In this talk I will describe how to construct global in time classical solutions to the master equation arising in mean field games. Our method works for a general class of non-separable Hamiltonians and final data that satisfy a suitable monotonicity condition. This stems from the so-called displacement convexity condition introduced and used successfully in the theory of optimal mass transportation. Our results hold true independently of the intensity of the idiosyncratic noise.
In space dimension larger or equal to two, the non-linear Klein-Gordon equation with small, smooth, decaying initial data has global in time solutions. This no longer holds true in one space dimension, where examples of blowing up solutions are known. On the other hand, it has been proved that if the non-linearity satisfies a convenient compatibility condition, the 'null condition', one recovers global existence and that the solutions satisfy the same dispersive bounds as linear solutions. The goal of this talk is to show that, in the case of cubic semi-linear nonlinearities, this null condition is optimal, in the sense that, when it does not hold, one may construct small, smooth, decaying initial data giving rise to solutions that display inflation of their L∞ and L2 norms in finite time.
The first seminar included an overview of rewilding, highlighting some of the areas in which we expect that the mathematical sciences could have an impact. This was followed by a response/discussion led by mathematical scientists. There were then opportunities for informal discussion as well. The session closed with a wrap-up in which we captured the most promising lines to follow up at the next seminar.
The Euler equation does not possess a unique solution for the flow over a 2-dimensional object. This problem has serious repercussions in aerodynamics; it implies that the inviscid aero-hydrodynamic lift force over a 2-dimensional object cannot be determined from first principles; a closure condition must be provided. The Kutta condition has been ubiquitously considered for such a closure in the literature, even in cases where it is not applicable (e.g. unsteady). In this talk, I will present a special variational principle that we revived from the history of analytical mechanics: Hertz’s principle of least curvature. Using this principle, we developed a novel variational formulation of Euler’s dynamics of ideal fluids that is fundamentally different from the previously developed variational formulations based on Hamilton’s principle of least action. Applying this new variational formulation to the century-old problem of the ideal flow over an airfoil, we developed a general (dynamical) closure condition that is, unlike the Kutta condition, derived from first principles. In contrast to the classical theory, the proposed variational theory is not confined to sharp edged airfoils; i.e., it allows, for the first time, theoretical computation of lift over arbitrarily smooth shapes, thereby generalizing the century-old lift theory of Kutta and Zhukovsky. Moreover, the new variational condition reduces to the Kutta condition in the special case of a sharp-edged airfoil, which challenges the widely accepted wisdom about the viscous nature of the Kutta condition. We also generalized this variational principle to Navier-Stokes’s via Gauss’s principle of least constraint, thereby discovering the fundamental quantity that Nature minimizes in every incompressible flow. We proved that the magnitude of the pressure gradient over the field is minimum at every instant! We call it the Principle of Minimum Pressure Gradient (PMPG). We proved that the Navier-Stokes equation is the necessary condition for minimizing the pressure gradient subject to the continuity constraint. Hence, the PMPG turns any fluid mechanics problem into a minimization one where fluid mechanicians need not to apply Navier-Stokes equations, but merely need to minimize the proposed action.
Climate change is having profound effects on the incidence of vector borne disease, such as dengue, chikungunya and West Nile virus. However, developing effective measures of disease risk on a global scale are challenged by the complex ways in which environmental variation acts in vector-host-pathogen systems. One way in which insect vectors, such as mosquitos, respond to environmental variation is to change their traits. For example, if food was scarce for juvenile mosquitos then when they become adults they are smaller, and lay fewer eggs to ensure there is less competition for food in the next generation. So the environment of the juvenile determines the trait the individual has as an adult. In this way the individuals adapt to the environment. Current models over-simplify the interaction between vector traits and environmental variation and so risk misestimating disease risk. Here, we derive a mathematical framework for capturing the interaction of vector traits and population dynamics. I show how this new mathematical framework leads to both interesting mathematical questions and can be used to help explain the location, magnitude and timing of historical dengue outbreaks.
Liquid crystal elastomers (LCEs) are advanced multifunctional materials that combine elasticity with orientational order. Specifically, mechanical strains give rise to changes in liquid crystalline order and, conversely, changes in the orientational order generate mechanical stresses and strains. The quest for responsive materials with the ability to mimic living systems or to enable green energy production and conversion processes is one of the major challenges for modern materials design. Because of their large reversible deformations and complex material responses in the presence of natural stimuli like heat or light, and electric or magnetic fields, LCEs are suitable for a wide range of applications in science, manufacturing, and medical research. Moreover, biodegradable, recyclable and reprocessable LCEs can also been achieved. This talk will offer an introduction to core concepts in the mathematical modelling of LCEs by linking non-linear elasticity with liquid crystal theory.
A planar incompressible and electrically conducting fluid can be described by the 2D Navier-Stokes-MHD system. One simple yet physically relevant laminar state is the Couette flow with a constant homogeneous magnetic field, given by uE=(y,0), BE=(b,0) in the domain T×R. The goal is to estimate how large can be a perturbation of this state while still resulting in a solution close to the laminar regime, thereby preventing the onset of turbulence. We prove that Sobolev regular initial perturbations of size O(Re-2/3), with Re being the Reynolds number, remain close to uE, BE and exhibit dissipation enhancement. The latter quantifies the convergence towards an x-independent state on a time-scale O(Re-1/3), much faster than the standard diffusive one O(Re-1).
In this talk, we present new findings on the regularity of first-order mean field games systems with a local coupling. We focus on systems where the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function for small densities. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, when f(m) = mθ with θ > 0, we prove that the speed of propagation is finite. In this case, we establish that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys C1,1 regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.
Starting from a simple deterministic model, we show that the asymptotic outcomes (as time goes to infinity) of both shallow and deep neural networks such as those used in BloombergGPT to generate economic time series are exactly the Nash equilibria of a non-potential game. We then analyse deep neural network algorithms that converge to these equilibria. The approach is extended to federated deep neural networks between clusters of regional servers and on-device clients. Finally, the variational inequalities behind large language models including encoder-decoder related transformers are established.
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