Differential Geometry, PDE and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
The Kuramoto-Sivashinsky equation (KSE) draws the attention of many mathematicians and scientists for its chaotic behavior and its similarity to Navier-Stokes equations. In this present study, Serrin-type regularity of two-dimensional Kuramoto-Sivashinsky equation is studied. PDF available Watch online on Tuesday the 1st at 4 PM via this Zoom link.
Motivated by the long-standing question of how precisely classical mechanics relates to quantum mechanics (QM), I show how equations discovered in 1927 by Erwin Madelung mathematically relate to the Schrödinger equation and Newtonian mechanics. Historically, Madelung's equations were constitutive for the de-Broglie-Bohm interpretation of QM, considered its first major `hidden variable' formulation. However, even if one takes the less stringent philosophical position of viewing QM as a statistical theory describing ensembles of particles, Madelung's equations provide a deep insight into quantum dynamics, the origin of `(first) quantization' and complex numbers in QM, as well as the classical limit. This provides the basis for viewing the Madelung equations as physically more fundamental than the Schrödinger equation.
Furthermore, if one compares the expectation values of the fundamental observables of position, momentum, energy, as well as angular momentum with the expectation values of the respective, naturally chosen functions in the framework of Kolmogovian probability theory, one is naturally lead to the question whether von Neumann's functional-analytic axiomatization of QM was a consequence of the historical development of QM - rather than empirical necessity - so that an axiomatization in terms of Kolmogorov's framework and differential-geometric evolution equations - such as Madelung's - would be more appropriate. By potentially depriving quantum probability theory of its empirical basis, the question is thus of foundational importance for mathematical probability theory in general. Finally, I will argue for a stochastic interpretation of the Schrödinger theory, as pioneered by Fenyes, Nelson, Bohm and Vigier, placing it in the same class of physical theories as the theory of diffusion.
This talk is primarily based on one of my articles, published in Foundations of Physics 47, 1317–1367 (2017).
Watch online via this Zoom link.
Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models. In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of the underlying population dynamics. Watch online Tuesday the 10th at 3 PM via this Zoom link -- passcode 427144
Watch online Tuesday the 1st at 4 PM via this Zoom link -- meeting ID 5370 7942, passcode 311075