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The Provost's Distinguished Women Scholar Lecture is supported by the Provost’s Distinguished Women Scholars Lecture Committee and the Pacific Institute for the Mathematical Sciences.

• Tea reception 6:00-6:30pm
• Public talk 6:30-9:00pm

Abstract

New kinds of data are giving rise to exciting new opportunities in public health. The recent revolution in DNA sequencing means that for the first time, we can read the sequences of viruses and bacteria that cause human disease. This allows us to track their evolution through time, and even to track how they are spreading through communities and environments. However, the best ways to control infectious disease do not leap off the pages of these data. Instead, we need to build ways to model and understand these rich sources of data, and we need to connect the information we gain to our efforts in public health. These tasks require new mathematics. In my group, we have developed a way to connect the DNA sequences from bacterial infections to 'outbreak stories' -- who infected whom and when. We combine genomic data, locations and clinical histories, allowing us to estimate how the infection spreads in a community. I will describe how the method works and some of its applications for public health, and I will give a broader picture of some of the coming challenges that healthcare brings to the mathematical sciences.

The Provost's Distinguished Women Scholar Lecture is supported by the Provost’s Distinguished Women Scholars Lecture Committee and the Pacific Institute for the Mathematical Sciences.

• Tea reception 3:00-3:20pm
• Academic Seminar 3:20-4:20pm

Abstract

Improvements in sequencing technology mean that we have rich data on how infections evolve and spread. In this talk I will describe two settings where this calls for new mathematics. Trees -- in the sense of graphs with no cycles -- are a cornerstone of how we understand these data. Trees can represent patterns of ancestry through time, showing evolutionary stories that are supported by data and revealing how consistent models are with data. In the first part of the talk, I will describe a metric on trees that is based on polynomial representations of trees. Since each tree has a unique polynomial, comparing the coefficients yields a suite of tree comparison tools. We illustrate these with data from cetacean evolution and infectious disease. In the second part of the talk, I will describe how we can use trees, together with a type of colouring, to do Bayesian inference of who infected whom in an infectious disease outbreak. I will apply this tool to data from tuberculosis infections.

ABSTRACT: Spatial capture-recapture can be applied to acoustic monitoring using a grid of microphones where the call is "recaptured" across the grid. However, currently it is only able to estimate "call" and not "caller" density due to the uncertainty in individual ID between calls. For species that stay relatively close to the same location throughout the study, we observe clusters of calls around their activity centre. Using this spatial information can help differentiate the unique individuals in the population. I will introduce some new ideas using Dirichlet Process Mixture Models to partition the set of latent capture histories to the most likely group of calls in the context of terrestrial acoustics.

Abstract: The classical Perron-Frobenius theorem can be applied to Markov chains with a single primitive transition matrix to show that there is a unique stationary distribution for the chain, and that distributions relax exponentially quickly to that stationary distribution, where the rate is determined by the second-largest eigenvalue. We can generalize Markov chains: first to chains with randomly chosen transition matrices, then to cocycles of transfer operators, which describe how densities move around according to random underlying dynamics on a state space. I will describe a generalization of the classical Perron-Frobenius theorem that can be applied in this setting to give an analog of a stationary distribution and a relaxation rate, along with some of the definitions and background required to understand the theorem statement.

Abstract: In the previous talk we introduced a class of hyperbolic dynamical systems called Smale spaces . In this talk, following the philosophy of noncommutative topology, we will encode certain dynamical aspects of Smale spaces into topological groupoids and then construct their C*-algebras. A particularly nice class of such C*-algebras are the Ruelle algebras introduced by Ian Putnam. They can be thought as higher-dimensional analogues of Cuntz-Krieger algebras. Ruelle algebras have the following features:

• They are separable, simple, nuclear, purely infinite, stable and in the UCT class. Hence, they can be classified by their K-theory.
• They satisfy a noncommutative version of Poincare duality; i.e. there are natural isomorphisms between the K-theory and the K-homology of the Ruelle algebras.

