October 5, 2018
Marilyn Carlson, Arizona State University
A research-based approach for improving precalculus teaching and learning
The function concept is a central idea of precalculus and beginning calculus and is used for modeling in the sciences and engineering, yet many students complete courses in precalculus and calculus with weak understandings of this concept. Students who are unable to construct meaningful function formulas to relate two varying quantities have little chance of responding to novel applied problems, or understanding key ideas of calculus such as derivative, accumulation and the Fundamental Theorem of Calculus. I will share data that reveals how students might construct these and other critical reasoning abilities and understandings for learning calculus. I will share the research developed Pathways Precalculus student materials and teacher resources that provide the context for this research, and are resulting in large gains in student learning of the function concept and other foundational ideas for learning calculus. Results from using Pathways materials at 5 large universities will be shared and contrasted with other popular approaches to teaching precalculus mathematics.
October 12, 2018
Meir Pachter, Air Force Institute of Technology
Many-on-one pursuit
We consider swarm pursuit-evasion dierential games in the Euclidean plane where an evader is engaged by multiple pursuers and point capture is required. All the players have simple motion à la Isaacs and the pursuers are faster than the evader. It is shown that in group/swarm pursuit, when the players are in general position, capture is eected by one, two, or by three critical pursuers, and this irrespective of the size N (> 3) of the pursuit pack. Thus, group pursuit devolves into pure pursuit by one of the pursuers, into a pincer movement pursuit by two pursuers, or cornering by three pursuers, who isochronously capture the evader, a mènage à trois. The solution of the Game of Kind is obtained and critical pursuers are identified. Concerning the Game of Degree, the players' state feedback optimal strategies are synthesized and the Value of the game is derived.
October 19, 2018
Ksenija Simic-Muller, Pacific Lutheran University
Teaching mathematics for social justice: the promise and practice
Teaching mathematics for social justice centers mathematics as a tool for understanding the sociopolitical forces that shape the world around us. It addresses complex and sometimes controversial real-world issues (especially those related to economic and racial justice) through open-ended investigations. This approach to teaching is sometimes criticized as not being rigorous enough; and some also believe that mathematics is neutral and should not deal with controversial issues. In this talk, I will argue that addressing social justice issues in mathematics courses is timely and important; and that it can be rigorous and result in significant learning both of mathematics and the issues investigated. I will share examples of activities, assignments, and projects I have developed and used, and give some recommendations for beginning and sustaining this work.
November 2, 2018
Lalit Jain, University of Washington
Ordinal embedding
The standard problem of metric ordinal embedding concerns learning the embedding of n objects into a d dimensional Euclidean space by asking questions of the form "Is object i closer to object j than object k?" Ordinal embedding is a classical technique with roots in psychometrics. However, even though it has been in use for over 70 years, the proper theoretical foundations were lacking. In this talk, I'll discuss some recent results, algebraic questions that arise from this problem, various algorithms, and connections to standard matrix completion problems.
November 16, 2018
Luka Grubišić, University of Zagreb
Constrained PDE models on metric graphs: should we let the linear algebra solver do the heavy lifting?
In this talk we present two 1D models of an endovascular stent. Endovascular stents are biomedical devices made of struts used for treating arterial stenosis. The state of the system in both models is described by a vector valued function on a metric graph which satisfies a system of ODEs and a set of algebraic constraints. Both models are obtained by Γ-convergence from 3D nonlinear elasticity. As a result of asymptotic analysis, solutions are contained in a set of functions which are constrained by a set of algebraic constraints in the nodes of the graph and by requiring that the middle line of a strut does not extend. In the first model we place all constraints in the variational product space and build a finite element approximation there, whereas in the second model we study the problem in a large “free” product space and leave all of the constraints as a part of the system matrix to be removed by a linear algebra solver in a search of the solution. We will present convergence results for both models, but the much more puzzling question is which of the models will yield more efficient numerical methods. Namely, the second model yields a system matrix which is more than three times larger than in the first model. We present results of empirical comparison of the solution methods. We will further also study properties of the eigenvalue and the dynamical problem on a metric graph and discuss the solution methods and their efficiency. Finally, we will present validation experiments for the method by comparing it empirically to the 3D model solved by the standard legacy finite element code. This is a joint work with M. Ljulj, V. Mehrmann and J. Tambaca.
February 1, 2019
Bernardo Cockburn, University of Minnesota
Variational principles for hybridizable discontinuous Galerkin methods: A short story
The so-called hybridizable discontinuous Galerkin (HDG) methods were originally introduced about a decade ago in the framework of steady-state diffusion. The guiding principle was to make it sure that the only globally-coupled unknowns were those associated with the approximation in the internet-element boundaries. Here, we present a different point of view and show how these methods can be obtained by the variational principles which were essential in the early 60s for the devising of finite element methods for solid mechanics: The principle of minimal potential energy, and the principle of minimal complementary energy. We describe these two principles and then show that each of them is associated with four types of numerical methods. We describe the various corresponding finite element methods and show how to obtain the HDG methods. Finally, using this framework, we briefly discuss several historical approaches leading to the HDG methods.
