• Montie Avery (Boston University) Predicting front invasion speeds via marginal stability
  Front propagation into unstable states often mediates state transitions in spatially extended systems, in biological models and across the sciences. Classical examples include the Fisher-KPP equation for population genetics, Lotka-Volterra models for competing species, and Keller-Segel models for bacterial motion in the presence of chemotaxis. A fundamental question is to predict the speed of the propagating front as well as which new state is selected in its wake. In some cases, the propagation speed in a full nonlinear PDE model agrees with that predicted by its linearization about the unstable state, in which case we say the speed is linearly determined, and the fronts are pulled. If the nonlinear speed is faster than the linear spreading speed, we say the fronts are pushed. The marginal stability conjecture asserts that front invasion speeds are determined by the spectrum of the linearization about traveling wave solutions of the PDE model. We present a formulation and proof of the marginal stability conjecture, with a focus on explaining why marginal stability is the right concept to predict front speeds, as well as practical applications of the theory.


• Oluwatosin Babasola (U Georgia) Epidemiological approach for mitigating Mpox transmission dynamics
  Mpox (formerly known as monkeypox) is a zoonotic disease that has experienced sporadic outbreaks in various regions, with recent significant outbreaks in West Africa. This talk focuses on the epidemiological dynamics of Mpox to enhance the understanding of its spread and to examine targeted control strategies aimed at mitigating its impact on public health. In light of this, a mathematical model was developed to assess the transmission risks and the effectiveness of control measures in critical hotspots. This is then incorporated with Mpox cases data to estimate the reproduction number and analyze the efficacy of interventions in the most affected regions. Finally, the work projects the potential outcomes of various intervention scenarios, offering vital insights for informed decision-making in Mpox management and control efforts.


• Chenglong Bao (Tsinghua) On the global R-linear convergence of NAG method and beyond
  The Nesterov Accelerated Gradient (NAG) method is a widely-used extrapolation-based algorithm that accelerates the convergence of the gradient descent method in convex optimization problems. In this talk, we will explore the global linear convergence properties of the NAG method when applied to strongly convex functions by assuming that the extrapolation coefficient is independent of the strong convexity parameter. Moreover, we will provide a mathematical analysis that demonstrates the advantages of restart schemes in the NAG method. Finally, we will compare these results with the continuous understanding of the NAG method from the perspective of ordinary differential equations.


• Margaret Beck (BU) An overview of methods for determining the point spectrum of certain linear operators
  This talk is intended to be an accessible introduction to some techniques that can be utilized to determine the point spectra of linear differential operators in the context of stability theory for certain classes of PDEs, such as reaction diffusion systems. Such techniques are useful for determining whether or not key stationary solutions of interest, such as stationary fronts and pulses, are spectrally stable. Since it is typically only the stable solutions that are observable in real-world applications, such analysis can shed light on the types of phenomena that one could expect to observe in physical and biological systems.


• Ryan Goh (Boston University) Fronts in the wake of a parameter ramp: slow passage through pitchfork and fold bifurcations
  We discuss front solutions in the presence of a parameter ramp which slowly varies in space, rigidly propagates in time, and moderates the (in)stability of a spatially-homogeneous equilibrium, nucleating a traveling wave in its wake. For moving ramps, the front location is governed by a slow passage between convective and absolute instability; a projectivized fold. For stationary ramps, fronts are governed by slow-passage through a pitchfork and a special connecting solution of the Painléve-II equation found by Hastings and Mcleod. Joint with Tasso Kaper, Arnd Scheel, and Theodore Vo.


• Masato Hara (Kyoto University) A scenario of learning dynamics by reservoir computing
  Reservoir computing is a kind of machine learning, which can learn and predict time series generated by a dynamical system. We think that a reservoir can predict time series because it becomes a (semi) conjugate system with a system behind the time series. In this talk, I will sketch a scenario of learning dynamics by reservoir computing. This is a joint work with Professor Hiroshi Kokubu (Kyoto University).


• Alanna Haslam-Hyde (BU) Exponential Dichotomies for Spatial Evolutionary Systems in Elliptic PDEs (Poster)
  Exponential dichotomies, when they exist, provide powerful information about the structure of bounded solutions even in the case of an ill-posed evolutionary equation. The method of spatial dynamics, in which one views a spatial variable as a time-like evolutionary variable, allows for the use of classical dynamical systems techniques, such as exponential dichotomies, in broader contexts. This has been utilized to study stationary solutions of PDEs on spatial domains with a distinguished unbounded direction (e.g. the real line or a channel of the form ). Recent work has shown how to extend the spatial dynamics framework to elliptic PDEs posed on general multi-dimensional spatial domains. In this poster we show that, in the same context, exponential dichotomies do exist, thus allowing for their use in future analyses of coherent structures, such as spatial patterns in reaction-diffusion equations on more general domains.


