Applied Mathematics Seminars in 2024

Abstract: This seminar will offer a review of recent studies on the entanglement of free-fermion systems on graphs where advantage is taken of methods pertaining to signal processing and algebraic combinatorics. On the one hand, a parallel with time and band limiting problems will be made to obtain a tridiagonal matrix that commutes with the entanglement Hamiltonian for a broad class of situations and on the other hand, the irreducible decomposition of the Terwilliger algebra arising in the context of P-polynomial association schemes will be shown to bring significant computational insights.

Abstract: We systematically derive a normal form for the emergence of radially symmetric blowup solutions from stationary ones in the nonlinear Schrodinger equation. The derivation uses the methodology of asymptotics beyond all orders, and the resulting normal form applies when either the power law or dimension is used as the bifurcation parameter. It yields excellent agreement with numerics in both leading and higher-order effects, is applicable to both infinite and finite domains, and is valid in both critical and supercritical regimes.

Abstract: We would like to solve difficult optimization problems -- perhaps to allocate resources, perhaps to train a machine learning model – and we would like to do it quickly. Is it reasonable to hope that quantum computers can help us do that? In this talk we discuss the current state of quantum optimization and quantum machine learning research, for problems that are classically defined: the problems are described on a classical (i.e., non-quantum) computer, and the solution that we are looking for is also classical. We will see that for some problems quantum algorithms are promising, even if only as an alternative to classical algorithms with different tradeoffs, but in other cases there has been no convincing evidence of their usefulness so far.

Abstract: The prevailing notion of a tipping point (TP) in a complex system is the saddle-node bifurcation. This makes idealizing assumptions, such that the system displays fixed-point dynamics, is homogeneous and "well-mixed", and posesses one dominant positive feedback leading to bi-stability. A benefit of these assumptions is the existence of early-warning signals (EWS) due to the phenomenon of critical slowing down, which in principle allow the detection of an impending TP during a quasi-adiabatic parameter shift.

Many recent efforts exploit this and determine from historical observations whether and when TPs in real-world systems, such as the climate, are being crossed. While the statistical issue of EWS is continuously refined, the underlying assumptions remain uncontested due to the complex and unknown nature of real-world systems. In ongoing research, we try to close this gap by scrutinizing the TP of a collapse of the Atlantic Meridional Overturning Circulation (AMOC) in a climate model using brute-force computing power and various other techniques.

This led to the observation of rate-induced tipping before crossing the bifurcation in the model for non-adiabatic rates of a parameter shift, with a loss of predictability of final state due to chaos. Moreover, due to its spatially-extendedness and multiple feedbacks, the system is far beyond bi-stable, displaying a multitude of coexisting states differing subtly in their spatio-temporal variability. Hence, the TP is not a singular event easily targeted by EWS based on local stability. Finally, we argue that even with solid physical knowledge it may be impossible to know what degrees of freedom carry the critical dynamics of a TP. Thus, to really anticipate a TP it may be necessary to determine the real tipping mechanism by knowledge of the unstable edge state, and if possible mean action paths (instantons) between attractors. This is explored here using edge tracking and rare-event algorithms.

Abstract: Given a point cloud in Euclidean space, and a fixed length scale, we can create simplicial complexes (called Rips complexes) to represent that point cloud using the pairwise distances between the points. By tracking how the homology classes as we increase that length scale we summarise the topology and the geometry of the "shape" of the point cloud, in what is called the persistent homology of its Rips filtration. A major obstacle to more widespread take up of persistent homology as a data analysis tool is the long computation time and, more importantly, the large memory requirements needed to store the filtrations of Rips complexes and compute its persistent homology. We bypass these issues by finding a "Reduced Rips Filtration" which has the same degree-1 persistent homology but with dramatically fewer simplices. The talk is based off joint work is with Musashi Koyama, Facundo Memoli and Vanessa Robins.

Abstract: This talk will serve as an introduction to the (theoretical computer science) area of distribution testing, a subfield of property testing, and discuss some tricks, techniques, and ideas useful to design or analyze distribution testing algorithms. Focusing on the simple yet fundamental example of uniformity testing, the talk will then venture into differential private testing, and discuss a few tools allowing one to (easily?) obtain testing algorithms under various privacy threat models.

