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

FREDDIE MULLIN

BEd (University of Victoria, 2013)
BSc (University of Victoria, 2011)
BA (University of Victoria, 2011)

“Studies on Math Education and Trigraph Homomorphisms”

Department of Mathematics and Statistics
Thursday, December 14, 2023
3:30 P.M.
Virtual Defence

Supervisory Committee:
Dr. Gary MacGillivray, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Jane Butterfield, Department of Mathematics and Statistics, UVic (Co-Supervisor)
Dr. Richard Brewster, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Jacobus Swarts, Mathematics Department, Vancouver Island University

Chair of Oral Examination:
Dr. Marlea Clarke, Department of Political Science, UVic

Abstract
This thesis is comprised of two parts: (i) a study of homomorphisms of weak trigraphs and (ii) an analysis of the effectiveness of the University of Victoria (UVic) Department of Mathematics and Statistics’ Pretest. In the first part of the thesis, we study homomorphisms of weak trigraphs. Results analogous to those for graph homomorphisms are developed. In particular, we determine the complexity of deciding whether there is a weak trigraph homomorphism of a weak trigraph G to a weak trigraph H, the complexity of deciding whether a given weak trigraph has a weak trigraph homomorphism to a proper subgraph (the complexity of deciding whether it is not a core) and describe an efficient algorithm based on Consistency Checking that determines whether there is a weak trigraph homomorphism from a given cactus weak trigraph to a fixed weak trigraph H. In the second part of the thesis, we analyse the effectiveness of the UVic Pretest. First we compare the current online UVic Pretest with the past paper Pretest in terms of their respective effectiveness in identifying students who are ready for Calculus I. We also analyse the current online UVic Pretest in greater detail, to identify which precalculus skills are most likely to predict success on that test itself. Finally, we use odd ratios to categorize each question on the online UVic Pretest and identify questions that are particularly useful to the test.

Zoom link.

This is our 3rd talk of the PIMS Data Science Seminar Series. PIMS requests all seminar participants to complete the demographics form online.

Abstract:
In this data explosion era, machine learning has accelerated research in biology and medicine at an unprecedent speed and in multiple dimensions. The increased data volume and capacity for data aggregation and analytics power, along with decreasing costs of genome sequencing has spurred the growth in bioinformatics and need for novel tools to integrate the highly heterogenous data from multiple sources and of varying types, and extract meaningful patterns. The big data analytics and AI tools have already created significant impact in many fields of life sciences including medicine. However, the data complexity and multi-dimensionality have led to technical challenges in developing and validating AI solutions that generalize to diverse populations and imped the progress in their implementation in clinical practice due to imbalance in data distribution across population demography and data sparsity. This leads to the unconscious biases in the generated models and algorithms. In this talk, the speaker will discuss applications of AI in biology and medical research, advances and major challenges.

Short bio of the speaker:
Dr. Youlian Pan is an international expert in integrative pattern recognition from big data in Life Sciences. He has authored and co-authored over 80 refereed articles and created significant applications of data mining, machine learning, AI and bioinformatics in genomics, transcriptomics and systems biology with various medical applications, such as cancers, infectious diseases and neurodegenerative diseases. He also has extensive research interest in plants’ pathogenesis and embryogenesis, their interaction with environment, and biological oceanography specifically in marine pollution. Dr. Pan is a Senior Research Scientist at the National Research Council Canada and an Adjunct Professor at the University of Victoria and Brock University. He received his PhD in Biology and Master of Computer Science from Dalhousie University. He has served at various capacities in editorial board of six international journals, such as Journal of Computations & Modeling, Open Medical Informatics, and Frontiers in Genetics, Microbiology and Plant Sciences; and various national and international grant evaluation panels such as Natural Sciences and Engineering Research Council (NSERC) of Canada, National Science Foundation (NSF) of US.

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

J. ETHAN WILLIAMS
BSc (University of Victoria, 2021)

“Eternal Domination Problems”

Department of Mathematics and Statistics

Wednesday, December 13, 2023
10:30 A.M.
David Turpin Building
Room A203

Supervisory Committee:
Dr. Gary MacGillivray, Department of Mathematics and Statistics, University of Victoria
(Co-Supervisor) Dr. Richard Brewster, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:
Dr. William Klostermeyer, School of Computing, University of North Florida

Chair of Oral Examination:
Dr. Matt Moffitt, Department of Chemistry, UVic

Abstract
Consider placing mobile guards on the vertices of a graph. The vertices are then attacked by an assailant, requiring you to move guards to the attacked vertices. What is the minimum number of guards you need in order to be able to defend against any sequence of attacks? This question is the basis for the eternal domination problem. In this thesis we investigate this problem and introduce new parameters related to it.

These new parameters arise from changing three of the assumptions made when defining the game. Specifically we assume that any number of guards can move when defending against an attack; only one attack needs to be defended against at a time; and that any number of guards can occupy a vertex. Changing these assumptions gives rise to the maneuver, invasion, and stacking numbers respectively. We investigate these parameters throughout this thesis, especially as they relate to trees.

Additionally, we tackle the related problem of eternal Roman domination, which is based on the topic which originally gave rise to the eternal domination problem. We establish a best possible upper bound for this parameter over all graphs. Finally, we present exponential time algorithms for solving all of these problems, as well as a host of other related problems.

Interested students, please contact Peter Dukes at dukes@uvic.ca for an invitation link.

Abstract: Zero forcing is a type of graph propagation based on the color-change rule: Given graph $G$, if each vertex of $G$ is colored either white or blue, and vertex $v$ is a blue vertex with only one white neighbor $w$, then change the color of $w$ to blue. In this talk we prove a conjecture formulated by the automated conjecturing program called \emph{TxGraffiti}. The conjecture states that in a claw-free graph, the vertex cover number of the graph is at least the zero forcing number of the graph. We also prove a relationships about the zero forcing and independence number of a connected subcubic graph.

Register for Zoom link.

