Mathematics

Colloquia 2023

The department colloquia are held throughout the fall and spring semesters. Check back to this site for more information. Events are held at 3:35 p.m. in SC 323. Refreshments are served in SC 313F beginning at 3:15 p.m., unless otherwise noted. All are invited to attend.

Graduate Students: The mathematics colloquium is a valuable resource and you should make every effort to attend regularly.

May

May 10: Mohammadreza Bidar, Michigan State University; Blocking problems and the derivative of the exponential maps in connected Lie groups

Abstract: In the first part of the research presentation, I talk about the connection blocking property which originates from a billiard orbit problem. I present a brief preliminary and a brief summary of my Ph.D. dissertation work. Then I proceed to my more recent work on Connection blocking in a particular Lie group called Sol and the differential of the exponential map on Lie groups.

May 8: Tavish Dunn, Oxford College of Emory University; Properties of Generalized Inverse Limits

Abstract: Inverse limits are an interesting object that allows us to glue compact metric spaces together via continuous functions to form more exotic spaces. Properties like compactness and connectedness of the resulting space, as well as the structure of closed sets, can be determined by the component spaces and the individual bonding functions. When a single bonding function is used, the dynamics of the function determines qualities of the resulting inverse limit and vice-versa. More recently, set-valued functions have been employed to construct a wider variety of inverse limits. However, many basic results, such as the inverse limit of continua being itself a continua, do not generalize to this new setting. We discuss conditions on set-valued bonding functions that are sufficient for the inverse limit to be connected and have the full-projection property. Next, we show that the existence of points of period not a power of 2 implies the existence of an indecomposable subcontinuum of the inverse limit. We show the conditions are sharp by way of constructing a function that does not meet all the criteria yet does have points of all periods and whose inverse limit is hereditarily decomposable.

April

April 28: Mohammed Almalki, University of Dayton; Exponential Stability and Instability Via Lyapunov Functionals

Abstract: In this project we display a Lyapunov functional to obtain exponential  stability of the zero solution of an integro-diffetential equation with multiple delays. We extend our method to show instability of the zero solution under slightly different conditions.

April 28: Sharmina Yasmin, University of Dayton; Forecasting Categorical Time Series Using a Combination of Logistic Regression and ARIMA Models

Abstract: In this research, we explore a categorical time series data that changes with time and other input variables using a combination of Logistic Regression, and ARIMA model. We use an Electroencephalogram (EEG) dataset with two states of the response variable (closed or open state of the eye). Using EEG sensor values as input, we use Logistic Regression model to obtain the predictive probability to classify the eye state. Due to the autocorrelation among the residuals and to time dependence, the Logistic Regression model can be improved using the ARIMA model to produce better results. This will help making the residuals a white noise. This work is developed further using a Transfer Function model that produces an even better result.

April 27: Nur Saglam, School of Mathematics, Georgia Tech; Geography and Botany Problem of Symplectic 4-Manifolds

Abstract: The world of 4-manifolds is very interesting in many senses. For example, a closed topological n-manifold X^n has exactly one smooth structure if n≤3 and has at most finitely many smooth structures if n>4. However, there are many simply-connected closed 4-manifolds which admit infinitely many smooth structures. Also, classification problems for smooth, simply connected 4-manifolds have still not fully been understood. For example, the generalized Poincaré conjecture is still open for dimension 4. In this talk, we will talk about the construction of symplectic 4-manifolds homeomorphic but not diffeomorphic to some well-known manifolds with given topological invariants. We will mostly focus on the simply-connected symplectic 4-manifolds, describe the topological invariants, and give examples.

April 25: Challita Jabbour, University of Dayton; Using neural networks for European option pricing in Heston model and for implied volatility

Abstract: This project aims to investigate option pricing in the Black-Scholes-Merton (BSM) model and the Heston model using neural networks. These two models are well-known for their significance in modern finance theory, particularly when it comes to option pricing. Analytical formulas were examined, and neural networks were tweaked to efficiently produce accurate results. Apple stock was used for the BSM model. The Heston model takes stochastic volatility into consideration, and a combining technique between the two models was utilized to generate implied volatility (both smile and surface). MatLab was used  for the implementation.

April 25: Ibrahim Guediri, University of Dayton; A Numerical Solution of Coupled Drinfeld-Sokolov-Wilson System Using Meshless Method of Lines

Abstract: For this project, we reproduce the meshless method of lines (MOL) numerical solution of coupled Drinfeld-Sokolov-Wilson system. This method uses radial basis functions (RBFs) for spatial collocation. Time integration of the resulting system of ODEs will be solved using fourth order of Runge-Kutta method. Accuracy (L2 and L∞) will be compared with the results from other methods available in the literature.

April 25: Richard Buckalew, Wilmington College; From the cell cycle to the redistricting cycle, feedback mechanisms in complex systems lead to emergent dynamics (5 p.m.)

Abstract: A common theme of my research is studying feedback mechanisms in complex dynamical systems: small-scale interactions between individuals that lead to interesting, and sometimes surprising, behavior at the population scale. A Dynamical Systems approach provides a common framework for understanding many different phenomena, including the metabolic cell cycle in yeast, the early development of the fruit fly embryo, potassium homeostasis in the brain, the movement of actors during a warm-up exercise, and the self-sorting of voters.

