Abstracts for Mathematical Sciences Colloquia

---

Schedule and Abstracts:

[Hide Abstracts]

• 2 September Balgobin Nandram, WPI

Title: Bayesian logistic regression for numerous small areas
Abstract: We analyze binary data and covariates, which are available for numerous small areas. Pre- dictive inference is required for the finite population proportion of individuals with a specific character for each area.We use a standard hierarchical Bayesian logistic regression model with each area, rather than groups of areas, having its own random effect. This modeling helps to correct for overshrinkage so common in small area estimation and to warn against using a single random effect for a group of areas. Because there are numerous areas, the computational time of the joint posterior density using standard Markov chain Monte Carlo (MCMC) methods is prohibitive and tuning is time-consuming. Therefore, the joint pos- terior density of the hyper-parameters is approximated using an integrated nested normal approximation (INNA) via the multiplication rule. This approach provides a sampling-based method that permits fast computation, thereby avoiding very time-consuming MCMC meth- ods. Then, the random effects are obtained from the exact conditional posterior density using parallel computing and nonsampled covariates are obtained using the Bayesian bootstrap. After presenting some basic statistical concepts, we discuss the theory of this method and an example on health severity using Nepal's Living Standards Survey, the households being the small areas.

• 16 September Simone Cassani, Worcester Polytechnic Institute

Title:Blood circulation in the eye: multi-scale modeling and clinical applications
Abstract: Several ocular diseases, including glaucoma and age-related macular degeneration, have been associated with impaired retinal perfusion. In glaucoma, many risk factors contribute to ocular damage, including elevated intraocular pressure, age, genetics, and other diseases such as diabetes and systemic hypertension. Interestingly, alterations in retinal hemodynamics have also been associated with glaucoma. A better understanding of the factors that contribute to these hemodynamic alterations could lead to improved and more appropriate clinical approaches to manage and hopefully treat glaucoma patients. However the interplay among these factor is complex and not always easy to interpret in a clinical setting. Mathematical modeling can be used to investigate the complex relationship among these factors.

In this talk several lumped compartments mathematical models aimed at describing ocular hemodynamics and oxygenation in health and disease will be described. The main objective of this work is to study the relationship between intraocular pressure systemic blood pressure, and the functionality of vascular autoregulation, and to investigate the transport and exchange of oxygen in the retinal vasculature and tissue.

The model results show that the insight provided by mathematical modeling alongside clinical studies can improve the understanding of diseases and potentially contribute to the clinical development of new treatments.

• 22 September Anthony Nixon, Lancaster University

Title:Unique realisations of graphs on surfaces
Abstract: Consider a graph as a physical object whose vertices are revolute joints and whose edges are stiff bars. When is such a generic realisation of a graph on a fixed surface S in R³ unique (up to isometries)? When the surface is a plane or a sphere then the answer depends only on the graph and the properties of the graph guaranteeing uniqueness (or global rigidity) can be tested efficiently. I will survey these results and then talk about recent work to extend these characterisations to other surfaces.

• 23 September Derek Olson, Rensselaer Polytechnic Institute

Title:Regularity and Locality for Point Defects in Multilattices
Abstract: Crystal defects play an important role in determining the mechanical and electrical properties of crystalline materials. In this talk, we formulate a model for a point defect in a multilattice crystal with an empirical interatomic potential interaction and prove an estimate for the decay of the long-range elastic fields with increasing distance from the defect.

These decay estimates are essential in quantifying approximation errors in coarse-grained models, and we present an example of how they are used in the construction of an optimal numerical method for approximating a Stone-Wales defect in graphene using the blended force-based quasicontinuum method.

• 30 September Robert Neel, Lehigh University

Title: Minimal surfaces and associated martingales
Abstract: We first discuss a class of degenerate martingales that arises naturally as the diffusion associated with minimal submanifolds, mean curvature flow, and some sub-Riemannian structures. This provides a unified approach to "coarse" properties, such as transience, of such structures. We then specialize to minimal surfaces in R³, in which case the associated martingale (which is just Brownian motion on the surface, viewed as a process in R³) has the additional property that the tangent plane also evolves as a martingale. Taking advantage of this extra structure, we develop an extrinsic analogue of the mirror coupling of two Brownian motions. This allows us to study finer geometric and analytic properties of minimal surfaces, such as intersection results (strong halfspace-type theorems) and Liouville properties.