In 1962, Tutte proposed a simple method to produce a straight-line embedding of a planar graph in the plane, known as Tutte’s spring theorem. It leads to a surprisingly simple proof of a classical theorem proved by Bloch, Connelly, and Henderson in 1984, which states that the space of geodesic triangulations of a convex polygon is contractible. In this talk, I will introduce spaces of geodesic triangulations of surfaces, review Tutte’s spring theorem, and present this short proof. It time permits, I will briefly report the recent progress in identifying the homotopy types of spaces of geodesic triangulations of general surfaces.

Zoom link.

Abstract: The random interlacement point process (introduced by Sznitman, generalized by Teixeira) is a Poisson point process on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift on a transient graph G. We show that the random interlacement point process on any transient transitive graph G is a factor of i.i.d., i.e., it can be constructed from a family of i.i.d. random variables indexed by vertices of the graph via an equivariant measurable map. Our proof uses a variant of the soft local time method (introduced by Popov and Teixeira) to construct the interlacement point process as the almost sure limit of a sequence of finite-length variants of the model with increasing length. We also discuss a more direct method of proving that the interlacement point process is a factor of i.i.d. which works if and only if G is non-unimodular. Based on joint work with Márton Borbényi and Sándor Rokob.

See announcement and abstract (PDF)

Zoom link.

PIMS requests all seminar participants to complete the demographics form online at https://ubc.ca1.qualtrics.com/jfe/form/SV_6QcNr2rQcIlQGyy

Abstract: In this talk, we will share a new evolutionary but ensemble method, that enable us to track different regimes of behavior and identify when the changes occurred. As opposed to traditional methods that relies on statistical change tracking to detect intrusions, we use the performance, and the predictive power of evolving models to detect when and which models can or cannot describe the observations anymore. By doing so, we obtain a much fine-grain, more adaptive outlier detection algorithm that can reliably model data while being robust to its variations. The algorithm can be viewed as an evolutionary algorithm with growth and new generation capabilities, but it is special in the sense that it includes ensemble of models with performance measures and age decay corrections to evolve and compare models. For some special cases, the algorithm can be related to Bayesian Change Point techniques.

Bio: Belaid Moa received the B.Sc. degree in electrical engineering from École Hassania des Travaux Publics, Casablanca, Morocco, the M.Eng. degree in electronics and signal processing from École Nationale Supérieure d'électronique, d'électrotechnique, d'informatique, d'hydraulique et des Télécommunications, Toulouse, France, the DEA Diploma degree in Telecommunications and Networks from the Institute National Polytechnique de Toulouse, Toulouse, and the Ph.D. degree in computer science from the University of Victoria. He is currently an adjunct faculty with ECE Dept., and Advanced Research Computing Specialist with the Digital Research Alliance of Canada/BCDRI /University Systems, at the University of Victoria. He has authored or co-authored many research articles and conference proceedings in various journals and multi-disciplinary research areas

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

Hanan Abousaleh
BSc (University of Victoria, 2021)

Examining Committee

Supervisory Committee
Dr. Julie Zhou, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Michelle Miranda, Department of Mathematics and Statistics, UVic (Member)

External Examiner
Dr. Nilanjana Roy, Department of Economics, UVic

Chair of Oral Examination
Dr. Hong-Chuan Yang, Department of Electrical and Computer Engineering, UVic

Abstract
We focus on a class of optimization problems known as optimal design problems, where the goal is to select design points optimally with respect to some criterion of interest. For regression models, the optimality criterion is based on the statistical model itself and is often a function of the information matrix. We solve A-, D-, and EI-optimal design problems in this thesis. The CVX program in MATLAB is a modelling tool and solver for convex optimization problems. As with other numerical methods in the literature, formulating an optimal design problem in a CVX-compatible way requires a discrete design space. We develop a CVX-based algorithm to solve optimal design problems on large and irregular discrete spaces for multiple regression models. The algorithm uses innovative rules to add several design points at each iteration, and clusters nearby points together at the end of iteration. Furthermore, we provide useful guidelines for discretizing irregular regions. These are based on derived theoretical properties which relate optimal designs on continuous and discrete design spaces. Several numerical examples and their MATLAB codes are presented for A-, D-, and EI-optimal designs for both linear and generalized linear models. The optimal designs found via the CVX solver are better than those presented in the literature. In addition, our guidelines to discretizing design spaces improve the efficiency of optimal designs, especially over irregular regions. We find that our iterative procedure overcomes the bottlenecks of typical sequential and multiplicative algorithms.

Abstract: Eternal domination is a graph game where mobile guards are placed on vertices and then moved in order to defend against sequences of attacks. If only one guard is allowed to move in response to an attack, then it has been shown that there is no benefit to allowing multiple guards to occupy a vertex. It was conjectured that if all guards can move there would similarly be no benefit from multiple guards occupying a vertex. Finbow et al. showed this was untrue, and provided a construction demonstrating that there are graphs which need fewer guards when 2 guards are allowed on the same vertex. We extend these results to show that for any $k$ and $s$ there are $k$-connected graphs which need fewer guards when up to $s$ guards are allowed on any vertex.

This is joint work with Georgia Penner.

PCIC is pleased to announce an upcoming talk on Wednesday, November 22nd, titled, Clustering for Climate Science Insights, as part of our Pacific Climate Seminar Series.

This talk will be delivered by Dr. John R.J. Thompson, an Assistant Professor at the University of British Columbia (Okanagan campus) whose areas of expertise are nonparametric and applied statistics and machine learning. This talk will be held between 3 p.m. and 4 p.m. Pacific Time, via Zoom meetings. For more on this talk, including registration information and an abstract, see the talk’s page on our site.

Register via Zoom

Abstract: In this talk we will first discuss this soon to be 100 years old conjecture, which states that the set of primes for which an integer $a$ different from $-1$ or a perfect square is a primitive root admits an asymptotic density among all primes. In 1967 Hooley proved this conjecture under the Generalized Riemann Hypothesis.

