Department of Mathematics and Statistics Colloquium Series

The transcriptome-wide association study (TWAS) has emerged as one of several promising techniques for integrating multi-scale ‘omics’ data into traditional genome-wide association studies (GWAS). Unlike GWAS, which associates phenotypic variance directly with genetic variants, TWAS uses a reference dataset to train a predictive model for gene expressions, which allows it to associate phenotype with variants through the mediating effect of expressions. Although effective, this core innovation of TWAS is poorly understood, since the predictive accuracy of the genotype-expression model is generally low and further bounded by expression heritability. This raises the question: to what degree does the accuracy of the expression model affect the power of TWAS? Furthermore, would replacing predictions with actual, experimentally determined expressions improve power? To answer these questions, we compared the power of GWAS, TWAS, and a hypothetical protocol utilizing real expression data. We derived non-centrality parameters (NCPs) for linear mixed models (LMMs) to enable closed-form calculations of statistical power that do not rely on specific protocol implementations. We examined two representative scenarios: causality (genotype contributes to phenotype through expression) and pleiotropy (genotype contributes directly to both phenotype and expression), and also tested the effects of various properties including expression heritability. Our analysis reveals two main outcomes: (1) Under pleiotropy, the use of predicted expressions in TWAS is superior to actual expressions. This explains why TWAS can function with weak expression models, and shows that TWAS remains relevant even when real expressions are available. (2) GWAS outperforms TWAS when expression heritability is below a threshold of 0.04 under causality, or 0.06 under pleiotropy. Analysis of existing publications suggests that TWAS has been misapplied in place of GWAS, in situations where expression heritability is low.

In this colloquium, we will be investigating the sustainability and optimality of delayed impulsive harvesting. Impulses describe an instantaneous change in a system due to some external effect (like harvesting in a fishery), which has a duration that is negligible compared to the overall time scale of the process. These impulses can then be combined with differential equations (DEs) to form impulsive DEs.

Including delays within the impulsive conditions is a topic of current interest, since delays within harvesting can represent delayed data collection, or targeted harvesting based on age. Our question will be whether the inclusion of a delay will affect the sustainability and/or optimality of an impulsive harvest. To answer this question, we will reduce the problem from an impulsive DE to a non-linear difference equation. While appearing simple, non-linear difference equations can exhibit rich and oftentimes complex dynamics. By analyzing the difference equation, we will show that while optimality is generally unaffected by the delay, the sustainability can be highly affected.

To many analysts and applied mathematicians, establishing a Poincaré inequality is a fundamental problem of mathematics. In recent joint work with D. Cruz-Uribe (U. Al.) and M. Penrod (U. Al. graduate student), we show the equivalence between the validity of a variable exponent matrix weighted Poincar ́e inequality and the existence of regular weak solutions to a variable exponent matrix weighted Neumann problem for the p(·)-Laplacian. This work has it’s beginnings in the analogous context of non-variable exponent Poincar ́e inequalities and p-Laplacians; joint work from 2018 with D. Cruz-Uribe and E. Rosta (CBU honours student). To make an accessible talk for all, I will focus on this 2018 result and will point to generalizations and ideas for the future.

Dr. Rodney is originally from Montreal QC, earned a BSc with Honours in Mathematics at Concordia University, and got his Ph.D. at McMaster University. He held a postdoctoral position in Rutgers University where he worked in collaboration with Richard Wheeden. Dr. Rodney's research concentrates on degenerate partial differential equations. This includes questions concerning existence, continuity and other properties of weak solutions including maximum principles and spectral theory. Furthermore, he studies functional and analytic properties of degenerate Sobolev spaces. He is currently an Assistant Professor at Cape Breton University, and holds adjunct positions at Dalhousie University and McMaster University.

