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Abstract: Given a finite simple graph $G$, an {\em odd cover of $G$} is a collection of complete bipartite graphs, or bicliques, in which each edge of $G$ appears in an odd number of bicliques and each non-edge of $G$ appears in an even number of bicliques. We denote the minimum cardinality of an odd cover of $G$ by $b_2(G)$ and prove that $b_2(G)$ is bounded below by half of the rank over $\mathbb{F}_2$ of the adjacency matrix of $G$. We show that this lower bound is tight in the case when $G$ is a bipartite graph and almost tight when $G$ is an odd cycle. However, we also present an infinite family of graphs which shows that this lower bound can be arbitrarily far away from $b_2(G)$. Babai and Frankl (1992) proposed the ``odd cover problem," which in our language is equivalent to determining $b_2(K_n)$. Radhakrishnan, Sen, and Vishwanathan (2000) determined $b_2(K_n)$ for an infinite but density zero subset of positive integers $n$. In this paper, we determine $b_2(K_n)$ for a density $3/8$ subset of the positive integers. This is joint work with Calum Buchanan, Alexander Clifton, Jason O'Neil, Puck Rombach, and Mei Yin.

Zoom link: https://uvic.zoom.us/j/83114822200?pwd=bDY1RnFmb05wZXJRZk52THBGbDFYZz09

High-dimensional data pose unique challenges for data processing in an era of ever-increasing amounts of data availability. Graph theory can provide a structure of high-dimensional data. We introduce two key properties desirable for graphs in testing homogeneity. Roughly speaking, these properties may be described as: unboundedness of edge counts under the same distribution and boundedness of edge counts under different distributions. It turns out that the minimum spanning tree violates these properties but the shortest Hamiltonian path posses them. Based on the shortest Hamiltonian path, we propose two combinations of edge counts in multiple samples to test the homogeneity. We give the permutation null distributions of proposed statistics when sample sizes go to infinity. The power is analyzed by assuming both sample sizes and dimensionality tend to infinity. Simulations show that our new tests behave very well overall in comparison with various competitors. Real data analysis of tumors and images further convince the value of our proposed tests. Software implementing the test is available in the R package Relevance.

Zoom link: https://uvic.zoom.us/j/83114822200?pwd=bDY1RnFmb05wZXJRZk52THBGbDFYZz09

ABSTRACT: Although log-likelihood is widely used in model selection, the log-likelihood ratio has had few applications in this area. In this talk, I present a log-likelihood ratio based method for selecting regression models which focuses on the set of models deemed plausible by the likelihood ratio test. I show that when the sample size is large and the significance level of the test is small, there is a high probability that the smallest model in the set is the true model; thus, the method selects this smallest model. The significance level of the test serves as a tuning parameter that controls the balance between the false active rate and false inactive rate of the selected model. I consider three levels of this parameter in a simulation study and compare this method with the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) to demonstrate its excellent accuracy and adaptability to different sample sizes.

Model selection is an active area of research with a long history, a wide range of perspectives, and a rich collection of methods. For students unfamiliar with this area, this talk includes a review of key methods including the AIC, BIC and modern Lp penalty methods. The new method presented in this talk offers a frequentist perspective on the model selection problem. It is an alternative and a strong competitor to the AIC and BIC for selecting regression models.

Abstract: Content Moderation is the key and fundamental component for any content services and platforms to operate and to offer friendly, meaningful and non-toxic content for consumers to enjoy and engage with, and for users to build an online community to interact and communicate as well. Moderation service is the safety gatekeeper for other features to build on top of, such as content recommendations and personalization, targeted advertisement and so on. However, there are many big practical challenges we are facing on a daily basis. In this talk, we will present the trending solutions for content moderation in the Tech Industry, by leveraging both Artificial Intelligence (AI) and Human Intelligence (HI) to overcome multi-dimensional obstacles and to achieve the goals from multiple perspectives in real practices.

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This talk is aimed at a general audience and does not require a mathematics background. The talk will start by discussing the popular card game SET and in particular the question of how many cards one can have in this game without creating a so-called “SET”. Considering this question for extended versions of the game, we will find a connection to a recent breakthrough result of Ellenberg and Gijswijt on a famous problem in the area of extremal combinatorics. The talk will also discuss related results due to the speaker, some of which are of a geometric flavor.

Our Distinguished Women Scholars Lecture series was established by the Vice-President Academic and Provost to bring distinguished women scholars to the University of Victoria.

Download the poster (PDF file)

Abstract: Single-cell RNA sequencing (scRNA-seq) has transformed our understanding of cellular heterogeneity in health and disease, but the lack of physical relationships among dissociated cells has limited its applications. In this talk, we present CeLEry, a supervised deep learning algorithm to recover the spatial origins of cells in scRNA-seq by leveraging gene expression and spatial location relationships learned from spatial transcriptomics. CeLEry has a data augmentation procedure via a variational autoencoder to enlarge the training sample size, which improves the robustness of the method and overcomes noise in scRNA-seq. CeLEry can infer the spatial origins of cells in scRNA-seq at multiple levels, including 2D location as well as the spatial domain or tissue layer of a cell. CeLEry also provides uncertainty estimates for the recovered locations. This framework can be applied to study the changing of cell distribution in cerebral cortex layers during the progression of Alzheimer's disease.