April 20: Heshan Aravinda Pathirannehelage, University of Florida; Discrete Log-Concave Distributions, Properties, and Applications

Abstract: Log-concavity appears naturally in combinatorics, algebra, analysis, geometry, computer science, probability and statistics. In the context of probability, log-concave assumption provides a broad and flexible, yet natural, convolution-stable class of distributions on integers. Examples include Bernoulli, independent Bernoulli sums, geometric, binomial, negative binomial and Poisson. While log-concave measures, their geometry, and properties are well understood in the classical setting, consideration of discrete log-concavity in the probabilistic setting is very limited. In this talk, I will talk about several results concerning these distributions. First, we study a structured class within discrete log-concave distributions, namely ultra-log concave, and prove that all ultra log-concave sequences exhibit Poisson-type concentration. As an application, we derive concentration bounds for so-called “intrinsic volumes” of a convex body, which generalizes and improves a result of Lotz-McCoy-Nourdin-Peccati-Tropp (2019). Next, we show that a strengthened version of a conjecture of Feige holds for the class of log-concave distributions. Finally, we explore information-theoretic properties. More specifically, we show that the geometric distribution minimizes the min-entropy within the class of log-concave probability sequences with fixed variance, improving the work of Bobkov-Marsiglietti-Melbourne (2022). Our approach is based on a localization-type machinery, a technique that reduces these problems to some extremal cases, which we manage to identify. This talk is based on joint works with Arnaud Marsiglietti, James Melbourne, and Abdulmajeed Alqasem.

April 18: Charbel Al Bacha and Soulayma Saba, University of Dayton; Neural Networks for European Option Pricing Using Jump Diffusion Models

Abstract: This project aims to examine Neural Networks (NN) for pricing European options and compare to other numerical methods. We consider European options written on one and two stocks where the stock prices follow jump-diffusion models. The work intends to investigate the efficiency and time needed to calculate the option price. The analytical formula of the jump diffusion model for one stock and the Monte-Carlo simulation for two stocks are used respectively to generate a large data set which is used to train the Neural Network. Neural Networks efficiency is directly related to the size of the data set, the number of nodes and layers, the optimizer model, and the number of inputs. Thus, the same data sets have been analyzed in different ways to compare the results with the analytical formula and the Monte-Carlo simulation. Our experiments show that the well trained NN can reduce the computational time significantly.

April 11: Charles Destefani, University of Dayton; Using Neural Networks for Option Pricing in Regime-Switching Models

Abstract: We studied the application of neural networks (NNs) to European-style regime-switching option pricing models. We discuss this model's construction and operation in detail (including a comparison to the Black-Scholes model) and summarize the analytical and semi-Monte Carlo simulation approaches used for pricing. Further, we produced Matlab codes that implement these methods efficiently and generally. We conclude that NNs are appropriate for this pricing model and begin to expand our consideration to American-style options.

April 4: Serge Alhalbi, University of Dayton; Multivariate Time Series Transfer Function Models

Abstract: A time series is a sequence of data points that occur over time. Time series forecasting is the process of using a model to predict future values based on past observations. While regression analysis is commonly used to examine the relationship between time series, it does not analyze the relationship between different time points within a single series. Time series examples include weather data, temperature readings, heart rate monitoring, and stock prices. The number of inputs studied determines whether the time series is univariate or multivariate. Each case may require different models and theories. Multivariate time series problems can be addressed using transfer function models, which explain the output variable in terms of the input variables using cross-correlation. In this work, a multivariate transfer function model will be constructed and applied to forecast the Vanguard Real Estate Index Fund.

March

March 28: Jun Li, University of Dayton; Isotopy of the Base Class of a Ruled 4-Manifold

Abstract: In this talk, we will begin by reviewing the geometry and topology of closed 2-dimensional surfaces. We will then move on to discuss the importance of studying 4-dimensional manifolds, which are much more complex and difficult to understand than their lower-dimensional counterparts. Specifically, we will focus on ruled 4-manifolds, which can be constructed by fibering a 2-sphere over a base, which is a 2-dimensional surface. We will also introduce a symplectic form on that and explain why they are of particular interest to mathematicians. Our main result in this talk is the symplectic isotopy of the base class of a ruled 4-manifold, which describes a continuous transformation of the base class that preserves certain important properties.

March 21: Kyle Helfrich, University of Dayton; Applications of DNNs

Abstract: Deep Neural Networks (DNNs) have grown in popularity over the last decade. Despite the popularity of DNNs, there are still misconceptions on what a DNN is and how they work. The purpose of this talk is to provide a high-level overview of DNNs and is designed to be accessible for those unfamiliar with DNNs. I will also be focusing on several applications of DNNs including Natural Language Processing (NLP) in the context of ChatGPT, solving PDEs, and image infill.

March 7: Ying-Ju (Tessa) Chen, Jun Li, George Todd, Matthew Wascher (Facilitator, Aparna Higgins); Exploring ChatGPT

Abstract: According to ChatGPT itself, “ChatGPT is an AI language model developed by OpenAI, which is capable of generating human-like text based on the input it is given.” ChatGPT can create some impressive paragraphs of writing, but it can also give completely incorrect answers to some factual questions. Especially since anyone can explore ChatGPT freely at this time, it will surely influence our teaching – including what questions we ask and whether we can give unsupervised assignments with access to the internet. However, ChatGPT may also provide us with opportunities in many of our teaching and research activities. Four of our departmental colleagues will share with us some of their explorations on ChatGPT, giving us an idea of what they tried to do, and how satisfied they were with the results. Dr. Ying-Ju (Tessa) Chen will tell us about using ChatGPT to brainstorm ideas for the Data Analytics Major and how ChatGPT can be (mis)used in SPC (statistical process control) Practice, Education, and Research; Dr. Jun Li will show us how he may use some of ChatGPT’s responses in his MTH 342, Set Theory, course; Dr. George Todd will tell us about trying to get ChatGPT to write an algorithm and how that went; and Dr. Mathew Wascher will tell us about trying to get ChatGPT to answer test questions from his courses. We hope to have some time at the end for questions and comments from the audience.

FEBRUARY

Feb. 28: David Sivakoff, Ohio State University; Recurrent epidemics on networks

Abstract: The contact process is a stochastic dynamical system modeling a recurring disease in a population represented by a network or a spatial grid. I will introduce the model and highlight some of its history and recent results. As time allows, I will also discuss a variant of the model wherein healthy individuals attempt to avoid those who are infected. I intend for this talk to be accessible to students, and will give a brief advertisement for the Statistics Ph.D. program at Ohio State. Based on joint work with Shirshendu Chatterjee and Matthew Wascher.

