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Nonlinear mixed effects (NLME) models are commonly used in modelling many longitudinal data such as pharmacokinetics and HIV viral dynamics. These models are often derived based on the underlying data generation mechanisms, so the parameters in these models often have meaningful physical interpretations and natural restrictions such as some parameters being positive. Hypothesis testing for these parameters should incorporate these restrictions, leading to one-sided or constrained tests. Motivated from HIV viral dynamic models, in this article we propose multi-parameter one-sided or constrained tests for NLME models with censored responses, e.g., viral dynamic models with viral loads below detection limits. We propose approximate likelihood-based tests which are computationally efficient. We evaluate the tests via simulations and show that the proposed tests are more powerful than the corresponding two-sided or unrestricted tests. We apply the proposed tests to two AIDS datasets with new findings.

See announcement and abstract PDF.

Interested students, please contact Peter Dukes at dukes@uvic.ca or register directly via the MAA website.

The William Lowell Putnam Mathematical Competition is the preeminent mathematics competition for undergraduate college students in the United States and Canada. The Putnam Competition takes place annually on the first Saturday of December. The competition consists of two 3-hour sessions, one in the morning and one in the afternoon. During each session, participants work individually on 6 challenging mathematical problems.

The Putnam began in 1938 as a competition between mathematics departments at colleges and universities. Now the competition has grown to be the leading university-level mathematics examination in the world. Although participants work independently on the problems, there is a team aspect to the competition as well. Prizes are awarded to the participants with the highest scores and to the departments of mathematics of the five institutions, the sum of whose top three scores is the greatest.

Abstract: Given a graph H, an edge-coloured graph G is H-rainbow saturated if it does not contain a rainbow copy of H, but the addition of any non-edge in any colour creates a rainbow copy of H. The rainbow saturation number, denoted by rsat(n,H), is the minimum number of edges among all H-rainbow saturated edge-coloured graphs on n vertices. We prove that, for any non-empty graph H, the rainbow saturation number is linear in n. This confirms a recent conjecture of Girão, Lewis, and Popielarz. Based on joint work with Natalie Behague, Tom Johnston, Shoham Letzter, and Natasha Morrison.

Abstract: Imprimitivity theorems establish the Morita equivalence between C*-algebras arising from certain group/groupoid dynamics. For example, Green's imprimitivity theorem arises when two groups act on a space by commuting free and proper actions. Recently, we introduced the notion of self-similar actions (also known as Zappa-Szep product) of groupoids on other groupoids and Fell bundles. These self-similar actions encode two-way actions between the groupoids, in contrast to the one-way action encoded by the classical group/groupoid actions. We prove a generalized imprimitivity theorem arising from such self-similar actions, which establishes a new way of constructing Morita equivalent groupoid and Fell bundles. This is joint work with Anna Duwenig.

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Abstract: The log-rank test is considered the criterion standard for comparing 2 survival curves in pivotal registrational oncology trials. However, with novel immunotherapies that often violate the proportional hazards assumptions over time, log-rank can lose power and may fail to detect treatment benefit. We performed systematic review and meta-analysis of 63 studies between the log-rank, maxcombo and dRMST. The findings of this review show that MaxCombo may provide a pragmatic alternative to log-rank when departure from proportional hazards is anticipated. Both tests resulted in the same statistical decision in most comparisons. Discordant studies had modest to meaningful improvements in treatment effect. The dRMST test provided no added sensitivity for detecting treatment differences over log-rank.

Bio: Jiabu Ye is principal scientist of Biostatistics at MSD. He is endometrial indication lead statistician over see several endometrial late phase clinical trials. Before joining MSD, Jiabu worked at late development trial statistician in AstraZeneca and lead multiple late phase trial development. He is also a member of NPH cross-pharma working group. He received PhD in biostatistics from University of Texas Health Science Center at Houston.

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.

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Abstract: The Schwarzschild potential is defined as U(r)=-A/r-B/r^3, where r is the relative distance and A, B>0. In my talk I will present qualitative results on the scattering solutions of 3 body systems interacting with Schwarzschild's potential, where the 3 point masses form an isosceles triangle at all times. We first provide collision vs singularity criterion and convexities of scattering solutions. Then, we approximate the scattering and partial scattering solutions in various cases such that the error term converges to 0 as t goes to infinity. The scattering dynamics are also classified based on the value of the angular momentum. We also investigated some topological properties of the configuration space by applying the collision criterion and the scattering classification. This is a joint work with S. Ibrahim.

