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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.

Download poster (PDF)

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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Download programme (PDF)

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

Download poster (PDF)

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/.

Abstract In models for networks of regulatory interactions in biological molecules, the sigmoid relationship between concentration of regulating bodies and the production rates they control has lead to the use of continuous time ‘switching’ ordinary differential equations (ODEs), sometimes referred to as Glass networks. These Glass networks are the result of a simplifying assumption that the switching behaviour occurs instantaneously at particular threshold values. Though this assumption produces highly tractable models, it also causes analytic difficulties in certain cases due to the discontinuities of the system, such as non-uniqueness. In this thesis we explore the use of ‘ramp’ functions as an alternative approximation to the sigmoid, which restores continuity to the ODE and removes the assumption of infinitely fast switching by linearly interpolating the focal point values used in a corresponding Glass network. A general framework for producing a ramp system from a certain Glass network is given. Solutions are explored in two dimensions, and then in higher dimensions under two different restrictions. Periodic behaviour is explored in both cases using mappings between threshold boundaries. Limitations in these method are explored, and a general proof of the existence of periodic solutions in negative feedback loops is given.

• Mike Bailey, Latin squares with holes
• Etienne Leclerc, Switching homomorphisms on mixed graphs

Abstract: I will discuss transport of passive scalars by incompressible flows (such as a die in a fluid) and measures of optimal mixing and stirring under physical constraint on the flow, such as under an energy or an enstrophy budget. In particular, I will present recent results concerning examples of flows that achieve the optimal mixing rate, using methods from geometric analysis and partial differential equations. These examples are related to examples of (instantaneous) loss of regularity for solutions to linear transport equations with non-Lipschitz velocity.

Abstract: Cuntz, Deninger, and Laca initiated the study of left regular C*-algebras of full $ax+b$-semigroups over rings of integers in number fields. A more general class of C*-algebras is obtained by considering the left regular C*-algebras of semi-direct products formed from actions of a congruence monoids on a ring of integers. I will discuss how to use K-theoretic invariants to recover information about the underlying number field and the congruence monoid from these C*-algebras. This is joint work with Xin Li.

Abstract: A topological dynamical system $(X,S)$ is an "h-universal model for ergodic actions" if it's ergodic measures include an isomorphic copy of any free measure preserving action of entropy strictly lower than then h. In this talk, which is based on recent joint work with Nishant Chandgotia I will describe a new sufficient condition for a topological dynamical system to be a universal model in the above sense (and also in the stronger "almost Borel" since ). Our main result extends a line of previous work starting from Krieger in the 1970's to Quas and Soo from several years back, to very recently published work of David Burguet. I will explain try to outline the main ideas of the proof and also (as time premits) how we use our main result (combined with previously known results) to deduce various conclusions and answer a number of questions. Among these applications:

• A ``generic'' homeomorphisms of a compact manifold of topological dimension at least two can model any ergodic transformation,
• Non-uniform specification implies almost Borel universality (This is an independent and extended proof of Burguet's recent strengthening of the Quas-Soo result).
• $3$-colorings in $\mathbb{Z}^d$ and dimers in $\mathbb{Z}^2$ are almost Borel universal.

This talk will essentially consist of two or three mini-talks. First and foremost, I will answer a couple of questions from the last two talks that I responded to with "I don't know" at the time. In particular, I'm going to actually construct a few orthogonal Latin squares via Wilson's construction. Then I will finally move on from that paper and present a proof (of Douglas Stinson) that there do not exist two MOLS(6). Finally, if there is time, I will present one more construction which produces a pair of MOLS of order 10.

Abstract: The category of real C*-algebras is equivalent to the category of complex C*-algebras with anti-automorphism. One of the main questions driving work in this field is this: given a complex C*-algebra B, can we identify up to isomorphism all the real C*-algebras A such that the complexification of A is B? I will review K-theory in the context of real C*-algebras, the state of classification results for real C*-algebras, and how they are used to address the main question. Examples will include finite real C*-algebras, commutative real C*-algebras, AF real C*-algebras, purely infinite simple real C*-algebras, and graph real C*-algebras.

Abstract: Adaptive time stepping methods for meta-stable dynamics of the Allen Cahn and Cahn Hilliard equations are investigated in the spatially continuous, semi-discrete setting. Predicted behaviour of the number of time steps with specified local error tolerance $\sigma$ based on the local truncation error and its dependence on the order parameter $\epsilon$ in the equations is verified in computational studies. A number of first and second order methods are considered. It is seen that in this setting, some methods, including but not limited to so-called energy stable methods, require asymptotically (as $\epsilon \rightarrow 0$) more time steps. It is shown that having the dominant local truncation error a pure time derivative is necessary for optimal number of time steps. Further, we introduce the concept of asymptotic consistency and show in this context that first order Backward (Implicit) Euler has a particularly desirable property. This last result is obtained with a formal asymptotic argument backed by computational evidence. The talk will include some more basic ideas in numerical analysis for non-experts.