Abstract:
The set of stable matchings induces a distributive lattice. The supremum of the stable matching lattice is the boy-optimal (girl-pessimal) stable matching and the infimum is the girl-optimal (boy-pessimal) stable matching. The classical boy-proposal deferred-acceptance algorithm returns the supremum of the lattice, that is, the boy-optimal stable matching. In this paper, we study the smallest group of girls, called the minimum winning coalition of girls, that can act strategically, but independently, to force the boy-proposal deferred-acceptance algorithm to output the girl-optimal stable matching. We characterize the minimum winning coalition in terms of stable matching rotations and show that its cardinality can take on any value between $0$ and $\floor{\frac{n}{2}}$, for instances with $n$ boys and $n$ girls. Our two main results concern the random matching model. First, the expected cardinality of the minimum winning coalition is small, specifically $(\frac{1}{2}+o(1))\log{n}$. This resolves a conjecture of Kupfer. Second, in contrast, a randomly selected coalition must contain nearly every girl to ensure it is a winning coalition asymptotically almost surely. Equivalently, for any $\varepsilon>0$, the probability a random group of $(1-\varepsilon)n$ girls is not a winning coalition is at least $\delta(\varepsilon)>0$. This is joint work with Sergey Norin and Adrian Vetta.

Join us for this talk in the Seminar Series: Mathematics of Ethical Decision-making Systems

Talk at 3:30 PM
Reception at 4:30 PM

Fairness in machine learning classification has been a topic of great interest given the increasing use of such classifiers in critical settings.

There are many possible definitions of fairness and many potential sources of unfairness. Given this complex landscape, most research has focused on studying single classifiers in isolation.

In reality an individual is subjected to a network of classifiers: for example, one is classified at each stage of life (school, college, employment to name a few), and one may also be classified in parallel by many classifiers (such as when seeking college admissions). In addition, individuals may modify their behavior based on their knowledge of the classifier, leading to equilibrium phenomena. Another feedback effect is that the result of the classifier may affect the features of an individual (or of the next generation) for future classifications.

In this talk we present work that takes the first steps in exploring questions of fairness in networks of classifiers and in systems with feedback. Given the inherent complexity of the analysis, our models are very stylized, but it is our belief that some of the qualitative conclusions apply to real-world situations.

***For those unable to attend this talk in person, we have a Zoom alternative. For the Zoom meeting ID/Passcode, please send an email to pims@uvic.ca. Thank you ***

Download Poster (PDF)

Zoom link: https://uvic.zoom.us/j/83114822200?pwd=bDY1RnFmb05wZXJRZk52THBGbDFYZz09

Abstract: Like traditional meta-analysis that pools effect sizes across studies to improve statistical power, it is of increasing interest to conduct clustering jointly across datasets to identify disease subtypes for bulk genomic data and discover cell types for single-cell RNA-sequencing (scRNA-seq) data. Unfortunately, due to the prevalence of technical batch effects among high-throughput experiments, directly clustering samples from multiple datasets can lead to wrong results. The recent emerging meta-clustering approaches require all datasets to contain all subtypes, which is not feasible for many experimental designs.

In this talk, I will present our Batch-effects-correction-with-Unknown-Subtypes (BUS) framework. BUS is capable of correcting batch effects explicitly, grouping samples that share similar characteristics into subtypes, identifying features that distinguish subtypes, and enjoying a linear-order computational complexity. We prove the identifiability of BUS for not only bulk data but also scRNA-seq data whose dropout events suffer from missing not at random. We mathematically show that under two very flexible and realistic experimental designs—the “reference panel” and the “chain-type” designs—true biological variability can also be separated from batch effects. Moreover, despite the active research on analysis methods for scRNA-seq data, rigorous statistical methods to estimate treatment effects for scRNA-seq data—how an intervention or exposure alters the cellular composition and gene expression levels—are still lacking. Building upon our BUS framework, we further develop statistical methods to quantify treatment effects for scRNA-seq data.

Abstract: This is joint work with Sonae Hadama (Kyoto). We consider a system of Schrodinger equations with the Hartree interaction, which is a simplified mean-field model for fermions. The equation may be rewritten for operators, where the trace class corresponds to the case of finite total mass (L^2). Lewin and Sabin proved stability of some translation-invariant stationary solutions from physics, using the Strichartz estimate for the free equation in the Schatten class, where the mass is merely p-th power summable with respect to the number of particles for some p>1. In this case, the perturbation argument for the Duhamel integral is not so easy as in the scalar case, because the Schatten class is not simply embedded into or interpolated with the space-time Lebesgue norms for the Strichartz estimate. We propose some framework to solve the equation in the Schatten class. The main novelties are norms for propagators corresponding to the best constants of the Strichartz estimate in the Schatten class, and a Schatten version of the Christ-Kiselev lemma for the Duhamel integral on operators.