Abstract: Many systems whose physics is generally well understood are modeled with differential equations. However, many of these differential equations have the property that they are sensitive to the choice of initial conditions. If one instead has snapshots of a system, i.e. data, one can make a more educated guess at the true state by incorporating the data via data assimilation. Many of the most popular data assimilation methods were developed for general physical systems. However, in the context of fluids, data assimilation works better than would be anticipated for a general physical system. In particular, turbulent fluid flows have been proven to have the property that, given enough perfect observations, one can recover the full state irrespective of the choice of initial condition. This property is surprisingly unique to turbulent fluid flows, a consequence of their finite dimensionality. Unfortunately, the development of data assimilation methods has not actively take this finite dimensionality into account, as it has been developed independent of a considered system. This will be an informal presentation of what I currently understand with regards to data assimilation, both statistical and continuous methods. In particular, I will highlight the difficulties in equating the existing methods, and where these difficulties are presenting themselves in current research directions.

The speaker, Beiyan Ou, is a MSc alumnus of our department, who is now a senior manager in the BC Ministry of Agriculture and Food. In this seminar talk, students will learn some of the analyses that she’s done to inform policy development in her career in the BC Public Service. Beiyan will also offer a glimpse into the typical work day for a statistician who is part of a multidisciplinary team of government agencies. This special talk is not focused on the development of novel methods. However, it will offer helpful insights for students considering working in government agencies after graduation.

It's time to revive SUMS, our student society for undergraduates in Mathematics and Statistics. I feel that a good way to do this is to hold an initial meeting of interested students, and various others who want to help get the ball rolling.

SUMS Gathering: Friday, November 18 from 2:30 to 3:20 in DSB C116

If you're an instructor, especially of a 300 or 400-level course, please pass on the word to your students. If you're interested in outreach and getting to know our students, feel free to attend yourself. You might even be interested in acting as a faculty member liaison.

If you're involved in UVic's AWM Chapter, you're already a step ahead on academic and social activities involving our students. Please consider attending if you can. Assuming we can revive SUMS, a partnership with the AWM Chapter will be great to have in place.

If you're a grad student but a former UVic undergrad, please consider attending if you're able. This will help achieve some continuity from the pre-pandemic SUMS activities to the present/future.

If you're a grad student with experience from SIGMAS, our *graduate* student society, you may have valuable advice/insight for structural or administrative aspects of the society. And SIGMAS would be another good partner society.

Thank you for your help in this effort, even if it's just to get the word out.

Peter Dukes
Acting Chair - Math and Stats

A broadcast on a connected graph G with vertex set V(G) is a function f:V(G) ➛ {0, 1, ..., diam(G)} such that f(v) ≤ e(v), where e(v) denotes the eccentricity of v. If f(v) > 0, the vertex v is said to broadcast at strength f(v). In generalizing independent sets to broadcasts, Erwin (2001) introduced the concept of hearing independent broadcasts, in which no broadcasting vertex u lies within distance f(v) of another broadcasting vertex v. If instead we require that the distance d(u, v) is at least f(u)+f(v) for all u, v with f(u), f(v) > 1, we obtain the definition of boundary independence introduced by Mynhardt and Neilson in 2019.

In this talk, we consider the minimum costs of maximal hearing independent and boundary independent broadcasts on graphs and show these parameters are comparable, solving an open problem posed by Marchessault and Mynhardt in 2021. We further prove that the minimum cost of a maximal boundary independent broadcast on a connected graph equals the minimum among those of its spanning trees, and that the same is true for hearing independent broadcasts.

