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

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

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

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

See poster (PDF file).

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

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

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

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

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

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

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

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

See poster (PDF file).

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

The Final Oral Examination for the Degree of Master of Science

(Department of Mathematics and Statistics)

ZIYI LYU

University of Waterloo, 2021 (B.Math)

Investigating the Relationship between Bayes Factors and Credible Intervals

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

Online

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

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

Register for talk.

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

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

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

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

This is joint work with Gary MacGillivray and Kieka Mynhardt.

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

See abstract PDF file.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

TUAN VIET DAO
BA (Gustavus Adolphus College, 2017)

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

Department of Mathematics and Statistics

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

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

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

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

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

See announcement and abstract (PDF).

BSc (University of Victoria, 2021)

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

Mathematics and Statistics

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

Reviewers

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

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

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

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

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

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

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

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

Department of Mathematics and Statistics

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

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

See announcement (PDF)

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

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

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

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

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

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

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

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

Abstract: A classic result of Chvatál and Erdős (1972) asserts that, if the vertex-connectivity of a graph G is at least as large as its independence number, then G has a Hamilton cycle. We prove a similar result, implying that a graph G is pancyclic, namely it contains cycles of all lengths between 3 and |G|: we show that if |G| is large and the vertex-connectivity of G is larger than its independence number, then G is pancyclic. This confirms a conjecture of Jackson and Ordaz (1990) for large graphs.

Abstract: A famous result of Rado characterises those integer matrices $A$ which are partition regular, that is, for which any finite colouring of $[n]:=\{1,2,\dots,n\}$ (for $n$ sufficiently large) gives rise to a monochromatic solution to the equation $Ax=0$. The probability threshold for when the binomial random set $[n]_p$ has the property above was established via results of R\"odl and Ruci\'nski (for the 0-statement), and Friedgut, R\"odl and Schacht (for the 1-statement). Collectively these results form the Random Rado Theorem.

There has also been interest in Rado-type results in the setting of abelian groups. In this talk we prove an analogue of the Random Rado Theorem in the setting of sequences of (subsets of) abelian groups.

Joint work with Andrea Freschi and Andrew Treglown.

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

MIKAYLA HOLMES
BSc (University of Victoria, 2021)

Background Connectivity: Understanding the brain's functional organization

Department of Mathematics and Statistics

Friday, May 19, 2023 10:00 A.M.
David Strong Building Room C128

Supervisory Committee:
Dr. Michelle Miranda, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Mary Lesperance, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Jodie Gawryluk, Department of Psychology, UVic

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

Abstract
Task-state fMRI (tfMRI) and rest-state fMRI (rfMRI) surface data from the Human Connectome Project (HCP) was examined with the goal of better understanding the nature of background activation signatures and how they compare to the functional connectivity of a brain at rest. In this paper we use a hybrid-decomposition and seed-based approach to calculate functional connectivity of both rfMRI data and the estimated residual data from a Bayesian spatiotemporal model. This model accounts for local and global spatial correlations within the brain by applying two levels of data decomposition methods. Moreover, long memory temporal correlations are taken into account by using the Haar discrete wavelet transform. Motor task data from the HCP is modelled, followed by an analysis of the residuals, which provide details regarding the brain's background functional connectivity. These residual connectivity patterns are assessed using a manual procedure and through studying the induced covariance matrix of the model's error term. When we compare these activation signatures to those found for the same subject at rest we found that regions within the subcortex displayed strong connections in both states. Regions associated with the default mode network also displayed statistically significant connectivity while the subject was at rest. In contrast, the pre-central ventral and mid-cingulate regions had strong functional patterns in the background activation signatures that were not present in the rest-state data. This modelling technique combined with a hybrid approach to assessing functional activation signatures provides valuable insights into the role background connections play in the brain. Moreover, it is easily adaptable which allows for this research to be extended across a variety of tasks and at a multi-subject level.

Abstract: I will discuss finite-time blowup for a model of the 3D Euler equation where the Helmholtz projection is replaced by a projection onto a more restrictive constraint space. The model preserves the structure of the self-advection nonlinearity entirely, and also conserves energy and helicity.