We consider a model for collective behaviour with intrinsic interactions on Riemannian manifolds. We establish the well-posedness of measure solutions, defined via optimal mass transport, on several specific manifolds (sphere, hypercylinder, rotation group SO(3)), and investigate the mean-field particle approximation. We study the long-time behaviour of solutions, where the primary goal is to establish sufficient conditions for a consensus state to form asymptotically. The analytical results are illustrated with numerical experiments that exhibit various asymptotic patterns.

Dr. Razvan Fetecau is now a professor in the Department of Mathematics at Simon Fraser University (SFU). Before joining SFU in 2006, Dr. Fetecau received his Ph.D. in Applied and Computational Mathematics from the California Institute of Technology in 2003 and held the Szegö Assistant Professorship position at Stanford University from 2003 through 2006.

Dr. Fetecau’s research falls in the general area of nonlinear differential equations, with particular interests in mathematical models for self-collective and swarming behavior. Such models can capture complex aggregation behaviours that emerge from simple interaction rules, in the absence of a leader or external coordination. Dr. Fetecau has worked on the mathematical analysis for various nonlocal aggregation models, and also on their applications to population biology, robotics, and opinion formation. Besides, in previous research, Dr. Fetecau studied the regularizations of fluid dynamics equations, wave propagation in stratified media, and symplectic integrators for non-smooth Lagrangian mechanics.

We prove the existence in the sense of sequences of
solutions for some integro-differential type equations involving the drift
term in the appropriate H^2 spaces using the fixed point technique when the
elliptic problems contain second order differential operators with and
without
Fredholm property. It is known that, under the reasonable technical
conditions, the convergence in L^1 of the integral kernels yields the
existence and convergence in H^2 of solutions.

Dr. Vitali Vougalter obtained a Bachelor Degree with Honours in 1995 from Nizhni Novgorod State University, Russia, and his Ph.D. in 2000 at the Georgia Institute of Technology.

Dr. Voulgater held temporary positions at the University dos British Columbia, McMaster University, the University of Notre Dame, Indiana, the University of Toronto, and the University of Missouri, Columbia. He also held a permanent position as lecturer in Mathematics and Applied Mathematics at the University of Cape Town, and currently is a Visiting professor at the Department of Mathematics of the University of Toronto.

Dr Voulger has over 90 publications in his areas of research, which include Nonlinear Analysis and Partial Differential Equations, Mathematical Biology, Functional, Real and Harmonic Analysis, Statistical Mechanics, Quantum Mechanics and Quantum Field Theory and Dynamical Systems.

We discuss recent progress on singularity of random matrices with i.i.d. Bernoulli entries. The talk is partially based on a joint work with K. Tikhomirov.

Alexander Litvak has a Master of Science in Mathematics from Saint Petersburg State University (1991) and a Ph.D. in Mathematics from Tel Aviv University (1998). He held post doctorate positions at the University of Alberta and at Technion - Israel Institute of Technology. For the past two decades he held an academic position at the University of Alberta where he is currently a full professor. His research interests are in the areas of Asymptotic Geometric Analysis and related topics in Convex Geometry, Approximation Theory, and Probability. Professor Litvak has over seventy publications in peer review journals.  

The power of genotype-phenotype association mapping studies increases greatly when contributions from multiple variants in a focal region are meaningfully aggregated. Currently, there are two popular categories of variant aggregation methods. Transcriptome-wide association studies (TWAS) represent a category of emerging methods that select variants based on their effect on gene expressions, providing pretrained linear combinations of variants for downstream association mapping. In contrast, kernel methods such as SKAT model genotypic and phenotypic variance using various kernel functions that capture genetic similarity between subjects, allowing non-linear effects to be included. From the perspective of machine learning, these two methods cover two complementary aspects of feature engineering: feature selection/pruning, and feature modeling. Thus far, no thorough comparison has been made between these categories, and no methods exist which incorporate the advantages of TWAS and kernel-based methods. In this work we developed a novel method called kTWAS that applies TWAS-like feature selection to a SKAT-like kernel association test, combining the strengths of both approaches. Through extensive simulations, we demonstrate that kTWAS has higher power than TWAS and multiple SKAT-based protocols, and we identify novel disease-associated genes in WTCCC genotyping array data and MSSNG (Autism) sequence data. The source code for kTWAS and our simulations are available in our GitHub repository (https://github.com/theLongLab/kTWAS)