Abstract: An irrational rotation algebra is a C*-algebra defined as the crossed product of the continuous functions on the circle, T=R/Z, by the Z-action induced from translation on T by integer multiples of a real number ℏ (modulo Z). Such irrational rotation algebras serve as some of the first motivating examples in noncommutative geometry; In particular, the early 90’s saw Connes use an adaptation of the Dirac cycle on the 2-torus to obtain a K-homology class (and representative cycle) over the tensor product of two identical irrational rotation algebras which induces a self-dialuty in the sense of KK-theory. In this presentation, I will introduce higher dimensional analogues of irrational rotation algebras constructed by considering lattices in Euclidean space of dimension possibly greater than one. Until recently, the aim in considering such higher dimensional rotation algebras was to provide a class of geometrically defined C*-algebras exhibiting KK-duality, with the particular duality classes being constructed in a manner involving a pair of (suitably) transverse groupoids arising from appropriately chosen lattices. However, the current goal is now to construct interesting spectral triples using our notion of transverse groupoids and, as in the work of Butler, Emerson, and Schulz, determine when one can meromorphically extend the resulting zeta functions. After some basic setup and notation, I will discuss the new goal, and a spectral triple of interest. Time permitting, I may also touch on the previous goal and, in particular, our candidate unit and co-unit for a KK-duality between pairs of these higher dimensional rotation algebras.

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Abstract: Optimal transport (OT) is a principled approach for matching, having achieved success in diverse applications such as tracking and cluster alignment. It is also the core computation problem for solving the Wasserstein metric between probabilistic distributions, which has been increasingly used in machine learning. In this talk, I will present a new distance called Minimized Aggregated Wasserstein (MAW) for Gaussian mixture models. The definition of MAW exploits OT as a fundamental matching principle. We then extend OT by a new optimization formulation called Optimal Transport with Relaxed Marginal Constraints (OT-RMC). Specifically, we relax the marginal constraints by introducing a penalty on the deviation from the constraints. We demonstrate how MAW and OT-RMC can easily adapt to various tasks by single-cell data analysis.

See Zoom link, schedule and abstracts (PDF)

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Abstract: Contact tracing is an important intervention measure to control infectious diseases. We present a new approach that borrows the edge dynamics idea from network models to track contacts included in a compartmental SIR model for an epidemic spreading in a randomly mixed population. Unlike network models, our approach does not require statistical information of the contact network, data that are usually not readily available. The model resulting from this new approach allows us to study the effect of contact tracing and isolation of diagnosed patients on the control reproduction number and the number of infected individuals. We also estimate the effects of tracing coverage and capacity on the effectiveness of contact tracing. Finally, we show an extension of this model that incorporates a latent/exposed compartment.

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.

Recently, Mohar introduced a variant of the cops and robber game that is played on geodesic spaces. The game combines properties of pursuit-evasion games with the classical cops and robber game played on graphs. In this talk we discuss the rules of the game and strategies for the players when the game is played on compact surfaces of constant curvature. Joint work with Vesna Iršič and Bojan Mohar.

Talk is at 3:30 pm.

Reception is at 4:30 pm.

Download poster (PDF).

Abstract: Prediction systems face exogenous and endogenous distribution shift -- the world constantly changes, and the predictions the system makes change the environment in which it operates. For example, a music recommender observes exogeneous changes in the user distribution as different communities have increased access to high speed internet. If users under the age of 18 enjoy their recommendations, the proportion of the user base comprised of those under 18 may endogeneously increase. Most of the study of endogenous shifts has focused on the single decision-maker setting, where there is one learner that users either choose to use or not. In this talk, I'll describe several settings where user preferences may cause changes in distributions over the life of an ML system, and how these changes will affect the long-term performance of such systems. Joint work with Sarah Dean, Mihaela Curmei, Maryam Fazhel and Lillian Ratliff.

Bio: Jamie Morgenstern is an assistant professor in the Paul G. Allen School of Computer Science and Engineering at the University of Washington. She was previously an assistant professor in the School of Computer Science at Georgia Tech. Prior to starting as faculty, she was fortunate to be hosted by Michael Kearns, Aaron Roth, and Rakesh Vohra as a Warren Center fellow at the University of Pennsylvania. She completed her PhD working with Avrim Blum at Carnegie Mellon University. She studies the social impact of machine learning and the impact of social behavior on ML's guarantees. How should machine learning be made robust to behavior of the people generating training or test data for it? How should ensure that the models we design do not exacerbate inequalities already present in society?

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

Abstract: The relative Baum-Connes conjecture claims that a certain relative Baum-Connes assembly map is an isomorphism. It provides an algorithm of the computation of the K-theory of relative group C*-algebras. In my talk, I will present several cases when the relative assembly maps are isomorphic (or injective). The strong relative Novikov conjecture states that the relative assembly map is injective. I will also talk about the applications of the strong Novikov conjecture in geometry and topology, especially about the relative higher signatures of manifolds with boundary.