Feb. 21: Paul Eloe, University of Dayton (emeritus); Maximum and anti-maximum principles in neighborhoods of simple eigenvalues

Abstract: Under suitable hypotheses, for boundary value problems of the form Ly+ay=f, BCy=0, where L is a linear ordinary or partial differential operator, a is real and BC denotes a linear boundary operator, it can be shown that there exists A > 0 such that if 0< |a| < A, then Ly +ay =f, BCy = 0 has a unique solution for each f in an appropriate space. Moreover, f > 0 implies ay > 0. In particular, for -A < a < 0, y < 0, and for 0 < a < A, y > 0. We shall provide suitable hypotheses so that this behavior is valid for a linear fractional differential operator L of Riemann Liouville type. Two examples will be presented. If time permits, an application of this behavior to a boundary value problem for a nonlinear fractional differential equation will be given.

January

Jan. 31: Yanxi Li, University of Kentucky; Some Modeling Considerations Involving the Exponentially-Modified Gaussian (EMG) Distribution

Abstract: Fitts' law is often employed as a predictive model for human movement, especially in the field of humancomputer interaction. Models with an assumed Gaussian error structure are usually adequate when applied to data collected from controlled studies. However, observational data (often referred to as data gathered "in the wild") typically display noticeable positive skewness relative to a mean trend as users do not routinely try to minimize their task completion time. As such, the exponentially-modified Gaussian (EMG) regression model has been applied to aimed movements data. However, it is also of interest to reasonably characterize those regions where a user likely was not trying to minimize their task completion time. In this paper, we propose a novel model with a two-component mixture structure -- one Gaussian and one exponential -- on the errors to identify such a region. An expectation-conditional-maximization (ECM) algorithm is developed for estimation of such a model and some properties of the algorithm are established. The efficacy of the proposed model, as well as its ability to inform model-based clustering, are addressed in this work through extensive simulations and an insightful analysis of a human aiming performance study.

Jan. 24: Dr. Thilini Jayasinghe, Wittenberg University; Regression models using the LINEX loss to predict lower bounds for the number of points for approximating planar contour shapes and LINEX loss to fit SIR model

Abstract: Researchers in statistical shape analysis often analyze outlines of objects in two dimensions, which can be modeled as planar contours. However, even though these objects are infinite-dimensional in theory, they must be discretized in practice. When discretizing, it is important to reduce the number of sampling points considerably to reduce computational costs but not to use too few sampling points to result in too much approximation error. Unfortunately, determining the minimum number of sampling points needed to achieve sufficiently low approximation error is computationally expensive. As such, we fit linear regression models to predict these lower bounds using characteristics of the contours that are easy to calculate as predictor variables. However, least squares regression is inadequate for this task because it treats overestimation and underestimation equally, but underestimation is far more serious since the response variable is a lower bound. Thus, the LINEX loss function was used to fit the regression models, which allows penalizing underestimation at an exponential rate while penalizing overestimation only linearly. A novel, data-driven approach to select the shape parameter of the loss function and tools for analyzing how well the model fits the data that are analogous to least squares regression. Through validation methods, we show that the LINEX regression models work well for limiting the amount of underestimation for the lower bounds for the number of sampling points. The usage of the LINEX loss function is not only limited to regression analysis. We further used the LINEX loss function in dynamical data fitting in different applications rather than the Least squares method to reduce the underestimation. A basic SIR model was used to expand the studies on the usage of the LINEX loss function for infectious disease data to reduce the underestimation of the model parameters based on the data.

Jan. 19: Yafan Guo, University of Kentucky; Approximate Tolerance Intervals for Semiparametric Regression Models

Abstract: Tolerance intervals in regression allow the user to quantify, with a specified degree of confidence, bounds for a specified proportion of the sampled population when conditioned on a set of covariate values. While methods are available for tolerance intervals in fully-parametric regression settings, the construction of tolerance intervals for semiparametric regression models has been treated in a limited capacity. This paper fills this gap and develops likelihood-based approaches for the construction of pointwise one-sided and two-sided tolerance intervals for semiparametric regression models. A numerical approach is also presented for constructing simultaneous tolerance intervals. An appealing facet of this work is that the resulting methodology is consistent with what is done for fully-parametric regression tolerance intervals. Extensive coverage studies are presented, which demonstrate very good performance of the proposed methods. The proposed tolerance intervals are calculated and interpreted for analyses involving a fertility dataset and a triceps measurement dataset.

Past Colloquia Events

Spring 2022

January

Jan. 20: Kyle Helfrich, Ohio Wesleyan University; Orthogonal Recurrent Neural Networks and Batch Normalization in Deep Neural Networks

Abstract: Despite the recent success of various machine learning techniques, there are still numerous obstacles that must be overcome. One obstacle is known as the vanishing/exploding gradient problem. This problem refers to gradients that either become zero or unbounded. This is a well-known problem that commonly occurs in Recurrent Neural Networks (RNNs). In this talk, we will describe how this problem can be mitigated, establish three different architectures that are designed to avoid this issue and derive update schemes for each architecture. Another portion of this talk will focus on the often used technique of batch normalization. Although found to be successful in decreasing training times and in preventing overfitting, it is still unknown why this technique works.

Jan. 25: Kelly Buch, University of Tennessee, Knoxville; Multi-Disciplinary Research at the Interface of Mathematics and Biology

Abstract: Mathematical models of real-life situations are powerful tools that can be used in a variety of disciplines when research questions cannot be answered with experiments or field study alone. The process of creating and interpreting mathematical models of real-world systems is a multi-disciplinary effort. It requires collaboration across disciplines in all steps: translating relevant information about the system into mathematical objects, performing mathematical analysis on the model, and translating the results of the analysis back into the original discipline. As a mathematical modeler, I love this translation process and I collaborate with scientists to model biological phenomena. In this talk, I will present two of my recent multi-disciplinary modeling projects to display both my approach to mathematical modeling and the breadth of applications I’m interested in. In both projects, I will highlight model development of the model and the utility of model analysis in the application discipline. We will begin with an application in ecology, introducing a mathematical model for vector-borne tree diseases and interpret it to make recommendations for wildlife managers attempting to eradicate such diseases. We will also discuss an application in physiology, introducing a mathematical model for Reactive Scope, a conceptual framework for the long-term impacts of short-term stress responses. While in its infancy, this modeling approach shows promise in its ability to produce testable outcomes to advance experimental physiology.