Abstract: I will discuss recent constructions of finite time singularity for solutions to the 3d Euler equation living in certain regularity classes.

Abstract: Cuntz, Deninger, and Laca studied the left regular C*-algebra of the full $ax+b$-semigroup over a ring of integers in a number field. A more general class of C*-algebras is obtained by considering the left regular C*-algebras of semi-direct products formed from actions of a congruence monoids on a ring of integers. I will give an overview of several properties of these C*-algebras for the special case of the ring $\mathbb{Z}$. In this situation, every congruence monoid is the disjoint union of certain arithmetic progressions.

Abstract: Nonzero Lyapunov exponents carry crucial information on the dynamics of a system. Positive exponents are considered a hallmark of chaos, while negative exponents are often associated to regular behaviour such as synchronisation. Deciding whether a given system admits nonzero exponent is, in general, a very hard task, and providing analytic estimates is an even harder endeavour. The subtlety of this problem is attested by the fact that L. exponents are often not stable under small perturbations of the dynamics.

During the first talk, I will introduce basic notions on Lyapunov exponents and random systems, and will review some relevant literature for the topic of the second talk. This talk will target undergraduate and graduate students with basic background in ergodic theory. During the second talk, I will describe recent results on stability of Lyapunov exponents for Morse-Smale diffeomorphisms of the unit circle with a sink and a source that are perturbed by a a fixed rotation that occurs rarely, but that can be large in magnitude sending the sink in the vicinity of the source.

Jesse Johnson will talk about the paper “Long binary patterns are Abelian 2-avoidable” by James Currie and Terry Visentin.

Abstract: We formulate a strengthening of the strong Novikov conjecture and show that it holds in almost all cases where we currently can verify the strong Novikov conjecture.

Abstract: Emerson and Meyer constructed their 'coarse co-assembly' map as a dual to the well-known coarse assembly map. Its domain is the reduced $K$-theory of the stable Higson corona and not --as one might expect-- the $K$-homology of the Roe algebra. In this talk I will give an index theoretic interpretation of this duality that indicates, that the reduced $K$-theory of the stable Higson corona is indeed a good replacement for the $K$-homology of the Roe algebra.

Abstract: Nonzero Lyapunov exponents carry crucial information on the dynamics of a system. Positive exponents are considered a hallmark of chaos, while negative exponents are often associated to regular behaviour such as synchronisation. Deciding whether a given system admits nonzero exponent is, in general, a very hard task, and providing analytic estimates is an even harder endeavour. The subtlety of this problem is attested by the fact that L. exponents are often not stable under small perturbations of the dynamics.

During the first talk, I will introduce basic notions on Lyapunov exponents and random systems, and will review some relevant literature for the topic of the second talk. This talk will target undergraduate and graduate students with basic background in ergodic theory. During the second talk, I will describe recent results on stability of Lyapunov exponents for Morse-Smale diffeomorphisms of the unit circle with a sink and a source that are perturbed by a a fixed rotation that occurs rarely, but that can be large in magnitude sending the sink in the vicinity of the source.

Abstract: Given a partially oriented graph G and a class C of oriented graphs, the orientation completion problem asks whether it is possible to complete the partial orientation of G to obtain an oriented graph in the class C. A local tournament is an oriented graph G where the in-neighbourhood and the out-neighbourhood of each vertex each induces a tournament. The talk will focus on the case of the orientation completion problem for the class of local tournaments. Particularly, we will explore some structural properties of the minimal obstructions and see an example of an infinite family of minimal obstructions.

We describe a point of view on geometry with wide applications in geometric group theory and topology, called 'coarse geometry.' This talk will be introductory, in preparation for several coarse geometry talks by visitors next week. I will define the coarse geometric category, and and talk about some work of myself and Meyer on coarse boundaries and connections to index theory and K-theory.

Abstract: We will recall the notion of Lyapunov exponents, and discuss them in a context of Perron-Frobenius for some Blaschke products. The key motivating question is to determine circumstances under which the Lyapunov exponents are stable under perturbations of the system.

Note: This will be part TWO a two part presentation.

