Applied Mathematics Seminar, Fall 2023

- Please direct
**questions or comments**about the seminar to**Will Feldman**(`feldman (at) math.utah.edu`

), or**Akil Narayan**(`akil (at) sci.utah.edu`

), or to**Kshiteej Deshmukh**(`kjdeshmu (at) math.utah.edu`

) - Talks are announced through the applied-math
**mailing list**. Please ask the seminar organizers for information about how to subscribe to this list.

**Monday, September 11 at 4pm. In-person, LCB 219**

Speaker: Omri Azencot,
Computer Science Department, Ben-Gurion University of the Negev

**Title: **Koopman-based Causal Representation Learning

**Abstract: **Using deep learning approaches, among the most powerful statistical technologies at our disposal, to solve causal inference tasks is challenging. In comparison, casual structural models naturally support causality but struggle with learning from data. A fundamental question in this context, how to combine the advantages of deep learning with those of causal approaches, is motivated by two open problems in representation learning: 1) learning disentangled representations, i.e., extracting from a given dataset its factors of variation, e.g., a collection of images that differ in their rotation angle and scale is associated with two disentangled factors. 2) learning interventional world models, i.e., data-driven models whose predictions depend on external inputs, e.g., a patientâ€™s prescribed treatment is personalized based on patient health measures as treatment is administered.

To fill the open problem gaps, the proposed study will develop a new causal representation learning framework based on Koopman theory and practice. Koopman showed that, theoretically, nonlinear dynamical systems can be represented with an (infinite-dimensional) linear operator, which, in practice, can be approximated via a medium-size matrix. Therefore, our approach to disentanglement learning exploits spectral properties of the Koopman operator (e.g., its eigenvalues) to identify factors of variation in an unsupervised fashion. Further, we formulate interventional models using Koopman optimal control problems.

Introducing Koopman-based causal modeling in representation learning is expected to have major theoretical and practical implications across different areas. In particular, it will improve interpretability and analysis of machine learning tools and help healthcare specialists in administering individual treatment plans.

**Monday, September 18 at 4pm. In-person, LCB 219**

Speaker: Bohyun Kim,
Department of Mathematics, University of Utah

**Title: **A positivity-preserving numerical method for a thin liquid film on a vertical cylindrical fiber

**Abstract: **When a thin liquid film flows down on a vertical fiber, one can observe the complex and captivating interfacial dynamics of an unsteady flow. Such dynamics are applicable in various fluid experiments due to their high surface area-to-volume ratio. Recent studies verified that when the flow undergoes regime transitions, the magnitude of the film thickness changes dramatically, making numerical simulations challenging. In this paper, we present a computationally efficient numerical method that can maintain the positivity of the film thickness as well as conserve the volume of the fluid under the coarse mesh setting. A series of comparisons to laboratory experiments and previously proposed numerical methods supports the validity of our numerical method. We also prove that our method is second-order consistent in space and satisfies the entropy estimate.

**Monday, September 25 at 4pm. In-person, LCB 219**

Speaker: Tyler Evans,
Department of Mathematics, University of Utah

**Title: **Viscous thin-film models of nanoscale self-organization under ion bombardment

**Abstract: **For decades, it has been observed that broad-beam irradiation of semiconductor surfaces can lead to spontaneous self-organization into highly regular patterns, sometimes at length scales of only a few nanometers. Initial theory was largely based on erosion and redistribution of material occurring on fast time scales, which are able to produce good agreement with certain aspects of surface evolution. However, further experimental and theoretical work eventually led to the realization that numerous effects are important in the irradiated target, including stresses associated with ion-implantation and the accumulation of damage leading to the development of a disordered, amorphous layer atop the substrate. We develop a continuum model based on viscous thin-film flow and ion-induced stresses within the amorphous layer which leads to good agreement with a broad range of experimental results and suggest that future advances may originate from the stress-based, viscous flow perspective.