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Abstract: Mountain pine beetle (MPB) are a bark beetle native to western North America that lay their eggs in pine trees, killing them in the process. In the last few decades, MPB have spread well beyond their historical range into Alberta and they threaten further spread North and East. Existing population models of MPB focus on capturing a single outbreak, but management goals have moved into the longer term and are also concerned with changing tree resilience. In this talk, I will present a discrete time model that couples key biological aspects of MPB population dynamics to an age structured tree model to understand how MPB dynamics will change on longer time scales. I will show that as forest resilience decreases, a fold bifurcation occurs and there is a stable fixed point with a non-zero MPB population. If the number of beetles is above the Allee threshold, the population approaches this fixed point over a long time scale with transient outbreaks driven by the age structure of the forest. I will then discuss a biologically motivated vignette and show how adding a small number of lower vigor trees can add an additional stable fixed point with a low number of beetles, and how decreasing resilience can result in outbreaks from this endemic population.

Abstract: Schrödinger operators with delta potentials are of longstanding interest for admitting solutions expressible in closed analytic forms. By some duality, these operators also receive new interest for solving moments of the continuum directed random polymers and the Kardar–Parisi–Zhang equation for interface growth. Along with a review of the above background of up to 3D, this talk will discuss a Feynman–Kac-type path integral representation for a standard model of Schrödinger operator in 2D with delta potential. In particular, it will be explained why this representation falls outside the scope of standard Feynman–Kac formulas and is probabilistically of “two-minus dimensions” by construction and nature.

In this talk, I will describe an "embarrassingly parallel'' method for Bayesian inference, annealed Sequential Monte Carlo (ASMC) with an adaptive determination of annealing parameters. The ASMC method can efficiently provide an approximate posterior distribution and an unbiased estimator for the marginal likelihood. We can use this unbiasedness property to test the correctness of posterior simulation and the estimated marginal likelihood to conduct Bayesian model comparison. We have applied the annealed SMC method to two non-standard applications with complex statistical models: 1) Bayesian inference of phylogenetic trees and evolutionary parameters from biological sequence data; 2) Estimation of parameters in nonlinear ordinary differential equations and model selection. We illustrate our method by comparing it with other methods such as standard Markov chain Monte Carlo algorithms using simulation studies and real data analysis.

Short bio: Liangliang Wang is an Associate Professor and Graduate Program Chair in the Department of Statistics and Actuarial Science at Simon Fraser University, where she has been a faculty member since 2013. Dr. Wang completed her Ph.D. in statistics at the University of British Columbia and her master's degree in statistics at McGill University. Her research interests focus on computational statistics and statistical machine learning. Her favourite applications come from important scientific questions raised in genetics, biology, public health, and environmetrics. Dr. Wang is interested in tackling the computational issues in complex statistical models applied to large-scale data. She has published 50 papers in statistical journals and machine learning conferences.

The first two problems are concerning edge-colorings of complete graphs. Erd\H{o}s and Tuza asked in 1993 whether for any graph F on l edges and any completely balanced coloring of any sufficiently large complete graph using l colors contains a rainbow copy of F. We answer this and a related question. This is joint work with Maria Axenovich.

The third problem concerns point sets in the plane. Bárány and Füredi asked to determine the maximum number of triangles being almost similar to a given triangle in a planar point set of fixed size. Exploring connections to hypergraph Turán problems, we answer this question for almost all triangles. This is joint work with József Balogh and Bernard Lidick\'{y}.

Abstract: We study actions of right LCM semigroups on C*-algebras by endomorphisms that respect the ideal structure of the semigroup, and prove an equivariant Stinespring dilation theorem that combines a Naimark dilation of a semigroup of contractions with a Stinespring dilation of a completely positive unital *-linear map. This requires that we identify a class of contractive covariant representations and show they can be dilated to isometric covariant representations if and only if the C*-algebra map is unital and completely positive. We also analyze what happens for the boundary quotient. This is joint work with Boyu Li.

