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Robust estimators to build reliable disease trajectories from short longitudinal data

Title: Robust estimators to build reliable disease trajectories from short longitudinal data

Speaker: Professor Tanya P Garcia, University of North Carolina, Chapel Hill

Abstract: Discovering therapies for neurodegenerative diseases is notoriously difficult, and made worse without accurate disease trajectories to identify when interventions will best prevent or delay irreparable damage. Modeling a disease trajectory is not easy. These diseases progress slowly over decades, and no study covers the full disease course due to time and cost constraints. To compensate, researchers model disease trajectories by piecing together short longitudinal data from patients at different disease stages. The challenge is how to piece together the data to create realistic disease trajectories. One promising way pieces together the short longitudinal data to show changes before and after major events on the disease timeline, like when disease onset occurs. This approach has helped produce realistic disease trajectories, but has shortcomings when the time of the disease event is unknown since without these times, we don’t know where to place the data on the disease timeline. To overcome this issue, researchers currently replace all unknown times with predicted times. Despite efforts to predict the time of disease events without bias using various models, the assumptions these models make often do not hold in practice and result in inaccurate predictions. This leads to an incorrect model of the disease trajectory, producing misleading conclusions about how quickly impairments change as the disease advances. We propose a series of estimators to model the disease trajectory around times of disease events without the need to predict times that are unknown. We show that our estimators produce accurate estimates of the trajectory around times of disease events even when we completely misspecified the distribution model of that time of disease event. We apply our methods to studies of Huntington disease where we model trajectories of motor impairments before and after times of major disease events, to help pinpoint when interventions will best prevent or delay irreparable damage.

Time: Apr 8, 2022 03:30 PM Central Time (US and Canada)


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Colloquium Lecture

Title: Trees and Subtrees

Speaker: Professor Gary Gordon,  Lafayette College
Abstract: Choose a subtree of a complete graph. What is the probability that your subtree is a spanning tree? This question has a very nice answer, and leads to related questions about subtree density (the average size of a subtree) in other graph families. We give examples where adding an edge to a graph decreases the subtree density, and conjecture that for any non-complete graph, there is an edge that can be added that increases this density. We also give a procedure for creating examples of non-isomorphic trees with the same subtree data, i.e., the same number of subtrees having k edges and m leaves for all k and m.

Bio of speaker:   Gary Gordon is the Marshall R. Metzgar Professor of Mathematics at Lafayette College, where he’s been teaching since 1986. His research interests include combinatorics and finite geometry, but he likes math problems of all kinds. He served as the editor for The Playground, the problem solving column of Math Horizons, and he is also involved with the AMC10-12 contest. He has won multiple awards from the Math Association of America (MAA), and has co-authored two books. His favorite mathematical activity – writing The Joy of SET about the math behind the card game SET, with co-authors Liz McMahon, Hannah Gordon, and Rebecca Gordon, his wife and two daughters.

Topic: Colloquium lectures
Time: Mar 25, 2022 04:00 PM Central Time (US and Canada)
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Meeting ID: 875 6057 6865
Passcode: 137492
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Combinatorics Seminar

Time: Friday, March 25, 2022, 1:00pm-2:00pm.

Location: DH 245

Title: A Chevalley Warning Theorem Modulo Differing Prime Powers

Speaker: David J. Grynkiewicz, U. Memphis

Abstract: Polynomial  methods are a powerful algebraic tool that can be used to prove many interesting and otherwise difficult combinatorial results. One such tool is the Chevalley-Warning Theorem. It states that if $p$ is a prime, $f_1,\ldots,f_s\in \mathbb Z[X_1,\ldots, X_n]$ are polynomials and $$V=\{\textbf x\in [0,p-1]^n:\; f_1(\textbf x)\equiv 0,\;\ldots,\;f_s(\textbf{x})\equiv 0\mod p\}$$ is the set of all common roots of the polynomials $f_1,\ldots,f_s$ (modulo $p$), then $$|V|\equiv 0\mod p$$ provided $n> \sum_{i=1}^{s}\mathsf{deg} f_i$. In particular, if the polynomials have no constant term and the number of variables is larger than the sum of degrees, then there is guaranteed to be a nontrivial common root to all the polynomials.