• 7 October Xiaodan Zhou, Worcester Polytechnic Institute

Title: Discrete game theory and nonlinear partial differential equations
Abstract: In this talk, we will discuss game-theoretic approaches to various partial differential equations. We first describe the connection between discrete games and nonlinear equations involving the mean curvature operator and normalized p-Laplace operators. We then discuss applications of these game interpretations, especially to the study of convexity preserving properties for nonlinear parabolic equations. We present a new proof based on selecting appropriate game strategies and iterating the corresponding dynamic programming principles.

• 28 October Uri Shafrir, University of Toronto

Title: Enhancing Learning Outcomes in STEM with Pedagogy for Conceptual Thinking
Abstract: Ongoing changes in certification requirements in engineering associations, and recently announced change of funding formula for Ontario universities, bring strong emphasis on enhancing learning outcomes as an important measure in STEM teaching and learning. Pedagogy for Conceptual Thinking evolved since 2002 at the Department of Architectural Science, Ryerson University, Toronto, through sequential stages of development, testing, validation, and implementation in collaboration with Ontario Institute for Studies in Education (OISE) at University of Toronto, as well as other educational institutions. This novel pedagogy guides sequential teaching/learning episodes in a STEM course by focusing learners' attention on conceptual meaning with Meaning Equivalence Reusable Learning Objects (MERLO). This pedagogy is designed to motivate and engage learners, encourage peer cooperation, and allow the instructor to assess deep comprehension of conceptual content by eliciting responses that signal learners' ability to recognize, and to produce, multiple representations, in multiple sign-systems (text; equations; diagrams; etc.) that share equivalence-of-meaning.

On-going analysis of course data collected in weekly MERLO formative assessments, as well as summative midterm and final exams, provide continuous, detailed and reliable statistics of learning outcomes of different concepts for each individual student, as well as class means. This statistical data allow instructors to provide timely feedback to individual students that support corrective measures and enhance learning outcomes. Recordings of small group sessions in weekly MERLO formative quizzes reveal enhanced students' engagement and peer cooperation. It often show members of a group listening intently and arguing with individuals presenting a convincing point-of-view.

Pedagogy for conceptual thinking in the classroom enhance learning outcomes in STEM courses and higher-order thinking skills. These results are particularly evident when observing and analyzing learning outcomes in large undergraduate STEM classes guided by pedagogy for conceptual thinking.

Certification workshop provide instructors with experiential learning of the implementation and use of pedagogy for conceptual thinking in K-12 and postsecondary institutions. The workshop consists of 72 instruction hours, including: lectures; individual and group work; preparation of MERLO assessment items for individual instructors' courses; and final project presentations. Following the workshop, certified participants are invited to take part in weekly, post-workshop F2F and online meetings to share with colleagues their experience and new educational and technological developments.

• 4 November F. Patricia Medina, Worcester Polytechnic Institute

Title: Hybrid Modeling and Analysis of Multicomponent Adsorption with Applications to Coalbed Methane
Abstract: We consider a non-standard model of multicomponent adsorption with applications to gas adsorption process in coalbeds. In particular, we follow thermodynamically consistent approaches, both at macroscale, via the Ideal Adsorbate Solution (IAS) theory, as well ad the pore-scale (built with statistical mechanics and specifically with mean-field approach). In this talk, I will be focusing more on the macroscale approach. The models we consider do not have a simple algebraic form, and therefore their analysis results and numerical simulation have challenges. We present several mathematical analysis results and numerical solutions to illustrate the issues. This is joint work with Malgorzata Peszynska from the Mathematics department at Oregon State University.