We will discuss the above features with a focus on the second one.

Abstract: The spread of invasive species is a serious problem for ecosystem integrity in many regions of the world. Mathematical theory has made great contributions to understanding the mechanisms behind such invasions and to estimate spread rates in homogeneous landscapes. However, most landscapes are heterogeneous at many levels, and individuals adjust their movement behaviour to local environmental conditions. Including such movement details into tractable mathematical models is a challenge. In this talk, I will present some background on models for spatial spread and a relatively recent approach to model movement in ``patchy'' landscapes. I will show that these models allow for some explicit calculations of spread rates, and that including the individual level details is crucial for obtaining biologically reasonable results. I will also point out mechanisms by which management strategies aimed at reducing the spread rate of invasives could backfire and increase the spread rate.

Abstract: This will be part one of a series of two talks. In the first part we will discuss about a class of hyperbolic dynamical systems called Smale spaces. This part will not have a C*-algebraic flavour. However, we will give examples and try to convince the audience that they are interesting.

ABSTRACT: The problem of testing for the order of finite mixture models has a long history and remains a hot research topic for a foreseeable future. Since Ghosh (1985) pointed out the undesirable asymptotic properties of the likelihood ratio test in this context, there have been marked progresses toward addressing this challenging problem. The seemingly neat and well-defined problem is challenging numerically, mathematically and statistically. The most proud development of of my research team have been the modified likelihood ratio test and the EM-test. The idea of EM-test has been extensively explored for independent and identically distributed observations, for data from hidden Markov model, and for data from regression mixtures. It possesses neat solutions to finite mixture of univariate normal distributions and several other mixtures. At the same time, it seems to be less powerful when handling mixtures of general multi-parameter distributions. In this talk, we present our recent work on finite mixture of distributions from location-scale families. We show that the EM-test has manageable limiting distributions in general, and neat chi-square limiting distributions for some important examples.

Abstract: I will explain how to compute the partition functions of C*-dynamical systems arising from actions of congruence monoids on rings of algebraic integers in number fields. Then I will show how to use these partition functions to recover information from the C*-dynamical system about the underlying number field and a certain class field that is naturally associated with the initial data. This talk is based on joint work with Marcelo Laca (University of Victoria) and Takuya Takeishi (Kyoto Institute of Technology), and it builds on Marcelo's talk from last week.

Abstract: The C*-dynamical systems from number theory introduced in joint work with Cuntz and Deninger have been generalized by Bruce by considering affine semigroups whose multiplicative part is a congruence monoid. In these systems, the phase transition of equlibrium states at low temperature does not respect the energy levels, and this calls for a closer look at what is really meant by `partition function' in the context of general C*-dynamical systems. The talk is about this closer look and what it reveals.

This research is part of current joint work with Chris Bruce (UVic) and Takuya Takeishi (Kyoto Inst. Tech.).

In a forthcoming talk, Chris will discuss the phase transition of systems from congruence monoids, and how the associated partition functions give rise to C*-dynamical invariants that can be used to retrieve information about the number field.

Abstract: The TFDW (Thomas-Fermi-Dirac-Weizacker) model has been used to describe electron configurations of molecules. On the other hand, the Ohta-Kawasaki model arises in the context of diblock copolymer melts. In this talk, we discuss results concerning compactness of minimizing sequences of the TFDW model and a variant of the Ohta-Kawasaki model.

The UVic Mathematics Competition is held annually in the fall. The competition will be held on Tuesday, September 24, 2019 between 4:30-6:30 pm in ECS 130. There are monetary prizes. To participate, just show up.

Here are some recent question papers: 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006.

Abstract: Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an i.i.d. process. Joint work with Omer Angel.

PIMS Distinguished Lecture

Join us for refreshments at a meet and greet at 3:00 pm, the talk will follow at 3:30 pm.