Please email the organizer for the Zoom link.

Abstract: In this talk we will give an exact characterization for the asymptotic distribution of quadratic forms in IID random variables with finite second moment, where the underlying matrix is the adjacency matrix of a graph. In particular we will show that the limit distribution of such a quadratic form can always be expressed as the sum of three independent components: a Gaussian, a (possibly) infinite sum of centered chi-squares, and a Gaussian with a random variance. As a consequence, we derive necessary and sufficient conditions for asymptotic normality, and universality of the limiting distribution.

Abstract: A Gromov hyperbolic group is a group with a certain large-scale negative curvature property. Almost all groups are hyperbolic (e.g. fundamental groups of a compact Riemann surface of genus g is hyperbolic unless g=0 or 1. The theory of hyperbolicity is important in topology in the classification of manifolds. Hyperbolic groups also are interesting from the point of view of dynamical systems. Any hyperbolic group can be compactified by adding a boundary to it. The boundary is a compact metrizable space on which the group acts by homeomorphisms. These boundary actions of hyperbolic groups code in special cases, asymptotic behaviour of geodesics on negatively curved surfaces, and determine simple purely infinite C*-algebras with Type III von Neumann closures. This means that the traditional tools of Noncommutative Geometry cannot be used to endow the corresponding `noncommutative spaces' with geometric structure. In these talks we report on progress on developing a `twisted' NCG for them, building on previous work of the author and Bogdan Nica.

Abstract: A Glass network is a system of first order ODEs with discontinuous right hand side coming from step function terms. The "ON/OFF" switching dynamics from the step functions makes Glass networks effective at modelling switching behaviour typical of gene and neural networks. They also have potential application as models of true random number generators (TRNGs) in electronic circuits. As random number generators, it is desirable for networks to behave as irregularly as possible to thwart potential hacking attempts. Thus, a measure of irregularity is necessary for analysis of proposed circuit designs. The cybersecurity industry wants bit sequences generated by the circuit to have positive entropy. The nature of the discontinuities allows for Glass networks to be transformed into discrete time dynamical systems, where discrete maps represent transitions through boxes in phase space, where all possible box transitions are represented using a directed graph called the transition Graph (TG). Dynamics on the TG naturally allows for the network dynamics to be represented by shift spaces with an alphabet of symbols representing boxes. For shift spaces, entropy is used to gauge dynamical irregularity. As a result it is a perfect measure for the application to TRNGs. Previously it was shown that the entropy of the TG acts as an upper bound for the entropy of the actual dynamics realized by the network. By considering more dynamical information from the continuous system we have shown that the TG can be reduced to achieve more accurate entropy upper bounds. We demonstrate this by considering examples and use numerical simulations to gauge the accuracy of our improved upper bounds.

Abstract: Statistical detection of a rare class of objects in a two-class classification problem can pose several challenges. Because the class of interest is rare in the training data, there is relatively little information in the known class response labels for model building. At the same time the available explanatory variables are often moderately high dimensional. In the four assays of our drug-discovery application, compounds are active or not against a specific biological target, such as lung cancer tumor cells, and active compounds are rare. Several sets of chemical descriptor variables from computational chemistry are available to classify the active versus inactive class; each can have up to thousands of variables characterizing molecular structure of the compounds. The statistical challenge is to make use of the richness of the explanatory variables in the presence of scant response information. Our algorithm divides the explanatory variables into subsets adaptively and passes each subset to a base classifier. The various base classifiers are then ensembled to produce one model to rank new objects by their estimated probabilities of belonging to the rare class of interest. The essence of the algorithm is to choose the subsets such that variables in the same group work well together; we call such groups phalanxes.

The UVic Mathematics Competition is held annually in the fall. This year it will be held on Monday, September 26, 2022 between 3:00-5:00 pm in CLE C108. There are monetary prizes. To participate, just show up. The competition is open to all undergraduate students at UVic, including first year students. In the past some prizes were won by first year students.

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

Abstract: A spanning tree is a connected subgraph of a graph with no cycles. I will explain some well-known connection between such collection of trees and some Poissonian collection of loops, and a magical algorithm (known as Wilson's alorithm) to sample such trees very fast. Then I will try to explain a recent extension to the multiply connected setting, and pose an open question in the end. Joint work with N. Berestycki and B. Laslier.

See the abstract (PDF file).

Please contact the organizer for the Zoom link.