Abstract: The COVID-19 pandemic has placed an unprecedented strain on healthcare systems worldwide, making it difficult to accurately predict demand surge for acute care and allocate resources effectively. This paper proposes a novel predictive learning model that combines a differential equation model with delays and a multi-objective learning genetic algorithm to forecast hospitalizations and intensive care admissions based on predicted infections in a given population. Using data from Canada, Alberta, and Edmonton, we demonstrate the effectiveness of our model in significantly improving the ability to predict acute care capacity and manage resources during emergent infectious disease outbreaks such as COVID-19. Our approach is tailored to different population segments and considers various factors, such as variant strains of the virus and vaccination campaigns. Our model represents a significant contribution to the field, providing healthcare managers and decision-makers with a powerful tool to plan resources in the face of epidemics and major disasters. We contribute to the conversation on how we respond to public health emergencies, enabling more accurate predictions of demand for care and allocation of resources, ultimately saving lives and limiting the impact of these crises on our communities.

This is a joint work with Y. Bahri, N. Gilani, S. Ibrahim, A. Park and O. Chakroun

The Final Oral Examination
for the Degree of

Master of Science
(Department of Mathematics and Statistics)

Jianping Yu
Phd in Math, Chinese Academy of Sciences

“Hierarchical Model for Evaluation of Physician Care in the ICU”

April 26th, 2023
10:00am
On zoom

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

Chair of Oral Examination:
Dr. Laura Cowen, Department of Mathematics and Statistics, UVic

Abstract: One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete, which is a significant difficulty when modeling chaotic systems whose solutions are sensitive to initial conditions. If one instead has snapshots of a system, i.e. data, one can make a better 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, not just turbulent flows or chaotic 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 unique to turbulent fluid flows, a consequence of their finite dimensionality. In this presentation, we will discuss the continuous data assimilation algorithm that was used to prove the convergence in the original, perfect data setting, present various robustness results of the continuous data assimilation algorithm, and discuss how continuous data assimilation can be used to identify and correct model error. Moreover, we will highlight the efforts of our other current research into long-standing problems in the stability of fluid flows, and how we have discovered some very interesting connections to our data assimilation research.

See announcement (pdf).

Abstract: Negative amphicheiral knots provide torsion elements in the knot concordance group, and torsion elements are less well understood than infinite-order elements. In this talk, I will introduce an equivariant version of the Alexander polynomial for strongly negative amphicheiral knots called the half-Alexander polynomial, focusing on its applications to knot concordance. In particular, we will show how understanding the geography behavior of the half-Alexander polynomial led to the construction of the first examples of non-slice amphichiral knots of determinant 1. This talk is based on joint work with Keegan Boyle.

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

HANS OERI

MSc (Simon Fraser University, 2018)
BSc (Simon Fraser University, 2016)

“Convex Optimization Methods for Bounding Lyapunov Exponents”

Department of Mathematics and Statistics

Friday, April 14, 2023
10:00 A.M.
Virtual Defence

Supervisory Committee:
Dr. David Goluskin, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Anthony Quas, Department of Mathematics and Statistics, UVic (Member)
Dr. Nishant Mehta, Department of Computer Science, UVic (Outside Member)

External Examiner:
Dr. Milan Korda, Laboratoire d'analyse et d'architectures des systèmes, Centre national de la recherche scientifique, France

Chair of Oral Examination:
Dr. Daniela Constantinescu, Department of Mechanical Engineering, UVic

Abstract
In dynamical systems, the stability of orbits is quantified by Lyapunov exponents, which are computed from the average rate of divergence of trajectories. We develop techniques for computing sharp upper bounds on the maximal LE using methods from convex optimization, which have previously been used to compute sharp bounds on the time averages of scalar quantities on bounded orbits of dynamical systems. For discrete-time dynamics we develop an optimization-based approach for computing sharp bounds on the geometric mean of scalar quantities. We therefore express LEs as infinite-time averages and as geometric means in continuous-time systems and discrete-time systems, respectively, and then derive optimization problems whose solutions give sharp bounds on LEs. When the system’s dynamics is governed by a polynomial vector field, the problems can be relaxed to computationally tractable SOS programs whose solutions also give sharp bounds on LEs. An approach for the practical implementation of a sequence of SOS feasibility problems whose solutions converge to the maximal LE of discrete systems is provided. We explain how symmetries can be used to simplify and generalize the optimization problems in both continuous-time and discrete-time systems. We conclude by discussing the extension of the techniques developed here to the problem of bounding the sum of the leading LEs. Tractable SOS programs are derived for some special cases of this problem.

