In Many-Body physics and QFT one often encounters tedious computations of commutators involving creation and annihilation operators. A diagrammatic language introduced by Friedrichs in 1965 allows for cutting down these computations tremendously, while representing the occurring operators in a particularly convenient visual form. We revisit a formula for bosonic commutators in terms of Friedrichs diagrams and prove its fermionic analogue. The talk is based on joint work with Morris Brooks from IST Vienna.

A tipping point presents perhaps the single most significant threat to an ecological system as it can lead to abrupt species extinction on a massive scale. Climate changes leading to the species decay parameter drifts can drive various ecological systems towards a tipping point. We investigate the tipping-point dynamics in multi-layer ecological networks supported by mutualism. We unveil a natural mechanism by which the occurrence of tipping points can be delayed by multiplexity that broadly describes the diversity of the species abundances, the complexity of the interspecific relationships, and the topology of linkages in ecological networks. For a double-layer system of pollinators and plants, coupling between the network layers occurs when there is dispersal of pollinator species. Multiplexity emerges as the dispersing species establish their presence in the destination layer and have a simultaneous presence in both. We demonstrate that the new mutualistic links induced by the dispersing species with the residence species have fundamental benefits to the wellbeing of the ecosystem in delaying the tipping point and facilitating species recovery. Articulating and implementing control mechanisms to induce multiplexity can thus help sustain certain types of ecosystems that are in danger of extinction as the result of environmental changes.

This talk will provide an overview of the recent progress made in exploring Sourav Chatterjee's newly introduced rank correlation. The objective is to elaborate on its practical utility and present several new findings pertaining to (a) the asymptotic normality and limiting variance of Chatterjee's rank correlation, (b) its statistical efficiency for testing independence, and (c) the issue of its bootstrap inconsistency. Notably, the presentation will reveal that Chatterjee's rank correlation is root-n consistent, asymptotically normal, but bootstrap inconsistent - an unusual phenomenon in the literature.

A holomorphic map is locally given by its power series expansion. Thus, the local dynamics of the an iterated holomorphic map near a fixed point can often be determined from a finite number of terms of the expansion at the fixed point. Local invariant sets with locally stable dynamics can then be extended via backward images to global objects with the same long-term dynamics. For polynomials on C, stable dynamics near fixed points are well understood and all stable dynamics arise from fixed points. In C2 both local and global stable dynamics still pose many open questions. New types of dynamics at neutral fixed points in C2 arise from non-trivial multiplicative relations of eigenvalues in the linear part, called resonances. In this talk I will present a construction of examples in C2 with a product of eigenvalues equal to 1. In this so-called one-resonant case, we obtain a non-linear projection to one variable, allowing us to construct a doubly connected attracting open set. To show the “hole” cannot be filled to obtain a larger simply connected attracting set, we impose a small divisor condition on the eigenvalues and show that our attracting set is the whole basin of attraction of the fixed point.

In environmental science applications, extreme events frequently exhibit a complex spatio-temporal structure, which is difficult to describe flexibly and estimate in a computationally efficient way using state-of-art parametric extreme-value models. In this talk, we propose a computationally-cheap non-parametric approach to investigate the probability distribution of temporal clusters of spatial extremes, and study within-cluster patterns with respect to various characteristics. These include risk functionals describing the overall event magnitude, spatial risk measures such as the size of the affected area, and measures representing the location of the extreme event. Under the framework of functional regular variation, we verify the existence of the corresponding limit distributions as the considered events become increasingly extreme. Furthermore, we develop non-parametric estimators for the limiting expressions of interest and show their asymptotic normality under appropriate mixing conditions. Uncertainty is assessed using a multiplier block bootstrap. The finite-sample behavior of our estimators and the bootstrap scheme is demonstrated in a spatio-temporal simulated example. Our methodology is then applied to study the spatio-temporal dependence structure of high-dimensional sea surface temperature data for the southern Red Sea. Our analysis reveals new insights into the temporal persistence, and the complex hydrodynamic patterns of extreme sea temperature events in this region. This is joint work with Raphael Huser.

Multiple stochastic integrals with respect to Brownian motion is a classical topic while its version with respect to stable processes has created minor interest. Their distributions can be simulated using U-statistics. This will be discussed in the first part of the talk. On the other hand this representation allows for statistical applications for observations with slowly decaying tail distributions. I shall present some simulations and give an application from neuroscience.

