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

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

External Examiner
Dr. David Muraki, Department of Mathematics, Simon Fraser University

Chair of Oral Examination
Dr. Graham Voss, Department of Economics, UVic

Abstract
In this thesis, we discuss several numerical methods to approximate singular solutions for some partial differential equations such as Burgers’ equation, Prandtl’s equations, and the inviscid primitive equations. The numerical solutions we obtain for Burgers’ equation and Prandtl’s equations are compared with the existing analytical and numerical solutions in the literature. We observe the singularity formation in the numerical solutions to Burgers’ equation and Prandtl’s equations in finite time. For the inviscid primitive equations with the initial data are close to a suitable rescale of a smooth blowup profile proven by Collot, Ibrahim, and Lin, we compare the numerical solution to the theoretical blowup profile. The solution we obtain from the numerical scheme follows the profile, but the difference between the numerical and analytical profiles is quite significant closer to the blowup time. We then examine the stability of the numerical solutions by considering a small perturbation for the initial data. The gap between the perturbed and unperturbed solutions reduces as we choose smaller perturbation. However, this gap grows as it approaches the blowup time, and the stability of the numerical solutions remains in doubt.

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See announcement and abstract.

See announcement and abstract (PDF).

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.

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.