2011-2012 Colloquia

Martin Golubitsky (Ohio State University)

Networks and Synchrony

Abstract. This talk will focus on patterns of synchrony (balanced colorings) in networks of systems of differential equations and their associated quotient networks. Examples that illustrate the relationship and differences with symmetry will be given. These ideas will be applied to a general model for rivalry introduced by Hugh Wilson.

Martin Golubitsky (Ohio State University)

Pattern Formation and Symmetry-Breaking

Abstract. The equivariant branching lemma (EBL) allows us to find certain equilibria of symmetric differential equations by completing algebraic calculations. The transition from Couette flow to Taylor vortices in the Taylor-Couette experiment is one of the basic pattern forming transitions in fluid mechanics. The first half of this talk will introduce the lemma and show how it predicts the spatial characteristics of Taylor vortices from the symmetries present in the experimental apparatus. The second half will apply EBL to the study of geometric visual hallucinations, which originated with Cowan and Ermentrout.

Martin Golubitsky (Ohio State University)

Patterns Patterns Everywhere

Abstract. Regular patterns appear all around us: from vast geological formations to the ripples in a vibrating coffee cup, from the gaits of trotting horses to lapping tongues of flames, and even in visual hallucinations. The mathematical notion of symmetry is a key to understanding how and why these patterns form. This lecture will show some of these fascinating patterns and explain how mathematical symmetry enters the picture.

Marcus Khuri (SUNY, Stony Brook)

The Yamabe Problem on Manifolds with Boundary

Abstract: We study conformal deformation of Riemannian structure to constant scalar curvature and zero mean curvature on the boundary. It is shown that if the boundary is umbilic, then the full set of solutions is compact for dimensions $n\leq 24$. Counterexamples to compactness will be constructed for dimensions $n\geq 25$. Our methods also indicate an approach for the nonumbilic case.

This is joint work with Marcelo Disconzi, and generalizes previous results of Brendle, Khuri, Marques, and Schoen from the case of manifolds without boundary.

Host: Mao-Pei Tsui

Ralph B. D'Agostino, Sr. (Boston University)

The Framingham Study: The development and validation of risk prediction functions.

Abstract: The Framingham Study has been generating cardiovascular disease (CVD) risk functions for over 60 years. These have been used in a number of settings such as treatment guidelines, clinical trials (for example, entry criteria and covariate adjustments) and general consumer evaluations of their own CVD risks. In this presentation we will review the history of the Framingham Study and how these functions were a natural extension of the objectives of the study. We will also review the development, validation and transportability of these functions and the statistical concepts that were developed to address these.

Host: Paul Hewitt

Brain Yurk (Hope College)

Mathematics and the Beetles: Using Mathematical Models to Understand Insect Development in a Changing Climate

Abstract: Mathematical modeling can play an important role in understanding and predicting biological phenomena. In this talk I will discuss the role of mathematical models in understanding the development of cold-blooded organisms (poikilotherms). Because their body temperatures are not internally regulated, poikilotherm development rates depend on environmental temperatures. By modeling this dependence we can predict how insect populations might respond to global climate change.

I will focus on our work with two species of beetles-mountain pine beetles and bean beetles. Mountain pine beetles are responsible for recent massive forest loss in western North America and have undergone recent range expansion linked to climate change. Bean beetles are an emerging model insect species that infests stored products (legumes) in Africa and Asia.

Host: Paul Hewitt.

Brian Yurk (Hope College)

Wind Blows, Sand Moves, We Model: Modeling Physical Processes in Sand Dunes

Abstract: Since 1941, researchers have worked to develop mathematical models to describe and predict the growth and movement of sand dunes in deserts and coastal regions. If wind transfers sufficient momentum to a sand surface, grains become entrained in the air and bounce along the surface, ejecting additional grains (a process called saltation). As grains are deposited on or eroded from the surface, the elevation of the surface changes. This introduces an interesting feedback mechanism, because dune topography also affects wind flow-the dune shapes the wind, and the wind shapes the dune.

In this talk I will discuss the use of mathematical models to understand physical processes in sand dunes, including fluid and saltation models. I will also discuss the problem of connecting the models to measurements made in the field.

Guanqun Cao (Michigan State University)

Simultaneous Inference for Dense Functional Data

A polynomial spline estimator is proposed for the mean function of dense functional data together with a simultaneous confidence band which is asymptotically correct. In addition, the spline estimator and its accompanying confidence band enjoy “oracle” efficiency in the sense that they are asymptotically the same as if all random trajectories are observed entirely and without errors. The confidence band is also extended to the difference of mean functions of two populations of functional data. We also consider nonparametric estimation of the covariance function for dense functional data using tensor product B-splines.

