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