• 11 November Lucia Carichino, Worcester Polytechnic Institute

Title: Multiscale Mathematical Modeling of Ocular Blood Flow and Oxygenation and their Relevance to Glaucoma
Abstract: Retinal blood flow and oxygenation play a crucial role in glaucoma, the second cause of blindness worldwide. Clinical observations show significant correlations between alterations in retinal hemodynamics and vision impairment. However, the mechanisms giving rise to these correlations are not yet fully understood. In this talk I will present the main challenges encountered in developing a mathematical model that describes the fundamental mechanisms governing the blood flow in the retina, that couples retinal hemodynamics with ocular structure deformation, and that can be used to interpret clinical data. The model requires sophisticated mathematical techniques, including fluid structure interaction and multi-scale coupling. Moreover, I will present patient-specific theoretical interpretations of clinical measurements to propose possible explanations to clinically observed correlations.

• 18 November John Little, College of the Holy Cross

Title: Continua of central configurations in the Newtonian n-body problem with a negative mass
Abstract: A central configuration in the Newtonian n-body problem in celestial mechanics occurs when the acceleration vector of each body is proportional to its displacement from the center of mass of the system and the proportionality constants are all equal. If the initial positions of n bodies form a central configuration and the initial velocities are suitably chosen, then it is possible to derive explicit analytic solutions of Newton's equations of practical significance. The physically interesting central configurations occur in 2 or 3 dimensions, but the problem of determining such configurations of bodies can also be posed in Euclidean spaces of any dimension. The number of equivalence classes of central configurations of bodies of positive mass is known to be finite for configurations of up to 5 bodies, but it remains to be shown if this is true in general and the problem can be seen as an extremely delicate question about real solutions of a system of polynomial equations. By allowing one mass to be negative, Gareth Roberts constructed a continuum of inequivalent planar central configurations of n = 5 bodies. We reinterpret Roberts' example and generalize the construction of his continuum to produce a family of continua of central configurations, each with a single negative mass. These new examples exist in all even-dimensional Euclidean spaces of dimension at least 4.

• 2 December Moon Duchin, Tufts University

Title: Sprawl and other geometric statistics
Abstract: I'll define a statistic called the "sprawl" of a metric measure space which gives an indication of how efficiently one can move around. Related statistics come up across geometry, in group theory, in category theory, and in applications from biodiversity to gerrymandering. In this talk I'll spend some time discussing the case of Banach spaces.

• 9 December Mallikarjunaiah Muddamallappa, WPI

Title: : On an adaptive finite element approximation of a time discrete phase-field model for dynamic fracture
Abstract: In this talk we describe an efficient finite element treatment of a variational, time-discrete model for dynamic brittle fracture. We start by providing an overview of an existing dynamic fracture model that stems from Griffith's theory and based on the Ambrosio-Tortorelli crack regularization. Further, we propose an efficient numerical scheme based on the bilinear finite elements. We use a primal-dual active set strategy, which can be identified as a semi-smooth Newton's method. A delicate issue in models such as this concerns irreversibility and there is no general agreement on how to enforce this. In particular, we will discuss two irreversibility criteria and show that these could result in unphysical crack-widening effect. In addition, it is well known that to resolve the crack-path accurately, the mesh near the crack needs to be very fine, so it is common to use adaptive meshes. We propose a simple, robust, local mesh-refinement criterion to reduce the computational cost. Finally, I will show some numerical results for anti-plane crack propagation in elastic media.

*This is an ongoing joint work with Professors Larsen and Sarkis.

• 13 January Noah Daleo, Worcester State University

Title: Numerical algebraic geometry and the Kuramoto model
Abstract: In 1996, Sommese and Wampler coined the term numerical algebraic geometry to describe a new research area focused on the numerical solution of systems of polynomial equations. In recent years, this has been a productive area due to the development of new algorithms and software. We'll give an introduction to numerical algebraic geometry and discuss some its advantages and disadvantages compared to traditional symbolic methods. In particular, we'll look at an application to the Kuramoto model, which is a tool for studying synchronization among coupled phase oscillators.

