Mathematics Colloquia and Seminars
Title: A Comprehensive Review of Circular Scan Method
Speaker: Chamodi Wijenayake
Date: 4/22/2024
Abstract: A “disease cluster” is an unusual aggregation of health events that are grouped together in space and/or time. Detecting clusters can provide insights into the dynamics of diseases, helping us prevent outbreaks and discover diseases. Data related to health, crime, and other events of interest are frequently reported as counts within pre-specified enumeration regions such as counties or census tracts. This preserves the privacy of individuals associated with the event of interest while allowing for the detection of patterns. Spatial scan methods are popular for identifying disease clusters of points and aggregated data. The original circular scan method proposed by Kulldorff addressed two important weaknesses prevalent in early cluster detection techniques: (i) global tests that identified a general discrepancy between observed and expected incidence rates but failed to identify a specific set of regions with an unusually high incidence rate, and (ii) did not satisfactorily address the problem of multiple comparisons. In this presentation, I will describe the two main components of the circular scan method: (i) constructing the set of candidate zones and (ii) the likelihood ratio test statistics. Lastly, I will demonstrate the detection of clusters using the smerc R package.
Title: Bridging from single to collective cell migration: Repolarization, Cell entrainment, and migrating clusters
Speaker: Dr. Andreas Buttenschoen, Assistant Professor in the Department of Mathematics and Statistics, University of Massachusetts Amherst
Date: 4/19/2024
Abstract: Eukaryotic cell motility requires coordination across three spatial size scales: intracellular signaling that regulates cell shape and movement, single cells motility (e.g. cells responding to extracellular signals), and collective cell behavior from a few cells to tissues (e.g. cells working together). Mathematical and computational models can assist in interpreting experiments and developing an understanding of cell behavior at many levels of organization. In this talk, I will focus on three collective phenomena: (1) repolarization, (2) cell entrainment, and (3) cell cluster migration. At each of these spatial scales, a different modeling approach is most suitable. At the intra-cellular scale, I will use systems of reaction diffusion equations and bifurcation analysis to elucidate repolarization of small GTPases, which are central regulators of cell morphology and motility. Agent-based models inspired by colloidal physics are employed at the cellular scale to understand cell entrainment, and finally I will use non-local partial differential equations to elucidate cell cluster stability at the “tissue level”. At the tissue level, I will demonstrate how Morse potentials (a commonly used cell-cell interaction potential) can be derived and how analysis of non-local PDEs can inform agent-based models.
Title: Dirichlet distribution parameter estimation and statistical testing with applications in microbiome analyses (Part II)
Speaker: Thevasha Sathiyakumar
Date: 4/15/2024
Abstract: Understanding microbiomes is of paramount importance for understanding human and environmental health. Quantifying this importance relies on statistical modeling of either the raw taxonomic abundances or the transformed relative abundances. There is currently no ideal distribution or method for carrying out this modeling, despite that numerous studies have promoted the idea that relative abundance is the most accurate method to interpret microbiome data (as the absolute abundance estimates are noninformative). In this work, the Dirichlet distribution is proposed to model the relative abundances of taxa directly without the use of any further transformation. Several Method of Moments Estimators (MMEs) of the Dirichlet distribution are compared with the asymptotic properties of the Maximum Likelihood Estimator (MLE) in a simulation study. Comparison of various estimators is done through a comprehensive simulation study for different dimensions and sample sizes. We demonstrate the Dirichlet modeling methodology with several real environmental soil and human body microbiome datasets and compare the performance of the Dirichletestimators to the popular Dirichlet-Multinomial Model (DMM) with nonparametric bootstrapping. We speculate that differences in sequencing techniques may lead to greater differences between modeling with the Dirichlet distribution compared to a Bayesian DMM. We also implement some initial statistical testing through both classical likelihood-based tests and parametric bootstrapping and through a simulation study compare the size and power of these tests across different regions of the Dirichlet parameter space. We explore properties of both simple and composite symmetric null hypothesis testing for the Dirichlet distribution and demonstrate the benefits of this novel methodology in an applied context.