Abstract: A vibrating nanomechanical device immersed in a viscous fluid is known to generate an oscillatory flow that can dictate its performance in a range of applications. This includes its sensitivity to mass adsorption and environmental changes. The commonly characterized small-amplitude motion of these devices may generate a nonlinear flow which can modify their response. In this talk, I will discuss experimental and theoretical work aimed at elucidating this nonlinear response, its underlying physical mechanism and exploring its application. This will include the fluid-structure interaction driving the autonomous propulsion of nanorods and the presence of this nonlinear phenomenon in nanomechanical sensing. I will also present a data-driven approach that enables mass measurements using advanced nanoelectromechanical devices with uncharacterized vibrational modes.

Abstract: Motivated by recent experimental developments in atomic physics, a large theoretical effort has been devoted to the analysis of the dynamics of quantum isolated systems after a sudden quench. In this talk I will describe the evolution of a family of classical many-body integrable (Neumann) models after instantaneous quenches of the same kind. The asymptotic dynamics of these models can be fully elucidated, and the stationary properties (in the thermodynamic limit) compared to the ones obtained exactly using a Generalised Gibbs Ensemble. The latter can not only be built but also used to evaluate analytically all relevant observables, a quite remarkable fact for an interacting integrable system with a non-trivial phase diagram.

Abstract: A neural surrogate model of a dynamical system learned from time series data may fail to reproduce its true long-term behavior. In other words, vanilla generalization does not determine the statistical accuracy of a neural surrogate model. When the Jacobian data of the true system is added to the regression problem however, the physical invariant distribution — ensemble/long term behavior — is reproduced by the neural dynamical model. We combine statistical learning theory with ergodic theory of dynamical systems to explain these observations. Our new generalization bounds characterize when a neural ODE model can learn the physical distribution as well as the short-term dynamics. Such a dynamics-aware generalization theory provides a principled basis for constructing new loss functions and implies that purely data-driven, as opposed to hybrid, approaches can also lead to accurate statistical modeling of chaotic systems. Further, we observe that these surrogate models, are able to sample the physical distribution, even though they are not explicitly trained to be generative models and rather use a supervised learning setup with a smaller sample complexity than dynamical generative models. In the second half, we present an infinite-dimensional Newton-Raphson method for transport of a tractable source distribution to a generic target distribution. The Newton-Raphson method finds a zero of a score-residual operator. This Score Operator Newton (SCONE) transport is a composition of transport maps obtained from each Newton iteration, which involves solving an elliptic PDE. The PDE requires a black box function that can evaluate the score of the target distribution. A natural application of SCONE is when the target distribution is a posterior in a Bayesian inference problem with a known likelihood and prior that can be sampled easily. We present another powerful application in Bayesian data assimilation in certain dynamical systems where the score of the target distribution can be computed without explicit knowledge of the prior distribution. Both in learning probability distributions and ODEs, we gather supporting evidence for our hypothesis that dynamical systems and ergodic theory can fruitfully intersect with statistical learning to improve our understanding and implementation of ML algorithms. The work on sampling is joint with Youssef Marzouk (MIT) and on learning dynamics is with Jeongjin Park (Georgia Tech).

Abstract: During an inflammatory response there is a complex cascade of reactions, which may lead to health or sustained inflammation during many diseases and processes. In order to understand how the immune cells involved in the inflammatory response contribute to the disease progression, we have developed various models for the immune cell dynamics. Using parameter estimation, sensitivity analysis, and/or classification methods we will explore predictors of outcome and how modulating the immune response dynamics can alter disease progression. In this talk, I will focus on 1) a model for sequential influx of immune cells following a bacterial stimulus and 2) the role of intestinal Alkaline Phosphatase (IAP) in the leaky gut phenomenon and the associated pro-inflammatory signal to the immune system.

Abstract: Recent events have highlighted that cybersecurity is very difficult, and a purported solution can itself become the problem. We need better tools to reason about security solutions. Formal, mathematical tools, including graph theoretical constructs, provide a means to support network managers to reason about their network policies to create secure-by-design networks. In this talk, I will present one such tool that we have been using recently -- metagraphs -- which are closely related to hypergraphs. I will show how they can be used to represent security policy, and some of the techniques that make them valuable for reasoning about cybersecurity policies.

Abstract: Eigenvalue interlacing is a tremendously useful tool in linear algebra and spectral analysis. In its simplest form, the interlacing inequality states that a rank-one positive perturbation shifts the eigenvalue up, but not further than the next unperturbed eigenvalue. For different types of perturbations, this idea is known as the "Weyl interlacing" (additive perturbations), "Cauchy interlacing" (for principal submatrices of Hermitian matrices), "Dirichlet-Neumann bracketing" and so on.

