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.
Quantum Computing (QC) has witnessed remarkable growth and has garnered significant attention in recent years, often accompanied by strong hype. This introductory talk aims to provide a comprehensive overview of the fundamental concepts in QC and its potential for Mathematicians. The talk begins by explaining the core principles of QC like entanglement and superposition. Furthermore, the talk delves into the mathematical underpinnings of QC, demonstrating how operations on a quantum computer can be elegantly described using linear algebra. This approach enables the manipulation and analysis of quantum states, paving the way for the development of quantum algorithms. As an illustration of the power of quantum computing, the concept of quantum teleportation will be introduced, showcasing the remarkable ability to transmit quantum information using classical channels. The talk also highlights two influential quantum algorithms: Shor's and Grover's algorithm. Shor's algorithm, renowned for its impact on cryptography, presents a polylogarithmic approach to factoring large numbers, thereby threatening conventional encryption methods. On the other hand, Grover's algorithm offers a powerful tool for database search, potentially providing a quadratic speedup compared to classical search algorithms. Lastly, the current state of the art in quantum computing is addressed.
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.
Pattern-forming systems with broken reflection symmetry $x \mapsto -x$ often exhibit periodic wavetrains with non-zero phase velocity, which bifurcate from a spatially homogeneous state as a bifurcation parameter increases. This type of instability is referred to as a Turing-Hopf or oscillatory instability and it can be found for example in a flow down a heated, inclined plane. In experiments, one observes that these periodic patterns typically arise in the wake of an invading heteroclinic front, which invades the unstable homogeneous state. Mathematically, this behavior can be modelled with a modulating traveling front solution, a traveling front connecting a periodic wavetrain to a spatially homogeneous state. I will present recent existence results for such solutions in a prototypical model with broken reflection symmetry.
The mean-field problem for interacting diffusions aims to derive a macroscopic continuum model to approximate the particle system when the number of particles is large. In this talk, we improve the mean-field convergence for interacting diffusions in two directions. The first part is to study the central limit theorem for interacting diffusions, which provides a better continuum approximation than the mean-field convergence. The second part is to study the mean-field problem for interacting diffusions with inhomogeneous interactions, i.e., the interactions are weighted, therefore these particles are not exchangeable. For both topics, our results apply to the stochastic vortex model related to the 2D Navier-Stokes equation. This talk is based on joint work with Zhenfu Wang (Peking University) and Rongchan Zhu (Beijing Institute of Technology).
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.
Propagation and generation of "chaos" is an important ingredient in rigorous control of applicability of kinetic theory, in general. Chaos can here be understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system. Cumulant hierarchy of these random variables thus often gives a way of controlling the evolution and the degree of such independence, i.e., the amount of "chaos" in the system. In this talk, we will consider two, qualitatively different, example cases for which kinetic theory is believed to be applicable: the discrete nonlinear Schrodinger evolution (DNLS) with suitable random, spatially homogeneous initial data, and the stochastic Kac model. In both cases, we set up suitable random variables and propose methods to control the evolution of their cumulant hierarchies. The talk is based on joint work with Aleksis Vuoksenmaa, and earlier works with Matteo Marcozzi, Alessia Nota, and Herbert Spohn.
Optimal balance is a numerical method to compute a point on the slow manifold of a two-scale dynamical system, where the fast time scale is characterized by rapid oscillations. We describe how the method is based on the concept of adiabatic invariance of the fast degrees of freedom, show that an implementation by backward-forward nudging is quasi-convergent - convergent up to exponentially small residuals - and give examples of the use of optimal balance in geophysical fluid dynamics.
In this talk, we describe the kinetic equation for the Bogoliubov excitations of the Bose-Einstein Condensate. We find three collisional processes: One of them describes the 1↔2 interactions between the condensate and the excited atoms. The other two describe the 2↔2 and 1↔3 interactions between the excited atoms themselves. This is a joint work with Yves Pomeau.
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
Propagation and generation of "chaos" is an important ingredient for rigorous control of applicability of kinetic theory. Chaos can here be understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system. In this talk based on a joint work with Aleksis Vuoksenmaa, we consider the stochastic Kac model with random velocity exchange. In earlier pioneering works by Janvresse, Carlen, Carvalho and Loss, Mischler and Mouhot, properties such as spectral gap and propagation of chaos starting from certain permutation invariant states have been determined. Here, we focus on generation of chaos, in the sense of approach to statistical independence and permutation invariance. We set up suitable random variables and propose methods to control the evolution of their cumulants.
