to the FitzHugh Nagumo model. This talk is linked to the one given by my co-author Eric Luçon the following week, in which he will speak about the long time behavior of the population of particles when the population is finite.

Large-time dynamics of mean-field interacting diffusions along a limit cycleVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

This talk is the natural continuation of the previous talk of Christophe Poquet which concerned the existence of periodic solutions to nonlinear Fokker-Planck equations. We are here interested in the microscopic counterpart of the same problem: nonlinear Fokker-Planck equations are natural limits of the empirical measure of N mean-field interacting diffusions as N goes to infinity. Standard propagation of chaos estimates show that this limit remains relevant only up to times that remains bounded in N. A natural question is then to ask about the dynamics of the empirical measure of the system on a larger time scale. We answer to this question in the case the FP limit possess a smooth and stable limit cycle. The main result of the talk will be to show that, on a time scale of order N, the empirical measure remains with high probability close to the periodic orbit with a diffusive dynamics along the limit cycle.

A new state space of algebraic measure trees for stochastic processesOnline: attendB 252 (Theresienstr. 39, 80333 München)

In the talk, I present a new topological space of ``continuum'' trees, which extends the set of finite graph-theoretic trees to uncountable structures, which can be seen as limits of finite trees. Unlike previous approaches, we do not use the graph-metric but formalize the tree-structure by a tertiary operation on the tree, namely the branch-point map. The resulting space of algebraic measure trees has coarser equivalence classes than the more classical space of metric measure trees, but the topology preserves more of the tree-structure in limits, so that it is incomparable to, and not coarser than, the standard topologies on metric measure trees. With the example of the Aldous chain on cladograms, I also illustrate that our new space can be very useful as state-space for stochastic processes in order to obtain path-space diffusion limits of tree-valued Markov chains.

What is Mathematical Consciousness Science?Online: attend (Code 101816)MI 03.04.011 (Boltzmannstr. 3, 85748 Garching)

In the last three decades, the problem of consciousness - how and why physical systems such as the brain have conscious experiences - has received increasing attention among neuroscientists, psychologists, and philosophers. Recently, a decidedly mathematical perspective has emerged as well, which is now called Mathematical Consciousness Science. In this talk, I will give an introduction and overview of Mathematical Consciousness Science for mathematicians, including a bottom-up introduction to the problem of consciousness and how it is amenable to mathematical tools and methods.

Link and Passcode: https://tum-conf.zoom.us/j/96536097137 Code 101816