Power Operations in Higher Semiadditive CategoriesOnline (Boltzmannstr. 3, 85748 Garching)

In a 2013 paper, J. Lurie and M. Hopkins introduced the notion of ambidexterity, a kind of higher semiadditivity, and showed that it applies to certain categories of spectra arising in chromatic homotopy theory. Specifically the limit and colimit over a \pi-finite space in this category both exist and are actually equivalent. This is as a homotopy theoretic generalisation of the classical notion of a semiadditive category, where all finite products and coproducts - i.e. limit and colimit over a set - exist and agree. In the classical case this implies that there is an addition on the morphism sets, whereas higher semiadditivity lets us add morphisms "over spaces".

Building on this framework, S. Carmeli, M. Schlank and L. Yanovski in 2018 proved that when a morphism space in such a category is a (homotopy theoretic) ring, then the interplay between the multiplication of the ring and the addition over a classifying space generates a p-derviation: a kind of power operation, which they successfully used to prove new properties about the aforementioned categories of spectra.

We plan to construct a whole system of power-operations on these rings which we suspect gives them the structure of beta-rings.

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

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

Analysis of Fractal Characteristics of Complex Networks and Its ApplicationVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

The fractal property is one of the most important properties in complex networks. It describes the power law relationship between characteristics of the box and the box size. There are numerous research studies focusing on the fractal property in networks through different dimensions. In order to study the problems across various disciplines, fractal dimension and local dimension are proposed to study network and node properties respectively. The different dimensions used to describe the fractal property of networks and their applications will be discussed in this talk. Through these studies, we emphasize that the fractal property is an important tool for understanding network characteristics. We will give our conclusion and discuss possible future directions for fractal dimension research.

A Simple Measure Of Conditional Dependence(using Zoom, see http://go.tum.de/410163 for more details) (Parkring 13, 85748 Garching)

We propose a coefficient of conditional dependence between two random variables $Y$ and $Z$, given a set of other variables $X_1, \cdots , X_p$, based on an i.i.d. sample. The coefficient has a long list of desirable properties, the most important of which is that under absolutely no distributional assumptions, it converges to a limit in $[0, 1]$, where the limit is 0 if and only if $Y$ and $Z$ are conditionally independent given $X_1, \cdots , X_p$, and is 1 if and only if Y is equal to a measurable function of $Z$ given $X_1, \cdots , X_p$. Moreover, it has a natural interpretation as a nonlinear generalization of the familiar partial $R^2$ statistic for measuring conditional dependence by regression. Using this statistic, we devise a new variable selection algorithm, called Feature Ordering by Conditional Independence (FOCI), which is model-free, has no tuning parameters, and is provably consistent under sparsity assumptions. A number of applications to synthetic and real datasets are worked out.

Latent-variable modeling: causal inference and false discovery control(using Zoom, see http://go.tum.de/410163 for more details) (Parkring 13, 85748 Garching)

Many driving factors of physical systems are latent or unobserved. Thus, understanding such systems and producing robust predictions crucially relies on accounting for the influence of the latent structure. I will discuss methodological and theoretical advances in two important problems in latent-variable modeling. The first problem focuses on developing false discovery methods for latent-variable models that are parameterized by low-rank matrices, where the traditional perspective on false discovery control is ill-suited due to the non-discrete nature of the underlying decision spaces. To overcome this challenge, I will present a geometric reformulation of the notion of a discovery as well as a specific algorithm to control false discoveries in these settings. The second problem aims to estimate causal relations among a collection of observed variables with latent effects. Given access to data arising from perturbations (interventions), I will introduce a regularized maximum-likelihood framework that provably identifies the underlying causal structure and improves robustness to distributional changes. Throughout, I will explore the utility of the proposed methodologies for real-world applications such as water resource management.

Synchronization in adaptive networksVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

Adaptive networks are found in numerous phenomena from electrical power grids to neural systems in the brain. A common approach is to model the nodes of the network as Kuramoto phase oscillators. Here we address the question whether and how a dynamical network exhibits a different synchronization threshold and behaviour than the original Kuramoto model. To this end, we propose and study an adaptive network system where the weights of the links between two nodes evolve according to a measure of similarity between these nodes. Partial synchronization is accompanied by the formation of a two-population network, reducing the effective coupling strength that drives synchronization in the Kuramoto model.