**Monday, October 2 at 4pm. In-person, LCB 219 [Special double
feature! Two 25 minute faculty talks]**

Speaker 1: Andrej Cherkaev,
Department of Mathematics, University of Utah

**Title: **Structure of gradient vector fields in optimal multiphase composites

Speaker 2: Braxton Osting,
Department of Mathematics, University of Utah

**Title: **Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

**Monday, October 16 at 4pm. In-person, LCB 219**

Speaker: Peter Morfe,
Max Planck Institute, Leipzig

**Title: **Anomalous Diffusion in the Curl of the Gaussian Free Field

**Abstract: **I will describe recent work on anomalous diffusion asymptotics for diffusions advected by turbulent velocity fields. Precisely, the model of interest involves a passive tracer subjected to Brownian diffusion and advection by the curl of the Gaussian free field (or divergence-free white noise). Recent work of Cannizzaro, Haunschmidt-Sibitz, and Toninelli (2022) established that the mean-square displacement grows like t \sqrt{\log(t)}, confirming earlier predictions of the physics literature and a conjecture of Toth and Valko (2011). In joint work with Chatzigeorgiou, Otto, and Wang, we give an alternative proof built around ideas from stochastic homogenization, with a slightly stronger conclusion.

**Monday, November 6 at 4pm. In-person, LCB 219**

Speaker: Siting Liu,
Department of Mathematics, UCLA

**Title: **An inverse problem in mean field game from partial boundary measurement

**Abstract: **Mean-field game (MFG) systems offer a framework for modeling multi-agent dynamics, but unknown parameters pose challenges. In this work, we tackle an inverse problem, recovering MFG parameters from limited, noisy boundary observations. Despite the problem's ill-posed nature, we aim to efficiently retrieve these parameters to understand population dynamics. Our focus is on recovering running cost and interaction energy in MFG equations from boundary measurements. We formalize the problem as an constrained optimization problem with L1 regularization. We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method. Numerical experiments illustrate the effectiveness and robustness of the algorithm.

This is a joint work with Yat Tin Chow (UCR), Samy Wu Fung (Colorado School of Mines), Levon Nurbekyan (Emory), and Stanley J. Osher (UCLA).

**Monday, November 13 at 4pm. In-person, LCB 219**

Speaker: Guillaume Bal,
Department of Statistics and Mathematics, U Chicago

**Title: **Asymmetric transport in topological insulators

**Abstract: **A salient feature of two-dimensional topological insulators is the surprising robustness to perturbation of the asymmetric transport observed along interfaces separating distinct insulating bulks. Such a robustness has a topological origin and may be interpreted as an unexpected obstruction to Anderson localization. This talk reviews recent analysis and classification of partial differential Hamiltonians modeling such systems. We present a general bulk-edge correspondence relating a hard-to-compute analytical index describing the asymmetry to an easy-to-compute topological index naturally associated to the Hamiltonian. The theory finds applications in many wave phenomena in electronics, photonics, and geophysics.

**Monday, November 20 at 4pm. In-person, LCB 219**

Speaker: Knut Solna,
Department of Mathematics, UC Irvine

**Title: **Source Imaging and the Shower Curtain Effect

**Abstract: **An interesting phenomenon in optics is that it is possible to see a person behind a shower curtain better than that person can see us. This effect has been referred to as the shower curtain effect. We address the challenge of giving a precise mathematical description of this phenomenon. In addition we identify what governs the effect and discuss how imaging algorithms can be designed and analyzed when the objective is to image a source hidden behind a complex section.

**Monday, December 4 at 4pm. In-person, LCB 219**

Speaker: Michael Perlmutter,
Department of Mathematics, Boise State University

**Title: **Geometric Deep Learning on Graphs and Manifolds

**Abstract: **Many of the most successful machine learning methods are designed to incorporate the intrinsic structure of the data. The success of Convolutional Neural Networks for image classification is due, in part, to their ability to leverage the grid-like structure of the pixels, and the success of Recurrent Neural Networks is based on their ability to leverage the sequential nature of time-series data. This motivates the new field of geometric deep learning which aims to extend these successes to domains with more irregular geometry, such as graphs and manifolds, in a manner that still utilizes the intrinsic structure of the data. In my talk, I will describe popular methods for extending deep learning to graphs via graph neural networks (GNNs) as well as the strengths and limitations of popular GNNs. I then discuss how we may overcome these limitations and also how to extend these methods from graphs to the much less explored setting of manifolds.

`feldman (at) math.utah.edu`

),
`akil (at) sci.utah.edu`

), and
`kjdeshmu (at) math.utah.edu`

).
**Past lectures:**
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Fall 1995.

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