Title: COVID-19 in New York state: Effects of demographics and air quality on infection and fatality
Speaker: Prof. Sumona Mondal, Department of Mathematics, Clarkson University
Date: 4/12/2024
Abstract: The coronavirus disease 2019 (COVID-19) has had a global impact that has been unevenly distributed among and even within countries. Multiple demographic and environmental factors have been associated with the risk of COVID-19 spread and fatality, including age, gender, ethnicity, poverty, and air quality, among others. However, the specific contributions of these factors are yet to be understood. Here, we attempted to explain the variability in infection, death, and fatality rates by understanding the contributions of a few selected factors. We compared the incidence of COVID-19 in New York State (NYS) counties during the first wave of infection and analyzed how different demographic and environmental variables associated with the variation observed across the counties. We observed that infection and death rates, two important COVID-19 metrics, are highly correlated with both being highest in counties located near New York City, considered as one of the epicenters of the infection in the US. In contrast, disease fatality was found to be highest in a different set of counties despite registering a low infection rate. To investigate this apparent discrepancy, we divide the counties into three clusters based on COVID-19 infection, death, or fatality and compared the differences in the demographic and environmental variables such as ethnicity, age, population density, poverty, temperature, and air quality in each of these clusters. Furthermore, a regression model built on this data reveals PM2.5 and distance from the epicenter are significant risk factors for infection, while disease fatality has a strong association with age and PM2.5. Our results demonstrate that for the NYS, demographic components distinctly associate with specific aspects of COVID-19 burden and also highlight the detrimental impact of poor air quality. These results could help design and direct location-specific control and mitigation strategies.
Title: Mathematical models and control strategies related to the COVID-19 pandemic and infectious diseases generally
Speaker: Prof. James Greene, Department of Mathematics, Clarkson University
Date: 4/12/2024
Abstract: Since the onset of the COVID-19 pandemic, there has been much scientific interest in the ability of mathematical models to both predict disease dynamics, as well as their use in designing intervention strategies that can mitigate disease burden on medical infrastructure, reduce transmission, minimize negative economic and psychological impacts, etc. In this talk, we will present a number of recent modeling projects which address different questions of interest related to the COVID-19 pandemic and infectious diseases generally. Specifically, we will discuss early novel models of the spread of COVID-19, which capture both the effect of asymptomatic transmission and social distancing via explicit compartments. We will then discuss the role of non-pharmaceutical interventions in both reducing peak infection numbers ( " flattening the curve" ) while simultaneously minimizing time spent in strict lockdowns; general optimal design strategies can be numerically seen to exist throughout a large class of epidemic models, which we show to be rigorously justified in the SIR model. Opening/closing strategies in schools/universities will also be studied, where we analyze robust feedback laws which maximize in-person instruction while keeping infections below a critical threshold. Furthermore, as mutations lead to new viral strains, such as the Omicron and Delta variants, important questions related to evolutionary fitness/competition exist, such as the effect of selection with respect to infectivity vs. disease severity. Again, utilizing relatively simple mathematical models, we study the impact of selection on mutant variants and characterize necessary parameter changes yielding a fitness advantage. We note that in almost all of the scientific questions of interest addressed here, transient disease dynamics are of fundamental importance, which will be a theme throughout this talk.
Title: Ergodic Quotient Representation of Dynamical System Phase Space
Speaker: Lucas Reynolds
Date: 4/1/2024
Abstract: The ergodic quotient is a structure that simplifies the study of a dynamical system’s phase space by mapping unique ergodic sets (think, orbits) to unique points. However, its construction necessarily places the ergodic quotient into infinite dimensional (sequential) space. Being a true quotient, this structure has no larger intrinsic dimension than the phase space it represents. The work presented here briefly states the ergodic quotient construction algorithm, followed by a foray through several existing dimensionality reduction algorithms, and ends with some examples showing how a low dimensional representation of a numerical approximation of the Chirikov Standard Map’s ergodic quotient reflects the structure of said map’s phase space.
Title: A mathematical model for investigating the role of membrane-to-cortex binding proteins in bleb expansion
Speaker: S. H. Dinuka S. de Silva
Date: 4/1/2024
Abstract: Cells utilize two primary structures for migration: blebs and F-actin driven protrusions (FDP). Blebs are spherical cell membrane protrusions driven by intracellular fluid pressure, while FDP utilize rapid actin polymerization to propel the membrane forward. Cells can dynamically switch between FDP and bleb-based movement depending on their environment. Although FDP are well-characterized, many questions remain unanswered regarding the physical and chemical mechanisms governing bleb-driven motility. Particularly intriguing is the role of membrane-to-cortex binding proteins (such as talin) in regulating bleb size and frequency. Previous studies have suggested that the absence of membrane-to-cortex linker proteins produce larger and more frequent blebs. However, recent experimental data from Dictyostelium discoideum cells show that the loss of talin reduces the size and frequency of blebs. In this work, we present a mathematical model of bleb expansion in confined Dictyostelium discoideum cells, simulated with the level set method, and employ this model to clarify the role of talin in bleb expansion.
Title: Reconstruction of Crystallographic Grain Orientation Data using U-nets
Speaker: Emmanuel A. Atindama
Date: 3/25/2024
Abstract: This talk aims to reduce the time it takes to obtain crystallographic data using a Scanning Electron Microscope.