We investigate the value function of the deterministic and stochastic shallow lake problem, which turns out to be a uniquely determined viscosity solution to a Hamilton-Jacobi-Bellman equation. We present a verification and a comparison theorem, based on which we prove the existence of optimal control. We numerically investigate the equilibrium distribution and the path properties of the optimally controlled lake, particularly when the system of the lake has an indifference (Skiba) point. This case corresponds to a non-smooth double-well potential and the computation of the mean transition times between the wells is based on a generalisation of the Arrhenius law.
In this talk we discuss the homogenisation of integral functionals with q growth in a bounded domain of ℝ^n, n>q>1, which is perforated by a random number of small spherical holes with random radii and centers. We show that for a class of stationary short-range correlated measures for the centres and radii of the holes, in the homogenised limit we obtained a nonlinear averaged analogue of the “strange term” obtained by Cioranescu and Murat in 1982 in the periodic case. In our case we only require that the random radii have finite (n - q)-moment, which is the minimal assumption to ensure that the expectation of the nonlinear capacity of the balls is finite. Although under this assumption there are holes which overlap with probability one, we can prove that the clustering holes do not have any impact on the homogenisation procedure and the limit functional. This is a joint work with K. Zemas (University of Muenster) and L. Scardia (Heriot Watt University).
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).
Mathematical and computational tools are ubiquitous nowadays, and decision-making in various disciplines leads to optimization problems in very different settings. In this talk we discuss finite-dimensional constrained structured programming in the fully nonconvex setting, capturing a variety of problems that include nonsmooth objectives, combinatorial structures, and nonlinear, set-membership constraints. We will discuss examples that highlight the modeling versatility of nonstandard formulations, with applications in vehicle control, robotics, and data science. Then, the augmented Lagrangian framework is extended to cover this broad problem class, with established asymptotic properties and convergence guarantees under mild assumptions. Finally, seeking to develop algorithms with even wider scope, yet without trading-off performances, we will indicate the theoretical challenges that arise in this unexplored territory.
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.
This talk is concerned with Triebel-Lizorkin Spaces, which are advanced generalizations of Sobolev Spaces. We will see that this function spaces can be used to measure the smoothness and integrability properties of functions and distributions in a very precise way. Since Triebel- Lizorkin Spaces can be described via several equivalent quasi-norms using the Fourier transform, higher order differences, oscillations, wavelets, atoms or quarklets, they are a powerful tool when solving problems stemming from very different areas of analysis. For example they can be used within regularity theory for PDEs to describe the smoothness and integrability properties of unknown solutions. Here it is also possible to investigate demanding PDEs such as the Schrödinger equation or to deal with SPDEs. Using equivalent quarklet characterizations Triebel-Lizorkin Spaces also can be applied to approximate the unknown solution of PDEs by adaptive numerical methods inspired by finite element methods. Finally we will discuss how Triebel-Lizorkin Spaces can be used within the theory of dynamical systems and in connection with neural networks.
Many phenomena can be modeled as network dynamics with punctuate interactions. However, most relevant dynamics do not allow for computational tractability. To circumvent this difficulty, the Poisson Hypothesis regime replaces interaction times between nodes by independent Poisson processes, allowing for tractability in several cases, such as intensity-based models from computational neuroscience. This hypothesis is usually only conjectured, or numerically validated. In this work, we introduce a class of processes in continuous time called continuous fragmentation-interaction-aggregation processes. The state of each node, described by the stochastic intensity of an associated point process, aggregates arrivals from its neighbors and is fragmented upon departure. We consider the replica-mean-field version of such a process, which is a physical system consisting of randomly interacting copies of the network of interest. We prove that the Poisson Hypothesis holds at the limit of an infinite number of replicas.
Packing problems arise in many settings, for example, when loading cargo into a truck or a shop, when assigning virtual servers to real servers, and in general, when limited resources play a role. Packing problems are typically computationally hard; therefore, we do not expect to find efficient algorithms that solve them optimally. Thus, we study approximation algorithms which are efficient algorithms that compute solutions that are provably close to the optimal solution. In this talk, we will see recent results in approximation algorithms for multi-dimensional geometric packing problems and the unsplittable flow on a path problem, an important packing problem that arises in throughput maximization and scheduling.
