The Schrodinger equations as inspiration of beautiful mathematics

 Gigliola Staffilani (MIT)

In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrodinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on the concept of energy transfer and its connection to dynamical systems, and I will end with some results on the derivation of a wave kinetic equation.

Rate-induced collapse in evolutionary systems

Constantin Arnscheidt (MIT)

Recent work on dynamical systems has highlighted the possibility of “rate-induced tipping”, in which a system undergoes an abrupt transition when a perturbation exceeds a critical rate of change. Here we argue that rate-induced tipping towards extinction is likely a ubiquitous feature of evolutionary systems. We demonstrate the emergence of rate-induced extinction in two general evolutionary-ecological models, and connect these results with the established literature on “evolutionary rescue” as well as recent work on mass extinctions.

Rates of homogenization for fully-coupled McKean-Vlasov SDEs via the Cauchy-Problem on Wasserstein Space

Zachary William Bezemek (BU)

In many applications, an accurate model will have important features at multiple scales of time and/or space. There is a long history of analyzing such systems’ effective behavior as the separation between these scales tends to infinity. The process of obtaining an effective equation in this limit is known as homogenization. It is well known that in the fully-coupled setting for stochastic differential equations (SDEs), one can obtain only convergence in distribution of the multiscale system to the homogenized system. There is great interest in obtaining rates of said convergence, and the prevailing method of doing so involves the study of certain Poisson equations and the Cauchy problem (Kolmogorov backward equation) associated to the homogenized system (R\”ockner and Xie 2021).
In this talk, we discuss extending this analysis to obtain rates of homogenization for fully-coupled multiscale McKean-Vlasov processes, i.e. stochastic differential equations whose coefficients depend on the distribution of the solution itself at each time. Equations of this form arise naturally from the many-particle limit of weakly-interacting diffusions, and there is immense interest in their properties in recent years due in part to the exploding popularity of Mean-field Games. Owing to the non-linearity of the Markov semigroup associated to McKean-Vlasov processes, the appropriate tool for studying rates of convergence is the Cauchy-Problem on the metric space of probability measures with finite second moment (Wasserstein Space), originally posed by Buckdahn, Li, Peng, and Rainer (2017). Hence, in the course of the proof we gain results on the regularity of Poisson-type of equations in their measure parameter and of the Cauchy-Problem on Wasserstein space that are of independent interest. We also gain as a corollary an extension of existing results on rates of homogenization for standard SDEs which allows for possibly non-linear test functionals of their Laws.

Uniform Asymptotic Stability for Convection-Reaction-Diffusion Equations in the Inviscid Limit Towards Riemann Shocks

Paul Blochas (University of Rennes 1)

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In this talk, I will present a result obtained in a recent paper about the study of the stability in time of a family  of traveling waves solutions to

that approximate a given Riemann shock, and we aim at showing some uniform asymptotic orbital stability result of these waves under some conditions that guarantee the asympotic orbital stability of the corresponding Riemann shock, as proved in a previous work of V. Duchêne and L. M. Rodrigues.

Even at the linear level, to ensure uniformity in  the decomposition of the Green function associated with the (fast-variable) linearization about of the above equation into a decreasing part and a phase modulation is carried out in a highly non-standard way.

Furthermore, we introduce a multi-scale norm depending in ϵ that is the usual norm when restricted to functions supported away from the shock location. To avoid the use of arguments based on parabolic regularization that would preclude a result uniform in, we close nonlinear estimates on this norm trough some suitable maximum principle.

Superharmonic Instability of Stokes Waves

Anastassiya Semenova(ICERM, Brown)

We consider the classical problem of water waves on the surface of an ideal fluid in 2D. This work is concentrated on the study of Stokes waves at finite and infinite depths. We consider the stability of nearly limiting Stokes waves at infinite depth to superharmonic perturbations. We identify previously inaccessible branches of instability in the equations of motion for fluid, and find that real positive eigenvalues of the linearized problem converge to a selfsimilar curve as a function of steepness. The power law is suggested for unstable eigenmodes in the immediate vicinity of the limiting Stokes wave. The behaviour of Stokes wave in finite depth is also considered.

Improved bounds on entropy production in living systems

Dominic Skinner (MIT)

Living systems maintain or increase local order by working against the second law of thermodynamics. Thermodynamic consistency is restored as they consume free energy, thereby increasing the net entropy of their environment. Recently introduced estimators for the entropy production rate have provided major insights into the efficiency of important cellular processes. In experiments, however, many degrees of freedom typically remain hidden to the observer, and, in these cases, existing methods are not optimal. Here, by reformulating the problem within an optimization framework, we are able to infer improved bounds on the rate of entropy production from partial measurements of biological systems. In contrast to prevailing methods, the improved estimator reveals nonzero entropy production rates even when non-equilibrium processes appear time symmetric and therefore may pretend to obey detailed balance, and can bound entropy production from waiting time statistics alone. We demonstrate the broad applicability of this framework by providing improved bounds on the energy consumption rates using experimental trajectory data from a diverse range of biological systems including bacterial flagella motors, growing microtubules, and calcium oscillations within human embryonic kidney cells. We are also able to gain bounds on the energy consumption rate of gene regulatory networks, mammalian behavioral dynamics and bacterial sensors from waiting time distribution alone.

Turing bifurcation in systems with conservation laws

Aric Wheeler (Indiana University)

Generalizing results of Matthews-Cox/Sukhtayev for a model reaction-diffusion equation, we derive and rigorously justify weakly nonlinear amplitude equations governing general Turing bifurcation in the presence of conservation laws. In the nonconvective, reaction-diffusion case, this is seen similarly as in Matthews-Cox, Sukhtayev to be a real Ginsburg-Landau equation weakly coupled with a diffusion equation in a large-scale mean-mode vector comprising variables associated with conservation laws. In the general, convective case, by contrast, the amplitude equations consist of a complex Ginsburg-Landau equation weakly coupled with a singular convection-diffusion equation featuring rapidly-propagating modes with speed ∼1/ϵ where ϵ measures amplitude of the wave as a disturbance from a background steady state. Applications are to biological morphogenesis, in particular vasculogenesis, as described by the Murray-Oster and other mechanochemical/hydrodynamical models. This work is joint with Kevin Zumbrun.