ABSTRACT: Optimization and optimal dynamical control are used to investigate the accuracy of analytical estimates for solutions of some basic nonlinear partial differential equations of mathematical hydrodynamics. Even though many mathematical estimates are demonstrably sharp, the result of a sequence of applications of such estimates need not be, leaving uncertainty in the precision of the ultimate result of the analysis. We examine the classical analysis bounding enstrophy and palinstrophy amplification in Burgers’ and the Navier-Stokes equations and discover that the best known instantaneous growth rates estimates for these quantities are indeed sharp. Integrating the estimates in time may (as in the 2D Navier-Stokes case) but does not always (as in the Burgers case) produce sharp estimates in which case optimal control techniques must be brought to bear to determine the actual extreme behavior of the nonlinear dynamics. The question for the 3D Navier-Stokes equations remains unresolved although work is in progress to apply these tools to address it. The most recent work reported here is joint with Diego Ayala and Thilo Simon, Journal of Fluid Mechanics 837, 839–857 (2018).

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Abstract: We consider three types of dynamical systems: shifts of finite type (SFT), substitution tilings spaces (STS), and hyperbolic toral automorphisms (HTA). A Markov partition on a HTA gives a finite-to-one factor map from a SFT onto the HTA. We investigate when such a factor map can be split as a composition of $u$-and $s$-bijective maps. It has already been shown that if the tiling system, given by the Markov partition, satisfies the forcing the border condition, then a splitting exists. We will show a partial converse, namely, if a splitting exists then the Markov partition must satisfy a certain condition. Finally, we will show that the existence of a splitting does not imply that the tiling system forces its border.

Title: The phase transition of planar random cluster model for q>4 is discontinuous: two short proofs and a new graphical representation of the six-vertex model

Abstract: It is a well-known conjecture due to Baxter that the phase transition of the critical planar random cluster and Potts model for q>4 is discontinuous, meaning that at criticality, there exist multiple Gibbs states. This conjecture was recently solved by Duminil--Copin et. al (https://arxiv.org/abs/1611.09909). using the so-called Bethe Ansatz techniques. We present two new (depending on time), short, soft and probabilistic proofs of this fact. One of the proofs introduce a new graphical representation of the six-vertex model which we believe is of independent interest. Joint with Yinon Spinka (UBC).

Abstract:
Solvers for the Boolean satisfiability problem have been increasingly used to solve hard problems from many fields and now routinely solve problems with millions of variables. Combinatorial problems are a natural target, as SAT solvers contain excellent combinatorial search algorithms. Despite this, SAT solvers can fail on small problems, for example when properties of the problem cannot be concisely expressed in Boolean logic. We describe a new combinatorial search method that allows properties to be specified using a computer algebra system (CAS), thereby combining the expressiveness of a CAS with the search power of SAT solvers.

In this talk we describe how our SAT+CAS system MathCheck has verified, partially verified, or found new counterexamples to conjectures from design theory, graph theory, and number theory. In particular, we have classified Williamson matrices up to order 70, quaternary Golay sequence pairs up to length 28, best matrices up to order 7, verified the Ruskey–Savage and Norine conjectures up to larger bounds than had previously been verified, found the smallest counterexample of the Williamson conjecture, and found three new counterexamples to a conjecture on good matrices. Currently we are using the system to verify the nonexistence of projective planes of order 10.

Notice of the Final Oral Examination for the Degree of Master of Science of

MACKENZIE WHEELER

BSc (University of Victoria, 2017)

“Chromatic Polynomials of Mixed Graphs”

Department of Mathematics and Statistics

Wednesday, August 21, 2019 11:00 A.M.
Clearihue Building Room B007

Supervisory Committee:
Dr. Gary MacGillivray, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Peter Dukes, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Danielle Cox, Department of Mathematics, Mount Saint Vincent University

Chair of Oral Examination:
Dr. Michel Lefebvre, Department of Physics and Astronomy, UVic

Abstract (PDF)

Notice of the Final Oral Examination for the

Degree of Master of Science of

YING YING YE

MSc (University of Victoria, 2008)
BSc (University of Victoria, 2007)

“On Chordal Digraphs and Semi-Strict Chordal Digraphs”

Department of Mathematics and Statistics

Wednesday, August 14, 2019 10:30 A.M.