After that, we will look into a generalization of this conjecture, where we don't restrain ourselves to look for primes for which $a$ is a primitive root but instead elements of an infinite subset of $\N$ for which $a$ is a generalized primitive root. In particular, we will take this infinite subset to be either $\N$ itself or integers with few prime factors.

Speaker Biography: Paul Péringuey is a PIMS Postdoctoral fellow at the University of British Columbia where he is working with Prof. Greg Martin, under the sponsorship of the PIMS Collaborative Research Group "L-Functions in Analytic Number Theory". He obtained his PhD in 2022 at Université de Lorraine under the supervision of Cécile Dartyge. His area of research is in Analytic Number Theory, Comparative Prime Number Theory, as well as Additive Combinatorics.

Poster: The poster for the talk is available.

Zoom link

Abstract. We study a mean-field spin glass model whose coupling distribution has a power-law tail with exponent \alpha \in (0, 2). This is known as Levy spin glasses in literature. Though it is a fully connected model, many of its important characteristics are driven by the presence of strong bonds that have a sparse structure. In this sense, the Levy model sits between the widely studied Sherrington-Kirkpatrick model (with Gaussian couplings) and diluted spin glass models, which are more realistic but harder to understand. In this talk, I will report a number of rigorous results on the Levy model. For example, when 1< \alpha < 2, in the high-temperature regime, we obtain the limit and the fluctuation of the free energy. Also, we can determine the behaviors of the site and bond overlaps at the high temperature. Furthermore, we establish a variational formula of the limiting free energy that holds at any temperature. Interestingly, when 0< \alpha < 1, the effect of the strong bonds becomes more pronounced, which significantly changes the behavior of the model. For example, the free energy requires a different normalization (N^{1/\alpha} vs N), and its limit has a simple description via a Poisson point process at any temperature.

This is a joint work with Wei-Kuo Chen and Heejune Kim.

Speaker Bio:
Kristin Lauter is an American mathematician and cryptographer whose research interest is broadly in application of number theory and algebraic geometry in cryptography. She is particularly known for her work in the area of elliptic curve cryptography. She was a researcher at Microsoft Research in Redmond, Washington, from 1999 - 2021, and the head of the Cryptography Group from 2008 - 2021; her group developed Microsoft SEAL. In April 2021, Lauter joined Facebook AI Research (FAIR) as the West Coast Head of Research Science. She became the President-Elect of the Association for Women in Mathematics in February 2014 and served as President from 2015 - 2017.

Registration:
Participants register once on Zoom and can attend any of the Colloquium talks. Please remember to download the calendar information to save the dates on your calendar. PIMS will resend the confirmation from Zoom prior to the event date.

Zoom link.

Abstract: For every $k\ge 2$ and $\Delta$, we prove that there exists a constant $C_{\Delta,k}$ such that the following holds. For every graph $H$ with $\chi(H)=k$ and every tree with at least $C_{\Delta,k}|H|$ vertices and maximum degree at most $\Delta$, the Ramsey number $R(T,H)$ is $(k-1)(|T|-1)+\sigma(H)$, where $\sigma(H)$ is the size of a smallest colour class across all proper $k$-colourings of $H$. This is tight up to the value of $C_{\Delta,k}$, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.

Abstract: In this talk I will be speaking about the dimer model sampled on a general Riemann surface. Dimer model on a graph consists of sampling a random perfect matching with the probability proportional to the product of edge weights. In the case when the graph is planar, perfect matchings are in correspondence with their height functions defined on faces of the graph. Given a sequence of graphs approximating a given planar domain in a small mesh size, one can ask whether the underlying sequence of dimer height functions has a scaling limit and how to describe it. The landmark result of Kenyon asserts that in the case of a simply-connected domain approximated by Temperley polygons on the square grid the fluctuations of the height functions around their means converge to the (properly normalized) Gaussian free field in the target domain. The same result in the case of general Temperley graphs was established by Berestycki, Laslier and Ray 15 years later.

The dimer height function can still be defined when the graph is not planar, but is embedded into a general Riemann surface. In this case the height function becomes additively multivalued and is expected to converge to the compactified free field on the surface in the scaling limit. Recently, this problem was studied by Berestycki, Laslier and Ray in the case of general Temperley graphs embedded into the Riemann surface. Using soft probabilistic methods they proved the scaling limit exists, is conformally invariant and does not depend on a particular sequence of graphs. However, its identification with the compactified free field was missing. My goal is to fill this gap by studying the same problem from the perspective of discrete complex analysis. For this I consider graphs embedded into locally flat Riemann surfaces with conical singularities and satisfying certain geometric assumptions with respect to the local Euclidean structure. Using various analytic methods (both discrete and continuous counterparts are non-trivial) I obtain convergence to the compactified free field when the Riemann surface is chosen generically. Moreover, I am able to prove that the tightness of the underlying height fluctuations always implies their convergence to the compactified free field.

The Coast Combinatorics Conferences are informal, low-key meetings that welcome 30-minute (or so) talks on any topic in discrete mathematics and/or theoretical computer science. There is no registration fee, no main speaker, and you may on your own for coffee etc. at a nearby cafe. There is often an informal social event at the end of the day.

Everyone is welcome to attend the CCC. Please let us know if you are coming by sending email to Gary MacGillivray: gmacgill at UVic dot ca. It makes planning easier.

For more information and a schedule see the conference website.

This talk is co-sponsored by Pacific Institute for the Mathematical Sciences (PIMS).

This seminar will be in person and available through Zoom.

PIMS requests all seminar participants to complete the demographics form online at https://ubc.ca1.qualtrics.com/jfe/form/SV_6QcNr2rQcIlQGyy

Abstract: In life history studies one often encounters situations where individuals in a population are eligible to enroll only if the response time does not exceed an associated censoring time, which leads to the so called left truncated lifetime data. While auxiliary information for the truncated individuals from the same or similar cohorts may be available, challenges arise due to the practical issue of accessibility of individual-level data and taking account of various sampling conditions for different cohorts. We propose a likelihood-based method for incorporating auxiliary data to eliminate the bias due to left-truncation and improve efficiency. Simulation results and an application to data from a longitudinal study of aging are given.

