UCSB Applied Math/PDE/Data Science Seminar

Abstract: The surface quasi-geostrophic (SQG) equation is a fundamental model in fluid dynamics that describes how temperature variations on the surface of a rotating fluid (like the atmosphere or ocean) can influence motion. In this talk, we study both the inviscid (no friction) and α-dissipative (with fractional diffusion) versions of this equation, collectively known as the α-SQG equations. To better understand these models, we first introduce a related system called the Double Dissipative SQG (DSQG) equations and show that solutions to this system behave nicely under certain conditions. We then investigate what happens as the effects of viscosity (or dissipation) become very small—a process known as taking the inviscid limit. Using energy estimates, we prove that solutions remain stable and converge as this limit is taken. Finally, we show that the limiting solution satisfies the weak form of the α-SQG equation, and with enough smoothness, it becomes a classical solution. Throughout, I’ll give an overview of the mathematical tools involved and explain how these results contribute to our broader understanding of fluid motion and turbulence models.

Host: Michael Gulas

Abstract: For a class of Schrodinger equations with locally analytic nonlinear terms, we treat the inverse problem of theoretically determining the potential and nonlinear interaction strength by measurements of the normal derivative on an arbitrary part of the boundary. In particular, we establish unique determination and stability of the coefficients with respect to the Neumann data provided a priori knowledge of the coefficients on some neighborhood of the boundary. This is joint work with my advisor, Hanming Zhou.

Host: Hanming Zhou

Abstract: (This is joint work with Karina Behera, Tron Omland, and Nicole Wu) Quantum metric spaces were originally developed by Marc Rieffel to address some statements in the particle physics literature, and they provide a notion of distance between quantum states, and therefore, a notion of distance between elements of an object in quantum information theory known as the density space. However, there already exist well-known metrics that provide distances between elements of a density space such as the Bures metric. Our work establishes comparisons of these metrics in a topological and metric sense. This work is partially supported by NSF grant DMS-2316892.

Host: Therese Landry

Abstract: This talk presents a control-oriented perspective on Scientific Machine Learning (SciML) for modeling, optimization, and control of dynamical systems. SciML provides a unifying computational paradigm that integrates physics-based models, optimization algorithms, and control policies within a differentiable programming framework. This synthesis enables computation of structured gradients for constrained system identification, learning-to-optimize, and learning-based control while preserving interpretability, stability, and physical consistency. Three recent advances will be highlighted. First, differentiable predictive control, a SciML approach that merges model predictive control with gradient-based learning to enable scalable, self-supervised training of explicit control policies suitable for real-time deployment on embedded hardware. Second, an operator-splitting formulation for neural differential-algebraic equations that integrates mechanistic dynamics with neural components to achieve robust extrapolation in systems with implicit constraints and conservation laws. Third, a self-supervised learning-to-optimize framework for mixed-integer nonlinear programs that provides feasibility guarantees and high-quality approximate solutions in milliseconds. Together, these advances demonstrate how SciML can unlock new capabilities for the modeling, optimization, and control of complex dynamical systems, with applications in power grid and building energy management.

Host: Paul J. Atzberger

Abstract: The quest to describe quantum states in a way that parallels classical probability theory goes back to the early days of quantum theory. Central to this effort has been the development of non-commutative versions of functional calculus and corresponding quasi-probability distributions, most notably, the Weyl functional calculus and Wigner distributions. A derivative classical probability distribution, known as the Husimi function, can be obtained from the Wigner distribution via Gaussian smearing in phase space. In this talk we explain the origin of the smearing in terms of continuous quantum measurements, lifting it from a convolution in phase space to a convolution in Hilbert space, via a novel notion of a Gaussian semigroup. This formalism yields a practical and canonical way to represent quantum states with probability distributions on Euclidean space for an arbitrary choice of self-adjoint observables.

Host: Therese Landry

Abstract: With a single circulating tick-borne pathogen in a population of ticks and vertebrate hosts, the basic reproduction number incorporates contributions from tick-to-tick, tick-to-vertebrate host, and vertebrate host-to-tick transmission routes. With two different co-circulating pathogen strains, resident and invasive, and under the assumption that tick-to- tick is the only transmission route in a tick population feeding on vertebrate hosts, the invasion reproduction number depends on whether the model system of ordinary differential equations describing the system possesses Alizon’s neutrality property. 

We show that a simple (intuitive) model, with two populations of ticks infected with a single strain, resident or invasive, one population of co-infected (infected with both strains) ticks, and one population of susceptible (not infected) ones, does not have such neutrality property. We prove that, depending on the choice of model parameters, the invasion reproduction number can be equal to, larger than, or smaller than one. However, in the limit when both resident and invasive strains are equal (so that co-infected ticks transmit at the same rate as singly-infected ones), the invasion reproduction number is greater than one. 

We propose a new mathematical model (the two-slot model) for tick-borne pathogen transmission with tick co-feeding, co-infection, and co-transmission, which is an extension of Alizon’s model. We show that the neutral two-slot model is capable of representing the invasion potential of a novel pathogen strain by including populations of ticks doubly-infected with the same strain (resident or invasive). The invasion reproduction number is analysed with the next-generation method and via numerical simulations. We apply the two-slot model to compute the fraction of co-infected questing ticks at the end of a season as a function of the co-transmission probability, and evaluate the potential risk to humans (https://pmc.ncbi.nlm.nih.gov/articles/PMC10983997/). 

The second story involves HIV-1 co-infection. In this instance, we are interested in quantifying the potential for a viral recombinant to get established in a population in which two viral HIV-1 subtypes co-circulate. We make use of epidemiological data in Brazil and China, and together with a mathematical model of co-infection and recombination describe and quantify infection dynamics. A feature of the model is an essential co-infection step. This allows to include in the model both population and within-host parameters that characterize infection dynamics, since recombi- nation takes place in an individual host, and transmission of the recombinant is a population-level rate. We describe how the ability of the viral recombinant to get stablished depends on both types of parameters. We conclude with a perspective on how to improve the current model and the data sets required for its calibration.

Host: Therese Landry