Cytoplasmic mRNA localization is an important biological strategy for establishing cell and developmental polarity in a variety of organisms and cell types. In diverse examples including oocytes, fibroblasts and neurons, mRNA localization results in spatially restricted gene expression through local protein synthesis. In Xenopus laevis oocytes, maternal mRNAs are transported to the vegetal hemisphere of developing oocytes where their local translation is critical for proper embryonic patterning. Motor-based vegetal transport of such mRNAs relies on the assembly of RNP transport cargos, a conserved feature of RNA localization pathways. However, the molecular and physical nature of these structures has remained largely unknown. Our recent work has revealed that that that the RNA cargos are phase-separated biomolecular condensates that contain a solid- or gel-like RNA phase and a dynamic protein phase.

We are estimated to spend one third of our life asleep, and it is increasingly apparent that good sleep is essential for overall health. Yet many people suffer from insufficient sleep or sleep disorders. Even though there is only a partial understanding about the why and how of sleep, mathematical models do exist that capture the broad features of sleep-wake regulation and are widely used in safety-critical industries to model fatigue risk. In this talk we will discuss some mathematical models of sleep-wake regulation and their multi-time scale features. Most mathematical models consider two states: a sleep and a wake state. However, the two state models do not account for the fact that during the night we cycle between two main sleep states (rapid eye movement (REM) and non-rapid eye movement (NREM) sleep). We will also discuss a model that considers three states: these two sleep states and a wake state. We will show that this model can be considered as a three-timescale problem. This three-timescale decomposition reveals additional geometric structure which acts to organise oscillations between REM and NREM states. This deeper geometric understanding of the generation of REM-NREM cycles brings insight into the relationship between model predicted and observed patterns of REM-NREM cycles and suggests ways in which models could be modified to more accurately reflect patterns of human sleep. This is joint research with Anne Skeldon, Derk Jan Dijk, Rachel Bernasconi, Matthew Bailey, Panos Kaklamanos, and Paul Glendinning.

Piecewise smooth stochastic differential equations (SDEs) describe systems where randomness interacts with abrupt changes or discontinuities in the underlying dynamics. These systems appear in diverse applications, such as neural activity, stock market fluctuations, climate transitions, and phase changes. In this talk, we present a framework for computing the most probable transition paths and, in some cases, expected tipping times for piecewise smooth SDEs. Specifically, we focus on systems where dynamics switch across a discontinuity set, modeled as a co-dimension one manifold called the switching manifold. Using variational tools like gamma convergence, we develop rigorous effective theories for dynamics constrained to this manifold which also yields surprising phenomena such as noise-induced sliding. Throughout the presentation, we demonstrate the implications of this work through case studies, including Arctic sea ice dynamics, transitions between bull and bear markets, and the polarization of online opinions.

Biological pattern formation has been extensively studied using reaction-diffusion and agent-based models. In this talk we will discuss nonlocal pattern forming mechanisms in the context of bacterial colony formation with an emphasis on arrested fronts. This will lead to a novel nonlocal framework to understand the interfacial motion in biological systems. We will then use this approach to model experiments for an interesting bacterial phenomenon.

We consider the impact of stochastic forcing on the behaviour of travelling fronts and pulses in various settings. We discuss a framework that is able to capture the timescales over which the patterns retain their stability and allows the expected corrections to the speed and shapes to be quantified. The governing equations lead to interesting follow-up questions involving singular perturbation theory.