Abstract: A broad class of problems in extremal geometry can be characterized as follows. Fix a positive integer k and a property of k-tuples of points in R^2. The problem is then to determine how many k-tuples in an n-point set in R^2 can have the chosen property. One of the most famous instances of this form of problem is the unit distance problem, asked by Erdős in 1946. Here, the special property is pairs of points being precisely distance one apart. Another pair of well-studied open problems is to determine the maximum number of triples that determine a triangle of unit area or unit perimeter. In this presentation we will discuss recent progress on these triangle problems, particularly our new bounds on the number of unit perimeter triangles from earlier this year.

This talk is based on a joint work with Ritesh Goenka and Ethan White.

Zoom link: https://uvic.zoom.us/j/5826187847?pwd=RlVrb0RoU0xDWTlLUDVkZW54ZThyQT09

Abstract: Tulips have captivated human interest for centuries, with their vibrant colors and unique shapes. Particularly striped tulips have been highly popular, leading to the “tulipomania” in the Dutch Golden Age. But how do the tulips get their stripes? Maybe Turing can help?

Abstract: Physics-Informed Neural Networks (PINNs) have received much attention recently due to their potential for high-performance computations for complex physical systems, including data-based computing, systems with unknown parameters, and others. The idea of PINNs is to approximate the equations and boundary and initial conditions through a loss function for a neural network. PINNs combine the efficiency of data-based prediction with the accuracy and insights provided by the physical models. However, applications of these methods to predict the long-term evolution of systems with little friction, such as many systems encountered in space exploration, oceanography/climate, and many other fields, need extra care as the errors tend to accumulate, and the results may quickly become unreliable.

We provide a solution to the problem of data-based computation of Hamiltonian systems utilizing symmetry methods. Many Hamiltonian systems with symmetry can be written as a Lie-Poisson system, where the underlying symmetry defines the Poisson bracket. For data-based computing of such systems, we design the Lie-Poisson neural networks (LPNets). We consider the Poisson bracket structure primary and require it to be satisfied exactly, whereas the Hamiltonian, only known from physics, can be satisfied approximately. By design, the method preserves all special functions of the bracket (Casimirs) to machine precision. LPNets yield an efficient and promising computational method for many particular cases, such as rigid body or satellite motion (the case of SO(3) group), Kirchhoff's equations for an underwater vehicle (SE(3) group), and others.

Joint work with Chris Eldred (Sandia National Lab), Francois Gay-Balmaz (CNRS and ENS, France), and Sophia Huraka (U Alberta). The work was partially supported by an NSERC Discovery grant.

Abstract: Plant virus infection causes enormous economic loss, $350 billion worldwide in 2021. We focus on grapevine viruses, which caused Canadian grape growers’ annual loss of over $23 million. Once infected by a virus, the grapevine remains infected for life and must be replaced, since no therapeutic treatments are available. So, diagnosis of virus infection is critical for disease control. Collaborating with researchers from CFIA and other institutes, we are developing a genomic-sequencing-based diagnostic test to detect virus infections of grapevines. More information about this project can be found at https://www.bloomberg.com/press-releases/2020-10-27/-10m-in-funding-coming-to-bc-researchers-for-improved-grapevine-and-cannabis-management.

My role is to support the method/software development to decide if a sample is infected by any virus (from a list of ~1000 viruses) based on its genomic sequencing data. In this talk, I will briefly discuss our recent research results from this project.

A Latin square is an n by n grid filled with n symbols so that each symbol appears exactly once in every row and column. A transversal in a Latin square is a selection of entries with no row, column, or symbol repetition. The study of transversals in Latin squares is as old as combinatorics, going back to Euler's work in the 18th century. A lot of exciting progress has been made in this area in the past few years. In 2020, Kwan proved that, with high probability, a random Latin square contains a full transversal (hitting each row and column). In the other extreme, in 2022, Müyesser and Pokrovskiy characterised the "algebraic" Latin squares which contain full transversals. Very recently, Montgomery proved that all Latin squares contain a transversal hitting all but one row. A key technique that is used in each of these results is the "distributive absorption method", introduced in 2014 by Montgomery to find spanning trees in random graphs. In this talk, we will give a gentle introduction to this technique.

Abstract: We consider monoids M_k generated by the first k+1 generators of the infinite presentation of the Thompson group F

F=⟨x0,x1,x2,…∣ xjxi=xixj+1 for j>i⟩.

The standard normal form for F breaks down for these monoids and we develop a new reduced form that works in F and every Mk and allows us to analyze their constructible right ideal structure. We show that there exist embeddings Mk↪Mk+1 for which the associated Toeplitz algebras are functorial, and then we study the directed system of Toeplitz algebras λ(Mk)↪λ(Mk+1). We characterize faithful representations and show that the boundary quotients are purely infinite simple. This is joint work in progress with A. an Huef, B. Nucinkis, I. Raeburn and C. Sehnem.

Abstract: The Liouville quantum gravity (LQG) surface, formally defined as a 2-dimensional Riemannian manifold with conformal factor being the exponentiation of a Gaussian free field, is closely related to random planar geometry as well as scaling limits of models from statistical mechanics. In this talk we consider the bounded, simply connected setting, and explain the Weyl's law for the asymptotics of the eigenvalues of the (random) Laplace-Beltrami operator. We also discuss a few open problems about the spectral geometry of LQG. This is a joint work with Nathanael Berestycki.

Register for talk series.

Constructing and understanding the basic properties of Euclidean Yang-Mills theory is a fundamental problem in physics. It is also one of the Clay Institute's famous Millennium Prize problems in mathematics. The basic problem is not hard to understand. You can begin by describing a simple random function from a set of lattice edges to a group of matrices. Then you ask whether you can construct/understand a continuum analog of this object in one way or another. In addition to a truly enormous physics literature, this topic has inspired research within many major areas of mathematics: representation theory, random matrix theory, probability theory, differential geometry, stochastic partial differential equations, low-dimensional topology, graph theory and planar-map combinatorics.