March 8, 2019
Martin Levin, The Bridges Organization
The Platonic solids: geometric sculptures as an accessible introduction to group theory and projective geometry
Although the five Platonic solids may seem quite elementary, as one builds them up in one’s imagination and contemplates them, they become quite captivating. Since they represent all possible finite irreducible rotation groups of 3-space, they are fundamental to the structure of space. This talk will be accompanied by geometric sculptures that explore the relationships between the Platonic solids. They make concepts from group theory and projective geometry understandable by depicting them visually and vividly.
March 15, 2019
Jyotishka Datta, University of Arkansas
New directions in Bayesian sparse signal recovery
Sparse signal recovery remains an important challenge in large scale data analysis and global-local (G-L) shrinkage priors have emerged as the current state-of-the art Bayesian method handling sparsity. In the first half of this talk, I will survey some of the recent theoretical and methodological advances in this area, focusing on theoretical optimality of G-L priors in the context of multiple testing for both continuous as well as quasi-sparse count data. In the second half, I will discuss a few unexplored aspects of their behavior, such as, validity as a non-convex regularization method, performance in presence of correlated errors or extension to discrete data structures including sparse compositional data. I will offer some insights into some of these problems and point out future directions.
April 12, 2019
Alan Demlow, Texas A&M University
Geometric errors in surface finite element methods
Surface finite element methods (SFEM) are widely used to approximately solve partial differential equations posed on surfaces. Such PDE arise in a range of applications, from image processing to fluid dynamics. Typical SFEM involve first approximating the underlying surface and then formulating the finite element method on the approximate surface. In this talk we discuss how approximation of the underlying surface affects the overall quality of the finite element approximation. The talk includes discussion of the effects of surface smoothness on geometric errors and some surprising recent results on approximation of surface eigenvalue problems.
April 19, 2019
Gin McCollum, Portland State University
Mathematical cognition and the sensorimotor brain: Experiment on mental imagery
Not all mathematicians use mental imagery, but many do: 9 out of 11, in this study. Participants were asked, with eyes closed, to visualize graphs of quadratic equations and to wrap spirals around cylinders and cones. There were seven such mental imagery tasks performed in five different body positions, each of which set the head in a different direction with respect to gravity.
The mind's eye obeys much the same geometry as the physical eye, preferring usually to look at imagery in front of the face and right side up. But which way is right side up, when body vertical differs from gravitational vertical? One mathematician commented: "If you'd asked me in advance what I thought would be the outcome, I wouldn't have been anywhere close to the reality."
Under these conditions, participants related to imagery with a mixture of directiveness and curiosity. For example: "Plane starts on wall, rotates to floor, I 'bring it back up.'… Graph emerges, sort of breathes a bit, but doesn't oscillate." For another mathematician, the imagery was also semi-autonomous: "I felt as though my brain wanted to move up on the graph where the graph was wider so that my brain could grab onto the side of the graph in order to stabilize it and keep it from wobbling."
Examining participants' descriptions has led me to the conclusion that imagery is typically embedded in a three-dimensional, body-centered, multisensory space, probably not Euclidean. Such an interior space is called a "peripersonal space". A large body of research suggests that we all carry peripersonal spaces around with us for tool use and personal safety. Although placing imagery in an internal peripersonal space resembles physically placing a picture on a wall, the sensorimotor process in both cases is not Euclidean. Besides, imagery moves, seemingly autonomously.
Imagery is a creative tool used by many mathematicians, but not all. What does it suggest for education? What does it tell us about the nature of mathematical cognition and creativity? It connects to my sensorimotor research. There are many stories of solutions appearing autonomously in well-prepared minds. Is the semi-autonomy of imagery the humble kin of deep creativity?
April 26, 2019
Alexandre Tartakovsky, Pacific Northwest National Laboratory
Learning parameters and constitutive relationships with physics informed deep neural networks
We present a physics informed deep neural network (DNN) method for estimating parameters and unknown physics (constitutive relationships) in partial differential equation (PDE) models. We use PDEs in addition to measurements to train DNNs to approximate unknown parameters and constitutive relationships as well as states. The proposed approach increases the accuracy of DNN approximations of partially known functions when a limited number of measurements is available and allows for training DNNs when no direct measurements of the functions of interest are available. We employ physics informed DNNs to estimate the unknown space-dependent diffusion coefficient in a linear diffusion equation and an unknown constitutive relationship in a non-linear diffusion equation. For the parameter estimation problem, we assume that partial measurements of the coefficient and states are available and demonstrate that under these conditions, the proposed method is more accurate than state-of-the-art methods. For the non-linear diffusion PDE model with a fully unknown constitutive relationship (i.e., no measurements of constitutive relationship are available), the physics informed DNN method can accurately estimate the non-linear constitutive relationship based on state measurements only. Finally, we demonstrate that the proposed method remains accurate in the presence of measurement noise.