Jan. 27: Anataska Dobreva, Arizona State University; Studying vision and immune processes with mathematical modeling

Abstract: This talk will consist of two main parts, showcasing my work in applying mathematics in retina and immunology research. Photoreceptors, rods and cones are the cells of the eye responsible for vision. Photoreceptors mainly use glucose to create energy and to renew their light-absorbing outer segments (OS). In the first part of the talk, I will present the development and analysis of the first mathematical model for the metabolic dynamics of a cone that accounts for energy generation from external lactate and fatty acids oxidation of OS. With multiple parameter bifurcation analysis, we investigate how interactions among key mechanisms affect the cone’s metabolic vitality under glucose shortage, and with in-silico experiments we explore the possibility for recovery. Time-varying global sensitivity analysis is applied for both normal and nutrient stress conditions to assess in each case the dynamic impact of different processes to the model outputs of interest. Our work provides insight into the role of metabolites under glucose starvation conditions, which may elucidate pathways and disruptions resulting in degenerative retinal diseases such as age-related macular degeneration and retinitis pigmentosa whose progression has been linked to glucose deprivation and disruptions in cellular metabolism. In the second part of the talk, I will present my work on developing and analyzing the first mathematical models for the autoimmune hair loss disease alopecia areata (AA). In AA, the immune system attacks hair follicles and disrupts their natural cycle through phases of growth, regression, and rest. The disease manifests with distinct hair loss patterns, and what causes it and how to treat it are open questions. I will first present our ODE model for AA in follicles in stage of growth. Next, I will explain how we incorporate follicle cycling into the model and explore what processes have the greatest impact on the duration of hair growth in healthy versus diseased follicles. Finally, I will highlight the global and marginal linear stability analysis results from our PDE model, which captures the patterns that characterize hair loss in AA.

February

Feb. 1: Sougata Dhar, University of Connecticut; Lyapunov-type inequalities for third-order linear and half-linear difference equations and extensions

Abstract: In this talk, we will focus on the Lyapunov-type inequalities for third-order difference equations, the discreet version of differential equations. Unlike their counterparts, these inequalities for difference equations were limited only to even-order cases prior to this work. The resulting inequalities utilize positive and negative parts of a function rather than the traditional absolute value of the function. We first discuss the linear case and then establish the subsequent results for the quasilinear case. In this process, we introduce a new approach to tackle these inequalities as the techniques for even order difference equations are not applicable to odd order difference equations. Moreover, we will discuss a few additional results for linear difference equations as which are further extended to more general linear equations. We will conclude after a brief discussion for the third order backward difference equations and dynamic equations on time scales.

Feb. 3: Sabrina Streipert, McMaster University; Discrete Delay Population Models

Abstract: The continuous Hutchinson model is a delay logistic growth model, where a delay was introduced in the per-capita growth rate. Despite its popularity, this delay differential equation exhibits some questionable behavior as the population persists independent of the delay. One of its discretizations, the so-called Pielou model, can be criticized for the same reason. To formulate an alternative discrete delay population model of logistic growth, we first distinguish the growth and decline processes before introducing a delay solely in the growth term. The obtained model differs from existing discrete delay population models and exhibits realistic behaviour. If the delay exceeds a certain critical threshold, then the population goes extinct. On the other hand, if the delay is below that threshold, then the population survives and converges to a positive asymptotically stable equilibrium that decreases as the delay increases. As the next step toward extending our discrete delay population model to interacting species, we derived a discrete predator-prey model without delay. The analysis of this predator-prey model led us to the formulation of a discrete phase plane approach that has been deemed ineffective for planar maps. By considering the direction field, the corresponding nullclines, and our “next iterate root curves associated with the nullclines”, the global dynamics of the discrete predator-prey model are discussed.

Feb. 8: Yulong Li, University of Nevada, Reno; Hello, fractional ODEs!

Abstract: The fractional-order ordinary differential equations (in short, fractional ODEs) may be considered an old and yet novel topic, which is an emerging interdiscipline, intertwined with fractional-order calculus, fractional power operators, modeling, singular integral equations and special functions. In this presentation, we will:

1. Introduce some open problems arising from the subject of fractional ODEs and use the double-sided fractional diffusion-advection reaction ordinary differential equation as the main example to present our recent accomplishments toward those open questions.
2. Illustrate how fractional ODEs can attract collaborations from faculties who are doing classic ODEs, numerical analysis, integral equations and how it can produce many interesting research projects for our undergraduate and graduate students.

Feb. 22: Dr. Youssef Raffoul, Department of Mathematics, University of Dayton; Introduction to Time Scales with Applications to Stability and Boundedness

Abstract: Lately, there have been few talks on Time Scales by candidates that we interviewed to fill one of our positions. In this talk, we present an introduction to Time Scales and then provide some recent materials related to the stability of the zero solution and boundedness of solutions of general Dynamical Systems.

March

March 1: Dr. Youssef Raffoul, Department of Mathematics, University of Dayton; The Case for Large Contraction

Abstract: We start our talk by displaying an example to motivate our motive for the need for Large Contraction. We will look at the existence of a periodic solution for a totally nonlinear functional delay differential equation by utilizing a modified version of the Krasnoselsski fixed point theorem in which regular Contraction is replaced with Large Contraction. Then, we prove a topological theorem that classifies functions that are Large Contraction.

March 8: Dr. Jacob Shapiro, Department of Mathematics, University of Dayton; Semiclassical resolvent bounds for compactly supported radial potentials

Abstract: We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 \Delta + V(|x|) - E$  in dimension $n \ge 2$, where, $h, \, E > 0$ and $V : [0, \infty) \to \mathbb{R}$ is $L^\infty$ and compactly supported. We show that the weighted resolvent estimate grows no faster than $\exp(Ch^{-1})$, and prove an exterior weighted estimate which grows $\sim h^{-1}$ . The analysis at small angular momenta proceeds by a Carleman estimate and the WKB approximation, while for large angular momenta we use Bessel function asymptotics. This is joint work with Kiril Datchev (Purdue University) and Jeffrey Galkowski (University College London).