Is it possible to arrange the integers 1, 2, … , n in a circle such that any two adjacent entries sum to a square? Cube? Fibonacci number? What if the word sum is changed to difference? Or the square, cube, Fibonacci number restriction is replaced by some other permissible finite set of values? If such arrangements are possible, how many are there? Are there infinitely many? Richard Guy loved these types of questions which live at the interface between number theory and combinatorics and sound like simple brain teasers, but for which coming up with proofs is frequently extremely difficult if not outright impossible.

Our protagonist in this talk is an (a,b)-difference necklace, which is a circular arrangement of the first n non-negative integers such that the absolute difference of any two neighbours takes on one of two possible values (a,b). We prove that such arrangements almost always exist for sufficiently large n, and provide explicit recurrence relations for the cases (a,b) = (1,2), (1,3), (2,4) and (1,4). Using the transfer matrix method from graph theory, we then prove that for any pair (a,b) – and in fact for any finite set of two or more difference values – the number of such arrangements satisfies a linear recurrence relation with fixed integer coefficients.

This talk is accessible to a general audience; all “proofs” are by picture. No knowledge of number theory or graph theory is required. This work began in summer 2017 as an NSERC USRA project undertaken by Ethan White, now a PhD candidate at UBC, and jointly supervised by Richard Guy and the speaker.

A series of events on October 1-4, 2020 which celebrates the life of Emeritus professor and mathematician Dr. Richard Guy.  He was an inspirational teacher, colleague, philanthropist, mountaineer, and treasured friend who will be greatly missed and fondly remembered.

Speaker:  Dr. Terence Tao, Professor and The James and Carol Collins Chair in the College of Letters and Sciences Mathematics, UCLA
Speaker bio and full lecture abstract 

Title:  The notorious Collatz conjecture
Date:  Thursday, October 1, 2020
Time:  5:00 – 6:15 p.m. (MST)
Location:  Online

Celebrating the living legacy of the mathematics of Richard Guy

Join us for a one-day conference in honour of Richard Guy and his contribution to mathematical research. There will be two parallel sessions, geared towards combinatorics and number theory, respectively.  Each session will feature three talks throughout the day. 

Session details will be posted here closer to the event.

Date: Friday, October 2, 2020
Time: 9:00 a.m. – 3:30 p.m. (MST)
Location: Online

Special Lecture: The Life and Numbers of Richard Guy (1916 – 2020)

Speaker: Dr. Hugh Williams, Professor Emeritus, University of Calgary
Speaker bio and full lecture abstract

Title: The Life and Numbers of Richard Guy (1916 - 2020)
Date: Friday, October 2, 2020
Time: 4:00 – 5:15 p.m. (MST)
Location: Online

Mathematical Play in honour of Richard Guy: A day of recreational math for high school students

Richard Guy was an enthusiastic participant in recreational mathematics.  We would like to invite you to propagate his love of doing math for fun by joining us to solve mathematical puzzles, problems and challenges!  Problems from a special issue of Crux Mathematicorum dedicated to Richard Guy will be distributed on Saturday morning by email to registered participants.  Mathematicians will be available via Zoom in drop in sessions to discuss the problems with students, and to offer additional hourly challenges.  Participants are invited to post their solutions on a discussion board, and notable contributions by students will be recognized during the closing ceremonies session. Organized by Dr. Bill Sands and Dr. Lauren DeDieu.

Session details will be posted here closer to the event.

Date: Saturday, October 3, 2020
Drop-in problem sessions: 10 a.m. – 3 p.m. (MST)
Closing Ceremony:  4:00 – 5:00 p.m. (MST)
Location:  Online

“Peace is a disarming concept”: a celebration of the life of Richard Guy

You’re invited to join Richard and Louise’s family and their many friends to raise a cup of hot chocolate together and celebrate the impact their lives have had beyond the world of mathematics. We have spent time celebrating Richard’s passion for mathematics and sharing this passion with others. It’s now time for family, friends, and colleagues to share their memories of Richard and reflect on the impact he has made to the community and in their lives.