Attempts to understand this problem in the 1970's and 1980's helped inspire the study of "random surfaces" including Liouville quantum gravity surfaces. Various relationships between Yang-Mills theory and random surface theory have been obtained over the years, but many of the most basic questions have remained out of reach. I will discuss our own recent work in this direction, as contained in two long recent papers relating "Wilson loop expectations" (the fundamental objects in Yang-Mills gauge theory) to "sums over spanning surfaces."

1. Wilson loop expectations as sums over surfaces on the plane (joint with Minjae Park, Joshua Pfeffer, Pu Yu)

2. Random surfaces and lattice Yang-Mills (joint with Sky Cao, Minjae Park)

The first paper explains how in 2D (where Yang-Mills theory is more tractable) one can interpret continuum Wilson loop expectations purely in terms of flat surfaces. The second explains a general-dimensional interpretation of the Wilson loop expectations in lattice Yang-Mills theory in terms of discrete-and not-necessarily-flat surfaces, a.k.a. embedded planar maps.

Speaker Biography: Scott Sheffield is the Leighton Family Professor of Mathematics at MIT. He is a leading figure at the interface of mathematical physics and probability. He has held positions at Microsoft Research, Berkeley, the Institute for Advanced Study and New York University. He received the Rollo Davidson and Loève prizes in probability and has twice spoken at the International Congress of Mathematicians

We consider a single-species system with distributed delay and constant inflow. First, we consider the case with time delay in which the constant inflow does not exist. The condition of Hopf bifurcation for the delay order of k = 2 is known, but is not given clearly for the delay order of k ≥ 3.　We have obtained the relationship among systems parameters for Hopf bifurcation. Next, we consider the case in which the constant inflow exists. In this case, the positive equilibrium changes to be unstable from being stable first, and return to be stable again by increasing with average time delay T for small intrinsic growth rate r. It is found that there exists important difference between the delay orders k = 2 and k ≥ 3. For k = 2, the equilibrium can be stable for large T and any r(> 0), but for k ≥ 3, the equilibrium is unstable for large T and r .

Abstract: Structural Ramsey theory generalizes classical Ramsey theory by considering classes of finite sets with additional structure (such as finite groups or finite partial orders) and colourings of their substructures. When the class satisfies an appropriate amalgamation condition it is possible to assemble all of the structures in the class into a countably infinite structure, which is known as the Fraisse limit of the class. The goal of this talk is to describe a deep connection (due to Kechris, Pestov, and Todorcevic) between the structural Ramsey theory of a class of finite structures and the actions of the automorphism group of the class' Fraisse limit. This connection will be illustrated with a Ramsey theorem for finite-dimensional algebras of matrices.

Abstract: We consider monoids M_k generated by the first k+1 generators of the infinite presentation of the Thompson group F F=⟨x0,x1,x2,…∣ xjxi=xixj+1 for j>i⟩. The standard normal form for F breaks down for these monoids and we develop a new reduced form that works in F and every Mk and allows us to analyze their constructible right ideal structure. We show that there exist embeddings Mk↪Mk+1 for which the associated Toeplitz algebras are functorial, and then we study the directed system of Toeplitz algebras λ(Mk)↪λ(Mk+1). We characterize faithful representations and show that the boundary quotients are purely infinite simple. This is joint work in progress with A. an Huef, B. Nucinkis, I. Raeburn and C. Sehnem.

Abstract: The homomorphism density of a graph H in a graph G is the probability that a random function from the vertex set of H to the vertex set of G is a graph homomorphism. A natural question studied by Leontovich and Sidorenko back in the 80s and 90s is: given two trees, say H and T, under what conditions does the homomorphism density function of H dominate the homomorphism density function of T? We apply a beautiful information-theoretic approach of Kopparty and Rossman to reduce this problem to solving a particular linear program. We then use this perspective to answer the question for various pairs H and T of trees. Roughly speaking, short bushy trees tend to be more numerous than tall skinny ones, but there are exceptions. Joint work with Natalie Behague, Gabriel Crudele and Lina M. Simbaqueba.

Abstract: Neural spike trains are sequences of consecutive electrochemical waves generated by the nerve cells through which they communicate. These waves are localized in time hence called spikes. Statistical analysis of simultaneously recorded neural spike trains from an ensemble of neurons is a challenging problem from both statistical and computational points of view. I will discuss some biologically inspired multivariate point process models for such data, namely- Skellam Process with Resetting (SPR) and a generalization of it under a continuous-time latent factor model (LFM). To the best of our knowledge, this generalization of the SPR is the first continuous-time multivariate LFM for studying neuronal interactions and functional connectivity. Leveraging the computational efficacy of approximate Bayesian inference, we show that our model can handle larger neuronal ensembles compared to alternative approaches. Using experimental data from a classical conditioning study on the prefrontal cortex in rats, we shed light on our understanding of cue and outcome value encoding.

Abstract: The line graph of a graph $G$ is the graph $L$ with vertex set $E(G)$ in which two vertices are adjacent if and only if the corresponding edges of $G$ share an endpoint. A famous theorem of Beineke characterizes the class of line graphs by a list of 9 forbidden induced subgraphs. In this talk we will be interested in the following question: Given an "explicit description" of a line graph $L$, when does the underlying graph $G$ also have an "explicit description"? To use the precise terminology (which I will define in the talk), when is a Borel graph $L$ a line graph of a Borel graph $G$? It turns out that to answer this question, we need to expand Beineke's list to include a tenth forbidden subgraph. This is joint work with James Anderson.

Abstract: I discuss a modest framework for doing Noncommutative Geometry for certain examples of `noncommutative manifolds,' namely crossed products of discrete groups acting smoothly on manifolds. The analogue of the `fundamental class' in topology for these C*-algebras is something I call a `fibre class.' I will discuss the numerous examples of these fibre classes, and explain why the definition is so great.