David Strong Building Room C126

Supervisory Committee:
Dr. Jing Huang, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Venkatesh Srinivasan, Department of Computer Science, UVic (Outside Member)

External Examiner:
Dr. César Hernández Cruz, Departamento de Computación, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional

Chair of Oral Examination:
Dr. Colin Bradley, Department of Mechanical Engineering, UVic

Abstract

Chordal graphs are an important class of perfect graphs. The beautiful theory surrounding their study varies from natural applications to elegant characterizations in terms of forbidden subgraphs, subtree representations, vertex orderings, and to linear time recognition algorithms. Haskins and Rose introduced the class of chordal digraphs as a digraph analogue of chordal graphs. Chordal digraphs can be defined in terms of vertex orderings and several results about chordal graphs can be extended to chordal digraphs. However, a forbidden subdigraph characterization of chordal digraphs is not known and finding such a characterization seems to be a difficult problem. Meister studied semi-complete chordal digraphs and gave a forbidden subdigraph characterization of this class of digraphs.

In this thesis, we study chordal digraphs within the classes of quasi-transitive, extended semi- complete, and locally semi-complete digraphs. For each of these classes we obtain a forbidden subdigraph characterization of digraphs which are chordal. We also introduce in this thesis a new variant of chordal digraphs called semi-strict chordal digraphs. We obtain a forbidden subdigraph characterization of semi-strict chordal digraphs for each of the classes of semi- complete, quasi-transitive, extended semi-complete, and locally semi-complete digraphs.

Notice of the Final Oral Examination for the Degree of Master of Science of

HYUNGEUN SHIN
MSc (Kyungpook National University, 2014)
BSc (Kumoh National Institute of Technology, 2009)

“Effect of Equatorially Trapped Waves on the Tropical Cyclone Drift”

Department of Mathematics and Statistics

Wednesday, August 7, 2019 10:00 A.M.
David Strong Building Room C128

Supervisory Committee: Dr. Boualem Khouider, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Slim Ibrahim, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Jody Klymak, School of Earth and Ocean Sciences, UVic

Chair of Oral Examination:
Dr. Viviene Temple, School of Exercise, Science, Physical and Health Education, UVic

Abstract

The movement of tropical cyclones (TC) is studied numerically based on a two-dimensional barotropic model, using a previously developed non-oscillatory balanced scheme. The model of TC used here takes an exponential form, and its size and strength are selected to be of a middle scale. Without a background flow, TCs move in the northwest direction due to the beta effect. In addition, the amplitudes of high wavenumber modes of the asymmetric flow, that are believed to be responsible for the TC drift, are computed using Fourier analysis. The amplitude of wavenumber one and two modes are dominant, so they are indicators of beta conversion of energy. Also, the effect of the monsoon trough on the TC movement is investigated. The results show a sudden change of the TC propagation path, consistent with earlier work. These two studies correspond to previous works. Here, the effect of equatorially trapped waves such as Kelvin, Rossby, and Mixed Rossby Gravity, on the TC path is newly studied by varying the wavenumber and wavespeed of the underlying waves. The effect of the waves is considered because they are believed to contribute to cyclogenesis. For studying the effect, the barotropic flow induced by these waves via momentum transport and its variation were simulated for 50 days, and some patterns are found in the change of maximum wind speed. At a given time during the simulation, a TC is injected and the effect of the background wave is obtained and analyzed. Using the wavefield of 11 cases from 10 days to 30 days, the trajectories are calculated, and their patterns appear to be stochastic. So, the patterns are identified by calculating the mean path and its spread. The trajectories of TCs are different for different developed time of the waves. Kelvin waves make small variations on the length and direction of trajectory of TC. On the contrary, Rossby waves cause a dramatic change in the TC path and yield longer trajectories. Meanwhile, TCs in MRG waves keep fairly the same direction and usually have longer traveling distance. These changes vary by wave conditions. Therefore, the three kinds of waves have different effects on the trajectories of the TC. For some peculiar cases, the movements are explained based on wavefields.