Zoom link: https://uvic.zoom.us/j/83114822200?pwd=bDY1RnFmb05wZXJRZk52THBGbDFYZz09

ABSTRACT: The advent of high-throughput next-generation sequencing (NGS) technologies has transformed our understanding of cell biology and human disease. As NGS has been adopted earliest by the scientific community, its use has now become widespread, and the technology has improved rapidly. At present, it is now common for laboratories to assay genome-wide transcriptomes of thousands of cells in a single scRNA-seq experiment. In addition, technologies that enable the measurement of new information, for example, chromatin accessibility, protein quantification, and spatial location, have been developed. In order to take full advantage of the multi-modality information when analyzing NGS data, new methods are demanded. This seminar will introduce several machine learning algorithms for NGS data analysis with different aims, including cell type classification, spatial domain detection, and tumor microenvironment annotation.

KEYWORDS: single cell RNA sequencing (scRNA-seq), Spatial transcriptomics (ST), tumor microenvironment, machine learning

Abstract: A classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that a graph with average degree O(k^2) guarantees the existence of a K_k-subdivision. We study two directions extending this result.

Firstly, Verstraëte conjectured in 2002 that the quadratic bound O(k^2) would guarantee already two vertex-disjoint isomorphic copies of a K_k-subdivision. Secondly, Thomassen conjectured in 1984 that for each k \in \mathbb{N} there is some d = d(k) such that every graph with average degree at least d contains a balanced subdivision of K_k. Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on d(k) remains open.

In this talk, we show that the quadratic bound O(k^2) suffices to force a balanced K_k-subdivision. This gives the optimal bound on d(k) needed in Thomassen's conjecture and implies the existence of O(1) many vertex-disjoint isomorphic K_k-subdivisions, confirming Verstraëte's conjecture in a strong sense.

Abstract: The construction of C*-algebras from groupoids is a very general method for constructing C*-algebras, including many of great importance. I will give a short review the construction for etale groupoids. The Elliott classification program for C*-algebras has been a huge undertaking over the past three decades and has given many new insights which were unimagined thirty years ago. I will give a short overview of the subject (from a non-expert). The obvious question which links these topics is: Which C*-algebras which are classified by Elliott arise from groupoids? I will discuss various results to answer this. I will try to keep the two talks at a fairly elementary level, although this will involve avoiding a lot of technical issues.

See abstract (PDF)

See announcement and abstract (PDF).

Download poster PDF

Naively implemented, statistical procedures are prone to leaking information about their training data, which can be problematic if the data is sensitive. Differential privacy, a rigorous notion of data privacy, offers a principled framework to dealing with these issues. I will survey recent results in differential private statistical estimation, presenting a few vignettes which highlight novel challenges for even the most fundamental problems, and suggesting solutions to address them. Along the way, I’ll mention connections to tools and techniques in a number of fields, including information theory and robust statistics.

Abstract:
In the biased Hamilton Cycle Maker-Breaker game, two players alternate choosing edges from a complete graph. In the game with bias b, Maker chooses one previously unchosen edge in each turn and Breaker chooses b. The game is Maker-win for the given bias if Maker can ensure she chooses the edges of a Hamilton Cycle and Breaker-win otherwise.

Letting n be the number of vertices, if the bias is 0 then Maker wins the game in n moves. On the other hand, if the bias b=b(n) is {n choose 2}-1 then Breaker wins. Furthermore, if Breaker wins for some bias b, then she also wins for bias b+1. We discuss the threshold at which the game becomes Breaker win, and the number of moves maker needs to ensure she wins for b below this threshold, which is called the speed of the game.

Warmup: What bounds can you obtain on the threshold and speed without a literature search? Any ideas how to proceed?

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

SHAOZE ZHOU
BSc (University of Victoria, 2015)

“Deep Learning Methods May Not Outperform Other Machine Learning Methods on Analyzing Genomic Studies”

Department of Mathematics and Statistics
Friday, August 19, 2022 4:00 P.M.
Virtual Defence

Supervisory Committee:
Dr. Xuekui Zhang, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Min Tsao, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:
Dr. Xiaojian Shao, Digital Technologies Research Centre, National Research Council Canada

Chair of Oral Examination:
Dr. Dennis Hore, Department of Chemistry, UVic

Abstract
Deep Learning (DL) has been broadly applied to solve big data problems in biomedical fields, which is most successful in image processing. Recently, many DL methods have been applied to analyze genomic studies. However, genomic data usually has too small a sample size to fit a complex network. They do not have common structural patterns like images to utilize pre-trained networks or take advantage of convolution layers. The concern of overusing DL methods motivates us to evaluate DL methods’ performance versus popular non-deep Machine Learning (ML) methods for analyzing genomic data with a wide range of sample sizes. In this paper, we conduct a benchmark study using the UK Biobank data and its many random subsets with different sample sizes. The original UK biobank data has about 500k patients. Each patient has comprehensive patient characteristics, disease histories, and genomic information, i.e., the genotypes of millions of Single-Nucleotide Polymorphism (SNPs). We are interested in predicting the risk of three lung diseases: asthma, COPD, and lung cancer. There are 205,238 patients who have recorded disease outcomes for these three diseases. Five prediction models are investigated in this benchmark study, including three non-deep machine learning methods (Elastic Net, XGBoost, and SVM) and two deep learning methods (DNN and LSTM). Besides the most popular performance metrics, such as the F1-score, we promote the hit curve, a visual tool to describe the performance of predicting rare events. We discovered that DL methods frequently fail to outperform non-deep ML in analyzing genomic data, even in large datasets with over 200k samples. The experiment results suggest not overusing DL methods in genomic studies, even with biobank-level sample sizes. The performance differences between DL and non-deep ML decrease as the sample size of data increases. This suggests when the sample size of data is significant, further increasing sample sizes leads to more performance gain in DL methods. Hence, DL methods could be better if we analyze genomic data bigger than this study.