Spatiotemporal modeling of real-world data poses a challenging problem due to inherent high-dimensionality, measurement noise, and expensive data collection procedures. We present \textbf{S}parse \textbf{I}dentification of \textbf{N}onlinear \textbf{Dy}namics with \textbf{SH}allow \textbf{RE}current \textbf{D}ecoder networks (SINDy-SHRED), a method to jointly solve the sensing and model identification problems with simple implementation, efficient computation, and robust performance. SINDy-SHRED uses Gated Recurrent Units (GRUs) to model the temporal sequence of sensor measurements along with a shallow decoder network to reconstruct the full spatiotemporal field from the latent state space using only a few available sensors. Our proposed algorithm introduces a SINDy-based regularization; beginning with an arbitrary latent state space, the dynamics of the latent space progressively converges to a SINDy-class functional, provided the projection remains within the set. Thus a dynamical system model is enforced in the latent space of the temporal sequence model. We conduct a systematic experimental study including synthetic PDE data, real-world sensor measurements for sea surface temperature, and direct video data. With no explicit encoder, SINDy-SHRED enables efficient training with minimal hyperparameter tuning and laptop-level computing; further, it demonstrates robust generalization in a variety of applications with minimal to no hyperparameter adjustments. Finally, the interpretable SINDy model of latent state dynamics enables accurate long-term video predictions, achieving state-of-the-art performance and outperforming all baseline methods considered, including Convolutional LSTM, PredRNN, ResNet, and SimVP.

Many important statements about dynamical systems can be proven by finding scalar-valued auxiliary functions whose time evolution along trajectories obeys certain pointwise inequality that imply the desired result. The most familiar of these auxiliary functions is a Lyapunov function to prove steady-state stability, but such functions can also be used to bound averages of ergodic systems, define trapping boundaries, and so much more. In this talk I will highlight a method of identifying auxiliary functions from data using polynomial optimization. The method leverages recent advances in approximating the Koopman operator from data, so-called extended dynamic mode decomposition, to provide system-level information without system identification. The result is a flexible, data-driven, model-agnostic computational method that does not require explicit model discovery. Furthermore, it can be applied to data generated through deterministic or stochastic processes with no prior adjustments to the implementation. It can be used to bound quantities of interest, develop optimal state-dependent feedback controllers, and discover invariant measures.

Mathematical models of opinion dynamics are an important tool to gain insight into the qualitative dynamics of the evolution of opinions or ideologies over time. The sigmoidal bounded-confidence model (SBCM) is a smooth, nonlinear model of opinion dynamics which is parameterized by a scalar γ. This parameter controls the steepness of a smooth influence function that encodes the relative weights that agents place on the opinions of other agents, and interpolates between two well-studied models of opinion dynamics. When γ = 0, this influence function exactly recovers Taylor’s averaging model; when γ → ∞, the influence function converges to that of a modified Hegselmann–Krause (HK) model. For several special graph topologies, we can give analytical descriptions of important features of the space of steady states. A notable result is a closed-form relationship between the stability of a polarized state and the graph topology in a simple model of echo chambers in social networks. Because the influence function of our BCM is smooth, we are able to study it with linear stability analysis, which is difficult to employ with the usual discontinuous influence functions in BCMs. We will discuss interesting extensions arising from SBCM, both analytic (deriving and studying a mean-field integro-differential equation from the full network model) and data-focused (parameter inference from synthetic data).

Over the last 16 years I have mentored about a dozen different summer undergraduate research projects in applied dynamical systems. Ten of the projects have led to publications, of which three were in peer-reviewed journals. In my talk I will touch upon some of the problems the students looked at, and some of the mathematics they needed to develop and refine as they did their work. I will discuss in more detail the most recent project, which is concerned with modeling and analyzing a nonlinear compartment model for opinion propagation in a closed society.

I will give an overview of data-driven questions that can be tackled using optimal-transport based algorithms. First, I will show that irregular patterns formed of interacting spots and stripes can be described by pattern statistics, which are probability measures that capture the distribution of certain features, such as the number of connected components or their areas. We use this approach with arclength predictor-corrector continuation to trace out transition and bifurcation curves in parameter space by maximizing the Wasserstein distance of the pattern statistics. Secondly, I will discuss the application of optimal-transport-based algorithms to infer dynamic gene-regulatory networks from time-stamped single-cell gene expression counts.