Date:  Sunday, October 4, 2020 
Time:  1:00 – 2:30 p.m. (MST)
Location: Online

Populations of neurons display an extraordinary diversity in the behaviors they affect and display. Both numerical and analytical techniques have recently emerged that allow us to create networks of model neurons that display behaviors of similar complexity. Here we demonstrate the direct applicability of one such technique, the FORCE method, to spiking neural networks. FORCE training relies on the reservoir computing framework and allows for rapid online training of spiking networks to perform arbitrary dynamics. We train these networks to mimic dynamical systems, classify inputs, and store and replay discrete sequences that correspond to the notes of a song. Finally, the analysis of synaptic weights after training leads to offline analytical and non-unique solutions for the synaptic weights.

In cancer radiomics, textural features evaluated from image intensity-derived gray-level co-occurrence matrices (GLCMs) have been studied to evaluate gray-level spatial dependence within the regions of interest in the brain. Most of these analysis work with summary statistics (or texture-based features) constructed using the GLCM entries, and potentially overlook other structural properties in the GLCM.  In our proposed Bayesian framework, we treat each GLCM  as a realization of a two-dimensional stochastic functional process observed with error at discrete time points. The latent process is then combined with the outcome model to evaluate the prediction performance. We use simulation studies to assess the performance of our method and apply it to data collected from individuals with lower grade gliomas. We found our approach to outperform competing methods that use only summary statistics to predict isocitrate dehydrogenase (IDH) mutation status.

The global coronavirus pandemic (COVID-19) reached Lagos, Nigeria on February 27, 2020. Since then, COVID-19 infections have been reported in the majority of Nigerian states. We present a spatial Susceptible-Exposed-Infectious-Recovered-Dead (SEIRD) compartmental model of epidemiology to capture the transmission dynamics of the spread of COVID-19 and provide insight that would support public health officials towards informed, data-driven decision making. 

 Data assimilation is a general category of statistical tracking techniques that incorporate and adapt to real-time data as they arrive by sequential statistical estimation.  Data assimilation applied to the SEIRD model receives daily aggregated epidemiological data from the Nigeria Centre for Disease Control (NCDC) and uses this data to perform corrections to the current state vector of the epidemic. In other words, it enhances the operation of the SEIRD model by periodically executing a Bayesian correction to the state vector, in a way that is, at least arguably, statistically optimal. We observe that the prediction improves as data is assimilated over time. It is essential to understand what future epidemic trends will be, as well as the effectiveness and potential impact of government disease intervention measures. Predictions for disease prevalence with and without mitigation efforts are presented via spatiotemporal disease maps.

The theory of D-modules studies linear partial differential equations using the tools of ring theory and homological algebra of the ring D of differential operators. Originally motivated by questions of analysis, it has had considerable applications in other fields of mathematics. After introducing some basics of the theory, we will review some of the speaker's results in local cohomology and the b-function. This serves as a motivation for the study of more general commutative subalgebras of the ring of differential operators in positive characteristic, which we will conclude with. 

A fundamental concept in algebra and geometry is the notion of symmetry of an object or some base space. It can frequently happen in nature that two distinct objects can have related symmetries that move them around (such as a circle having quarter rotations and a square having quarter rotations too). Equivariance os the study of moving two objects that have common symmetries acting upon them and seeing which functions respect these symmetries, as well as how these symmetries can be used to capture important geometric information. In this talk, I will explain precisely what I mean by symmetries acting upon objects, and use this to describe what it means to be equivariant more generally. After this I will give a brief (but gentle) description of what it means to be a sheaf, and finally explore how these sheafy techniques and farming metaphors can help us study geometry, representation theory, and algebra.