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

MITCHELL HASLEHURST
MMath (University of Waterloo, 2016)
BA Hons. (Nipissing University, 2015)

“C*-algebras constructed from factor groupoids and their analysis through relative K-theory and excision”

Department of Mathematics and Statistics

Wednesday, August 17, 2022
12:00 P.M.

David Strong Building Room C128

Supervisory Committee:
Dr. Ian Putnam, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Marcelo Laca, Department of Mathematics and Statistics, UVic (Member)
Dr. Heath Emerson, Department of Mathematics and Statistics, UVic (Member)
Dr. Michel Lefebvre, Department of Physics and Astronomy, UVic (Outside Member)

External Examiner:
Dr. Aaron Tikuisis, Department of Mathematics and Statistics, University of Ottawa

Chair of Oral Examination:
Dr. Terri Lacourse, Department of Biology, UVic

Abstract
We address the problem of finding groupoid models for C*-algebras given some prescribed K-theory data. This is a reasonable question because a groupoid model for a C*-algebra reveals much about the structure of the algebra. A great deal of progress towards solving this problem has been made using constructions with inductive limits, subgroupoids, and dynamical systems. This dissertation approaches the question with a more specific methodology in mind, with factor groupoids.

In the first part, we develop a portrait of relative K-theory for C*-algebras using the general framework of Banach categories and Banach functors due to Max Karoubi. The purpose of developing such a portrait is to provide a means of analyzing the K-theory of an inclusion of C*-algebras, or more generally of a *-homomorphism between two C*-algebras. Another portrait may be obtained using a mapping cone construction and standard techniques (it is shown that the two presentations are naturally and functorially isomorphic), but for many examples, including the ones considered in the second part, the portrait obtained by Karoubi’s construction is more convenient.

In the second part, we construct examples of factor groupoids and analyze the relationship between their C*-algebras. A factor groupoid setup (two groupoids with a surjective groupoid homomorphism between them) induces an inclusion of two C*-algebras, and therefore the portrait of relative K-theory developed in the first part, together with an excision theorem, can be used to elucidate the structure. The factor groupoids are obtained as quotients of AF-groupoids and certain extensions of Cantor minimal systems using iterated function systems. We describe the K-theory in both cases, and in the first case we show that the K-theory of the resulting C*-algebras can be prescribed through the factor groupoids.

Please see the conference website.

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

YICHEN YAN
BSc (Xi’an University of Technology, 2020)

“Sub-phenotypes of Macrophages and Monocytes in COPD and Molecular Pathways for Novel Drug Discovery”

Department of Mathematics and Statistics
Friday, August 12, 2022 6:00 P.M.

Virtual Defence

Supervisory Committee:
Dr. Xuekui Zhang, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Min Tsao, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Shijia Wang, School of Statistics and Data Science, Nankai University

Chair of Oral Examination:
Dr. Adam Con, School of Music, UVic

Abstract
Chronic obstructive pulmonary disease (COPD) is a common respiratory disorder and the third leading cause of mortality. In this thesis we performed a clustering analysis of four specific immune cells in the GSE136831 dataset, using the default recommended parameters of the Seurat package in R, and obtained 16 subclasses with various COPD and cell-type proportions. Clusters 3, 7 and 9 had more pronounced independence and were all composed of macrophage-dominated control samples. The results of the pseudo-time analysis based on Monocle 3 package in R showed three different patterns of cell evolution. All started with a high percentage of COPD states, one ended with a high rate of Control states, and the other two still finished with a high percentage of COPD states. The results of differentially expressed gene analysis corroborated the existence of finer clusters and provided support for their rational categorization based on the similar marker genes. The gene ontology (GO) enrichment analysis for cluster 0 and cluster 6 provided feedback on enriched biological process terms with significant and unique characteristics, which could help explore latent novel COPD treatment directions. Finally, some top-ranked potential pharmaceutical molecules were searched via the connectivity map (cMAP) database.