In this talk we go over the details of an adaptive platform trial for the treatment of recurrent Clostridium difficile infection (CDI) by fecal microbiota transplantation (FMT). The latter has garnered global interest for treating recurrent CDI as well as other forms of dysbiosis such as inflammatory bowel disease. This class of designs are very appropriate as traditional antibiotic therapies appear to be less effective in treating recurrent CDI. On the other hand, the remarkable success of FMT has garnered global interest, however, there are numerous FMT modalities with varying CDI resolution rates reported. To test all of these simultaneously would be costly, therefore we seek more hospitable trial designs that can efficiently test these multiple modalities. A multi-arm multi-stage drop-the-loser CDI-FMT clinical trial appears to be well suited for this important investigation. Such a trial is scheduled to take place shortly, but prior to patient recruitment, we have methodically put together this platform trial and performed extensive simulations to see what we could expect. We report on the various results cast in differing scenarios performed within a strict size-power framework.

PIMS - UVic Distinguished Lecture

(pre-lecture refreshments at 3:00 pm)

Abstract: Range expansion is a crucial population response to climate change. Genetic consequences are coupled to ecological dynamics that, in turn, are driven by shifting climate conditions. We model a population with a reaction–diffusion system, coupled to a heterogeneous environment that shifts with time due to climate change. We decompose the resulting traveling wave solution into neutral genetic components to analyze the spatio-temporal dynamics of its genetic structure. Our analysis shows that range expansion under slow climate change preserves genetic diversity. However, diversity is diminished when climate change occurs too quickly. We show that populations with intermediate dispersal ability are best for maintaining genetic diversity. Our study also provides new insight regarding traveling wave solutions in heterogeneous environments. This is joint with Jimmy Garnier (CNRS).

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Abstract: Last time, we presented some theorems that construct sets of orthogonal Latin squares. This time, we will use these theorem to show 1. that if n is not 2 or 6, there exist a pair of MOLS(n), and 2. for n large enough, N(n)>n^(1/14.8)-3.

Abstract: Many problems in science and engineering are described in terms of equilibrium shapes on which certain conditions are imposed and which separate regions where the solution may have different properties. A prototypical problem of this type involves inviscid vortex equilibria in 2D and axisymmetric 3D geometries characterized by compact vortex regions embedded in a potential flow. Computation of such equilibrium configurations is made difficult by the fact that it requires finding the shape of the boundary separating the two regions. Similarly, studying the linear stability of such free-boundary problems is also challenging as it requires characterization of the sensitivity of the equilibrium solutions with respect to suitable perturbations of the boundary. We will demonstrate that such questions can be in fact systematically addressed using techniques of "shape calculus" applied to the boundary-integral formulations of such problems, leading to elegant and accurate computational approaches. In the context of vortex dynamics we use these techniques to efficiently compute the family of inviscid vortex rings initially discovered by Norbury (1973). We also obtain an equation characterizing the stability of general vortex equilibria from which certain classical results of vortex stability can be derived as special cases. Finally, this approach is employed to solve open problems concerning the linear stability of Hill's and Norbury's vortices to 3D axisymmetric perturbations, which leads to some unexpected findings.

Abstract: For a compact manifold M, the Atiyah-Singer index map is the map from the K-theory of the cotangent bundle of M to the integers induced by collapsing T*M to a point. The construction of this map makes use of the Thom isomorphism and wrong-way funtoriality for open embeddings. In this talk I will discuss the Thom isomorphism in topological K-theory, and show how given a map f:M->N between smooth manifolds which satisfies a certain orientability assumption, called "K-orientability", one can construct a map f!:K*(M)->K*(N), which generalizes the index map of Atiyah and Singer. I will then define topological correspondences, which consist in part of a K-orientable map, and show one can define bivariant K-theory for smooth manifolds using topological correspondences in a way which agrees with Kasparov's definition of KK-theory.

Abstract: Some of the most fundamental quantities in population ecology describe the growth and spread of populations. Population dynamics are often characterized by the annual rate of increase, lambda or the generational increase, R0. Analyses involving R0 have deepened our understanding of disease dynamics and life-history complexities beyond that afforded by analysis of annual growth alone. While range expansion is quantified by the annual spreading speed, a spatial analog of lambda, an R0-like expression for the rate of spread is missing. Using integro-difference models, we derive the appropriate generational spreading speed for populations with complex stage-structured life histories. The resulting measure, relevant to locations near the expanding edge of a (re)colonizing population, incorporates both local population growth and explicit spatial dispersal. The calculations for generational spreading speed are often simpler than those for annual spreading speed, and analytic or partial analytic solutions can yield insight into the processes that facilitate or slow a population’s spatial spread. We analyze the spatial dynamics of teasel as an example to demonstrate the flexibility of our methods and the intuitive insights that they afford. This work is joint with Andrew Bateman, Marty Krkosek and Mike Neubert.