• Sam Isaacson (BU) Spatial Jump Process Models for Estimating Antibody-Antigen Interactions
  Surface Plasmon Resonance (SPR) assays are a standard approach for quantifying kinetic parameters in antibody-antigen binding reactions. Classical SPR approaches ignore the bivalent structure of antibodies, and use simplified ODE models to estimate effective reaction rates for such interactions. In this work we develop a new SPR protocol, coupling a model that explicitly accounts for the bivalent nature of such interactions and the limited spatial distance over which such interactions can occur, to a SPR assay that provides more features in the generated data. Our approach allows the estimation of bivalent binding kinetics and the spatial extent over which antibodies and antigens can interact, while also providing substantially more robust fits to experimental data compared to classical bivalent ODE models. I will present our new modeling and parameter estimation approach, and demonstrate how it is being used to study interactions between antibodies and spike protein. I will also explain how we make the overall parameter estimation problem computationally feasible via the construction of a surrogate approximation to the (computationally-expensive) particle model. The latter enables fitting of model parameters via standard optimization approaches.


• Weijia Jing (Tsinghua) Periodic high contrast environments: quantitative homogenization and wave propagation problems
  We consider elliptic operators with periodic high contrast coefficients which model small inclusions that have very different physical properties compared to the surrounding environment. We report on some recent developments on the asymptotic analysis of such structures, including quantitative homogenization, uniform (with respect to contrast parameter and small periodicity) regularity, dispersion relation and Dirac points of high contrast honeycomb structures, and wave propagation and its approximation in such structures. The talk is based on some joint works with Habib Ammari and Xin Fu.


• Inkee Jung (BU) Evolutions of Logifold Structures on Measure Spaces
  Considering a dataset as a measure space, we introduce the concept of a linear logical function to formulate a logifold structure on the dataset. This approach involves utilizing network models with restricted domains as local charts. Additionally, quiver representations effectively describe the space of logical functions. Subsequently, we employ non-Archimedean analysis to apply stochastic gradient descent to the space of linear logical functions.


• Tasso Kaper (BU) Strong symmetry breaking in pairs of coupled, identical fast-slow oscillators
  In this joint work with Nazir Awal (Brandeis), Irving Epstein (Brandeis), and Theo Vo (Monash), we study pairs of symmetrically-coupled, identical fast-slow oscillators. We find a plethora of strong symmetry breaking rhythms in which the two oscillators exhibit qualitatively different oscillations, and their amplitudes differ by as much as an order of magnitude. Analysis of the folded singularities in the coupled system shows that a key folded node is the primary mechanism responsible for creating these strong symmetry breaking attractors.


• Benjamin Krewson (Boston University) Transverse modulational dynamics of quenched patterns (Poster)
  We study transverse modulational dynamics of striped pattern formation in the wake of a directional quenching mechanism. Such mechanisms have been proposed to control pattern-forming systems and suppress defect formation in many different physical settings, such as light-sensing reaction-diffusion equations, solidification of alloys, and eutectic lamellar crystal growth. In the context of two prototypical pattern forming PDEs, the complex Ginzburg-Landau and Swift-Hohenberg equations, we show that long-wavelength and slowly varying modulations of striped patterns are governed by a one-dimensional viscous Burgers' equation, with viscous and nonlinear coefficients determined by the quenched stripe selection mechanism.


• Taiga Kurokawa (Kyoto University) Existence of transit orbits in the planar restricted 3-body problem via variational methods
  In the past, the spacecraft trajectory had been designed using the two-body problem commonly. However, recently, the trajectory design with the three (or more)-body problem has been studied actively. In the planar restricted 3-body problem (PR3BP), the first problem in considering low-energy orbits is the existence of transit orbits. Although many numerical studies suggest the existence of transit orbits, few mathematical results show their existence. Mathematical results can be divided into two categories: those based on the perturbation theory method and those based on the variational method. The former cannot verify their existence for concrete situations, but the latter is important in that it allows us to do so. For the case of two bodies in the circular motion (PCR3BP), by minimizing Maupertuis' functional, Moeckel(2005) provided a sufficient condition for their existence. On the other hand, for the case of the elliptic motion (PER3BP), this variational structure does not work because it isn't autonomous, and no variational results had been known. We provide, for PCR3BP, the different sufficient condition from Moeckel's result by minimizing Lagrange's functional. Furthermore, by finding that this variational structure also works for non-autonomous systems, we also provide a sufficient condition for PER3BP. We also confirm numerically that these sufficient conditions are satisfied for the concrete situation when the two bodies have the same mass. In this talk, we will present these results.