Notice of the Final Oral Examination for the Degree of Master of Science of

MEIXIN LIU
BSc (University of Victoria, 2017)

“Graph Decompositions and Variance Balanced Block Designs of Experiments”

Department of Mathematics and Statistics

Tuesday, August 6, 2019 10:00 A.M.

David Strong Building Room C128

Supervisory Committee:
Dr. Peter Dukes, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Julie Zhou, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:
Dr. Hong-Chuan Yang, Department of Electrical and Computer Engineering, UVic

Chair of Oral Examination:
Dr. Charles Curry, School of Earth and Ocean Sciences, UVic

Abstract

We study construction methods for variance balanced design (VBD) with both uncorrelated and correlated errors, where block designs are used to investigate several treatment effects. We begin with a review for the development of VBDs when the errors in the linear effects model are uncorrelated. There are several construction methods of VBDs for equal and unequal block sizes. When the errors are correlated, we introduce graph theory to study construction methods of VBDs. We develop new methods via graph decomposition. In addition, we construct block designs such that the covariance matrix of the least squares estimator of treatment effects is completely symmetric. Various applications are presented for certain specific error covariance matrices.

Notice of the Final Oral Examination for the Degree of Doctor of Philosophy of

LINDA NEILSON
B.ASc (University of British Columbia, 1989)
B.Ed (University of British Columbia, 1994)

“Broadcast Independence in Graphs”

Department of Mathematics and Statistics

Thursday, July 25, 2019 10:00 A.M.
Clearihue Building Room B021

Supervisory Committee:
Dr. Kieka Mynhardt, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Rick Brewster, Department of Mathematics and Statistics, UVic (Member)
Dr. Aaron Gulliver, Department of Electrical and Computer Engineering, UVic (Outside Member)

External Examiner:
Dr. David Erwin, Mathematics and Applied Mathematics, University of Cape Town

Chair of Oral Examination:
Dr. Tim Pelton, Department of Curriculum and Instruction, UVic

Abstract
The usual graph quantifiers related to independent and dominating sets can be adapted to broadcasts on graphs. We examine some possible definitions for an independent broadcast. We determine the minimum maximal and the maximum broadcast weight for all our independence parameters on both paths and grids. For graphs in general, we examine the relationships between these broadcast independence parameters and the existing minimum and maximum minimal broadcast domination weight (or cost). We also determine upper and lower bounds for maximum boundary independent broadcasts and a new upper bound for hearing independent broadcasts.

Prof. Gerta Keller’s major research and discoveries ranged from climate change and its effects on ocean circulation, ocean anoxic events, polar warming, Deccan volcanism, comet showers, extraterrestrial impacts, the dinosaur mass extinction, the age of the Chicxulub impact and the 6th mass extinction. Her research frequently challenged accepted scientific dogma and placed her at the center of acrimonious debates fighting for survival of truth-based evidence. All but the cause of the Chicxulub impact were soon accepted by scientists and integrated into new research. After four decades, impact proponents still fiercely defend the impact theory, deny contrary evidence and at best incorporate volcanism as ad hoc revisions, proclaiming the impact triggered volcanism that caused the mass extinction.

Abstract: For the past 40 years the demise of the dinosaurs has been attributed to an asteroid impact on Yucatan, a theory that is imaginative, popular and even sexy. From the very beginning, scientists who doubted this theory were threatened into silence or their careers destroyed by the main theory proponents. Thus began the Dinosaur wars in 1980 – and still continuing. As in any war, there are two sides to the Dinosaur wars. The majority believes an asteroid hit Yucatan and instantaneously wiped out 75% of all life including the dinosaurs in a global firestorm and nuclear type winter. A small minority tested this theory and found contrary evidence that supported Deccan volcanism in India that caused rapid climate warming due to greenhouse gases (CO2), environmental stress, acid rain and ocean acidification culminating in the mass extinctions. This lecture highlights the four decades of the dinosaur wars, the increasing acceptance of volcanism’s catastrophic effects and likely cause of the mass extinction and the resulting ad hoc revisions to accommodate the impact theory.

