Department of Mathematics

We prove nonemptyness of domains of proper discontinuity of Anosov groups of affine Lorentzian transformations of Rn.

Our previous multiscale graph basis dictionaries/graph signal transforms—Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives—were developed for analyzing data recorded on vertices of a given graph. In this article, we propose their generalization for analyzing data recorded on edges, faces (i.e., triangles), or more generally κ -dimensional simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The key idea is to use the Hodge Laplacians and their variants for hierarchical partitioning of a set of κ -dimensional simplices in a given simplicial complex, and then build localized basis functions on these partitioned subsets. We demonstrate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset.

Many microorganisms propel themselves through complex media by deforming their flagella. The beat is thought to emerge from interactions between forces of the surrounding fluid, the passive elastic response from deformations of the flagellum and active forces from internal molecular motors. The beat varies in response to changes in the fluid rheology, including elasticity, but there are limited data on how systematic changes in elasticity alter the beat. This work analyses a related problem with fixed-strength driving force: the emergence of beating of an elastic planar filament driven by a follower force at the tip of a viscoelastic fluid. This analysis examines how the onset of oscillations depends on the strength of the force and viscoelastic parameters. Compared to a Newtonian fluid, it takes more force to induce the instability in viscoelastic fluids, and the frequency of the oscillation is higher. The linear analysis predicts that the frequency increases with the fluid relaxation time. Using numerical simulations, the model predictions are compared with experimental data on frequency changes in the bi-flagellated alga Chlamydomonas reinhardtii. The model shows the same trends in response to changes in both fluid viscosity and Deborah number and thus provides a possible mechanistic understanding of the experimental observations.

The allometric trophic network (ATN) framework for modeling population dynamics has provided numerous insights into ecosystem functioning in recent years. Herein we extend ATN modeling of the intertidal ecosystem off central Chile to include empirical data on pelagic chlorophyll-a concentration. This intertidal community requires subsidy of primary productivity to support its rich ecosystem. Previous work models this subsidy using a constant rate of phytoplankton input to the system. However, data shows pelagic subsidies exhibit highly variable, pulse-like behavior. The primary contribution of our work is incorporating this variable input into ATN modeling to simulate how this ecosystem may respond to pulses of pelagic phytoplankton. Our model results show that: (1) closely related sea snails respond differently to phytoplankton variability, which is explained by the underlying network structure of the food web; (2) increasing the rate of pelagic-intertidal mixing increases fluctuations in species biomasses that may increase the risk of local extirpation; (3) predators are the most sensitive species to phytoplankton biomass fluctuations, putting these species at greater risk of extirpation than others. Finally, our work provides a straightforward way to incorporate empirical, time-series data into the ATN framework that will expand this powerful methodology to new applications.

In this article, we evaluate the challenges and best practices associated with the Markov bases approach to sampling from conditional distributions. We provide insights and clarifications after 25 years of the publication of the Fundamental theorem for Markov bases by Diaconis and Sturmfels. In addition to a literature review, we prove three new results on the complexity of Markov bases in hierarchical models, relaxations of the fibers in log-linear models, and limitations of partial sets of moves in providing an irreducible Markov chain. Supplementary materials for this article are available online.

Differential privacy is a mathematical concept that provides an information-theoretic security guarantee. While differential privacy has emerged as a de facto standard for guaranteeing privacy in data sharing, the known mechanisms to achieve it come with some serious limitations. Utility guarantees are usually provided only for a fixed, a priori specified set of queries. Moreover, there are no utility guarantees for more complex-but very common-machine learning tasks such as clustering or classification. In this paper we overcome some of these limitations. Working with metric privacy, a powerful generalization of differential privacy, we develop a polynomial-time algorithm that creates a private measure from a data set. This private measure allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools. Moreover, we prove an asymptotically sharp min-max result for private measures and synthetic data in general compact metric spaces, for any fixed privacy budget ε bounded away from zero. A key ingredient in our construction is a new superregular random walk, whose joint distribution of steps is as regular as that of independent random variables, yet which deviates from the origin logarithmically slowly.

Light microscopy methods have continued to advance allowing for unprecedented analysis of various cell types in tissues including the brain. Although the functional state of some cell types such as microglia can be determined by morphometric analysis, techniques to perform robust, quick, and accurate measurements have not kept pace with the amount of imaging data that can now be generated. Most of these image segmentation tools are further burdened by an inability to assess structures in three-dimensions. Despite the rise of machine learning techniques, the nature of some biological structures prevents the training of several current day implementations. Here we present PrestoCell, a novel use of persistence-based clustering to segment cells in light microscopy images, as a customized Python-based tool that leverages the free multidimensional image viewer Napari. In evaluating and comparing PrestoCell to several existing tools, including 3DMorph, Omipose, and Imaris, we demonstrate that PrestoCell produces image segmentations that rival these solutions. In particular, our use of cell nuclei information resulted in the ability to correctly segment individual cells that were interacting with one another to increase accuracy. These benefits are in addition to the simplified graphically based user refinement of cell masks that does not require expensive commercial software licenses. We further demonstrate that PrestoCell can complete image segmentation in large samples from light sheet microscopy, allowing quantitative analysis of these large datasets. As an open-source program that leverages freely available visualization software, with minimum computer requirements, we believe that PrestoCell can significantly increase the ability of users without data or computer science expertise to perform complex image analysis.

Abstract: We compute the Borel–Moore homology of unramified affine Springer fibers for $\textrm{Gr}_n$ under the assumption that they are equivariantly formal and relate them to certain ideals discussed by Haiman. For $n=3$, we give an explicit description of these ideals, compute their Hilbert series, generators, and relations, and compare them to generalized $(q,t)$-Catalan numbers. We also compare the homology to the Khovanov–Rozansky homology of the associated link, and prove a version of a conjecture of Oblomkov, Rasmussen, and Shende in this case.

The function of the smooth muscle cells lining the walls of mammalian systemic arteries and arterioles is to regulate the diameter of the vessels to control blood flow and blood pressure. Here, we describe an in silico model, which we call the 'Hernandez-Hernandez model', of electrical and Ca²⁺ signaling in arterial myocytes based on new experimental data indicating sex-specific differences in male and female arterial myocytes from murine resistance arteries. The model suggests the fundamental ionic mechanisms underlying membrane potential and intracellular Ca²⁺ signaling during the development of myogenic tone in arterial blood vessels. Although experimental data suggest that K_V1.5 channel currents have similar amplitudes, kinetics, and voltage dependencies in male and female myocytes, simulations suggest that the K_V1.5 current is the dominant current regulating membrane potential in male myocytes. In female cells, which have larger K_V2.1 channel expression and longer time constants for activation than male myocytes, predictions from simulated female myocytes suggest that K_V2.1 plays a primary role in the control of membrane potential. Over the physiological range of membrane potentials, the gating of a small number of voltage-gated K⁺ channels and L-type Ca²⁺ channels are predicted to drive sex-specific differences in intracellular Ca²⁺ and excitability. We also show that in an idealized computational model of a vessel, female arterial smooth muscle exhibits heightened sensitivity to commonly used Ca²⁺ channel blockers compared to male. In summary, we present a new model framework to investigate the potential sex-specific impact of antihypertensive drugs.