We develop both local and global asymptotic distributions for the proposed estimator, and show that our estimator is as efficient as an “oracle” estimator where the true mean function is known. Simultaneous confidence envelopes are developed based on asymptotic theory to quantify the variability in the covariance estimator and to make global inferences on the true covariance. Simulation experiments provide strong evidence that corroborates the asymptotic theory while computing is efficient. Several real data examples on near infrared spectroscopy data, remotely sensed data and speech recognition data are also provided to illustrate the proposed methods.

Yunpeng Zhao (University of Michigan)

Community detection and extraction in networks

Community detection is a fundamental problem in network analysis, with applications in many diverse areas, including computer science, social sciences and biology. This talk contains both theoretical and methodological works in this area. The stochastic block model is a common tool for model-based community detection, but is limited by its assumption that all nodes within a community are stochastically equivalent, and provides a poor fit to networks with hubs or highly varying node degrees within communities, which are common in practice. The degree-corrected block model was proposed to address this shortcoming, which allows variation in node degrees within a community.

The first part of the talk will present general theory for checking consistency of community detection under the degree-corrected block model. The second part is about a new community extraction framework, which allows both tight communities and weakly connected background nodes. The proposed extraction criterion performs well on simulated and real networks. We also establish its asymptotic consistency for the case of block model.

Gavino Puggioni (Emory University)

Bayesian Hierarchical Models with Dynamic Structures: Methodology and Environmental Applications

Abstract: Mathematical and statistical modelling offer different insights to understand environmental and biological systems. Often it is desirable to combine the two approaches in a unified model. A statistical model with a rich mathematical structure offers great advantages in terms of flexibility and parameter interpretation, however the estimation process can be rather challenging. Bayesian hierarchical models are in many instances an ideal framework to overcome some of these challenges. We propose some methodological innovations that involve the use of stochastic differential equations for spatio temporal data. Examples with both synthetic and real data are provided. In particular the focus will be on an application to the spread of sugarcane yellow leaf virus, the causal agent of yellow leaf disease. The example is particularly useful to illustrate other challenges that are often encountered in environmental data analysis: identification, missing observations, and data collection design.

The last part of the talk is dedicated to show other ongoing research projects.

Edward L. Boone (Virginia Commonwealth University)

A hierarchical zero-inflated Poisson regression model for stream fish distribution and abundance

Abstract: Ecologists are frequently confronted with the challenge of accurately modelling species abundance. However, this task requires one to deal with both presence/absence as well as abundance. Traditional Poisson regression models are not adequate when attempting to deal with both issues simultaneously. Zero-Inflated regression models have been proposed to deal with this problem with much success. We extend these models to incorporate two separate multilevel hierarchical structures and spatial correlation. We demonstrate on how to make inferences on the parameters in the model. The model is illustrated using a dataset concerning the Hypseleotris galii (Firetailed Gudgeon), a native species to eastern Australia.

Richard Aron (Kent State University)

Smooth surjections without surjective restriction

Abstract: Let $f:E \to F$ be a "decent" onto function between Banach spaces $E$ and $F.$ Let us assume that $E$ is "much bigger" than $F$. The question that we investigate is: When is there a "smaller" Banach space $G \subset E$ such that the restriction $f\vert_G: G \to F$ remains onto. We study this question in four cases: when the underlying spaces $E$ and $F$ are real or when they are complex, and when the dimension of the range space $F$ is 1 or when it is bigger than 1.

Joint work with J. A. Jaramillo, T. Ransford

Host: Zeljko Cuckovic

Robert Lund (Clemson University)

Changepoints in Climatology

Abstract: This talk overviews changepoint issues in climate studies. Changepoints are ubiquitous features in climatic time series, occurring whenever stations relocate or gauges are changed. Ignoring changepoints can produce spurious conclusions. Changepoint tests involving cumulative sums, likelihood ratio, and maximums of F statistics are introduced; the asymptotic distributions of these statistics are quantified under the changepoint-free null hypothesis. We find that cumulative sum procedures work best when the changepoint is near the center of the data record; otherwise, maximums of F statistics perform better.