• Minh Le (Michigan State U) Global existence of chemotaxis systems under nonlinear Neumann boundary conditions. (Poster)
  The occurrence of finite-time blow-up solutions is a well-known phenomenon in chemotaxis systems with homogeneous Neumann boundary condition in a smooth bounded domain, and the appearance of logistic source can effectively prevent such blow-ups. In this poster, we consider nonlinear Neumann boundary conditions for a chemotaxis system with quadratic logistic source and briefly explain how we can effectively utilize the quadratic logistic degradation to establish the local and global existence of solutions.


• Yuto Nakajima (Tokai University) Transversal family of non-autonomous conformal iterated function systems
  A Non-autonomous Iterated Function System (NIFS) is a sequence of collections of uniformly contracting maps. In comparison to usual iterated function systems, we allow the contractions applied at each step to vary. In this talk, we focus on a family of parameterized NIFSs. Here, we do not assume the open set condition. We show that if a parameter family of such systems satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of the dimension of the ambient space and the Bowen dimension. Moreover, we give an example of a family of parameterized NIFSs such that the family of NIFSs satisfies the transversality condition but each NIFS does not satisfy the open set condition.


• Gabriel Ocker (Boston University) Mean field dynamics of stochastic neuronal networks
  The neuronal dynamics generating sensory, motor, and cognitive functions are commonly understood through field theories for neural population activity. Classic neural field theories are derived from highly simplified models of individual neurons, while biological neurons are highly complex cells. We will discuss an approach towards bridging that divide, constructing analytically tractable neural field theories without discarding all the biophysical complexity of real neurons.


• Misato Ogawa (Ochanomizu University) The relation between the Gibbs measure for a potential which depends on the first coordinate and Double Variational Principle on XY model
  The minmax problem for the sum of rate distortion dimension and the integral of a potential function is called Double Variational Principle with a potential and XY-model is a typical example for which the principle holds. On the other hand, we can construct a Gibbs measure for a continuous function on XY-model by using the Ruelle operator. Hence it is natural to ask the minmax value is attained by the Gibbs measure as an analogy of the Variational Principle for a shift with finite alphabets. In this talk we consider a potential depending only on the first coordinate on XY model and calculate the rate distortion dimension for the Gibbs measure of the potential. This is a joint work with Mao Shinoda in Ochanomizu University.


• Chihiro Oguri (Ochanomizu University) Is the penalty function of the hard square shift stable?
  The fundamental difference between subshift of finite type(SFT) in d=1 and that in d=2 is revealed by Gonschorowski et.al., i.e., the penalty function of every SFT in d=1 has the stablity but there exists a SFT in d=2 whose penalty function is unstable. However, the SFT in d=2 in their paper is far from 'chaotic'. Hence it is natural to ask whether the stability of the penalty function of SFT in d=2 with 'sufficiently chaotic' property. In this talk we consider the penalty function of the hard square shift.


• Tomoki Ohsawa (UT-Dallas) Relative Dynamics and Stability of Point Vortices
  I will talk about a Hamiltonian formulation of the relative dynamics of the planar $N$-vortex problem as well as a stability condition for its relative equilibria. The relative dynamics is the dynamics of the ``shape'' formed by the point vortices: For example, if $N = 3$, it is the dynamics of the shape of the triangle formed by three vortices, regardless of their position and orientation of the triangle. A relative equilibrium is a solution in which the vortices move in such a manner that the shape formed by the vortices does not change in time. The main result is a sufficient condition for the stability of relative equilibria using the Hamiltonian formulation of the relative dynamics.