Abstract: In this talk, we will discuss how convex optimization can be used to study statistical mechanics, with a focus on the example of the lattice Ising model. After introducing the Gibbs measure that describes the system, we will explain how convex optimization produces results that asymptotically converge to the Gibbs measure. If time permits, we will also explore the relationship between Markov chain Monte-Carlo and convex optimization for the Ising model.

Abstract: Recent advances in single cell sequencing has led to the availability of gene expression data. We model the trajectories of cells in gene expression space using a diffusion and gradient drift process that also undergoes branching. In this talk, we show that one can recover the law of this process given time marginals using an entropy minimization problem. The optimization is over the set of all processes with the same branching mechanism and given marginals. We prove that among these the one which minimizes the relative entropy with respect to a reference branching Brownian motion is the ground truth. We conclude this talk with a general result showing that the solution to a entropy minimization problem with respect to a Markov reference measure with given marginals is also Markov.

Abstract (plain text):
Given a graph H, say that a graph G is properly rainbow H-saturated if there exists a rainbow H-free proper edge-colouring of G and, for any non-edge e of G, every proper edge-colouring of G+e contains a rainbow copy of H. The proper rainbow saturation number is the minimum number of edges in a properly rainbow H-saturated graph on n vertices. This is a natural variant of the graph saturation problem based on the rainbow extremal number. In this talk, we will prove bounds on the proper rainbow saturation number for specific classes of graphs, and we will prove general bounds on the proper rainbow saturation number using related saturation numbers.

Joint work with Natasha Morrison.

Abstract: The random connection model (RCM) is a random graph model where the vertices are given by a Poisson point process with a given intensity, $\lambda>0$, and the edges exist independently with a probability that depends upon the relative positions of the two vertices in question. A standard example would be the Gilbert disc model. As we vary $\lambda$, we observe a percolation phase transition at a critical intensity $\lambda_c>0$. Finding $\lambda_c$ is only possible in very exceptional cases, so here we investigate a high-dimension asymptotic expansion for the critical intensity that applies for a great variety of RCMs. This is based on arXiv:2309.08830 with Markus Heydenreich (Universität Augsburg).

As a statistical scientist working in epidemiology and biostatistics, Dr. Shelley Bull has been recognized for her research in medicine, public health, genetics, epidemiology and statistical methodology as well as her leadership, supervision and mentorship, by awards from the Statistical Society of Canada and the International Genetic Epidemiology Society.

Using a research project in statistical methods for region-level analysis of genome-wide association studies as an illustration, this statistically oriented talk will explain the motivation and development of an alternative approach to GWAS data analysis, and consider criteria and techniques for method evaluation.

See poster (PDF file).

For a simple graph $G$ on $n$ vertices, let $\mathcal{S}(G)$ be the set of all $n\times n$ real symmetric matrices $A=[a_{i,j}]$ with $a_{i,j}\neq 0$ if and only if $\{i,j\}$ is an edge of $G$. There are no restrictions on the diagonal entries of $A$. The inverse eigenvalue problem for a graph $G$ (IEP-G) seeks to determine all possible spectra of matrices in $\mathcal{S}(G)$.

One of the relaxations of the IEP-G is to determine the minimum number of distinct eigenvalues of a matrix in $\mathcal{S}(G)$ for a given graph $G$. This parameter is denoted by $q(G)$ and it is called the minimum number of distinct eigenvalues of $G$.

In this presentation, we will review some of the results and the techniques from recent developments about $q(G)$.

Abstract: We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included.

The UVic Mathematics Competition is held annually in the fall. The competition will be held on Tuesday, September 26, 2023 between  4:30-6:30 pm in DSB C130. There are monetary prizes. To participate, just show up.

Here are some recent question papers: 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006.

As a statistical scientist working in epidemiology and biostatistics, Dr. Shelley Bull has been recognized for her research in medicine, public health, genetics, epidemiology and statistical methodology as well as her leadership, supervision and mentorship, by awards from the Statistical Society of Canada and the International Genetic Epidemiology Society.

In this general audience lecture, Bull will talk about how genome-wide studies and methods have evolved, highlight landmark papers, and give examples of key contributions from statistical methodology. The talk will close with comments on recent developments in biobanks, multi-omics and global population studies.

See poster (PDF file).

Abstract: I will introduce the Monomer-Dimer model, a Gibbs probability measure on the space of (not necessarily perfect) weighted matchings on a graph. I will recap some known results of Heilmann and Lieb which establish the absence of a phase transition. I will then describe how many of these results carry over when we consider iid random weights to the edges and vertices. In certain special cases, we show how laws of large numbers and central limit theorems can be established for partition function as well as statistics such as the number of unpaired vertices.

The Final Oral Examination for the Degree of Master of Science

(Department of Mathematics and Statistics)

ZIYI LYU

University of Waterloo, 2021 (B.Math)

Investigating the Relationship between Bayes Factors and Credible Intervals

Friday, September 22nd, 2023 2:00 P.M

Online

Supervisory Committee:
Dr. Farouk Nathoo, Department of Mathematics and Statistics, UVic (Supervisor)
Dr. Min Tsao, Department of Mathematics and Statistics, UVic (Member)

Chair of Oral Examination:
Dr. Ryan Budney, Department of Mathematics and Statistics, UVic

Register for talk.

Euler’s famous formula tells us that (with appropriate caveats), a map on the sphere with f countries (faces), e borders (edges), and v border-ends (vertices) will satisfy v-e+f=2. And more generally, for a map on a surface with g holes, v-e+f=2-2g. Thus we can figure out the genus of a surface by cutting it into pieces (faces, edges, vertices), and just counting the pieces appropriately. This is an example of the topological maxim “think globally, act locally”. A starting point for modern algebraic geometry can be understood as the realization that when geometric objects are actually algebraic, then cutting and pasting tells you far more than it does in “usual” geometry. I will describe some easy-to-understand statements (with hard-to-understand proofs), as well as easy-to-understand conjectures (some with very clever counterexamples, by M. Larsen, V. Lunts, L. Borisov, and others). I may also discuss some joint work with Melanie Matchett Wood.