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

BEN XIAO
BSc (University of Victoria, 2020)

“Random Forests on Trees”

Department of Mathematics and Statistics
Friday, August 5, 2022
9:00 A.M.
Virtual Defence

Supervisory Committee:
Dr. Gourab Ray, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Anthony Quas, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Tyler Helmuth, Department of Mathematical Sciences, Durham University

Chair of Oral Examination:
Dr. David Harrington, Department of Chemistry, UVic

Abstract
This thesis focuses a mathematical model from statistical mechanics called the Arboreal gas. The Arboreal gas on a graph G is Bernoulli bond percolation on G with the conditioning that there are no “loops”. This model is related to other models such as the random cluster measure. We mainly study the Arboreal gas and a related model on the d-ary wired tree which is simply the d-ary wired tree with the leaves identified as a single vertex. Our first result is finding a distribution on the infinite d-ary tree that is the weak limit in height n of the Arboreal gas on the d-ary wired tree of height n. We then study a similar model on the infinite d-ary wired tree which is Bernoulli bond percolation with the conditioning that there is at most one loop. In this model, we only have a partial result which proves that the ratio of the partition function of the one loop model in the wired tree of height n and the Arboreal gas model in the wired tree of height n goes to 0 as n → ∞ This allows us to prove certain key quantities of this model is actually the same as analogues of that quantity in the Arboreal gas on the d-ary wired tree, under an additional assumption.

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

XIANGLING JI

BSc (Simon Fraser University, 2019)

“scAnnotate: An Automated Cell Type Annotation Tool for Single-cell RNA-Sequencing Data”

Department of Mathematics and Statistics

Wednesday, July 27, 2022 1:00 P.M.
Virtual Defence

Supervisory Committee:
Dr. Xuekui Zhang, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Min Tsao, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:
Dr. Longhai Li, Department of Mathematics and Statistics, University of Saskatchewan

Chair of Oral Examination:
Dr. Alison Murray, Department of Anthropology, UVic

Abstract
Single-cell RNA-sequencing (scRNA-seq) technology enables researchers to investigate a genome at the cellular level with unprecedented resolution. An organism consists of a heterogeneous collection of cell types, each of which plays a distinct role in various biological processes. Hence, the first step of scRNA-seq data analysis often is to distinguish cell types so that they can be investigated separately. Researchers have recently developed several automated cell type annotation tools based on supervised machine learning algorithms, requiring neither biological knowledge nor subjective human decisions. Dropout is a crucial characteristic of scRNA-seq data which is widely utilized in differential expression analysis but not by existing cell annotation methods. We present scAnnotate, a cell annotation tool that fully utilizes dropout information. We model every gene’s marginal distribution using a mixture model, which describes both the dropout proportion and the distribution of the non-dropout expression levels. Then, using an ensemble machine learning approach, we combine the mixture models of all genes into a single model for cell-type annotation. This combining approach can avoid estimating numerous parameters in the high-dimensional joint distribution of all genes. Using fourteen real scRNA-seq datasets, we demonstrate that scAnnotate is competitive against nine existing annotation methods, and that it accurately annotates cells when training and test data are (1) similar, (2) cross-platform, and (3) cross-species. Of the cells that are incorrectly annotated by scAnnotate, we find that a majority are different from those of other methods, which suggests that further ensembling scAnnotate with other methods may largely improve annotation precision.

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

ZHUOYU XIAO BSc (Jinan University, 2020)

“Optimality Conditions for Cardinality Constrained Optimization Problems”

Department of Mathematics and Statistics

Tuesday, July 19, 2022
10:00 A.M.
Virtual Defence

Supervisory Committee:
Dr. Jane Ye, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Yu-Ting Chen, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Tim Hoheisel, Department of Mathematics and Statistics, McGill University

Chair of Oral Examination:
Dr. Rogério de Sousa, Department of Physics and Astronomy, UVic

Abstract
Cardinality constrained optimization problems (CCOP) are a new class of optimization problems with many applications. In this thesis, we propose a new framework called mathematical programs with disjunctive subspaces constraints (MPDSC), a special case of mathematical programs with disjunctive constraints (MPDC), to investigate CCOP. Our method is different from the relaxed complementarity-type reformulation in the literature.

The first contribution of this thesis is that we study various stationarity conditions for MPDSC and apply them to CCOP. In particular, we obtain new strong (S-) stationarity and new Mordukhovich (M-) stationarity for CCOP, which are sharper than those obtained from the relaxed complementarity-type reformulation.

The second contribution of this thesis is that we obtain some new results for MPDSC, which do not hold for MPDC in general. We show that many constraint qualifications like relaxed constant positive linear dependence (RCPLD) coincide with their piecewise versions for MPDSC. Based on these results, we prove that RCPLD implies error bounds for MPDSC. These two results also hold for CCOP. All of these new constraint qualifications for CCOP derived from MPDSC are weaker than those from the relaxed complementarity-type reformulation.