Most metals consist of a crystalline structure made of "crystal grains." The orientation of these grains plays a crucial role in determining the metal's properties like elasticity, strength, and fracture points. These orientations are usually measured using techniques such as electron backscatter diffraction (EBSD). The orientations are recorded as Euler angles; however, these measurements can have missing areas due to errors in instrumentation. Such gaps can lead to incorrect assessments of the material's structural properties, which might cause dangerous situations. Carefully scanning these grains using an electron microscope takes tens of hours to obtain the crystal orientations. Our goal, therefore, is to scan quickly (in minutes), and then accurately complete the restoring orientation data using a process known in image processing as "denoising”. The primary methods for denoising are one-step methods, PDE-based methods, and machine-learning algorithms. The exemplar-based method finds a region with a similar neighborhood in the data. In contrast, the machine learning approach uses extensive training datasets to fill the missing area optimally. Our study introduces a deep learning model for EBSD data inpainting. We fill the unknown areas using a deep-learning model supported by a large set of simulated data. We use statistical tests to demonstrate that this technique yields better results than prior deterministic algorithms.
Title: Mathematical model for rod outer segment dynamics during retinal detachment and reattachment
Speaker: William Ebo Annan
Date: 3/25/2024
Abstract: Retinal detachment (RD) is the separation of the neural layer from the retinal pigmented epithelium, thereby preventing the supply of nutrients to the cells within the neural layer of the retina. In vertebrates, primary photoreceptor cells consisting of rods and cones undergo daily renewal of their outer segment through the addition of disc-like structures and shedding of these discs at their distal end. When the retina detaches, the outer segment of these cells begins to degenerate, and if surgical procedures for reattachment are not done promptly, the cells can die and lead to blindness. The precise effect of RD on the renewal process is not well understood. Additionally, a time frame within which reattachment of the retina can restore proper photoreceptor cell function is not known. Focusing on rod cells, we propose a mathematical model to clarify the influence of retinal detachment on the renewal process. Our model simulation and analysis suggest that RD stops or significantly reduces the formation of new discs and that an alternative removal mechanism is needed to explain the observed degeneration during RD. Sensitivity analysis of our model parameters points to the disc removal rate as the key regulator of the critical time within which retinal reattachment can restore proper photoreceptor cell function.
Title: Dirichlet distribution parameter estimation and statistical testing with applications in microbiome analyses (Part I)
Speaker: Daniel Fuller
Date: 3/11/2024
Abstract: Understanding microbiomes is of paramount importance for understanding human and environmental health. Quantifying this importance relies on statistical modeling of either the raw taxonomic abundances or the transformed relative abundances. There is currently no ideal distribution or method for carrying out this modeling, despite that numerous studies have promoted the idea that relative abundance is the most accurate method to interpret microbiome data (as the absolute abundance estimates are noninformative). In this work, the Dirichlet distribution is proposed to model the relative abundances of taxa directly without the use of any further transformation. Several Method of Moments Estimators (MMEs) of the Dirichlet distribution are compared with the asymptotic properties of the Maximum Likelihood Estimator (MLE) in a simulation study. Comparison of various estimators is done through a comprehensive simulation study for different dimensions and sample sizes. We demonstrate the Dirichlet modeling methodology with several real environmental soil and human body microbiome datasets and compare the performance of the Dirichlet estimators to the popular Dirichlet-Multinomial Model (DMM) with nonparametric bootstrapping. We speculate that differences in sequencing techniques may lead to greater differences between modeling with the Dirichlet distribution compared to a Bayesian DMM. We also implement some initial statistical testing through both classical likelihood-based tests and parametric bootstrapping and through a simulation study compare the size and power of these tests across different regions of the Dirichlet parameter space. We explore properties of both simple and composite symmetric null hypothesis testing for the Dirichlet distribution and demonstrate the benefits of this novel methodology in an applied context.
Title: Spine arthritis: Using medical image analysis to better understand disease progression.
Speaker: Prof. Dale E. Fournier
Date: 1/29/2024
Abstract: Chronic back pain continues to be the leading cause of years-lived with disability worldwide. The premise guiding my research agenda is that a deeper understanding of imaging features of the spine is needed to improve diagnosis and clinical management of back pain caused by arthritis, and specifically diffuse idiopathic skeletal hyperostosis (DISH). My studies of DISH using microcomputed tomography, histological, and physical analyses revealed remarkable heterogeneity in the appearance of the condition. The purpose of this talk with the Department of Mathematics is to showcase this work and seek advice on methods to better understand data retrieved from imaging data.