Next, time series aspects of the problem are addressed. Series with positive autocorrelation can have long sojourns above and below mean levels, hence mimicing a mean shift. We show how to modify the above methods to account for autocorrelation. The methods are illustrated in several applications, including temperature trends and Atlantic Basin tropical storm counts. Changes in tropical storm and hurricane counts has been controversially addressed in the media recently, with the debate even reaching US Senate floors.

Host: Qin Shao

Aihua Wood (The Air Force Institute of Technology)

Topics on electromagnetic scattering from cavities

The analysis of the electromagnetic scattering phenomenon induced by cavities embedded in an infinite ground plane is of high interest to the engineering community. Applications include the design of cavity-backed conformal antennas for civil and military use, the characterization of radar cross section (RCS) of vehicles with grooves, and the advancement of automatic target recognition. Due to the wide range of applications and the challenge of solutions, the problem has been the focus of much mathematical research in recent years.

This talk will provide a survey of mathematical research in this area. In addition I will describe the underlining mathematical formulation for this framework. Specifically, one seeks to determine the fields scattered by a cavity upon a given incident wave. The general way of approach involves decomposing the entire solution domain to two sub-domains via an artificial boundary enclosing the cavity: the infinite upper half plane over the infinite ground plane exterior to the boundary, and the cavity plus the interior region. The problem is solved exactly in the infinite sub-domain, while the other is solved numerically. The two regions are then coupled over the artificial boundary via the introduction of a boundary operator exploiting the field continuity over material interfaces. We will touch on both the Perfect Electric Conducting and Impedance ground planes. Results of numerical implementations will be presented.

Host: Biao Ou

Timothy Fisher and James M. Martin-Hayden (Department of Environmental Sciences , University of Toledo)

Statistical Analysis and Modeling of Heterogeneous Geologic Data

Part I—Are there climate signals in eolian sand in lake sediment?

Part II—Sampling groundwater flow from stacked lithologically diverse sediment.

Haonan Wang (Colorado State University)

Exploration and Modeling for Tree-structured Objects

Abstract: Object oriented data, such as tree-structured data, random graphs, manifold data and curve data, are frequently collected in many scientific studies. Analysis of complex data objects poses serious challenges since traditional statistical models for multivariate data are built under Euclidean space setting. For example, two blood vessel systems differ in terms of topological structures and geometric properties (i.e., overall length, number of branches, and branching orientation).

A mathematical framework for statistical analysis of object oriented data has been developed. This includes measures of centrality, variability, and regression analysis. Notions of one-dimensional representation, lines and curves, are also developed. Furthermore, nonparametric smoothing method is generalized to tree space. The methodology is illustrated through applications to the analysis of brain blood vessel data.

Host: Qin Shao

Michael Geline (Northern Illinois University)

Representations of finite groups and some geometry in characteristic $p$

Abstract: There are many questions about representations of finite groups which date back to Richard Brauer, yet remain open in spite of the classification of the finite simple groups. One of these, known as Brauer's height zero conjecture, has led me to an elementary problem about three-dimensional space in characteristic $p$. I will explain Brauer's conjecture, its relationship to this "geometry problem", as well as a solution to the problem when $p=3$.

Host: Akaki Tikaradze

Martin Golubitsky (Ohio State University)

Synchrony and Synchrony-Breaking

Abstract: A coupled cell system is a network of interacting dynamical systems. Coupled cell models assume that the output from each cell is important and that signals from two or more cells can be compared so that patterns of synchrony can emerge. We ask: which part of the qualitative dynamics observed in coupled systems is the product of network architecture and which part depends on the specific equations? In our theory, local network symmetries replace symmetry as a way of organizing network dynamics, and synchrony-breaking replaces symmetry-breaking as a basic way in which transitions to complicated dynamics occur. This talk will focus on patterns of phase-shift synchrony in time-periodic solutions of coupled cell systems.

Host: Maria Leite

Rieuwert J. Block (Bowling Green State University)

Amalgams of Groups

Abstract: The motivation for the study of amalgams is that it provides a means for "knowing" an often fantastically complicated group - the universal completion of the amalgam) by local data, that is, by a relatively small collection of small subgroups called an amalgam. The rank-2 amalgams for groups of Lie type resulting from Phan's theorem and the Curtis-Tits theorem, are used in the Gorenstein-Lyons-Solomon revision of what might well be called the most momentous theorem in group theory, the classification of finite simple groups. An attractive aspect of the theory of amalgams is that it employs the interplay between groups and geometric structures. As such, it belongs in the modern area of geometric group theory. I will give a concise introduction to the topic, add some historical perspective, and then focus on some interesting questions and current developments.

Host: Nate Iverson