• 3 February Mario Bonk, UCLA

Title: Uniformization: Past and present
Abstract: The uniformization problem in classical complex analysis has a long history and can be traced back to Gauss (conformal or isothermal coordinates on surfaces) and Riemann (conformal maps of simply connected domains). Questions in analysis and geometry have recently led to the investigation of related, more general problems in a metric space setting. I will give a survey of this topic discussing classical and modern developments. The talk will be accessible for a general mathematical audience.

• 24 February Tullia Dymarz, University of Wisconsin

Title: Non-rectifiable Delone sets in amenable groups
Abstract: In 1998 Burago-Kleiner and McMullen constructed the first examples of coarsely dense and uniformly discrete subsets of Rⁿ that are not biLipschitz equivalent to the standard lattice Zⁿ. We talk about extensions of these results to nilpotent Lie groups, certain solvable Lie groups and more generally to other amenable groups. The techniques involve ideas from Burago-Kleiner and quasi-isometric rigidity results from geometric group theory.

• 3 March Sebastian Cioaba, University of Delaware

Title: A brief tour of spectral graph theory
Abstract: Spectral graph is the study of eigenvalues of graphs and their connections to the graphs combinatorial properties. In this talk, I will present some of my favorite results in spectral graphs involving expanders, graph decomposition and addressings, strongly regular graphs and spectral characterization of graphs. The talk should be accessible to undergraduate students and I will present several open problems.

• 17 March Akil Narayan, University of Utah

Title: Simulation of parameterized differential equations with multi-fidelity models
Abstract: We present an algorithm for coupling inexpensive low-fidelity model simulations with high-fidelity simulation data of parameterized differential equations. The goal is to grab a "free lunch": simulation accuracy of the high-fidelity model with algorithmic complexity of only a simplified low-fidelity model. The procedure forms an approximation with sparsely available high-fidelity simulations, and is a simple linear algebraic construction with connections to kernel learning and column skeletonization of matrices. We discuss theoretical results that establish accuracy guarantees, and introduce recent analysis providing bounds on the error committed by the algorithm.

• 24 March Guozhen Lu, University of Connecticut

Title: Geometric and functional inequalities and applications to geometry and PDEs
Abstract: Geometric and functional inequalities have important applications to geometry and PDEs. In this talk, we will describe some important inequalities including Sobolev inequalities, Hardy-Sobolev inequalities, Caffarelli-Kohn-Nirenberg inequalities and Trudinger-Moser inequalities, and Adams inequalities, etc. Our main interests are their best constants and extremal functions. Difficulties arise in settings where symmetrization argument does not work (such as the Heisenberg groups and Riemannian manifolds, etc.). New techniques and ideas are required to attack these problems. This talk is intended to be for the general audience.

• 30 March Chad Topaz, Macalester College
(Room HL116, THURSDAY 4pm)

Title: Gender representation through a data science lens
Abstract: Concerned by the grievous underrepresentation of women in many scientific fields, Shilad Sen and I used techniques of data science to measure gender representation on mathematics journal editorial boards. In this talk, I will share some historical anecdotes about women mathematicians, discuss why editorial boards matter, and explain how we used crowdfunding and crowdsourcing to amass a database of over 13000 journal editors and deduce the group's gender breakdown. While women are known to comprise approximately 15% of tenure-stream faculty positions in doctoral-granting mathematical sciences departments in the United States, we find that 8.9% of the editorships in our study are held by women. We also describe group variations within the editorships by identifying specific journals, subfields, publishers, and countries that significantly exceed or fall short of this average.

• 7 April Fred Daum, Raytheon

Title: Gromov's method for stochastic particle flow nonlinear filters
Abstract: We derive a new stochastic particle flow algorithm, which is a concrete application of a very general theorem of Gromov. This nonlinear filter is many orders of magnitude faster than standard particle filters for the same accuracy for high dimensional problems, and our filter beats the extended Kalman filter accuracy by several orders of magnitude for difficult nonlinear problems. The purpose of particle flow is to mitigate particle degeneracy in particle filters. We derive a simple exact formula for the covariance matrix of the diffusion for the stochastic particle flow.