(This talk is based on joint work with Dock Staal.) The phenomenon of tipping is widely studied in ecology, it is predicted to be driven by slowly evolving environmental processes that may push an ecosystem from a favorable state into a less desirable state. Tipping typically takes place on a fast time scale, it is associated with 'catastrophic collapse'. In mathematical terms, tipping is commonly identified with a stable configuration (in a model) approaching a saddle-node bifurcation due to slowly varying parameters. As these pass through their critical value, this state disappears and the system evolves towards another state. This picture is overly simplified, predictions of tipping ecosystems are typically based on ODE models. For a dryland ecosystem model of PDE type, recent research has shown that the saddle-node bifurcation is preceded by a Turing bifurcation: vegetation patterns appear from the homogeneous state and the ecosystem 'evades' the tipping point. Thus, by taking spatial effects into account the ecosystem turns out to be more 'resilient' than expected: it has 'multi-stability', i.e. many distinct stable patterned states, by which it may adapt in a relatively smooth fashion to worsening environmental circumstances. Naturally, there is nothing special about dryland ecosystems: tipping may be evaded by this 'Turing before tipping' mechanism in 'generic' evolutionary PDEs (modeling all kinds of complex systems). However, a crucial issue that needs to be settled to enable pattern formation to evade tipping is: "Does the Turing bifurcation generate patterns that remain stable for parameter values beyond the tipping point?" More specifically: "Under which conditions do patterns generated near a co-dimension 2 Turing-fold bifurcation remain stable for parameter values for which the state from which the patterns emerged no longer exist?" To answer these questions the classical Ginzburg-Landau formalism must be adapted: the Ginzburg-Landau equation needs to be extended to a system of coupled equations. We first perform this analysis in the setting of a Swift-Hohenberg-type scalar example model and next show that this approach can be generalized to study co-dimension 2 Turing-fold bifurcations in a general class of N-component reaction-diffusion equations.

Modern biomorphology models such as Murray-Oster and Scianna-Bell-Preziosi involve pattern formation in systems with mechanical/hydrodynamical effects taking the form of convection-reaction-diffusion models with conservation laws. Here, extending previous work of Matt hews-Cox and H\""acker-Schneider-Zimmerman in pattern formation with conservation laws, and of Eckhaus, Mielke, and Schneider on stability of Turing patterns in reaction diffusion models, we investigate diffusive stability of Turing patterns for convection-reaction-diffusion models with conservation laws. Formal multiscale expansion yields a singular system of amplitude equations coupling Complex Ginzburg Landau with a singular convection-diffusion system, similar to parti ally coupled systems found by H\""acker-Schneider-Zimmerman in the context of thin film flow, but with the singular convection part now fully engaged in long term stability and behavior rather than transient as in the (triangular) parti ally coupled case. The resulting complicated two-parameter matrix perturbation problem governing spectral stability can nonetheless be solved, yielding (m+1) simple stability criteria analogous to the Eckhaus and Benjamin-Feier-Newell criteria of the classical (no conservation law) case, where m is the number of conservation laws. It is to be hoped that these can play the same important role in the study of biopattern formation as the classical ones in myriad other applications.

Kerr microresonators are microscopic ring-shaped cavities that confine light by circulating it in a closed path and enhance the interaction of light through resonance. It has been experimentally observed that the interplay of the Kerr nonlinearity and dispersion in the microresonator can lead to a stable optical signal consisting of a periodic sequence of highly localized ultra-short pulses, resulting in broad frequency spectrum. The discovery that stable broadband frequency combs can be generated in microresonators has unlocked a wide range of promising applications, particularly in optical communications, spectroscopy, and frequency metrology. In its simplest form, the physics in the microresonator is modeled by the Lugiato-Lefever equation, a damped nonlinear Schrödinger equation with forcing. In this talk I demonstrate that the Lugiato-Lefever equation indeed supports arbitrarily broad Kerr frequency combs by establishing the existence and stability of periodic solutions consisting of any number of well-separated, strongly localized bright solitons on a single periodicity interval. The existence and spectral stability analyses rely on a rich blend of mathematical tools, such as Lyapunov-Schmidt reduction, Evans-function techniques, bifurcation theory, exponential dichotomies, and high-frequency resolvent bounds. Our analysis confirms that two-mode forcing improves the stability properties of the generated frequency combs compared to one-mode forcing. This is joint work with Lukas Bengel (Karlsruhe Institute of Technology).