We discuss the extension of this idea to general "perturbations in boundary conditions", encoded as interlacing between eigenvalues of two self-adjoint extensions of a fixed symmetric operator with finite (and equal) defect numbers. In this context, even the terms such as "signature of the perturbation" are not immediately clear, since one cannot take the difference of two operators with different domains. However, it turns out that definitive answers can be obtained, and they are expressed most concisely in terms of the Duistermaat index, an integer-valued topological invariant describing the relative position of three Lagrangian planes in a symplectic space. Two of the planes describe the two self-adjoint extensions being compared, while the third one corresponds to the distinguished Friedrichs extension.

We will illustrate our general results with simple examples, avoiding technicalities as much as possible and giving intuitive explanations of the Duistermaat index, the rank and signature of the perturbation in the self-adjoint extension, and the curious role of the third extension (Friedrichs) appearing in the answers.

Based on a work in progress with Graham Cox, Yuri Latushkin and Selim Sukhtaiev.

Abstract: In this talk, we discuss a conical extension of averaged non-expansive operators and its role in analyzing the convergence of several splitting algorithms: the proximal point algorithm, the forward-backward algorithm, the adaptive Douglas-Rachford algorithm, the 3-operator algorithm, and the adaptive ADMM. We also present an inspiring application of splitting algorithms in spatial design problems.

Abstract: An important problem in modern applied science is to characterize the behaviour of systems with complex internal dynamics subjected to external forcings. Many existing approaches rely on ensembles to generate information from the external forcings, making them unsuitable to study natural systems where only a single realization is observed. A prominent example is climate dynamics, where an objective identification of signals in the observational record attributable to natural variability and climate change is crucial for making climate projections for the coming decades. I will show that the spectral theory of dynamical systems, combined with techniques from data science, provides an effective means for extracting slowly decaying modes of climate variability, nonlinear trends and persistent cycles, from a single trajectory of a high-dimensional model or observed time series. We apply our framework to real-world examples from climate dynamics: the El Nino Southern Oscillation and variability of sea surface temperature over the industrial era, and the mid-Pleistocene transition of Quaternary glaciation cycles.

Abstract: In this talk I will first give an overview of oscillations in the voltage of neuron assemblies, and models thereof. We use these to study neurons in the hippocampus, a part of the brain thought to be central in learning and memory functions. These neurons are connected via electrochemical synapses, which use neurotransmitter released from the presynaptic neuron to change the voltage of the postsynaptic neuron. Inhibitory neurons cause the voltage of their target to decrease. Oscillations in inhibitory-to-inhibitory (I-I) coupled neurons in the hippocampus have been studied extensively numerically, and with analytic continuation methods. Neuromodulation on short time scales, in the form of presynaptic short-term plasticity (STP), can dynamically alter the connectivity of neurons in such a microcircuit. I will discuss the mechanism of STP, and a model for it parameterized from experimental data for a specific synapse in the hippocampus. The goal of the project is to understand the effect of adding this plasticity to the (I-I) microcircuit, both through numerical simulation and bifurcation analysis of a discrete dynamical system.

Abstract: The surfaces of the high latitude oceans are frozen into a layer of ``sea ice”, which plays an important role in the global climate by reflecting the sun’s rays. Ocean surface waves propagate from the open ocean into the sea ice covered ocean and break up the ice cover, leaving it more vulnerable to melting. The ice cover attenuates wave energy over distance, so that the breakup is confined to a region known as the marginal ice zone. Field observations have been interpreted as indicating the non-intuitive behaviour of attenuation decreasing following breakup. I will present a mathematical model that explains the observations in terms of a combination of attenuation and ice-edge reflection, backed by laboratory experiments.

Abstract: Though Einstein’s equation is well studied, relatively few Einstein metrics have been written in terms of explicit formulae via radicals. In this talk we discuss many such examples that occur as anti-self dual Einstein metrics and describe their singularities. The construction heavily relies upon the theory of isomonodromic deformation and related algebraic geometry developed by N.J. Hitchin in the 1990s and the equivalence of the anti-self-dual Einstein equation to a certain Painlevé VI equation under some symmetry assumptions discovered by K.P. Tod. The solution to Painlevé VI is achieved through a relation of its solution to pairs of conics obeying the Poncelet’s porism by exploiting Cayley’s criterion. In this talk we discuss some important cases that are not well fleshed out in the literature, such as the solution of Painlevé VI associated with the Poncelet porism where the inscribing-circumscribing polygons have an even number of sides. Moreover, we provide some explicit metrics with neutral signature and others with unusual cone angle singularities along a singular real projective plane that were speculated about by Atiyah and LeBrun.