The UVic Department of Mathematics and Statistics, in conjunction with the Association for Women in Mathematics and the Pacific Institute for the Mathematical Sciences, is pleased to announce an exciting new program for high school students. IMAGINING UVic (Inspiring Mathematical Growth and Intuition in Girls) is a summer camp and seminar series aimed at encouraging young women to pursue STEM fields. More information, including how to apply, can be found on our website: https://onlineacademiccommunity.uvic.ca/imagininguvic/

See announcement and abstract (PDF).

See announcement and abstract (PDF).

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

Xiaolin Huang
BSc (Washington University, 2019)

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. You Liang, Department of Mathematics, Toronto Metropolitan University

Chair of Oral Examination
Dr. Kirstin Lane, School of Exercise Science, Physical and Health Education, UVic

Abstract
COVID-19 has affected the lives of people worldwide. This thesis includes two studies on the response to COVID-19 using statistical methods. The first study explores the impact of lockdown timing on COVID-19 transmission across US counties. We used Functional Principal Component Analysis to extract COVID-19 transmission patterns from county-wise case counts, and used machine learning methods to identify risk factors, with the timing of lockdowns being the most significant. In particular, we found a critical time point for lockdowns, as lockdowns implemented after this time point were associated with significantly more cases and faster spread. The second study proposes an adaptive sample pooling strategy for efficient COVID-19 diagnostic testing. When testing a cohort, our strategy dynamically updates the prevalence estimate after each test, and uses the updated information to choose the optimal pool size for the subsequent test. Simulation studies showed that our strategy reduces the number of tests required to test a cohort compared to traditional pooling strategies.

The semi-random graph process can be thought of as a one player game. Starting with an empty graph on n vertices, in each round a random vertex u is presented to the player, who chooses a vertex v and adds the edge uv to the graph (hence 'semi-random'). The goal of the player is to construct a small fixed graph G as a subgraph of the semi-random graph in as few steps as possible. I will discuss this process, and in particular the asympotically tight bounds we have found on how many steps the player needs to win. This is joint work with Trent Marbach, Pawel Pralat and Andrzej Rucinski.

Speaker Biography: Natalie completed her PhD in 2020 at Queen Mary University of London under the supervision of Robert Johnson. Prior to this, she completed both her Bachelors and Masters degrees at the University of Cambridge. After finishing her PhD she spent a year at the University of Ryerson in Toronto with the Graphs at Ryerson research group. She has worked on various problems under the broad umbrella of probabilistic and extremal combinatorics, including automata, graph saturation, graph factorization and probabilistic zero-forcing (a model for infection or rumour spreading across networks). Since the start of 2022 she has been a postdoctoral fellow at the University of Victoria, working with Natasha Morrison and Jonathan Noel.

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For more information and registration: https://www.pims.math.ca/seminars/PIMSPDF

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Join us for a free screening of Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani in celebration of Women in Math Day!

Examine the life and mathematical work of Maryam Mirzakhani, an Iranian immigrant to the United States who became a superstar in her field. In 2014, prior to her untimely death at the age of 40, she became both the first woman and the first Iranian to be awarded the Fields Medal, the most prestigious award in mathematics, often equated in stature with the Nobel Prize.

University of Victoria will host a one-day mini-conference focussing on recent developments in probability theory. The goal of this endeavour is to bring together probabilisits in the northwest pacific area and provide a platform for possible future collaborations.

For more information, list of speakers and abstracts see the conference website.

Reviewers

Supervisory Committee
Dr. Slim Ibrahim, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. David Goluskin, Department of Mathematics and Statistics, UVic(Co-Supervisor)

External Examiner
Dr. Quyuan Lin, Department of Mathematics, University of California Santa Barbara

Chair of Oral Examination
Dr. Raad Nashmi, Department of Biology, UVic

Abstract
This thesis is devoted to the motion of the incompressible and inviscid ow which is axisymmetric and swirl-free in a cylinder, where the hydrostatic approximation is made in the axial direction. It addresses the problem of local existence and uniqueness in the spaces of analytic functions for the Cauchy problem for the inviscid primitive equations, also called the hydrostatic incompressible Euler equations, on a cylinder, under some extra conditions. Following the method introduced by Kukavica-Temam-Vicol-Ziane in Int. J. Differ. Equ. 250 (2011) , we use the suitable extension of the Cauchy-Kowalewski theorem to construct locally in time, unique and real-analytic solution, and find the explicit rate of decay of the radius of real- analiticity. Furthermore, this thesis discusses the problem of finite-time blowup of the solution of the system of equations. Following a part of the method introduced by Wong in Proc Am Math Soc. 143 (2015), we prove that the first derivative of the radial velocity blows up in time, using primary functional analysis tools for a certain class of initial data. Taking the solution frozen at r = 0, we can apply an a priori estimate on the second derivative of the pressure term, to derive a Ricatti type inequality.