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This is a conference in honour of Professor Andrew J. Majda’s 70th birthday. A public lecture on dinosaurs and the theories of mass extinction will be given on July 24th by Gerta Keller. The local organizer of the conference is by Boualem Khouider. Registration and schedule can be found at https://events.eply.com/ScientificGrandChallenges

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Notice of the Final Oral Examination for the Degree of Master of Science of

SAM CHURCHILL
BSc (University of Victoria, 2013)

“3–Manifolds Algorithmically Bound 4–Manifolds”

Department of Mathematics and Statistics

Friday, July 19, 2019 1:00 P.M.
David Strong Building Room C124

Supervisory Committee:
Dr. Ryan Budney, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Alan Mehlenbacher, Department of Economics, UVic (Co-Supervisor)

External Examiner:
Dr. Sue Whitesides, Department of Computer Science, UVic

Chair of Oral Examination:
Dr. Sudhakar Ganti, Department of Computer Science, UVic

Abstract
This thesis presents an algorithm for producing 4–manifold triangulations with boundary an arbitrary orientable, closed, triangulated 3–manifold. The research is an extension of the work of Costantino and Thurston’s work on determining upper bounds on the number of 4–dimensional simplices necessary to construct such a triangulation. Our first step in this bordism construction is the geometrically partitioning of an initial 3–manifold M using smooth singularity theory. This partition provides handle attachment sites on the 4–manifold M ×[0,1] and the ensuing handle attachments eliminate one of the boundary components of M×[0,1], yielding a a 4-manifold with boundary exactly M. We first present the construction in the smooth case before extending the smooth singularity theory to triangulated 3–manifolds.

Each summer, CMS Math Camps provide students with an interest in mathematics with a unique and unforgettable experience. The camps take place in universities and CEGEPs across Canada and range from day camps to week long events. Students who attend the camps leave with new friends, new ideas, and a new outlook on mathematics.

By invitation only for Grades 8-11.

Organized by Amanda Malloch.

See announcement and abstract (PDF).

A Symposium at the Intersection of Mathematics and Biology

Keynote Speaker: Dr. Simon Levin
J.S. McDonnell Distinguished University Professor in Ecology and Evolutionary Biology
Princeton University

To celebrate the conferral of UVic Honorary Doctor of Science to Dr. Levin

To Register: Complete the Contact Info and Registration at https://events.eply.com/LevinFest2019 prior to May 15, 2019.

Contributed Presentations: Oral Presentations will be selected from submitted abstracts.

To Submit an Abstract for a Talk or Poster: Complete abstract submission section on registration site prior to May 1, 2019

Please direct enquiries to levinfest@uvic.ca

Sponsors:
PIMS
CAIMS/SCMAI
University of Victoria

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Download schedule and abstracts (PDF)

The subject of mathematical ecology is one of the oldest and most exciting in mathematical biology, and has helped in the management of natural systems and infectious diseases. Though many problems remain in those areas, we face new challenges today in finding ways to cooperate in managing our Global Commons. From behavioral and evolutionary perspectives, our societies display conflict of purpose or fitness across levels, leading to game-theoretic problems in understanding how cooperation emerges in Nature, and how it might be realized in dealing with problems of the Global Commons. This lecture will attempt to weave these topics together, tracing the evolution from earlier work to challenges for the future.