March 10: Mohammad Niksirat, University of Alberta; Topological approach to the Doi-Onsager equation

Abstract: The Doi-Onsager equation describes the behavior of rigid-like molecules in liquid crystals @f@t  rf + div (f rr U(f))where f: Sn! R is the directional density function in the direction r 2Sn, Z f(r)  = 1and SSnjr   r0j f(r0) d ; U(f)(r) = for > 0. The equation is sometimes called the Smoluchowski or a non-linear Fokker-Planck Equation. It turns out that the stationary solutions of the equation satisfy the Onsager equation proposed in 1949. Experiments show that the phase of solutions changes between isotropic and nematic based on the intensity of the molecules. The problem in its original form with the Onsager kernel remained open until recently, despite some partial results developed for the problem with Maier Saupe potential. In this talk, we introduce an innovative topological method for the problem. This method can not only reproduce all previous results in a unified way, but also solves the equation with more general potentials than the one studied by Onsager.

March 29: Mathew Wascher, University of Dayton, Department of Mathematics; A mechanistic framework for estimating SARS-CoV-2 prevalence from dust

Abstract: As we enter the third year of the pandemic and transition to strategies for managing endemic SARS-CoV-2, we will need to move away from mass testing and consider passive methods for monitoring potential outbreaks. One such avenue is observing viral RNA shed into the environment by infectious individuals. We develop a mechanistic modeling framework using Poisson processes to relate the amount of viral RNA observed at a particular location to the number of infectious individuals in that location. Our model is flexible enough to account for interindividual variation in viral shedding, and we demonstrate several useful theoretical properties. We fit our model to viral RNA observed in dust collected from vacuuming isolation rooms at The Ohio State University during the Fall 2020 semester using a Markov chain Monte Carlo algorithm. We further investigate the performance of our model on synthetic data. Although our example data consists of viral RNA collected from environmental dust, our modeling framework is general and can be applied to other modes of environmental shedding. This is joint work with Colin Klaus, Joe Tien, Ashleigh Bope. and Karen Dannemiller.

April

April 5: Steve Rosenberg, Boston University; hosted by Andre Larrain-Hubach; Geometric Aspects of Machine Learning

Abstract: From a mathematical viewpoint, machine learning involves gradient flow of a loss function on an infinite dimensional manifold, a situation familiar to geometers from gauge theory and Floer theory.  The loss function usually involves an error measurement term and a regularization term, which keeps the function from rapidly oscillating.  Regularization terms that reflect the geometry of the loss function seem to work well.  In a related direction, we can try to fit a data set in a high dimensional Euclidean space to a lower dimensional submanifold: this is so-called manifold learning.  Finding the best submanifold involves gradient flow on an infinite dimensional space of embeddings, and the geometry again controls how long the flow stays in the space of embeddings.

April 26: Xiangzhou Song; Advisor, Dr. Liu; How technical analysis improves stock performance

Abstract: In technical analysis, the trader assumes that the past trading activities in a market and the past prices changes provide adequate information to show future price movements that are more likely. To determine the impact of technical analysis on traders’ stock portfolio performance, in this research project we use the historical price and volume data from the Dow Jones Industrial Average index (DJIA). The DJIA comprises the 30 largest companies in the USA which include IBM, Microsoft, Walmart, Apple, etc.  We apply different strategies and back-test the data to determine how the strategies affect the market portfolio performance. In our analysis and back-testing, we use a few indicators to chart the performance of the companies’ data over time and forecast the portfolio’s future returns.

April 26: Chris Bingman, Advisor, Dr. Peter Hovey; Logistic regression and classification trees in special forces selection

Abstract: Regression is commonly used to predict outcomes in various fields of study and industry based on a number of predictor variables often called regressors. The type of response, among other factors, determines the type of regression: linear or non-linear. In non-linear regression, logistic regression can be used to predict the probability of a qualitative response. Some of the numerous methods used to model regression are forward selection, backward elimination, and stepwise regression. Classification and Regression Trees are also a tool that can aid in identifying correlations between response and predictor variables. Using a dataset from the U.S. Army Special Operations Forces, we will investigate the effectiveness of using information gained from classification trees to supplement a logistic model, hopefully improving accuracy and predictability.

Fall 2021

September

Sept. 28: Dr. Ayse Sahin, Wright State University.

Sept. 21: Dr. Paul Eloe, University of Dayton.

November

Nov. 9: Dr. Sam Brensinger, University of Dayton.

Nov. 16: Jonah Reeger, Senior Research Mathematician with AFRL.

Spring 2021

February

Feb. 12: Jeffrey Lyons, The Citadel; Smoothness of Solutions to Third Order Conjugate Boundary Value Problem

Abstract: Under certain conditions, the solution to the third order conjugate boundary value problem consisting of the differential equation y’’’=f(x,y,y’,y’’) and conjugate boundary conditions y(x_1)=y_1, y(x_2)=y_2, y(x_3)=y_3 may be differentiated with respect to the boundary points and values. The resulting function solves the associated variational equation. This work extends that which has been attributed to Peano for initial value problems to boundary value problems and relies upon a continuous dependence result connecting the two. The proof of the main result ties together topics from calculus, differential equations, and linear algebra.

Feb. 19: Sougata Dhar, University of Connecticut; Lyapunov type inequalities and applications

Abstract: In this talk, we will discuss Lyapunov-type inequalities for several integer and fractional order differential equations for both linear and quasilinear cases. We will primarily concentrate on the third-order case and briefly mention the structure of the higher order case. Most of these inequalities utilize the integrals of the positive and negative parts of a function rather than the traditional absolute value of the function as compared to the existing results in the literature. Furthermore, we discuss the extensions of these inequalities to the multivariate case for radially symmetric functions. Finally, we use them to study the nonexistence, uniqueness, and existence-uniqueness criteria for several classes of boundary value problems. We will conclude the discussion with some open problems in this area.