Abstract: What is old is new again. Machine learning can be understood as attempts to apply data driven techniques to uncover underlying deterministic dynamics, or to approximate it through stochastic methods. In this talk I will describe three approaches to understand machine learning from the perspective of dynamical systems. First, recurrent neural networks will be shown to perform an embedding of time series data in the sense of Takens' theorem. That is, the internal state of the neural network is diffeomorphic to the underlying (presumed determinsitic) dynamical system. Second, while generative Artificial Intelligence achieves sentient-like performance through a carefully orchestrated random walk we will see how this can be construed as a stochastic dynamical system represented by a walk on a graph (or Markov chain). Thirdly, I will describe the application of learning techniques to estimate the state of a network dynamical system from observation of the node dynamics. Along the way, the utility of these methods will be demonstrated with application to industrial maintenance, music and detection of ventricular fibrillation.

Abstract: The KdV-Burgers equation was proposed by Whitham as a model for the propagation of tidal bores, and represents one of the simplest partial differential equations to incorporate nonlinearity, dispersion and dissipation. The existence and uniqueness (modulo translation) of traveling wave was proven by Bona and Schonbek, and the stability to small perturbations was proven by Pego in the case where the traveling wave is monotone. We prove that under a certain spectral condition the traveling wave is a global attractor for a range of parameter values that includes the monotone case. For a portion of the parameter set the proof is purely analytic, over the rest of the range we rely on techniques of rigorous numerics. This is joint work with Blake Barker (BYU), Vera Hur (University of Illinois), and Zhao Yang (Chinese Academy of Sciences).

Abstract: We consider particle filters with weakly informative observations (or 'potentials') relative to the latent state dynamics. The particular focus of this work is on particle filters to approximate time-discretisations of continuous-time Feynman-Kac path integral models --- a scenario that naturally arises when addressing filtering and smoothing problems in continuous time --- but our findings are indicative about weakly informative settings beyond this context too. We study the performance of different resampling schemes, such as systematic resampling, SSP (Srinivasan sampling process) and stratified resampling, as the time-discretisation becomes finer and also identify their continuous-time limit, which is expressed as a suitably defined `infinitesimal generator.' By contrasting these generators, we find that (certain modifications of) systematic and SSP resampling `dominate' stratified and independent `killing' resampling in terms of their limiting overall resampling rate. The reduced intensity of resampling manifests itself in lower variance in our numerical experiment. This efficiency result, through an ordering of the resampling rate, is new to the literature. The second major contribution of this work concerns the analysis of the limiting behaviour of the entire population of particles of the particle filter as the time discretisation becomes finer. We provide the first proof, under general conditions, that the particle approximation of the discretised continuous-time Feynman-Kac path integral models converges to a (uniformly weighted) continuous-time particle system. Joint work with N. Chopin, T. Soto and M. Vihola. DOI: 10.1214/22-AOS2222.

Abstract: Synchronization phenomena on networks have attracted much attention in studies of neural, social, economic, and biological systems, yet we still lack a systematic understanding of how relative synchronizability relates to underlying network structure. Indeed, this question is of central importance to the key theme of how dynamics on networks relate to their structure more generally. We present an analytic technique to directly measure the relative synchronizability of noise-driven time-series processes on networks, in terms of the directed network structure. We consider both discrete-time autoregressive processes and continuous-time Ornstein–Uhlenbeck dynamics on networks, which can represent linearizations of nonlinear systems such as the Kuramoto model. Our technique builds on computation of the network covariance matrix in the space orthogonal to the synchronized state, enabling it to be more general than previous work in not requiring either symmetric (undirected) or diagonalizable connectivity matrices and allowing arbitrary self-link weights. More importantly, our approach quantifies the relative synchronization specifically in terms of the contribution of process motif (walk) structures. We demonstrate that in general the relative abundance of process motifs with convergent directed walks (including feedback and feedforward loops) hinders synchronizability. We also reveal subtle differences between the motifs involved for discrete or continuous-time dynamics. Our insights analytically explain several known general results regarding synchronizability of networks, including that small-world and regular networks are less synchronizable than random networks..

Abstract: Unlike most universities, Cornell University reopened its Ithaca campus for in-person instruction in the Fall of 2020 during the COVID period and did so safely through the use of pooled testing. This decision and many others at the top levels of the university were guided by our mathematical modeling group. I'll discuss some of the questions we explored, the models we built, the data that informed our models, and how we dealt with several central data issues. Our work "under fire" motivated our current work in epidemiological modeling.