Speaker Biography: Ravi Vakil is a Professor of Mathematics and the Robert K. Packard University Fellow at Stanford University, and was the David Huntington Faculty Scholar. He received the Dean's Award for Distinguished Teaching, an American Mathematical Society Centennial Fellowship, a Frederick E. Terman fellowship, an Alfred P. Sloan Research Fellowship, a National Science Foundation CAREER grant, the presidential award PECASE, and the Brown Faculty Fellowship. Vakil also received the Coxeter-James Prize from the Canadian Mathematical Society, and the AndrÉ-Aisenstadt Prize from the CRM in MontrÉal. He was the 2009 Earle Raymond Hedrick Lecturer at Mathfest, and a Mathematical Association of America's Pólya Lecturer 2012-2014. The article based on this lecture has won the Lester R. Ford Award in 2012 and the Chauvenet Prize in 2014. In 2013, he was a Simons Fellow in Mathematics.

Abstract: In 1954, Tutte made a conjecture that every graph without a cut-edge has a nowhere-zero 5-flow. A parallel conjecture to this, but for signed graphs is Bouchet’s Conjecture (1983) that every signed graph without the obvious obstruction has a nowhere-zero 6-flow. We prove that Bouchet’s Conjecture holds in the special case of 3-edge-connected graphs when 6 is replaced with 8.

Abstract: In the eternal eviction game, a set of guards placed on the vertices of (a dominating set of) a graph G must move to defend the graph against attacks on those of its vertices that contain guards, while maintaining a dominating set of G. The eternal eviction number of G is the minimum number of guards required to defend G against any sequence of attacks. In this talk, we will present some recent progress on the game. In particular, we will show that for any integer $k \geq 1$, there exists $f(k)$ such that any graph with independence number at most $k$ has eviction number at most $f(k)$.

This is joint work with Gary MacGillivray and Kieka Mynhardt.

Abstract: I will begin with some basics about Approximately Finite dimensional C*-algebras or AF-algebras. These have played a fundamental role in many aspects of the theory and I will mention some special classes of interest. I will briefly describe the Oseledets or Multiplicative Ergodic Theorem (warning! I am not an expert) and how this can be used to create a broad class of AF-algebras with interesting features.

See abstract PDF file.

Abstract: I introduce a method for approximate marginal likelihood inference via adaptive Gaussian quadrature in mixed models with a single grouping factor. The core technical contributions are (a) an algorithm for computing the exact gradient of the approximate log marginal likelihood and (b) a useful parameterization of the multivariate Gaussian. The former leads to efficient quasi-Newton optimization of the marginal likelihood that is several times faster than established methods; the latter gives Wald confidence intervals for random effects variances that attain nominal coverage and low bias if enough quadrature points are used. The Laplace approximation is a special case of the method and is shown in simulations to perform exceptionally poorly for binary random slopes models, but this is mitigated by just adding more quadrature points.

Reviewers
Supervisory Committee
Dr. Anthony Quas, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Christopher Bose, Department of Mathematics and Statistics, UVic (Member)
Dr. Sean Chester, Department of Computer Science, UVic (Outside Member)

External Examiner
Dr. Ian Morris, School of Mathematical Sciences, Queen Mary University of London

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

Abstract
A linear cocycle is an object that arises naturally in the study of dynamical systems and statistics. Oseledets’ theorem [21] guarantees a decomposition of X into fast and slow subspaces. This dissertaton is a study of strongly measurable cocycles over an invertible ergodic system acting on a separable Banach space, including a proof of this theorem in this infinite dimensional setting.

The Final Oral Examination
for the Degree of
Master of Science
(Department of Mathematics and Statistics)

Guojun Ma
2020 University of Alberta B.Sc.
A meta-analysis of genome-wide associations with body mass index
August 21, 2023
2:00pm
Online

Supervisory Committee:
Dr. Yu-Ting Chen, Department of Mathematics and Statistics, UVic (Supervisor)
Dr. Xuekui Zhang, Department of Mathematics and Statistics, UVic (Co-Supervisor)

Chair of Oral Examination:
Dr. Trefor Bazett, Department of Mathematics and Statistics, UVic

Reviewers
Supervisory Committee
Dr. Xuekui Zhang, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Li Xing, Department of Mathematics and Statistics, UVic (Member)

External Examiner
Dr. Ibrahim Numanagic, Department of Computer Science, UVic

Chair of Oral Examination
Dr. Catherine Leéger, Department of French, UVic

Abstract
Several workflows have been developed for the diagnostic testing of plant viruses using high-throughput sequencing methods. Most of these workflows require considerable expertise and input from the analyst to perform and interpret the data when deciding on a plant’s disease status. The most common detection methods use workflows based on de novo assembly and/or read mapping. Existing virus detection software mainly uses simple deterministic rules for decision-making, requiring a certain level of understanding of virology when interpreting the results. This can result in inconsistencies in data interpretation between analysts which can have serious ramifications.

To combat these challenges, we developed an automated workflow using machine-learning methods, decreasing human interaction while increasing recall, precision, and consistency. Our workflow involves sequence data mapping, feature extraction, and machine learning model training. Using real data, we compared the performance of our method with other popular approaches and showed that our approach increases recall and precision while decreasing the detection time for most types of sequencing data.

Flat Wieler solenoids are inverse limits of locally expanding affine maps. They appear in several contexts in dynamical systems, for example as tiling spaces of self-similar tilings, and they are very interesting because they are not manifolds but have an interesting local product structure. I will talk about several geometric and dynamical properties of these spaces, emphasizing the interplay between intrinsic topological properties and renormalization. This talk is for everyone as no prior knowledge of inverse limits will be assumed.