A spatially-dependent fragmentation process(using zoom) (Parkring 11, 85748 Garching-Hochbrück)

We define a spatially-dependent fragmentation process, which involves rectangles breaking up into progressively smaller pieces at rates that depend on their shape. Long, thin rectangles are more likely to break quickly, and are also more likely to split along their longest side. We are interested in how the system evolves over time: how many fragments are there of different shapes and sizes, and how did they reach that state? Our theorem gives an almost sure growth rate along paths, which does not match the growth rate in expectation - there are paths where the expected number of fragments of that shape and size is exponentially large, but in reality no such fragments exist at large times almost surely.

Model-Free Data-Driven Science: Cutting Out the Middlemanvia ZOOM (Boltzmannstr. 3, 85748 Garching)

We consider a new class of problems in elasticity, referred to as Data-Driven problems, defined on the space of strain-stress field pairs, or phase space. The problem consists of minimizing the distance between a given material data set and the subspace of compatible strain fields and stress fields in equilibrium. We find that the classical solutions are recovered in the case of linear elasticity. We identify conditions for convergence of Data-Driven solutions corresponding to sequences of approximating material data sets. Specialization to constant material data set sequences in turn establishes an appropriate notion of relaxation. We find that relaxation within this Data-Driven framework is fundamentally different from the classical relaxation of energy functions. For instance, we show that in the Data-Driven framework the relaxation of a bistable material leads to material data sets that are not graphs.

Shear-induced chaos and conditioned Lyapunov exponents: A case study of the Bautin bifurcation Virtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

To analyze local dynamical properties of random dynamical systems, Engel and his co- workers introduced the notion of conditioned Lyapunov exponents in the setup of killed Markov processes and discussed it for the Hopf normal form with additive noise. This talk builds up on this idea and investigates differences between the dynamical properties for bounded regions measured with the conditioned Lyapunov exponent and the dynamical properties of the whole state space studying the Bautin bifurcation with additive noise. Besides analytical results, such as proving that the Bautin normal form with additive noise is indeed a random dynamical system, finding an upper bound for the top Lyapunov exponent, and approximating the solution of its Fokker-Planck equation, this thesis also provides three different numerical methods, namely a direct sampling algorithm, a finite difference scheme, and a Fourier-Chebychev method, calculating the conditioned Lyapunov exponent.

TBa(using zoom) (Parkring 11, 85748 Garching-Hochbrück)

TBA

The firefighter problem on infinite groups and graphs(using zoom) (Parkring 11, 85748 Garching-Hochbrück)

In the Firefighter model, a fire erupts on some finite set X_0 and in every time step all vertices adjacent to the fire catch fire as well (burning vertices continue to burn indefinitely) . At turn n we are allowed to protect f(n) vertices so that they never catch fire. The firefighter problem, also known as the fire-containment problem, asks how large should f(n) be so that we will eventually contain any initial fire. We are mainly interested in the asymptotic behaviour of f in relation with the geometry of the graph, focusing on Cayley graphs. Dyer, Martinez-Pedroza and Thorne proved that the growth rate of f is quasi-isometry invariant. Develin and Hartke proved upper bounds on the containment function for Z^d and conjectured that the correct bounds is cn^{d-2}. In joint work with Rangel Baldasso and Gady Kozma we prove this conjecture and show that this actually holds for any polynomial growth group. We will also survey other results on the problem for larger groups and their relation to the growth rate branching numbers. In the 2nd part of the talk we will introduce another related problem - "fire-retainment" that asks for saving only a positive portion of the graph. This turns out to be much more complicated. We give a full answer for the polynomial growth case and some interesting examples for larger groups. This part is based on joint work with Rangel Baldasso, Maria Gerasimova and Gady Kozma.

Cerebral blood flow regulation using nonlinear control approachesvia ZOOM (Boltzmannstr. 3, 85748 Garching)

We consider the mathematical model of cerebral hemodynamics in the form of two ordinary differential equations suggested by M. Ursino and C.A. Lodi. The control problem in question is to keep the cerebral blood flow treated as the function of arterial-arteriolar blood volume, intracranial pressure and arterial pressure close to some basal value required for tissue metabolism. The rate of change of arterial-arteriolar compliance is taken as the control input. The integrator backstepping approach is used to find tracking control laws.