In this talk, we will discuss the results of a series of papers related to single-front dynamics in a certain class of multi-component reaction-diffusion systems, where one fast component governed by an Allen-Cahn equation is weakly coupled to a system of linear slow reaction-diffusion equations. By using geometric singular perturbation theory, Evans function analysis and center manifold reduction, we demonstrate that and how the complexity of the front motion can be controlled by the choice of coupling function and the dimension of the slow part of the multi-component reaction-diffusion system. On the one hand, we show how to imprint and unfold a given scalar singularity structure. On the other hand, we show how chaotic behaviour of the front speed arises from the unfolding of a nilpotent singularity via the breaking of a Shil'nikov homoclinic orbit. The analysis is complemented by a numerical study that is heavily guided by our analytic findings. This is joint work with M. Chirilus-Bruckner, J. Rademacher, A. Doelman, H. Ikeda

Studies of real-world complex systems, such as chemical, biological, and epidemiological systems, often involve dynamical systems on network structures. The diversity of network types and variable characteristics complicates the analysis of how patterns emerge in such network dynamics and poses challenges for effective modeling. Motivated by the need to understand the role of graph structures in these problems, we adopt both analytical and empirical perspectives to develop techniques that capture network structures in these dynamical systems problems and further enable us to understand their effects. In this talk, we will present investigations into the influence of networks on dynamics from these two perspectives. First, we will discuss studies of localized stationary patterns in bistable reaction-diffusion systems, focusing on the case of ring lattices and demonstrating how interaction length affects the connectivity of solution branches and how these systems can be studied analytically. Next, we will highlight a social system application to illustrate effective modeling approaches that integrate networks and create opportunities to connect with real-world data.

I will summarize somewhat selectively recent theoretical efforts at grappling with the notoriously intractable questions of existence, stability, and instability of spiral waves. I'll first give a brief summary of the conceptual viewpoint from work with Bjorn Sandstede, which views spiral waves as heteroclinic orbits asymptotic to a periodic orbit in an infinite-dimensional, ill-posed radial dynamical system. This view point allows one, to describe, with numerically verifiable assumptions, many intricate properties of the linearization at a spiral wave, the effects of boundaries, and predict qualitative features of potential instabilities. I'll then discuss more recent efforts towards a more complete theoretical understanding in a setting of anchored spirals, where the spiral arm attaches to a central disk-shaped hole in the domain. Models we analyze are both a sharp interface approximation and simple phase oscillator reaction-diffusion dynamics.

Time-independent spatially extended and spatially localized patterns are well studied, particularly in one spatial dimension, but much less is known about similar but time-dependent states. In this talk I will describe related structures that arise in turbulent flows, in both two and three spatial dimensions. The appearance of these structures poses fundamental mathematical questions since the structures develop from a statistically stationary dynamical state with many degrees of freedom.

The emergence of fully localised 2D patches of regular cellular patterns have been known to emerge from a pattern-forming or Turing instability for over 30years based on numerous numerical investigations. However, proving the bifurcation of 2D localised patches of pattern remains to be solved. In this talk, I present two different approaches to finding a formal approximation to the emergence of fully localised patches of pattern that may lead to a path to proving the existence of these patterns.

Understanding both the presence and implication of gap eigenvalues on the imaginary axis for solitary patterns in NLS is a challenging issue. The multi-scale nature of the problem considered comes from a separation of the localized 3D soliton from a small linear potential in the far field. Hamiltonian and geometric singular perturbation methods are combined to provide a count of the gap eigenvalues. A key part involves estimates on relevant Ricatti equations that arise naturally within a dynamical systems framework which brings out the multiscale structure. This is joint work with Emmanuel Fleurantin (GMU), Jeremy Marzuola and Dmitro Golovanich (UNC-CH)