Feb. 26: Bo Li, The Citadel; Simultaneous confidence intervals of estimable functions based on quasi-likelihood for over-dispersed data

Abstract: Two major problems besetting count data analysis in multiple comparisons are over-dispersion and violation of distributional assumptions of real data. In this talk, we describe the simultaneous confidence interval method to inference a collection of estimable functions in generalized linear models based on quasi-likelihood estimation. We assume that the independent observations have the variance proportional to a given function of the mean. We define the pivotal quantities in an asymptotic sense. We derive the joint limiting distribution of the pivotal quantities and the asymptotic distribution of the maximum modulus statistic. In the presence of over-dispersion, large-sample approximation method is shown to be liberal in multiple comparisons. We propose a percentile-t bootstrap method based on Pearson residuals as a robust alternative. It shows that the proposed method outperforms large-sample approximation method in the spirit of attaining the overall coverage probability, even when the working variance-mean structure moderately deviates from the real structure of the underlying distribution.

March

March 5: Steve Szabo, Eastern Kentucky University; A Taxonomy of Rings and Related Minimal Examples

Abstract: Many types of rings are defined using zero divisors or nilpotent elements. These ring classes fit nicely into a hierarchy which will be explored in this talk. Examples differentiating the classes will be given. Where possible, the example will be minimal with respect to order.

March 12: Ying-Ju Chen, University of Dayton; Bias Reduction of the Gini Index Estimation Based on Grouped Data using a computational approach

Abstract: Many government agencies still rely on the grouped data as the main source of information for calculation of the Gini index. Previous research showed that the Gini index based on the grouped data suffers the first and second-order correction bias compared to the Gini index computed based on the individual data. Since the accuracy of the estimated correction bias is subject to many underlying assumptions, we propose a new method, D-Gini, which reduces the bias in Gini coefficient based on grouped data. We investigate the performance of the D-Gini method on an open-ended tail interval of the income distribution. The results of the simulation study showed that our method is very effective in minimizing the first and second order-bias in the Gini index and outperforms other methods previously used for the bias-correction of the Gini index based on grouped data. Three data sets are used to illustrate the application of this method. I will talk about some ongoing work based on this study as well.

March 19: Edward Timko, University of Manitoba; Some Noncommutative Extensions of the F. and M. Riesz Theorem

Abstract: In this talk, we develop operator theoretic analogues of the F. and M. Riesz Theorem. We first re-cast the classical theorem as one relating ‘anti-analytic’ bounded linear functionals to a distinguished $*$-representation of the continuous functions on the unit circle. Next, we establish an analogous result relating similarly ‘anti-analytic’ bounded linear functionals to ‘completely non-singular’ $*$-representations of the Cuntz algebra and highlight notable differences. We then explore conditions for the existence of weak$^*$-continuous extensions of the aforementioned bounded linear functionals. We again find significant differences between the noncommutative setting and the classical, and we establish necessary and sufficient conditions for when weak ∗ -continuous extensions can be obtained. This is joint work with R. Clouâtre and R. Martin.

March 26: Jacob Shapiro, University of Dayton; Semiclassical resolvents estimates for long range Lipschitz potentials

Abstract: We give an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 \Delta + V(x) - E)$ in dimension $n \neq 2$, where $h, E > 0$. The potential is real-valued and $V$ and $\partial_rV$ exhibit long range decay. The resolvent norm grows exponentially in $h^{-1}$, but near infinity it grows linearly. When V is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{-1/2}$ for some $C > 0$. This $E$-dependence is sharp and answers a question of Datchev and Jin. This talk is based on joint work with Jeffrey Galkowski.

April

April 1: Matthew Wascher (Statistics) Modeling epidemics with behavioral considerations: two problems

Abstract: SIR-type models, where individuals are assumed to take on states, most commonly susceptible, infectious, and removed, that change over time due to infection, recovery, or behavioral events, are among the most common frameworks used for quantitative modeling of infectious disease epidemics. I am particularly interested in how behavior, including behavior taken by individuals and actions taken by authorities, affect epidemics and the tools we use to model them. I will discuss two such problems. The first is a probabilistic study of an agent-based network model. The second is a statistical approach inspired by COVID-19 data.

In the first problem, we consider an SIS epidemic modified to include avoidance behavior on an explicit network. In this model, each infected individual infects each of its healthy neighbors at rate λ, each infected individual recovers at rate 1, and each healthy individual avoids each of its infected neighbors at rate α by deactivating the edge from the infected neighbor to itself until the neighbor recovers. We study this process on the networks Z,Zn, and the star graph. We show that on Z and Zn, the asymptotic behavior of the epidemic changes qualitatively depending on α and λ, a phenomenon called a phase transition. On the star graph, we derive explicit bounds for the asymptotic survival time of the infection and show they differ substantially from the behavior of the SIS epidemic without avoidance on the star graph. This is joint work with Shirshendu Chatterjee and David Sivakoff.

In the second problem, we consider modeling an epidemic when data come from a population where all individuals in the population are tested at regular intervals. Such a strategy has been used by universities, including Ohio State, to attempt to mitigate the effects of COVID-19 among student populations. If the interval between repeated tests is smaller than the natural recovery time of the infection, then we expect that most infections will be observable in the data. However, this data will be interval censored: we know that an individual who tests positive must have become infected between the time of a positive test and the time that individual’s previous negative test but not the exact date of the infection. We develop a method we refer to as Interval Dynamic Survival Analysis (IDSA) that leverages this feature of the data and discuss its performance. This is joint work with Patrick Schnell, Greg Rempala, and Wasuir Rahman KhudaBukhsh.

