Interactions like communication, cooperation and competition abound in the microbial world. They play an important role in natural ecosystems, infection, and agriculture. This talk focuses on experimental and mathematical approaches to understand two types of interactions in Pseudomonas bacteria: Quorum sensing, a mechanism of cell-cell communication that coordinates other cooperative behaviors, and iron acquisition via diffusible siderophores as both a cooperative and competitive behavior within and between species.

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

The onset of pattern formation, where a localized perturbation of a destabilized ground state leads to an invading front leaving periodic patterns in its wake, is well-understood. In spatially homogeneous systems such pattern-forming fronts are unstable as any perturbation ahead of the front grows exponentially in time due to the instability of the ground state. Nevertheless, pattern-forming fronts are observed in various spatially inhomogeneous settings such as light-sensing reaction-diffusion systems, directional solidification of crystals or ion beam milling. In these settings the unstable state is only established in the wake of the heterogeneity after which patterns start to nucleate. Consequently, perturbations cannot grow far ahead of the interface of the pattern-forming front. This begs the question of whether stability can be rigorously established. In this talk, I answer this question affirmative by presenting a stability result for pattern-forming fronts against $L^2$-perturbations in the spatially inhomogeneous complex Ginzburg-Landau equation. A technical challenge is posed by the presence of unstable absolute spectrum which prohibits the use of standard tools such as exponential dichotomies. Instead, we projectivize the linear flow and study the associated matrix Riccati equation on the Grassmannian manifold. Eigenvalues can then be identified as the roots of the meromorphic Riccati-Evans function. This is joint work with Ryan Goh (Boston University).

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

We consider the XY-model, a model for continuous spins on the d-dimensional Euclidean lattice, and discuss a time evolution for this model. In this time evolution, every spin behaves like a Brownian motion drifting to a local minimum of the Hamiltonian. If the underlying lattice is one-dimensional or if the temperature is high (for d>1), then the process converges to the unique Gibbs measure in a uniform sense. This is proven by showing a logarithmic Sobolev inequality for the Gibbs measure.

We study the inverse problem for persistent homology: For a fixed simplicial complex, we analyse the fiber of the continuous map PH on the space of filters that assigns to a filter the barcode of its associated sublevel set filtration. We find that PH is best understood as a map of stratified spaces. Over each stratum of the barcode space the map PH restricts to a fiber bundle with fiber a polyhedral complex. We illustrate the theory on the example of the triangle, which is rich enough to have a Möbius band as one of its fibers.

Link: https://tum-conf.zoom.us/j/99397204146?pwd=VkVXMFVQUXRtVEt2RVJjMVNGR0RBQT09

Seminar page: https://wiki.tum.de/display/topology/Applied+Topology+Seminar