Abstract: We will recall the notion of Lyapunov exponents, and discuss them in a context of Perron-Frobenius for some Blaschke products. The key motivating question is to determine circumstances under which the Lyapunov exponents are stable under perturbations of the system.

Note: This will be part one a two part presentation.

Abstract: In this talk I will present a next generation model for the spread of sea lice on a network of salmon farms. Sea lice are marine parasites with a free living larval stage and multiple attached stages where the parasites are directly attached to salmon on a farm. The larval stage can spread between salmon farms, and this spread is modelled by an advection diffusion equation. Survival and maturation through the attached stages are modelled using arbitrary functions, which can accommodate any maturation or survival delays in a stage. A next generation matrix is then used to examine the effect of different environmental variables on the source-sink distribution and persistence of the population.

Abstract: Assuming stationarity is unrealistic in many time series applications. A more realistic alternative is to allow for piecewise stationarity, where the model is allowed to change at given time points. In this talk, we propose a three-stage procedure for consistent estimation of both structural change points and parameters of high- dimensional piecewise vector autoregressive (VAR) models. In the first step, we reformulate the change point detection problem as a high-dimensional variable selection one, and propose a penalized least square estimator using a total variation penalty. We show that the proposed penalized estimation method over- estimates the number of change points. We then propose a backward selection criterion in conjunction with a penalized least square estimator to tackle this issue. In the last step of our procedure, we estimate the VAR parameters in each of the segments. We prove that the proposed procedure consistently detects the number of change points and their locations. We also show that the procedure consistently estimates the VAR parameters. The performance of the method is illustrated through several simulation scenarios and real data examples. This is a joint work with Ali Shojaie.

Jesse Johnson will talk about the paper "On Abelian 1-avoidable binary patterns”, by James D. Currie and Terry I. Visentin.

Abstract: In this talk I will present a next generation model for the spread of sea lice on a network of salmon farms. Sea lice are marine parasites with a free living larval stage and multiple attached stages where the parasites are directly attached to salmon on a farm. The larval stage can spread between salmon farms, and this spread is modelled by an advection diffusion equation. Survival and maturation through the attached stages are modelled using arbitrary functions, which can accommodate any maturation or survival delays in a stage. A next generation matrix is then used to examine the effect of different environmental variables on the source-sink distribution and persistence of the population.

Abstract: The Wright-Fisher diffusion and the Feller diffusion are among the first models of diffusion processes and were originally constructed from diffusion approximations of one-dimensional Markov chains (Wright, 1945; Feller, 1951). To this day, the method has been extensively developed for questions including establishing universality classes of stochastic differential equations, or stochastic partial differential equations (SPDEs), through scaling limits of discrete spatial models.

In this talk, I will start with an introduction of the Wright-Fisher diffusion and the Feller diffusion. Then I will discuss the progress in the last decade for two questions arising from diffusion approximations: (1) uniqueness in certain spatial generalizations of the Feller diffusion as solutions to SPDEs, and (2) spatial universality of the Wright-Fisher diffusion for interacting particle systems defined by deaths and births. This talk will be concluded with questions for future research.

Abstract: Non-Euclidean data that are indexed with a scalar predictor such as time are increasingly encountered in data applications, while statistical methodology and theory for such random objects are not well developed yet. To address the need for new methodology in this area, we develop a total variation regularization technique for nonparametric Fréchet regression, which refers to a regression setting where a response residing in a generic metric space is paired with a scalar predictor and the target is a conditional Fréchet mean. We show that the resulting estimator is representable by a piece-wise constant function and investigate the convergence rate of the proposed estimator for data objects that reside in Hadamard spaces. The method can also be applied to the problem of estimating multiple change-points in a sequence of non-Euclidean data. This is illustrated via the application to modeling the dynamics of brain networks and the study of evolving mortality distributions endowed with the Wasserstein distance.

Abstract: Types of interactions between entities of a system limit the hopes for the stability of the system, regardless of the level of interaction. In this talk, I will briefly introduce a dynamical system, namely the Kuramoto model, and then restrict it to a graph where attraction and repulsion define positive and negative edges of the graph, respectively. Then I'll present some results on how some graph-theoretic properties of the graph put bounds on the number of positive and negative eigenvalues of the Jacobian at a fixed point. These results are mainly from the 2014 paper by Bronski and DeVille, "Spectral theory for dynamics on graphs containing attractive and repulsive interactions", SIAM Journal of Applied Mathematics, 74 (1) 83–105.