SIAM Student Chapter Seminar

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Spring 2024

Date Location Speaker Title
2/2 VV911 Thomas Chandler (UW-Madison) Fluid–body interactions in anisotropic fluids
3/8 Ingraham 214 Danyun He (Harvard) Energy-positive soaring using transient turbulent fluctuations
3/15 VV911&Zoom Xiaoyu Dong (UMich) Approximately Hadamard matrices and Riesz bases in frames
3/22 VV911&Zoom Mengjin Dong (UPenn) Advancing Alzheimer's Disease Research: Insights and Innovations in MRI-Based Progression Tracking
4/5 VV911 Sixu Li (UW-Madison) A Good Score Does not Lead to A Good Generative Model
4/12 VV911&Zoom Anjali Nair (UChicago) Some scaling limits for long distance wave propagation in random media
4/19 VV911 Jingyi Li (UW-Madison) Arrested development of active suspensions in anisotropic fluids
5/2 (Thursday!) VV911 David Keating(UW-Madison) A tour of domino tilings


February 2, Thomas Chandler (UW-Madison): Fluid anisotropy, or direction-dependent response to deformation, can be observed in biofluids like mucus or, at a larger scale, self-aligning swarms of active bacteria. A model fluid used to investigate such environments is a nematic liquid crystal. In this talk, we will use complex variables to analytically solve for the interaction between bodies immersed in liquid crystalline environments. This approach allows for the solution of a wide range of problems, opening the door to studying the role of body geometry, liquid crystal anchoring conditions, and deformability. Shape-dependent forces between bodies, surface tractions, and analogues to classical results in fluid dynamics will also be discussed.

March 8, Danyun He (Harvard University): The ability of birds to soar in the atmosphere is a fascinating scientific problem. It relies on an interplay between the physical processes governing atmospheric flows, and the capacity of birds to process cues from their environment and learn complex navigational strategies. Previous models for soaring have primarily taken advantage of thermals of ascending hot air to gain energy. Yet, it remains unclear whether energy loss due to drag can be overcome by extracting work from transient turbulent fluctuations. In this talk, I will present a recent work that we look at the alternative scenario of a glider navigating in an idealized model of a turbulent fluid where no thermals are present. First, I will show the numerical simulations of gliders navigating in a kinematic model that captures the spatio-temporal correlations of atmospheric turbulence. Energy extraction is enabled by an adaptive algorithm based on Monte Carlo tree search that dynamically filters acquired information about the flow to plan future paths. Then, I will demonstrate that for realistic parameter choices, a glider can navigate to gain height and extract energy from flow. Glider paths reflect patterns of foraging, where exploration of the flow is interspersed with bouts of energy extraction through localized spirals. As such, this work broadens our understanding of soaring, and extends the range of scenarios where soaring is known to be possible.

March 15, Xiaoyu Dong (University of Michigan, Ann Arbor): An $n \times n$ matrix with $\pm 1$ entries which acts on $\R^n$ as a scaled isometry is called Hadamard. Such matrices exist in some, but not all dimensions. Combining number-theoretic and probabilistic tools we construct matrices with $\pm 1$ entries which act as approximate scaled isometries in $\R^n$ for all $n \in \N$. More precisely, the matrices we construct have condition numbers bounded by a constant independent of $n$.

Using this construction, we establish a phase transition for the probability that a random frame contains a Riesz basis. Namely, we show that a random frame in $\R^n$ formed by $N$ vectors with  independent identically distributed coordinate having a non-degenerate symmetric distribution contains many Riesz bases with high probability provided that $N \ge \exp(Cn)$. On the other hand, we prove that if the entries are subgaussian, then a random frame fails to contain a Riesz basis with probability close to $1$ whenever $N \le \exp(cn)$, where $c<C$ are constants depending on the distribution of the entries.

March 22, Mengjin Dong (University of Pennsylvania): Alzheimer’s disease (AD) is a progressive neurodegenerative disorder characterized by memory loss, cognitive decline, and behavioral changes primarily in the elderly population. As the most prevalent form of dementia, it impacts millions of families globally. The pathological hallmarks of AD, such as abnormal protein build-up in the brain, can manifest decades before the onset of clinical symptoms. Neuroimaging modalities such as positron emission tomography (PET) and magnetic resonance imaging (MRI) play pivotal roles in studying disease progression and elucidating its underlying mechanisms.

In this presentation, I will commence with an overview of AD fundamentals and recent research advancements. Subsequently, I will delve into my research, which utilizes deep learning techniques to longitudinally monitor and localize AD progression using MRI data.

April 5, Sixu Li (UW-Madison): Score-based Generative Models (SGMs) is one leading method in generative modeling, renowned for their ability to generate high-quality samples from complex, high-dimensional data distributions. The method enjoys empirical success and is supported by rigorous theoretical convergence properties. In particular, it has been shown that SGMs can generate samples from a distribution that is close to the ground-truth if the underlying score function is learned well, suggesting the success of SGM as a generative model. We provide a counter-example in this paper. Through the sample complexity argument, we provide one specific setting where the score function is learned well. Yet, SGMs in this setting can only output samples that are Gaussian blurring of training data points, mimicking the effects of kernel density estimation. The finding resonates a series of recent finding that reveal that SGMs can demonstrate strong memorization effect and fail to generate.

This is a joint work with Shi Chen and Qin Li.

April 12, Anjali Nair (UChicago): Wave propagation in random media is a rather interesting and complex phenomenon owing to the interplay of multiple scales. While a complete understanding of wave fields propagating in reasonably arbitrary random media remains essentially out of reach, much progress has been made in the setting of paraxial beam propagation. The paraxial approximation aims at considering high-frequency waves propagating over long distances along a privileged direction with negligible backscattering. In particular, I will discuss the so-called white noise paraxial scaling, where different asymptotic regimes lead to very different statistical limits for the wave field.

April 19, Jingyi Li (UW-Madison): Active suspensions in anisotropic, viscoelastic fluids experience competing stresses on their orientational alignment. The orientational order of extensile-stress-generating particles is hydrodynamically unstable to a bend instability, but the elasticity of the fluid drives the system towards alignment. To study these competing effects, we examine a dilute suspension of active particles in an Ericksen-Leslie model nematic liquid crystal. Our first observation is that, a bifurcation emerges uniform alignment with no flow, to a steady flowing state of arrested development, as the particle activity cross a critical threshold. And a secondary instability to transverse perturbations is observed at larger activity, leading to arrested, flowing states with features emerging at smaller wavelengths. Finally, if the system is motile, the arrested states become traveling wave solutions of coupled nonlinear advection-diffusion equations with spatially varying particle concentration and orientation.

May 2, David Keating (UW-Madison):A classic problem asks whether or not an 8x8 checkerboard can be tiled by dominos where each domino takes up covers two adjacent squares of the checkerboard and we do not allow the dominos to overlap. It is easy to see that this is possible (construct a tiling yourself!) but it is much harder to count the number of possible tilings. In this talk, we will discuss this and related questions about domino tilings of checkerboard regions. We will focus on two main topics.

Enumeration: is it possible to tile a region by dominos, and how many possible tilings are there?

Randomness: how can one chose a tiling uniformly at random from the set of all possible tilings, and what does such a tiling look like?

A region that will be of particular interest to us is known as the Aztec diamond.  Random tilings of very large Aztec diamonds exhibit what is known as the limit shape phenomenon, a kind of law of large numbers in which random tilings exhibit interesting geometric features.  We will see examples of these limit shapes and mention connections to other areas in probability.

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