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

TUAN VIET DAO
BA (Gustavus Adolphus College, 2017)

“Estimating the Size of the COVID-19 Population in British Columbia Using the Stratified Petersen Estimator”

Department of Mathematics and Statistics

Monday, August 14, 2023
10:00 A.M.
Virtual Defence

Supervisory Committee:
Dr. Laura Cowen, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Junling Ma, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:
Dr. Lloyd Elliott, Department of Statistics and Actuarial Science, Simon Fraser University

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

Abstract
The presence of undetected COVID-19 cases is a known phenomenon. Mathematical modelling techniques, such as capture-recapture, provide a reliable method for estimating the true size of the infected population. Treating a positive SARS-CoV-2 diagnostic test result as the initial capture and a hospital admission with a COVID-19-related diagnosis code as the recapture, we developed a Lincoln-Petersen model with temporal stratification, taking into account factors that influence the occurrence of captures. Applying this model to repeated patient encounter data collected at the provincial level in British Columbia, we estimated the number of COVID-19 cases among males aged 35 or older during the first week of March 2021. Our analysis revealed that the true number of cases ranged from 4.94 to 9.18 times greater than the number of detected cases.

See announcement and abstract (PDF).

BSc (University of Victoria, 2021)

Notice of the Final Oral Examination for the Degree of Master of Science
Topic: Saturation Problems on Graphs

Mathematics and Statistics

Date and location:
Wednesday, July 19, 2023 10:00 A.M.
David Strong Building Room C108

Reviewers

Supervisory Committee
Dr. Natasha Morrison, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Kieka Mynhardt, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner
Dr. Antonio Girão, Mathematical Institute, University of Oxford

Chair of Oral Examination
Dr. Tao Lu, Department of Electrical and Computer Engineering, UVic

Abstract In this thesis, we consider two variations on classical saturation problems in extremal graph theory: rainbow saturation and weak saturation.

An edge-coloured graph G is rainbow if every edge in G receives a distinct colour. Given a graph H, an edge-coloured graph G is H-rainbow-saturated if G does not contain a rainbow copy of H, but the addition of any non-edge to G, in any colour from , creates a rainbow copy of H. The rainbow saturation number of H, denoted by rsat(n,H), is the minimum number of edges in an H-rainbow saturated graph on n vertices. In Chapter 2, we prove that, like ordinary saturation numbers, the rainbow saturation number of every graph H is linear in n. This result confirms a conjecture of Girão, Lewis, and Popielarz.

In Chapter 3, we consider a specific type of weak saturation known as r-bond bootstrap percolation. In the r-bond bootstrap percolation process on a graph G, we start with a set of initially infected edges of G, and consider all other edges in G to be healthy. At each subsequent step in the process, the infection spreads to a healthy edge if at least one of its endpoints is incident with at least r infected edges. Once an edge is infected, it remains infected indefinitely. If a set of initially infected edges will eventually infect all of E(G), we refer to it as an r-percolating set of G. Define m_e(G, r) to be the minimum number of edges in an r-percolating set of G.

Recently, Hambardzumyan, Hatami, and Qian introduced a clever new polynomial method, which they used to provide recursive formulas for m_e(G, r) when G is either a d-dimensional torus or a d-dimensional grid. We push this polynomial method further, in order to determine m_e(G, r) for certain other graphs G. In particular, we provide recursive formulas for m_e(G, r) when G is a Cartesian product of stars or a Cartesian product of joined cycles (cycles with a single chord). We also give upper and lower bounds on m_e(G, r) when G is a Cartesian product of a tree with any graph H, and examine the conditions under which these bounds match.

BSc (University of Regina, 2021)
Notice of the Final Oral Examination for the Degree of Master of Science
Topic: Cross-Sperner Systems

Department of Mathematics and Statistics

Date and location
Tuesday, July 18, 2023
10:00 A.M.
David Strong Building, Room C108

Reviewers
Supervisory Committee
Dr. Natasha Morrison, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Kieka Mynhardt, Department of Mathematics and Statistics, UVic (Member)
External Examiner
Dr. Karen Meagher, Department of Mathematics and Statistics, University of Regina
Chair of Oral Examination
Dr. Shemine Gulamhusein, School of Child and Youth Care, UVic

See announcement (PDF)

Fahim Alam
BSc (Shahjalal University of Science and Technology, Bangladesh, 2017)

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

Topic
A New Numerical Approach to Solve 1D Viscous Plastic Sea Ice Momentum Equation

Department of Mathematics and Statistics
Date and location
Monday, June 26, 2023
9:30 A.M.
David Strong Building, Room C128

Supervisory Committee
Dr. Boualem Khouider, Department of Mathematics and Statistics, University of Victoria(Supervisor)
Dr. David Goluskin, Department of Mathematics and Statistics, UVic (Member)

External Examiner
Dr. Alex Bihlo, Department of Mathematics and Statistics, Memorial University ofNewfoundland

Chair of Oral Examination
Dr. Randy Scharien, Department of Geography, UVic

Abstract
While there has been a colossal effort in the ongoing decades, the ability to simulate ocean icehas fallen behind various parts of the climate system and most Earth System Models areunable to capture the observed adversities of Arctic sea ice, which is, as it were, attributed toour frailty to determine sea ice dynamics. Viscous Plastic rheology is the most by and largerecognized model for sea ice dynamics and it is expressed as a set of partial differentialequations that are hard to tackle numerically. Using the 1D sea ice momentum equation as aprototype, we use the method of lines based on Euler’s backward method. This results in anonlinear PDE in space only. At that point, we apply the Damped Newton’s method which hasbeen introduced in Looper and Rapetti et al. and used and generalized to 2D in Saumier et al.to solve the Monge-Ampere equation. However, in our case, we need to solve 2nd order linearequation with discontinuous coeffi cients during Newton iteration. To overcome this diffi culty,we use the Finite element method to solve the linear PDE at each Newton iteration. In thispaper, we show that with the adequate smoothing and re-scaling of the linear equation,convergence can be guaranteed and the numerical solution indeed converges effi ciently to thecontinuum solution unlike other numerical approaches that typically solve an alternate set ofequations and avoid the diffi culty of the Newton method for a large nonlinear algebraicsystem. The fi nite element solver failed to converge when the original setting of the smoothedSIME with a smoothing constant K = 2.8 x 10⁸ was used. A much smaller constant of K=100 wasnecessary. The large smoothing constant K leads to an ill conditioned mass matrix.