Abstract: In this thesis, we study a two fluid system which describes the motion of two charged particles in a strict neutral incompressible plasma. We study the well-posedness of the system in both space dimensions two and three. Regardless of the size of the initial data, we prove the global well-posedness of the Cauchy problem when the space dimension is two. In space dimension three, we construct global weak-solutions, and we prove the local well-posedness of Kato-type solutions. These solutions turn out to be global when the initial data are sufficiently small. We also study the stability of the solution around zero given that the initial data is small and has sufficient regularity. It turns out that our system is a system of regularity-loss and the L2 norm of lower derivatives of the solution decays. At last, this two fluid system can derive the classic MHD at least formally. Arsenio, Ibrahim and Masmoudi (2015) proved that the two fluid system converges to MHD under some constrain. We showed numerically that the two fluid system converges to MHD with no such constrain and found the approximate converge rate.

Abstract
The goal of this thesis is to give an introduction to the geometric picture of bivariant K- theory developed by Emerson and Meyer building on the ideas Connes and Skandalis, and then applying the machinery to prove a result of Emerson stated, but not proved, in a recent article [8]. We begin by giving an overview of topological K-theory, necessary for developing bivariant K-theory. Then we discuss Kasparov’s analytic bivariant K-theory, and from there develop topological bivariant K-theory. In the final chapter we state and prove the result of Emerson.

Abstract: If G is a discrete group acting on a space X, then the algebra C(X) of continuous functions canonically includes in the crossed product of C(X) by G. Renault abstracted the properties of these (and similar) inclusions to give an abstract notion of when an inclusion of a C*-algebra B into another C*-algebra A has a dynamical origin, in some sense: Renault called such inclusions 'Cartan subalgebras’. There has been significant work on whether such algebras exist, and whether they are unique in some sense: more-or-less, this asks whether a given C*-algebras has a dynamical origin, and whether the dynamical origin is unique.

On the other hand, Roe algebras are C*-algebras that encode the large scale geometry of a metric space; they come equipped with canonical Cartan subalgebras. In this talk, I’ll address uniqueness questions for such subalgebras (roughly, whether the geometry of the underlying metric space is uniquely determined).

This is based on joint work with Stuart White.

Abstract
This thesis presents research focused on developing statistical methods with emphasis on tools that can be used for the analysis of data in neuroimaging studies and cognitive science. The first contribution addresses the problem of determining the location and dynamics of brain activity when electromagnetic signals are collected using magnetoencephalography (MEG) and electroencephalography (EEG). We formulate a new spatiotemporal model that jointly models MEG and EEG data as a function of unobserved neuronal activation. To fit this model we derive an efficient procedure for simultaneous point estimation and model selection based on the iterated conditional modes algorithm combined with local polynomial smoothing. The methodology is evaluated through extensive simulation studies and an application examining the visual response to scrambled faces.

In the second contribution we develop a Bayesian spatial model for imaging genetics developed for analyses examining the influence of genetics on brain structure as measured by MRI. We extend the recently developed regression model of Greenlaw et al. (Bioinformatics, 2017) to accommodate more realistic correlation structures typically seen in structural brain imaging data. We allow for spatial correlation in the imaging phenotypes obtained from neighbouring regions in the same hemisphere of the brain and we also allow for correlation in the same phenotypes obtained from different hemispheres (left/right) of the brain. This correlation structure is incorporated through the use of a bivariate conditional autoregressive spatial model. Both Markov chain Monte Carlo (MCMC) and variational Bayes approaches are developed to approximate the posterior distribution and Bayesian false discovery rate (FDR) procedures are developed to select SNPs using the posterior distribution while accounting for multiplicity. The methodology is evaluated through an analysis of MRI and genetic data obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) and we show that the new spatial model exhibits improved performance on real data when compared to the non-spatial model of Greenlaw et al. (2017).

In the third and final contribution we develop and investigate tools for the analysis of binary data arising from repeated measures designs. We propose a Bayesian approach for the mixed-effects analysis of accuracy studies using mixed binomial regression models and we investigate techniques for model selection. Software and examples in R and JAGS for implementing the proposed analysis are described and available at https://v2south.github.io/BinBayes/.