• Kayode Oshinubi (N. Ariz. U) Dynamical modeling of infectious disease spread using data-driven models
  The objective of this study is to develop a robust method for predicting the changes and transition between endemic and epidemic phases of an infectious disease dynamics, using COVID-19 outbreak as an example. We define indicators for detecting changes and transitions between endemic and epidemic phases using seven scalars calculated from daily reported news cases: variation coefficient, entropy, dominant/subdominant spectral ratio, skewness, kurtosis, uniformity index, and normality index. The indicators chosen are related to the form of the empirical distribution of new cases seen over a fourteen-day period chosen to smooth out the influence of weekends when fewer new cases are registered. We used the Principal Component Analysis (PCA) to create a score from the seven proposed indicators that allows an acceptable level of forecasting performance by providing a realistic retro-predicted date for the rupture of the stationary endemic model corresponding to the entry into the epidemic exponential growth phase. The first principal component (a linear combination of the seven indicators) explains a considerable portion of the observed variance and can thus be used as a predictor of the event studied (in this case, the presence of an epidemic wave). This score is used to forecast the limits of the several phases of the COVID-19 outbreak in various nations following endemic and epidemic transitions and changes. We were able to build a new forecasting strategy for predicting an epidemic wave that comes after an infectious disease’s endemic stationary period. This research offers a valuable tool for early epidemic detection, aiding in effective public health responses.


• Kohki Sakamoto (U Tokyo) Harmonic measures in percolation clusters on hyperbolic groups
  


• Vaibhava Srivastava (Iowa St. U) The "Fear" Effect in Competition Systems: Theory and Applications to Avian Invasions
  Non-consumptive effects, such as fear of depredation, can strongly influence predator-prey dynamics. There are several ecological and social motivations for these effects in competitive systems as well. In this work, we consider the classic two species ODE and PDE Lotka-Volterra competition models, where one of the competitors is "fearful" of the other. We find that the presence of fear can have several interesting dynamical effects on the classical competitive scenarios. Notably, for fear levels in certain regimes, we show novel bi-stability dynamics. Furthermore, in the spatially explicit setting, the effects of several spatially heterogeneous fear functions are investigated. In particular, we show that under certain integral restrictions on the fear function, a weak competition-type situation can change to competitive exclusion. Applications of these results to ecological as well as sociopolitical settings are discussed, which connect to the "landscape of fear" (LOF) concept in ecology. Using the test case of northern spotted and barred owl populations in the Pacific Northwest region of the United States, we evaluate if this fear (co-occurrence) model can generate more robust population estimates than previous models. We then evaluate if potential co-occurrence effects among barred and northern spotted owls are uni- or bi-directional. Lastly, we leverage the best-performing model to evaluate the degree to which a recently proposed barred owl culling program may help recover northern spotted owl populations.


• Yoshiki Takeguchi (Kyoto University) Design of low-thrust spacecraft trajectories with minimum costs by optimal control and dynamical systems approaches
  In this talk, we consider the problem of computing low-thrust spacecraft trajectories with minimum costs from the Earth to the Moon or Mars. The spacecraft is modeled by the restricted three- or four-body problem. Optimal control and dynamical systems approaches are used, so that low-thrust trajectories are computed by numerical continuation of boundary value problems of Hamiltonian systems obtained via Pontryagin's maximum principle. I will also give preliminary numerical results for low-thrust trajectories from a circular orbit to another around the Earth.


• Son Tu (Michigan State University) Rate of convergence for quasi-periodic homogeniation of Hamilton-Jacobi equation and application
  


• Takayuki Watanabe (Chubu University) On the stochastic bifurcations regarding random iterations of rational maps
  We consider random iterations of rational maps on the Riemann sphere and investigate the stochastic bifurcation of them. For example, adding independent noise to iteration of $z \mapsto z^2 - 1$ yeilds a dynamical system that is qualitatively different from the original dynamical system, even if the noise size is relatively small. We give some quantitative estimates of bifurcation parameters and present beautiful figures of random Julia sets.


• Jinxin Xue (Tsinghua) Generic dynamics of the mean curvature flows
  The mean curvature flow is to evolve a hypersurface in Euclidean space using the mean curvatures at each point as the velocity field. The flow has good smoothing property, but also develops singularities. The singularities are modeled on an object called shrinkers, which give homothetic solutions to the flows. As there are infinitely many shrinkers that seem impossible to classify, it is natural to explore the idea of generic mean curvature flows that is to introduce a generic perturbation of the initial conditions. In this talk, we shall explain our work on this topic, including perturbing away nonspherical and noncylindrical shrinkers, and generic isolatedness of cylindrical singularities. The talk is based on a series of works jointly with Ao Sun.