There are many generalizations of this result, including those quantifying just how large a prime power $p^m$ divides the cardinality $|V|$. We will discuss  these results and illustrate them by giving some examples of simple proofs of otherwise very difficult theorems. We will then talk about a seemingly un-noticed generalization of  the Chevalley-Warning Theorem. By a rather straightforward   modification of a  proof of R. Wilson, one can obtain a version of the Chevalley-Warning Theorem applicable when the equations $f_1(\textbf x)\equiv 0\mod p^{m_1},\;\ldots,\;f_n(\textbf x)\equiv 0\mod p^{m_s}$ are considered modulo \emph{differing prime powers} of the  common prime $p$. Using Hensel’s Lemma, which we will discuss in detail for those unfamiliar, we will explain how this result means any typical Chevalley-Warning argument modulo $p$ can effectively be implemented modulo varying primes powers $p^{m_1},\ldots,p^{m_s}$ of a common prime, with tight bounds for how large a prime power $p^m$ divides the cardinality of the corresponding set $V$.   As one simple example, this gives a new proof of the Davenport Constant of a finite abelian $p$-group without need of using group algebras.

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Basics of Abstract Harmonic Analysis

Speaker: Conner Griffin

Location: DH 313

Time: 3:00PM Fri. 3/25/2022

Abstract: Abstract harmonic analysis is an attractive theory that draws on many areas of mathematics. This talk will be an overview of some of the basics of harmonic analysis on locally compact abelian groups. It will lead into the definition of the Fourier transform for functions defined on such groups.

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Variable selection in mixture models: Uncovering cluster structures and relevant features

Speaker: Dr. Mahlet Tadesse

Time: March 4, 2022 04:00 PM Central Time (US and Canada)

Abstract: Identifying latent classes and component-specific relevant predictors can shed important insights when analyzing high-dimensional data. In this talk, I will present methods we have proposed to address this problem in a unified manner by  combining ideas of mixture models and variable selection in different contexts. In particular, I will discuss (1) a bi-clustering approach that allows clustering on subsets of variables by introducing latent variable selection indicators in finite or infinite mixture models, (2) an integrative model to relate two high-dimensional datasets by fitting multivariate mixture of regression models using stochastic partitioning, and (3) a mixture of regression trees approach to uncover homogeneous subgroups and their associated predictors accounting for non-linear relationships and interaction effects. I will illustrate the methods with various genomic applications.

Bio: Dr. Mahlet Tadesse is Professor and Chair of the Department of Mathematics and Statistics at Georgetown University. She is an elected member of the International Statistical Institute and an elected fellow of the American Statistical Association. Her research focuses on the development of statistical and computational tools for the analysis of high-dimensional data with an emphasis on -omic applications.

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Invariant ergodic measure, Mean dimension, and the small boundary property

Time: Fri 2/25 3:00 PM

Speaker: Xin Ma

Location: Dunn Hall 313

Title: Invariant ergodic measure, Mean dimension, and the small boundary property.

Abstract: In topological dynamics, the mean dimension can be used to classify dynamical systems. The small boundary property, related to mean dimension zero, was introduced by Weiss and Lindenstrauss, which is also found to have something to do with the structure theory of C*-algebras recently. In this talk, we mainly discuss the relationships among the three concepts in the title and see some applications. I will start from the very beginning of the topics. No specific background is assumed.

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Ergodic Th Seminar

Speaker: Szu Ting Kitty Yang

Time: Jan 28, 2022 01:00 PM Central Time (US and Canada)

Abstract: In the study of dynamical systems, an important class of examples consists of subshifts. By coding the orbits of points. you can model any dynamical system with a subshift. In this talk, we will focus on substitution subshifts, which arise by iterating a finite substitution. Given a subshift, we can associate various groups, including the automorphism group and the mapping class group, which reflect dynamical properties of its subshift. We will end the talk with a result that characterizes the mapping class group of a substitution subshifts, and look at various examples.