The applicability of all the techniques developed here is shown by applying them to various explicit examples. For some systems we numerically compute sharp bounds that agree with the the maximal LEs, and for some we prove analytic bounds on maximal LEs by solving the optimization problems by hand.

See announcement (pdf)

The aim of this talk is to present some theoretical and numerical results concerning Beltrami fields, called force-free fields in solar and fusion physics. The talk will be divided into three parts. In the first part, we will recall the physical context of the appearance of these fields, i .e. essentially the modeling of magnetic equilibria in solar physics. In the second part, we will recall some mathematical theoretical results concerning the existence and properties of fields in any three-dimensional domain. In the third and last part, we will expose some methods to compute these 3D fields and the associated physical equilibria, as well as some numerical results obtained with our codes.

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

Jack THOMAS
BSc (University of Victoria, 2019)

"Response to Improving abundance estimation by combining capture-recapture and presence-absence data: example with a large carnivore"

Tuesday, April 11, 2023
9:30 A.M.
David Strong Building C128

Supervisory Committee:
Dr. Laura Cowen, Department of Mathematics and Statistics, UVic (Co-Supervisor)
Dr. Simon Bonner, Department of Statistical and Actuarial Sciences, UWO (Co-Supervisor)

Chair of Oral Examination:
Dr. Farouk Nathoo, Department of Mathematics and Statistics, UVic

Given that Apr. 7th is a holiday, if the room in DSB is locked, we will have the seminar in the Math department.

Abstract: Cerf gave a not-well-known re-interpretation of Smale's proof that the diffeomorphism group of the 2-sphere has the homotopy-type of the isometry group. It fits into a family of proofs that allow for a partial description of the homotopy-fibres of the `scanning' maps Emb(D^{n-1}, S^1 x D^{n-1}) --> \Omega^j Emb(D^{n-1-j}, S^1 x D^{n-1}), as well as a proof that the monoid of co-dimension one spheres \pi_0 Emb(S^{n-1}, S^n) is a group, under the connect-sum operation. When n isn't 4 this is a well-known result due to the resolution of the generalized Schoenflies problem, but it would appear to be a new result when n=4. I will outline these observations.

Abstract: The Manickam-Miklós-Singhi Conjecture states that for positive integers n and k with n > (4k - 1), a multi-set X = {x_1,x_2, … ,x_n} with real entries and nonnegative sum has at least \binom{n-1}{k-1} subsets of size k with nonnegative sum.

Abstract: In the theory of harmonic analysis, one considers unitary representations of groups bearing some geometric significance. These may give rise to summable representations of an appropriate convolution algebra, and the trace can be computed either spectrally (in terms of irreducible factors) or geometrically, resulting in a "trace formula" that relates these two perspectives. Hallmark examples of this method include the Poisson summation formula and the Selberg trace formula. The work of Connes (1996, 1999) and Meyer (2004, 2007) considers geometric representations of groups arising from number theoretic data, and the trace formulas in their examples recover the explicit formulas of Weil relating prime numbers (or ideals) to the zeros of L-functions. In this talk, I will provide some background on trace formulas for groups, including one of the examples of Meyer, and introduce the group and representation that I am considering in my research.

For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a "threshold." Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures (e.g. random graphs and hypergraphs), with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. In 2006, Jeff Kahn and Gil Kalai conjectured that a natural (and often easy to calculate) lower bound q(F) (which we refer to as the “expectation-threshold”) for the threshold is in fact never far from its actual value. A positive answer to this conjecture enables one to narrow down the location of thresholds for any increasing properties in a tiny window. In particular, this easily implies several previously very difficult results in probabilistic combinatorics such as thresholds for perfect hypergraph matchings (Johansson–Kahn–Vu) and bounded-degree spanning trees (Montgomery). I will present recent progress on this topic in the first talk, and discuss more details about proof techniques in the second talk. Based on joint work with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.