Title: Continuous Quantum Walks on Infinite Graphs
Speaker: Prof. Chris Godsil, Department of Combinatorics and Optimization, University of Waterloo
Date: 8/3/2023
Abstract: A continuous quantum walk on a graph X with adjacency operator A is determined by the 1-parameter family of unitary
operators U(t) := exp(itA) (for real t). We have perfect state transfer from vertex a to vertex b at time t if |U(t)a,b| = 1, and
the vertex a is periodic if there is a positive time t such that |U(t)a,a| = 1. If there is perfect state transfer from a to b at time
t, the vertices a and b are each periodic at time 2t. For finite graphs, we have extensive theory about the occurrence of perfect
state transfer and periodicity. I will discuss some of the ideas and tools in a recent proof that connected infinite graphs with
bounded valency do not have periodic vertices.
Title: From Colorings to Virtual Links and Back
Speaker: Prof. Louis H Kauffman, University of Illinois, Chicago
Date: 6/21/2023
Abstract: We will discuss a generalization of virtual knot theory, Multiple Virtual Knot Theory, that includes an arbitrary number of distinct virtual crossing types in the context of a problem in graph coloring that motivates this theory.
To that end, we review the Penrose formula that counts the number of edge colorings (three distinct colors at a node) of trivalent plane graphs, and generalize the Penrose formula - once to obtain coloring counts for all cubic graphs - then to make a new perfect matching polynomial. The new perfect matching polynomial is mapped by a correspondence to virtual link theory to meet a generalized bracket polynomial. The appropriate virtual knot theory for this construction has two virtual crossings that detour over each other and over the classical crossings. A Reidemeister move configuration with the two distinct virtual crossings does not cancel. This form of Doubled Virtual Knot Theory is the correct image for the expanded Penrose evaluation and leads to new virtual invariants and new examples of virtualization phenomena between classical and virtual knot theory as well as new relationships between graph theory and virtual knot theory. With this motivation in hand we will also discuss the fully generalized multiple virtual knot theory.
Standard (one virtual crossing type) virtual knot theory can be interpreted in terms of knots and links embedded in thickened surfaces and so has relationships with a number of applications. We will indicate some of these connections as well.
Title: Improving mathematical models of population and disease dynamics by incorporating nonlocality and extreme weather variables
Speaker: Prof. Majid Bani Yaghoub, University of Missouri-Kansas City
Date: 1/15/2023
Abstract: Ecological and epidemiological studies have strongly benefited from analysis of ODE and PDE models for decades. The existence and stability of stationary solutions, speed and shape of traveling waves, and optimal control of invasive species are only a few examples. These models have also been successful in revealing the underlying mechanisms governing dynamics of infectious diseases. Accurate estimations of the basic reproduction number, prediction of endemic zones, estimation of eradication efforts and prediction of vaccination effects are good examples of model contributions to epidemiology. The interconnections between global issues such as epidemics and climate change necessitate mathematical models with enhanced predictive accuracies and more realistic dynamics. In this study, we first review recent advances in stationary and traveling wave solutions of PDE models with or without nonlocality. Then we present different strategies for incorporating weather and environmental factors in population models nested with disease models. Finally, we show that these models have a fair amount of agreement with the wildlife population data, and therefore they can potentially capture influences of extreme weather events on spread of infection from wildlife to humans.
Title: Can quantum transport occur on infinite networks?
Speaker: Prof. Christino Tamon, Dept. Computer Science, Clarkson University
Date: 11/28/2022
Abstract: Suppose Alice can send a quantum message to Bob on a quantum system modeled as a finite graph. But imagine there is an eavesdropper which can attach an infinite-dimensional quantum system to the finite graph used by Alice and Bob. Can Alice still send quantum messages to Bob in the presence of this external powerful probe? We explore this question in this talk (which assumes no background in
quantum theory but only some linear algebra). This is joint work with Pierre-Antoine Bernard, Luc Vinet, and Weichen Xie.
Title: Shapes and Geometries in Image Analysis
Speaker: Dr. Günay Doğan, National Institute of Standards and Technology, Gaithersburg, MD
Date: 10/17/2022
Abstract: In computer vision, medical imaging, and many other scientific applications, we are often interested in detecting significant regions
or objects in given images and making inferences about these objects. We may want to answer questions like: "Where are the objects in this image?", "Does this look like a pedestrian or a traffic sign?", "Is this mass in the CT scan benign or malignant?". The path to answering these questions often goes through the analysis of the geometry or shape of the image regions in consideration. In this talk, I will describe variational approaches for detecting shapes and geometries in images, then proceed to give a mathematical description of what we mean by "shape," which is an aspect of a region or boundary, invariant to scaling, rotation, and translation. Finally, I will describe efficient algorithms to compute shape distances for curves or boundaries in 2d. These algorithms can be used to quantify shape dissimilarity and be a building block for tasks like object retrieval from databases and statistical shape analysis, e.g., for clustering or classification of objects.