Nazim Acar

BSc (Uludağ Universitesi, 1998)

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

Reviewers

Supervisory Committee
Dr. Christopher Bose, Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Ahmed Sourour, Mathematics and Statistics, University of Victoria (Member)

External Examiner
Dr. Shafiqul Islam, School of Mathematical and Computational Sciences, University of Prince Edward Island

Chair of Oral Examination
Dr. Richard Keeler, Department of Physics, UVic

Abstract
A Pythagorean Triple (PT) is a triple of positive integers (a, b, c), that satisfies a² + b² = c². By requiring two of the entries being relatively prime, (a, b, c) becomes a Primitive Pythagorean Triple (PPT). This removes trivially equivalent PTs. Following up on the unpublished paper by D. Romik [1] we develop a sequence of mappings and show how each PPT has a unique path starting from one of the two initial nodes (3, 4, 5), (4, 3, 5). We explain a way of generating the PPTs through paper folding. Using a various techniques from dynamics we show how these mappings can be carried over to their conjugates in the first unit arc x² + y² = z², x, y ≥ 0 and the unit interval [0, 1]. Under these mappings and through the conjugacies we show that the PPTs, the pair of rational points on the first unit arc and the rational numbers on the unit interval correspond to each other with the forward orbits exhibiting similar behavior. We identify infinite, σ-finite invariant measures for one-dimensional systems. With the help of the developed conjugacies we extend the dynamics of the PPTs to the continued fraction expansion of the real numbers in the unit interval and show a connection to the Euclidean algorithm. We show that the dynamical system is conservative and ergodic.

Zoom link.

Abstract: Modern biomedical studies often collect multi-view data, that is, multiple types of data measured on the same set of objects. A typical approach to the joint analysis of two high-dimensional data views/sets is to decompose each data matrix into three parts: a low-rank common-source matrix that captures the shared information across data views, a low-rank distinctive-source matrix that characterizes the individual information within each single data view, and an additive noise matrix. Existing decomposition methods often focus on the orthogonality between the common-source and distinctive-source matrices, but inadequately consider the more necessary orthogonal relationship between the two distinctive-source matrices. The latter guarantees that no more shared information is extractable from the distinctive-source matrices. We propose a novel decomposition method that defines the common-source and distinctive-source matrices from the L2 space of random variables rather than the conventionally used Euclidean space, with a careful construction of the orthogonal relationship between distinctive-source matrices. The proposed estimators of common-source and distinctive-source matrices are shown to be asymptotically consistent and have reasonably better performance than some state-of-the-art methods in both simulated data and the real data analysis.

Please contact pims@uvic.ca for the Zoom link.

Large sets in Euclidean space should have large projections in most directions. Projection theorems in geometric measure theory make this intuition precise, by quantifying the words “large” and “most”.

How large can a planar set be if it contains a circle of every radius? This is the quintessential example of a curvilinear Kakeya problem, central to many areas of harmonic analysis and incidence geometry.

What do projections have to do with circles?

The talk will survey a few landmark results in these areas and point to a newly discovered connection between the two.

Abstract: Achievement games are a class of combinatorial games in which two players take turns selecting points from a set (the board), with the goal of being the first to occupy one of the previously designated "winning" subsets. In this talk, we will consider the avoidance variant, in which the first player to occupy such a set loses the game. As a strategy-stealing argument can be used to show that an achievement game cannot be a second-player win, one might expect that the avoidance variant cannot be a first-player win. However, it turns out that we can find transitive avoidance games that are first-player wins for all board sizes which are not primes or powers of two. This talk is based on the paper "Transitive Avoidance Games" by J. Robert Johnson, Imre Leader, and Mark Walters.

As studied in ​many examples, higher order correction added to the Newtonian potential often provides more realistic and accurate quasi-homogeneous models in astrophysics. Important examples include the Schwarzschild and the Manev potentials. The quasi-homogeneous N-body problem aims to study the interaction between N point particles under a prescribed potential. The classical (Newtonian) Hill’s lunar problem aims to improve the solution accuracy of the lunar motion obtained by solving the two-body (Earth-Moon) system. Hill's lunar equation under the Newtonian or homogeneous potentials has been derived from the Hamiltonian of the three-body problem in a uniform rotating coordinate system with angular speed $\omega$, by using symplectic scaling and heuristic arguments on various physical quantities. In this talk, we first introduce a new variational method characterizing relative equilibria with minimal energy. This enables us to classify the dynamic in terms of global existence and singularity for all possible ranges of the parameters. Then we derive Hill’s lunar problem with quasi-homogenous potential, and finally, we implement the same ideas to demonstrate the existence of ``black hole effect" for a certain range of the parameters: below and at some energy threshold, invariant sets (in the phase space) with non-zero Lebesgue measure that either contain global solutions or solutions with singularity are constructed.

Zoom link.

Abstract: I will present our recent results on estimating the Lyapunov exponents of weakly-damped, weakly-dissipated stochastic differential equations. Our primary tool is a new, mildly-quantitative version of Furstenberg’s criterion.