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

Meifan LIN
BMath, University of Waterloo, 2021

“On linear models selected by the constrained minimum criterion”

April 5th 10:00 a.m.
Conducted Virtually

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

Chair of Oral Examination:
Dr. Junling Ma, Department of Mathematics and Statistics, UVic

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ABSTRACT: Stochastic parameterizations (SMCM) are continuously providing promising simulations of unresolved atmospheric processes for global climate models (GCMs). One of the features of earlier SMCM is to mimic the life cycle of the three most common cloud types (congestus, deep, and stratiform) in tropical convective systems. In this present study, a new cloud type, namely shallow cloud, is included along with the existing three cloud types to make the model more realistic. Further, the cloud population statistics of four cloud types (shallow, congestus, deep, and stratiform) are taken from Indian (Mandhardev) radar observations. A Bayesian inference technique is used here to generate key time scale parameters required for the SMCM as SMCM is most sensitive to these time scale parameters as reported in many earlier studies. An attempt has been made here for better representing organized convection in GCMs, the SMCM parameterization is adopted in one of the state-of-art GCMs namely the Climate Forecast System version 2 (CFSv2) in lieu of the pre-existing simplified Arakawa–Schubert (default) cumulus scheme and has shown important improvements in key large-scale features of tropical convection such as intraseasonal wave disturbances, cloud statistics, and rainfall variability. This study also shows the need for further calibration the SMCM with rigorous observations for the betterment of the model’s performance in short term weather and climate scale predictions.

Kumar Roy is now working as a PIMS Postdoctoral Fellow with Prof. Boualem Khouider at Department of Mathematics and Statistics, University of Victoria. His current research topic is in the area of climate change modelling focusing on Stochastic models for clouds and tropical convection parameterization. He completed his PhD degree in Meteorology and Oceanography from Andhra University, India in March 2022 and his research has been carried out at Indian Institute of Tropical Meteorology (IITM), Pune, India. His research focuses on the development and application of sub-grid scale cloud models in numerical weather prediction (NWP) models, as well as how these models affect the forecasting abilities of NWP models.

Abstract: Probabilistic zero-forcing can be thought of as a model for rumour spreading, where a person is more likely to spread a rumour if several of their friends already believe it . We start with a graph that has one infected vertex. At each time step an infected vertex infects an uninfected neighbour with probability proportional to how many of its neighbours are already infected. I will focus in this talk on probabilistic zero-forcing on hypercubes and grids, and demonstrate tight bounds on how long it takes for every vertex to be infected (asymptotically almost surely). This is joint work with Trent Marbach and Pawel Pralat.

For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a "threshold." Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures (e.g. random graphs and hypergraphs), with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. In 2006, Jeff Kahn and Gil Kalai conjectured that a natural (and often easy to calculate) lower bound q(F) (which we refer to as the “expectation-threshold”) for the threshold is in fact never far from its actual value. A positive answer to this conjecture enables one to narrow down the location of thresholds for any increasing properties in a tiny window. In particular, this easily implies several previously very difficult results in probabilistic combinatorics such as thresholds for perfect hypergraph matchings (Johansson–Kahn–Vu) and bounded-degree spanning trees (Montgomery). I will present recent progress on this topic in the first talk, and discuss more details about proof techniques in the second talk. Based on joint work with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.

Abstract: In 2017, Ron Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most the ceiling of n/r.

I will begin with a summary of recent progress on Aharoni's conjecture based on a new survey article of Katie Clinch, Jackson Goerner, Freddie Illingworth, and myself. I will then sketch a proof that Aharoni's conjecture holds up to an additive constant for each fixed r. The last result is joint work with Patrick Hompe.