We study the weak convergence of conditional empirical copula processes indexed by general families of conditioning events that have non zero probabilities. Moreover, we also study the case where the conditioning events are chosen in a data-driven way. The validity of several bootstrap schemes is stated, including the exchangeable bootstrap. We define general multivariate measures of association, possibly given some fixed or random conditioning events. By applying our theoretical results, we prove the asymptotic normality of the estimators of such measures. We illustrate our results with financial data.

We consider Markovian open quantum dynamics (MOQD) in continuous space. We show that, up to small-probability tails, the supports of quantum states evolving under such dynamics propagate with finite speed in any finite-energy subspace. More precisely, we prove that if the initial quantum state is localized in space, then any finite-energy part of the solution of the von Neumann-Lindblad equation is approximately localized inside an energy-dependent light cone. We also obtain an explicit upper bound for the slope of this light cone. Joint work with S. Breteaux, J. Faupin, D.H. Ou Yang, I.M. Sigal, and J. Zhang.

Sample splitting is one of the most tried-and-true tools in the data scientist toolbox. It breaks a data set into two independent parts, allowing one to perform valid inference after an exploratory analysis or after training a model. A recent paper (Neufeld, et al. 2023) provided a remarkable alternative to sample splitting, which the authors showed to be attractive in situations where sample splitting is not possible. Their method, called convolution-closed data thinning, proceeds very differently from sample splitting, and yet it also produces two statistically independent data sets from the original. In this talk, we will show that sufficiency is the key underlying principle that makes their approach possible. This insight leads naturally to a new framework, which we call generalized data thinning. This generalization unifies both sample splitting and convolution-closed data thinning as different applications of the same procedure. Furthermore, we show that this generalization greatly widens the scope of distributions where thinning is possible. This work is a collaboration with Ameer Dharamshi, Anna Neufeld, Keshav Motwani, Lucy Gao, and Daniela Witten.

We consider an electric network on the $d$-dimensional integer lattice with an edge between every two points $x$ and $y$. The conductance of the edge $\{x,y\}$ equals $\|x-y\|^{-s}$, for some $s>d$. We show that the random walk on this network is recurrent if and only if $d \in \{1,2\}$ and $s\geq 2d$. We also discuss how this result relates to the return properties of random walks on percolation clusters, particularly on the two-dimensional weight-dependent random connection model.

Causal effect estimation from observational data is a fundamental task in empirical sciences. It becomes particularly challenging when unobserved confounders are involved in a system. This presentation provides an introduction to front-door adjustment – a classic technique which, using observed mediators, allows to identify causal effects even in the presence of unobserved confounding. Focusing on the algorithmic aspects, this talk presents recent results for finding front-door adjustment sets in linear-time in the size of the causal graph.

Link to technical report: https://arxiv.org/abs/2211.16468

Motivated by new nonlocal models in hyperelasticity, we discuss a class of variational problems with integral functionals depending on nonlocal gradients that correspond to truncated versions of the Riesz fractional gradient. We address several aspects regarding the existence theory of these problems and their asymptotic behavior. Our analysis relies on suitable translation operators that allow us to switch between the three types of gradients: classical, fractional, and nonlocal. These provide helpful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, the natural convexity notion in the classical calculus of variations, characterizes the weak lower semicontinuity also in the fractional and nonlocal setting. As a consequence of a general Gamma-convergence statement, we derive relaxation and homogenization results. The analysis of the limiting behavior as the fractional order tends to 1 yields localization to a classical model. This is joint work with Javier Cueto (University of Nebraska-Lincoln) and Hidde Schönberger (KU Eichstätt-Ingolstadt).

If explanatory variables and a response variable of interest are simultaneously observed, then multivariate models based on vine pair-copula constructions can be fit, from which inferences are based on the conditional distribution of the response variable given the explanatory variables.

For applications, there are things to consider when implementing this idea. Topics include: (a) inclusion of categorical predictors; (b) right-censored response variable; (c) for a pair with one ordinal and one continuous variable, diagnostics for copula choice and assessing fit of copula; (d) use of empirical beta copula; (e) performance metrics for prediction/classification and sensitivity to choice of vine structure and pair-copulas on edges of vine; (f) weighted log-likelihood for ordinal response variable; (g) comparisons with linear regression methods.

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