April 6: William Worden, Rice University (Topology) Hyperbolic 3-manifolds, polyhedral decompositions, and hidden symmetries

Abstract: In the early 1980s William Thurston revolutionalized the field of low-dimensional topology with his hyperbolization theorem and geometrization conjecture. His work brought to the forefront the study of hyperbolic 3-manifolds, and made possible computational tools like SnapPy that have led to further insight through experimentation. I will survey some of this history, explore some families of hyperbolic 3-manifolds that can be understood through their decompositions into polyhedra, and discuss some of my own work that leverages the combinatorial and geometric structure of such polyhedral decompositions.

April 8: Andrew Richards (Statistics) New Species Tree Inference Methods Under the Multispecies Coalescent Model

Abstract: Inference of the evolutionary histories of species, commonly represented by a species tree, is complicated by the divergent evolutionary history of different parts of the genome. Different loci on the genome can have different histories from the underlying species tree (and each other) due to processes such as incomplete lineage sorting (ILS), gene duplication and loss, and horizontal gene transfer. The multispecies coalescent is a commonly used model for performing inference on species and gene trees in the presence of ILS. (Li)kelihood-based assemb(ly), or Lily, is a new method for species tree inference under the multispecies coalescent. The method is presented and compared to two frequently used methods, SVDQuartets and ASTRAL, using simulated and empirical data. Generalizing the procedure to allow for differing mutation rates is briefly discussed.

April 9: George Todd, University of Dayton; Power Moments of Character Sums

Abstract: Character sums are ubiquitous in number theory. Kloosterman sums, for example, are character sums closely related to the fourier coefficients of modular forms. Due to a result of Katz, Kloosterman sums are asymptotically equidistributed with respect to Sato-Tate measure, suggesting that exact calculations are unlikely. One direction is to study power moments of Kloosterman sums. So far, the most successful method of computing these power moments is to calculate a related quantity realized as the trace of a symmetric power of the Kloosterman sheaf. This method, however, is quite involved. We discuss recent work on computing these power moments in a simpler way using supercharacters, as well as obstructions to further work on the topic. Along the way, we hope to show the utility of power moments of character sums more generally. This work is joint with Stephan Ramon Garcia, Ángel Chávez, and Brian Lorenz.

April 13: Won Chul Song (Statistics) Group Feature Screening via F-test

Abstract: Feature screening is crucial in analysis of ultrahigh dimensional data, where the number of variables (features) is exponentially larger than the number of observations. In various ultrahigh dimensional data, variables are naturally grouped with correlation structures within groups. In this article, we propose a group screening method via the F-test. The new method is in contrast with the existing literature on feature screening, which screens variables one by one. Under certain regularity conditions, we prove that the proposed group screening procedure possesses a sure screening property that selects all active groups with probability approaching to one at an exponential rate. We use simulations to demonstrate the advantages of the proposed method and show its application in a genome-wide association study. We conclude that grouping method is very effective in large-scale inference for high dimensional data and is a good choice, as an optimal test can detect true signals with desired properties.

April 15: Jun Li, University of Michigan (Topology) Symplectic mapping class groups in dimension 4

Abstract: Symplectic manifolds arise naturally in abstract formulations of classical mechanics, while symplectic topology is an interesting mixture of the “soft” and the “rigid.” In this talk, we will focus on rigidity phenomena in symplectic topology. In particular, we will discuss recent developments for the mapping class groups of 4- dimensional symplectic manifolds, and how these root in dynamics and topology in dimension 2.

April 20: Christoforos Neofytidis, Ohio State University (Topology) Topology, Geometry and Dynamics of aspherical manifolds

Abstract: The Borel conjecture asserts that the homeomorphism type of closed aspherical manifolds is determined by their fundamental groups. Manifolds satisfying the Borel conjecture are called topologically rigid. In this talk, we generalize in all dimensions (and give a new, uniform proof of) a rigidity theorem on aspherical fibered 3-manifolds, due to Gromov and Wang. A key ingredient for this new, mostly algebraic, approach is the non-vanishing of the Euler characteristic of the fiber (which is a hyperbolic surface in dimension three). We then explain how the Euler characteristic is expected to determine the topology of aspherical manifolds with respect to the Borel conjecture, as well as other long-standing problems, such as the Hopf problem (Topology) and the Anosov-Smale conjecture (Dynamics).

Fall 2020

September

Sept. 4: Joe Mashburn, University of Dayton; Microhomogeneous Spaces

Abstract: Since the 1960s the topological property of homogeneity has been useful in the study of cardinal bounds in topological spaces. We recently rediscovered a weaker and little known property called microhomogeneity which can be used in the place of homogeneity in many of these cardinal bounds. In this talk we will see how microhomogeneity can be used and compare the behavior of microhomogeneous spaces with that of homogeneous spaces.

Sept. 11: Paul Eloe, University of Dayton; Two point boundary value problems for ordinary differential equations, uniqueness implies existence

Abstract: We consider a family of two point $n-1 ,1$ boundary value problems for $n$th order nonlinear ordinary differential equations and obtain conditions in terms of uniqueness of solutions that imply existence of solutions. A standard hypothesis that has proved effective in uniqueness implies existence type results is to assume uniqueness of solutions of a large family of $n-$point boundary value problems. Here, we shall replace that standard hypothesis with one in which we assume uniqueness of solutions of a large family of two point boundary value problems. We then obtain readily verifiable conditions on the nonlinear term that in fact imply the uniqueness of solutions of the large family of two point boundary value problems.

Sept. 18: Jeffrey Neugebauer, Eastern Kentucky University; Positive Solutions of a Boundary Value Problem at Resonance

Abstract: In this talk, we will employ a shift method to study a boundary value problem that is at resonance. By proving the associated differential operator satisfies a maximum principle, we show the Green’s function of the shifted BVP is nonpositive. Upper and lower solutions and fixed point methods are used to show the existence of solutions of the shifted BVP.