Given three rings we can arrange them in a way that any two of the rings can be pulled apart, but altogether we cannot. This arrangement, called the Borromean rings, has been used in many cultures, but to prove this property mathematically is a non-trivial task. This linkedness can be shown by the non-vanishing of the “Massey product” in algebraic topology and exhibits one of the first examples of a “higher structure”. More generally, one of the central concepts in topology is to use loops to study the shape of a space. However, when doing this, a weak form of associativity appears, and certain higher structures precisely capture this behaviour. In the last 2 decades, these ideas have entered algebraic geometry and even number theory, for instance, counting the intersection points of algebraic curves which are not transverse correctly, it is helpful to understand the appearing formulas (which of course are much older) using a version of algebraic geometry built using higher structures; and in turn these techniques led to deep new results, most prominently by Scholze and Lurie.
Higher structures of a related but different nature appear when organizing spacetimes and spaces (modelled by manifolds and manifolds with boundaries) into a structure suitable for characterizing certain (mostly topological) quantum field theories. These so-called higher categories include information about the moduli space of spacetimes (varying the spacetime a bit) and can be used to encode the notion of locality of a field theory. Recent progress has been towards understanding symmetries of quantum field theory.
I will give a (gentle!) introduction to these structures by examples and explain how my own research fits into this picture.
We consider a quasistatic nonlinear model in thermoviscoelasticity at a finite-strain setting in the Kelvin-Voigt rheology where both the elastic and viscous stress tensors comply with the principle of frame indifference under rotations. The force balance is formulated in the reference configuration by resorting to the concept of nonsimple materials whereas the heat transfer equation is governed by the Fourier law in the deformed configurations. Weak solutions are obtained by means of a staggered in-time discretization where the deformation and the temperature are updated alternatingly. Our result refines a recent work by Mielke & Roubíček since our approximation does not require any regularization of the viscosity term.
Afterwards, we focus on the case of deformations near the identity and small temperatures, and we show by a rigorous linearization procedure that weak solutions of the nonlinear system converge in a suitable sense to solutions of a system in linearized thermoviscoelasticity. The same property holds for time-discrete approximations and we provide a corresponding commutativity result. The talk will be supplemented by several numerical experiments.
Deep neural networks are usually trained by minimizing a non-convex loss functional via (stochastic) gradient descent methods. Unfortunately, the convergence properties are not very well-understood. Moreover, a puzzling empirical observation is that learning neural networks with a number of parameters exceeding the number of training examples often leads to zero loss, i.e., the network exactly interpolates the data. Nevertheless, it generalizes very well to unseen data, which is in stark contrast to intuition from classical statistics which would predict a scenario of overfitting.
A current working hypothesis is that the chosen optimization algorithm has a significant influence on the selection of the learned network. In fact, in this overparameterized context there are many global minimizers so that the optimization method induces an implicit bias on the computed solution. It seems that gradient descent methods and their stochastic variants favor networks of low complexity (in a suitable sense to be understood), and, hence, appear to be very well suited for large classes of real data.
Initial attempts in understanding the implicit bias phenomen considers the simplified setting of linear networks, i.e., (deep) factorizations of matrices. This has revealed a surprising relation to the field of low rank matrix recovery (a variant of compressive sensing) in the sense that gradient descent favors low rank matrices in certain situations. Moreover, restricting further to diagonal matrices, or equivalently factorizing the entries of a vector to be recovered, shows connection to compressive sensing and l1-minimization.
Despite such initial theoretical results on simplified scenarios, the understanding of the implicit bias phenomenon in deep learning is widely open.
The talk will give a basic introduction to this topic, highlighting initial results and open problems. Based on Joint works with Bubacarr Bah, Hung-Hsu Chou, Johannes Maly, Ulrich Terstiege, Rachel Ward and Michael Westdickenberg
Dynamical low-rank approximation is a framework for time integration of matrix valued ODEs on a fixed-rank manifold based on a time dependent variational principle. Several applications arise from PDEs on product domains, but setting up a corresponding well-posed problem in function space (before discretization) may not be straightforward. Here we present a weak formulation of dynamical low-rank approximation for parabolic PDEs in two spatial dimensions. The existence and uniqueness of weak solutions is shown using a variational time-stepping scheme on the low-rank manifold which is related to practical methods for low-rank integration.