Our theory uses particle flow (like physics) to compute Bayes' rule, rather than a pointwise multiply. We do not use resampling of particles or proposal densities or importance sampling or any Markov chain Monte Carlo method. But rather, we design the particle flow with the solution of a linear first order highly underdetermined PDE, like the Gauss law in electromagnetics. We solve this PDE as a formula (rather than using numerical methods) to reduce computational complexity. Particle flow is similar to optimal transport theory, but it is much simpler and faster because we avoid solving a variational problem. Moreover, optimal transport researchers almost always assume deterministic rather than stochastic flow. Our numerical experiments show that stochastic flow is superior to deterministic flow in both accuracy and uncertainty quantification. The talk explains what a particle filter is, and why engineers like particle filters, but we also explain the curse of dimensionality. We explain particle degeneracy with a simple cartoon, and we show how to solve it. This theory can be applied to essentially any estimation or decision problem, including: tracking, guidance and navigation, control, robotics, and Bayesian decisions. We show many numerical results for various nonlinearities, with both stable and unstable plants, varying process noise, measurement noise, initial uncertainty of the state vector, and dimension of the state vector from d = 1 to 42. Particle filters generally suffer from the curse of dimensionality, and the computational complexity obviously depends on the coupling between components of the state vector and many other details of the problem. This talk is for normal people who do not have fiber bundles for breakfast.

• 14 April Maxim Bichuch, Johns Hopkins University

Title: Systemic Risk: the Effect of Market Confidence
Abstract: We postulate that market confidence is an important factor in a bank's ability to raise funds, which can be used to increase cash reserves. However, selling treasury stocks in order to raise money reduces market confidence and thus increases the overnight rate charged to the bank, which in turn makes it more costly to fund its daily operations. This puts an upper bound on the funds the bank can raise through selling treasury stocks. We calibrate the parameters of our model and calculate this upper limit. Additionally, we extend the work by Eisenberg and Noe and account for banks' ability to raise cash and increase reserves and find the social cost arising from systemic risk.

• 21 April Ronan Egan, University of Rijeka, Croatia

Title: Phased unitary Golay pairs and Butson Hadamard matrices
Abstract: Pairs of complementary binary or quaternary sequences of length v such as Golay pairs, complex Golay pairs and periodic Golay pairs may be used to construct Hadamard matrices and complex Hadamard matrices of order 2v. In this seminar I will define unitary Golay pairs and phased unitary Golay pairs of length v with entries in the k^th roots of unity for any k ≥ 2. This generalization will lead to a construction of Butson Hadamard matrices of order 2v over the k^th roots of unity for any even k. One motivation for developing this construction is that Butson Hadamard matrices have become increasingly valuable due to applications in, among other fields, quantum information theory. I will define equivalence for unitary Golay pairs and phased unitary Golay pairs, outline some constructions and discuss some computational results for small parameters. Finally, I will demonstrate how this construction strengthens a conjecture of Ito's, and consequently with this method we can construct a complex Hadamard matrix with order 2v for all v ≤ 46.

• 28 April Emily Evans, Brigham Young University

Title: Force-based models of cell-extracellular interaction
Abstract: To predict, alter and control wound healing and pathological conditions biologists need a better understanding of cell-cell and cell-extracellular interactions. We model these interactions as a system of stochastic differential equations representing the location of cell centers and extra-cellular matrix attachment sites. Numerical simulations and analysis show that when the duration of attachment to the extracellular matrix is a memoryless and force independent random process, the speed of the cell is independent of the force these attachment sites exert on the cell. Furthermore, understanding the dynamics of the attachment and detachment to the extra-cellular matrix is key to predicting cell speed. To better understand the relationship between attachment dynamics and cell speed, we consider the problem in the context of two related (but simpler) models of cell motion. In this talk, we will also show the full results showing that expected average cell speed is independent of force and dependent on attachment dynamics.

Return to main web page for Math colloquia at WPI.