Sept. 25: Johnny Henderson, Baylor University; Existence of local solutions for fractional difference equations with left focal boundary conditions

Abstract: For $1 < \nu \leq 2$ a real number and $T\geq 3$ a natural number, conditions are given for the existence of solutions of the $\nu$th order At{\i}c{\i}-Eloe fractional difference equation, $\Delta^\nu y(t) + f(t+ \nu -1, y(t+ \nu-1)) =0,$ $t \in \{0,1,\ldots, T\},$ and satisfying the left focal boundary conditions $\Delta y(\nu-2) = y(\nu +T) = 0$. The conditions involve growth conditions on $f$ and constraints on the size of $T$.

October

Oct. 2: Saleh Almuthaybiri, Qassim University; Sharper existence and uniqueness results for solutions to third-order boundary value problems

Abstract: The purpose of this talk is to present new results where we sharpen Smirnov's recent work on existence and uniqueness of solutions to third-order ordinary differential equations that are subjected to two-and three-point boundary conditions. The advancement is achieved in the following ways. Firstly, we provide sharp and sharpened estimates for integrals regarding various Green's functions. Secondly, we apply these sharper estimates to problems in conjunction with Banach's fixed point theorem. Thirdly, we apply Rus's contraction mapping theorem in a metric space, where two metrics are employed. Our new results improve those of Smirnov by showing that a larger class of boundary value problems admit a unique solution.

Oct. 9: Lynne Yengulalp, Wake Forest University; Topological completeness and the role of G-delta subsets

Abstract: We start with the Baire Category Theorem for complete metric spaces: The intersection of any countable sequence of open dense subsets of a complete metric space is dense in the space. The countable intersection of open sets is called a G-delta set. In this talk, we discuss some topological properties more general than complete metrizability and the role of G-delta sets in the study of such properties.

Oct: 16: Paul Eloe, University of Dayton; Three point boundary value problems for ordinary differential equations, uniqueness implies existence

Abstract: This is a continuation of the September 11 presentation in which uniqueness implies existence results for a family of two point $n-1 ,1$ boundary value problems for $n$th order nonlinear ordinary differential equations were presented. In this talk we move to a family of $n-2 ,1,1$ boundary value problems; again, an abstract result is obtained and again, a result with verifiable hypotheses, a monotonicity condition in the nonlinear term, is obtained.

Oct. 30: Kathryn McCormick, California State University Long Beach; Twisted Steinberg Algebras

Abstract: A groupoid is an algebraic object encompassing both groups and equivalence relations. For many years, algebras offunctions on groupoids, such as Steinberg algebras and groupoid C*-algebras, have provided diverse and tractable models for algebraists and analysts alike. Thirty years prior to the introduction of Steinberg algebras, Renault initiated the study of twisted groupoid C*-algebras, which have been valuable in modeling an even broader class of C*-algebras and in providing an inroad to difficult problems in C*-algebras such as the UCT problem. In this talk, I will describe twisted Steinberg algebras on ample Hausdorff groupoids, introduced by myself and coauthors, and some results on describing the ideal structure of a twisted Steinberg algebra when the underlying groupoid is effective.

November

Nov. 13: Benjamin F. Akers, Air Force Institute of Technology; Wave resonances: existence, analyticity and numerics

Abstract: Nonlinear waves occur in a wide variety of applications, including laser propagation, oceanography, combustion, and more. In special circumstances these waves resonate, resulting in bifurcatingsolution branches in the nonlinear problem. In this talk we consider the asymptotics, existence, andanalyticity of these solution branches. Two distinct proofs of existence are discussed: the first isbased on a perturbation series, the second is a contraction mapping argument. Both proofs spawnnumerical methods. The numerical methods are compared, and results from each are presented. The application of these methods to the spectral stability problem, where resonances occur ateigenvalue collisions, is discussed.

Nov. 20: Nathan Carlson, California Lutheran University; Cardinality bounds on topological spaces with a focus on homogeneous and microhomogeneous spaces

Abstract: The theory of cardinality bounds on topological spaces has played a prominent role in the field of set-theoretic topology. Early work by Alexandroff and Urysohn showed that the cardinality of a compact, perfectly normal space is at most c, the cardinality of the continuum. In 1969 Arhangel’skii answered a 50-year old question of Alexandroff and Urysohn by showing the cardinality of a compact, first countable, Hausdorff space is also at most c. In the 1970s it was noticed by van Douwen, Arhangel’skii and others that if a space is additionally homogeneous then bounds on the cardinality of the space can be improved. Roughly speaking, a space is topologically homogeneous if the topology acts the same at any point. Formally, X is homogeneous if for every x and y in X there exists a homeomorphism h from X onto X such that h(x) = y. We will survey results in this area, notably a result of Carlson and Ridderbos, as well as de la Vega’s Theorem, which states that the cardinality of a compact, homogeneous, Hausdorff space is at most 2^{t(X)}. t(X) is known as the tightness of X. We will also show that this result extends to the broader class of microhomogeneous spaces. This builds upon recent work by Anderson, Mashburn, and Yengulalp on cardinality bounds on microhomogeneous spaces.

December

Dec. 4: Jacob Shapiro, University of Dayton; Semiclassical resolvent estimates and wave decay in low regularity

Abstract: We study weighted resolvent bounds for semiclassical Schr\"{o}dinger operators. When the potential function is Lipschitz with long range decay, the resolvent norm grows exponentially in the inverse semiclassical parameter $h$. When the potential is merely bounded and has compact support, the resolvent still grows exponentially but with an extra polynomial loss in $h$. This extends the works of Burq and Cardoso-Vodev in the smooth case. Our main tool is a global Carleman estimate. We apply the resolvent estimates along with the resonance theory for blackbox perturbations to show local energy decay for the wave equation with a wavespeed that is an $L^\infty$ perturbation of unity.

Dec. 11: Alan Veliz-Cuba, University of Dayton; Designing multistability with AND gates

Abstract: Systems of differential equations have been used to model biological systems such as gene and neural networks. A problem of particular interest is to understand and control the number of stable steady states. Here we propose conjunctive networks (systems of differential equations equations created using AND gates) to achieve any desired number of stable steady states. Our approach uses combinatorial tools to easily predict the number of stable steady states from the structure of the wiring diagram.

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