• Kazuyuki Yagasaki (Kyoto University) Some Recent Results on Nonintegrability of Dynamical Systems
  In this talk, we review some recent results on nonintegrability of dynamical systems. I begin with a definition of integrability for general systems. After briefly reviewing the classical work of Poincare and Kozlov, the differential Galois theory, and the Morales-Ramis theory, I explain a new technique to prove the nonintegrability of nearly integrable systems. We apply the technique to the restricted three-body problem and time-periodic perturbations of single-degree-of-freedom Hamiltonian systems. The former result improves the work of Poincare and the latter is related to the subharmonic Melnikov method. I also briefly state my results for normal forms of codimension-two bifurcations and three- or four-dimensional systems with degenerate equilibria.


• Takumi Yagi (Kyoto University) Hyperbolicity for horocyclic perturbations of semi-parabolic Hénon maps
  We consider a family of dissipative quadratic complex Hénon maps $H_{a,t}$ with $a\in\mathbb{D}_{\delta}$ and $t\in [0,1]$, where $\delta>0$ is a small number. Suppose that $H_{a,t}$ has a fixed point $\textbf{q}_{a,t}\in\mathbb{C}^2$, depending continuously on $a$ and $t$, with one eigenvalue $\lambda_t$ such that $\lambda_t\to \lambda_0={\rm exp}(2\pi i p/r),(p,r)\in\mathbb{Z}\times\mathbb{N}$ as $t\to 0$. Let $\lambda_t/\lambda_0$ be expressed by ${\rm exp}(L_t+i\theta_t)$ and suppose $\theta_t\to 0$ as $t\to 0$. We say that $\{H_{a,t}\} is a horocyclic perturbation if $\theta_t^2=o(L_t)$. We see that $H_{a,t}$ is hyperbolic if $\theta_t=O(L_t)$ and $L_t\neq 0$. On the other hand, it is difficult to show the hyperbolicity for horocyclic perturbations. We introduce some ideas to compute the hyperbolicity for horocyclic perturbations.


• Caiwei Zhang (BU) Jump Process Simulation Algorithms for Modeling Biological Systems with Memory Optimization (Poster)
  When an unknown antigen infects an individual, it is a question how strongly the body’s immune system will respond to the viral infection. Quantifying the strength of the immune system’s reaction to antigens can aid biologists and pharmaceutical companies in understanding the severity of different viruses. In this project, we constructed and optimized a numerical method that can fit jump process models of antibody-antigen interactions to experimental surface plasmon resonance (SPR) assay data, allowing the quantification of different antibodies’ reaction efficacies. To further improve the memory usage of the surrogate that underlies the model, we designed an exponential series for fitting surrogates by piecewise interpolation, using variable projection methods to speed up the optimization process. This resulted in a 33x reduction in on-disk memory usage. We also adopted a global derivative-free optimizer that consistently gives the best, most robust, fits of the interpolated model compared to other derivative-free and derivative-based optimizers. This project was launched as preparation for extending our methodology in modeling biological systems to bispecific antibodies, where we will discover and update our model to account for the heterogeneity of the antibody binding to each antigen.


• Wenjun Zhao (Brown) Quantifying Patterns and Their Transitions in Spatially Extended Systems
  Tracing the transition curves for reaction-diffusion systems is an interesting question that arises in various applications. In this talk, we will discuss a framework to quantify patterns via alpha-shape statistics, utilizing the Wasserstein metric to quantify the differences between such statistics corresponding to different parameter values. The framework is then incorporated within a continuation algorithm that can trace transitions in a 2D parameter space. Notably, the algorithm is completely data-driven and requires limited prior knowledge about the underlying system. Finally, we will demonstrate our algorithm on examples such as spots/stripes and spiral waves.

Links to partial video recordings:
 May 28 morning  (Passcode: kT&Xa5w*)
 May 28 afternoon  (Passcode: q$y0#mm1)
 May 29 morning  (Passcode: C3QD$qs5)
 May 29 afternoon  (Passcode: j4D+Fe!$)
 May 31 morning  (Passcode: 9hndzs*L)
 May 31 morning 2nd part  (Passcode: q$y0#mm1)
 May 31 afternoon  (Passcode: kT&Xa5w*)

• Boston University Department of Mathematics and Statistics
• Boston University Graduate School of Arts and Sciences
• Keio University
• Tsinghua University
• National Science Foundation