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: A fundamental problem in statistical mechanics is the description of phase transitions. A useful method to show phase transitions in specific models is known as the Peierls argument, where it is shown that the energetic cost to create certain objects called contours is large and, in a certain regime of temperature, has a small probability of appearing, forcing the system to pass from a disordered to an ordered phase. We will review some results of phase transitions in deterministic as well as recent advances in random systems and, after that, will explain how a notion of multiscaled contours, inspired by previous articles of Fröhlich and Spencer, allowed us to show not only phase transition in a direct way for deterministic systems but also to study models where the usual argument for phase transition via correlation inequalities is not available. The talk will be based on recent works joint with Rodrigo Bissacot, Eric O. Endo, Satoshi Handa, and João Vitor Maia.

Abstract: In this talk, I will present some recent results of study on Burgers equations with time-delay and degeneracy of viscosity. The modelling equations arise from the nonlocal reaction-diffusion equations for population dynamics. When the Rankine-Hugoniot condition and the Lax's entropy condition hold, the dynamical equation possesses some sharper traveling waves, the so-called sharper viscous shock waves. The degeneracy of viscosity causes the traveling waves to be sharper, and the large time-delay makes the traveling waves to be oscillatory.

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Abstract: Malignant gliomas are highly invasive brain tumors. Recent attention has focused on their capacity for network-driven invasion, whereby mitotic events can be followed by the migration of nuclei along long thin cellular protrusions, termed tumour microtubes (TM). Here I develop a mathematical model that describes this microtube-driven invasion of gliomas. I show that scaling limits lead to well known glioma models as special cases such as go-or-grow models, the PI model of Swanson, and the anisotropic model of Swan.

Numerical simulations are used to compare between the models. (Joint work with N. Loy, K.J. Painter, R. Thiessen).

Abstract: I'll begin by describing shifts of finite: topological dynamical systems based on some simple combinatorial data. These have played a fundamental role in hyperbolic dynamics. I will then describe a new construction of factors (quotients) of such systems, also based on simple combinatorial data along with the idea of binary expansion of real numbers. This produces surprisingly elaborate geometric objects. I'll also briefly discuss applications to operator algebras.

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: The right-regular and left-regular representations of the ax+b monoid of the integers gives KK-equivalent C*-algebras with very different KMS structures under the natural gauge actions. A new feature which appears in the right-regular system is phase transition at high temperatures, which makes the analysis of these states more nuanced. I will start with a discussion of KMS states and their structure. I will then introduce the right ax+b system and its presentations, show that its KMS states can be identified with subconformal measures on the circle with respect to the wrapping action, and sketch the computation of these measures using twisted zeta functions.

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Abstract: In many biological systems, it is essential for individuals to gain information from their local environment before making decisions. In particular, through sight, hearing or smell, animals detect the presence of other individuals and adjust their behavior accordingly. Interestingly, this feature is not only restricted to higher level species, such as animals, but is also found in cells. For example, some human immune cells are able to interact non- locally by extending long thin protrusions to detect the presence of chemicals or signaling molecules. Indeed, the process of gaining information about the surrounding environment is intrinsically non-local and mathematically this leads to non-local advection terms in continuum models.

In this talk, I will focus on a class of nonlocal advection-diffusion equations modeling population movements generated by inter and intra-species interactions. I will show that the model supports a great variety of complex spatio-temporal patterns, including stationary aggregations, segregations, oscillatory patterns, and irregular spatio-temporal solutions. However, if populations respond to each other in a symmetric fashion, the system admits an energy functional that is decreasing and bounded below, suggesting that patterns will be asymptotically stable. I will describe novel techniques for using this functional to gain insight into the analytic structure of the stable steady state solutions. This process reveals a range of possible stationary patterns, including regions of multi-stability. These will be validated via comparison with numerical simulations.

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Abstract: The Non Linear Schrodinger Equation is a canonical example of a dispersive PDE that can also display stable, spatially localized solutions called solitons. Invariant measures for the flow of the equation have been used to study not only the well-posedness of the equation, but also the typicality of the long term behavior, whether dispersive or solitonic. In this talk, I will review some of the existing results and then describe joint work with Partha Dey and Kay Kirkpatrick on the corresponding Discrete PDE in dimension 3 and higher. In particular, I will define a family of invariant Gibbs measures for the discrete equations where the key parameter is the strength of the non linearity. We prove convergence of the associated free energy, and as the strength of the non linearity is varied, we establish existence of a phase transition. In the supercritical regime the support of the measure lies on very sharply peaked functions corresponding to a soliton phase, and resembles the Gaussian free field conditioned to have given L^2- norm in the subcritical regime.

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).