May 3, 2019
Katherine R. McLaughlin, Oregon State University
Visibility imputation for population size estimation using respondent-driven sampling
Respondent-driven sampling (RDS) is a network sampling method commonly used to access hidden populations, such as those at high risk for HIV/AIDS and related diseases, in situations where sampling frames do not exist and conventional sampling techniques are not possible. In RDS, participants recruit their peers into the study, which has proven effective as an enrollment strategy but requires careful statistical analysis when making inference about the population. Data from RDS surveys inform key policy and resource allocation decisions, and in particular population size estimates are essential to understand counts of at-risk individuals to develop counseling and treatment programs and monitor health needs and epidemics. Successive sampling population size estimation (SS-PSE) is a commonly used method to estimate population size from RDS surveys, in which the decrease in social network size of participants over the study period is used to gauge the sample fraction. However, SS-PSE relies on self-reported social network sizes, which are subject to missingness, misreporting, and bias, and it is not robust to extreme values. In this talk, we present a modification to the SS-PSE methodology that jointly models the effective social network size of each individual along with the population size in a Bayesian framework. The model for effective network size, which we call visibility to reflect its usage as a proxy for inclusion probability, incorporates a measurement error model for self-reported social network size, as well as the number of recruits an individual was able to enroll and the time they had to recruit. We present and assess the imputed visibility SS-PSE framework, and demonstrate its utility on three datasets of men who have sex with men (MSM) and people who inject drugs (PWID) from Kosovo.
May 10, 2019
Tavis Abrahamsen, Duke University
Convergence analysis of MCMC samplers for Bayesian linear mixed models with p > N
For the Bayesian version of the General Linear Mixed Model (GLMM), it is common to assign conditionally conjugate priors to the unknown model parameters. This results in a posterior density that can be explored using a simple two-block Gibbs sampler. It has been shown that when the priors are proper and the X matrix has full column rank, the Markov chains underlying these Gibbs samplers are almost always geometrically ergodic. We generalize this result by allowing for improper priors on the variance components, and, more importantly, by removing all assumptions on the X matrix. We also analyze a Bayesian GLMM where we replace the standard multivariate normal prior on the fixed effects coefficients with a Normal-Gamma shrinkage prior. In this case, our convergence results are for a hybrid sampler that utilizes both a deterministic step and a random scan step.
For both of these models, we derive easily satisfied conditions guaranteeing geometric ergodicity for the corresponding MCMC algorithms. Geometric ergodicity plays a key role in establishing central limit theorems (CLTs) for MCMC-based estimators. Thus, our results are important from a practical standpoint since all of the standard methods of calculating valid asymptotic standard errors are based on the existence of a CLT.
May 17, 2019
David Krumm, Reed College
A family of Galois groups in arithmetic dynamics
Given a polynomial map f:ℚ→ℚ, one can naturally associate to f a sequence of Galois groups Gn,f which encodes information about the dynamical properties of f. A precise understanding of how the structure of these groups changes as f varies (for instance, among all maps of a fixed degree d) would yield important results in arithmetic dynamics. In this talk we will discuss the family of groups Gn,f in the case where f has degree 2.
May 24, 2019
Robert Won, University of Washington
The card game SET, finite affine geometry, and combinatorial number theory
The game SET is a card game of pattern-recognition. To play the game, twelve cards are dealt face up and all players look for SETs, which are collections of three cards satisfying a certain property. When a SET is found, it is removed and three new cards are dealt. The player who finds the most SETs is the winner. When playing the game, a natural question arises: does every collection of twelve cards contain at least one SET? Or, perhaps more precisely: how many cards are needed to guarantee the presence of a SET?
This question is related to a problem that Terence Tao, in a blogpost from 2007, described as "perhaps [his] favourite open question." In this talk, we explore the connections between SET, finite affine geometry, and combinatorial number theory. We discuss recent breakthrough work of Ellenberg and Gijswijt which answers Tao's question. Finally, we introduce a generalization of this question and present some recent results.
May 31, 2019
Neha Prabhu, Queen's University
Equidistribution of sequences and modular forms
A sequence of numbers in a bounded interval can be distributed in many ways. When the proportion of terms falling in a subinterval tends to the length or any other measure of the subinterval, we say that the sequence is equidistributed. In the last few decades, there has been much activity in studying the equidistribution of Fourier coefficients of modular forms. We discuss some interesting results in this area, and my own work with K. Sinha on this topic.
June 7, 2019
Andrew Womack, Indiana University–Bloomington
Horseshoes with heavy tails
In high dimensional problems, the usual two groups problem of model selection is impossible due to the combinatorial complexity of the model space. In recent years, a set of one group models that approximates the two groups problem have been developed. Of these, the Horseshoe prior is perhaps the most famous and places a Beta(1/2,1/2) prior on the local shrinkage parameters.
There are many modifications and extensions of this framework, and we propose a new modification. Specifically, we model the local shrinkage parameter as a Beta(p,1-p) for each parameter under consideration in order to mimic the model selection problem. Placing priors on the p produces a prior distribution with extremely heavy tails that yields both very strong shrinkage of small signals and unbiased estimation of large signals, having overall better risk behavior. We also consider other prior specifications for p that provide superior inference in super-sparse settings.