Abstract: In noncommutative geometry, a spectral triple is a set of data which encodes a geometric phenomenon in an analytic way. The authors Butler, Emerson, and Schulz introduced a method of producing such spectral triples using the annihilation and creation operators of quantum mechanics. Their construction produced the Heisenberg cycle, a 2-dimensional spectral triple over the C*-algebra crossed-product of uniformly continuous, bounded functions on the real line crossed with the discretely topologized reals. In this talk, I will discuss a higher dimensional analogue of this Heisenberg cycle, properties of its zeta functions, as well as some promising examples of pullbacks for which the corresponding zeta functions can be meromorphically extended to the complex plane.

Abstract: A momentous legacy of twentieth-century mathematics is the realisation that deterministically evolving systems frequently exhibit, when observed for sufficiently extended periods of time, a statistical behaviour akin to the limiting behaviour of independent random variables. We shall explore a geometric incarnation of this surprising phenomenon, overviewing various kinds of statistical limit theorems for the free motion of a particle on a negatively curved surface. In order to emphasise the richness of possible asymptotic behaviours, as well as the variety of sources of randomness, we will further compare the free-motion dynamics with a closely related evolution on the same phase space, known as the horocycle flow.

In this talk, I will share our work on Single Cell RNA-Sequencing and cell-type annotation based on two recent projects. I'll start by introducing our BAL Single Cell RNA-seq data. Bronchoalveolar lavage (BAL) is a procedure that is sometimes done during a bronchoscopy. BAL is used to collect a sample from the lungs for testing, different from traditional lung tissue samples. After Covid-19, BAL has gotten more attention. However, it lacks significant consensus on annotating BAL scRNA-seq samples. The previous research mainly focused on annotating lung tissue samples. Consequently, we plan to construct an auto-annotation method for the BAL scRNA-seq sample, which can aid with the general cell type annotation. After that, I want to talk about exploring the Macrophages' and Alveolar macrophages' substructure. We want to identify their subtypes and how they affect the disease (COPD).

Description:
  The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth outside a set of singular points. However, explicit examples show that the singular set could be, in general, as large as the regular set. This talk aims to introduce this beautiful problem and describe some classical and recent results on the regularity of the free boundary.

Speaker Biography:
Alessio Figalli is a leading figure in the areas of Optimal Transport, partial differential equations and the calculus of variations. He received his Ph.D. from the Scuola Normale Superiore di Pisa and the Ecole Normale Superieur de Lyon and has held positions in Paris and Austin, Texas. He is currently a Professor at ETH Zurich. His work has been recognized with many awards including the Prize of the European Mathematical Society in 2012 and the Fields Medal in 2018.

Time:
All network wide colloquia take place at 1:30pm Pacific Time with a few exceptions.

Registration:
To attend this event please register here to receive the meeting link. Talks will be recorded and posted on the PIMS resource page www.mathtube.org.

Abstract: A collection of families $(\F_{1}, \F_{2} , \cdots , \F_{k}) \in \mathcal{P}([n])^k$ is \emph{cross-Sperner} if there is no pair $i \not= j$ for which some $F_i \in \F_i$ is comparable to some $F_j \in \F_j$. Two natural measures of the `size' of such a family are the sum $\sum_{i = 1}^k |\F_i|$ and the product $\prod_{i = 1}^k |\F_i|$. We prove upper and lower bounds on the sum measure for general $n$ and $k \ge 2$. In the process, we consider minimizing the number of sets comparable to a family $\F \subseteq \mathcal{P}([n])$ of a given size.

Abstract: Étale groupoid C*-algebras are ubiquitous and it is useful to have concrete methods for verifying that a given C*-algebra is isomorphic to a groupoid C*-algebra. It is a result, essentially due to Brown, Clark, Farthing, and Sims, that a Hausdorff second countable étale groupoid that is topologically principal and minimal has a simple reduced C*-algebra. This makes verifying faithfulness of its representations trivial. Removing the minimality condition one obtains an older result stating that faithfulness of a representation can be verified on the canonical commutative subalgebra of the groupoid. In this talk I will discuss the notion of an étale groupoid which is relatively topologically principal with respect to an open subgroupoid, as well as the consequences of this condition for representations of the groupoid C*-algebra.