Stochastic Black-Scholes Equation under Rough Volatility and Computations with Deep Learning-Based Methods

Rough volatility is a new paradigm in finance. We shall talk about the option pricing problems for rough volatility models. As the framework is non-Markovian, the value function for a European option is not deterministic; rather, it is random and satisfies a backward stochastic partial differential equation (BSPDE) or so-called stochastic Black-Scholes equation. The well-posedness of such kind of BSPDEs and associated Feynman-Kac representations will be discussed. These BSPDEs are also used to approximate American option prices. Moreover, a deep learning-based method will be investigated for the numerical approximations to such BSPDEs and associated non-Markovian pricing problems. Examples will be presented for both European and American options.

This talk is based on joint work with Dr. Christian Bayer from the Weierstrass Institute (Berlin, Germany) and the graduate student Yao Yao from the University of Calgary.

Clinical prediction models are important in clinical practice for the identification of patients at risk of developing an adverse outcome to initiate preventive measures.  An appropriate variable selection is fundamental in the development of a prediction model. It determines the set of variables providing the best fit for the model to make accurate predictions and is often considered one of the most difficult aspects of model building. However, variables are often measured with error in medical research. While it is well known that errors in the measurement of covariates introduce bias and imprecision in the quantification of exposure-outcome relationship, it remains under-studied to what extent misclassification in response causes in variable selection and prediction modeling. We investigate how misclassification affect the performance of variable selection when traditional techniques are used. We provide a general framework to deal with variable selection in the presence of misclassification in the response variable and absence of validation data. The key steps of a maximum likelihood estimation method are outline and performance of the proposed method is investigated in  simulation studies showing convincing results.   

\textbf{Transcendental Series of Reciprocals of Fibonacci Numbers} by Dang-Khoa Nguyen (University of Calgary)
    
    
    \medskip
    
    
    Let $F_1=1$, $F_2=1$, and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$ be the Fibonacci sequence. Motivated by
    the identity
    $\displaystyle\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$, Erd{\"o}s and Graham asked whether $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is irrational for
    any sequence of positive integers $n_1,n_2,\ldots$
    with $\displaystyle\frac{n_{k+1}}{n_k}\geq c>1$. We resolve the transcendence counterpart of their question: as a special case of our main theorem, we have that $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is transcendental when
    $\displaystyle\frac{n_{k+1}}{n_k}\geq c>2$. 
    The bound $c>2$ is best possible thanks to the identity at the beginning.   
    
    I will start from scratch: no knowledge of algebraic/transcendental numbers or diophantine approximation is required. A large part of this talk is accessible to those who know what a convergent series means

L₁-norm regularization is a widely used method in regression applications where shrinkage is afforded in addition to variable selection. In the field of bioinformatics where  is the norm rather than the exception, this has tended to the LASSO being applied to a multitude of datasets in both prediction and association studies. However, of note is that the LASSO can be inconsistent in its selection of variables especially in the presence of highly correlated variables. Since this is once more a norm in omics data, the use of obtaining a consensus gene set based on multiple LASSO runs in a bootstrap-like fashion is investigated. In multiple RNA-Seq datasets (The Cancer Genome Atlas, TCGA), biologically valid signatures are recovered, and the gene sets are validated as predictive in independent testing datasets of similar cancers. Furthermore, a co-expression-based pathway enrichment method finds pathways that have been previously experimentally identified as associated with the respective cancers. An overview of the field of biomarker identification, gene expression analysis and RNA-Seq techniques will also be introduced at the beginning of the presentation.

Ideal lattices form a powerful tool not only for computational number theory but also for cryptography and coding theory, thanks to their underlying structures that enable a variety of useful constructions. In this talk, we will first discuss ideal lattices and their application in computational number theory such as computing important invariants of a number field (the class number, the class group and the unit group). Then we will discuss an application of ideal lattices in cryptography: constructing cryptosystems that are conjectured to be secure under attacks by quantum computers. An application of ideal lattices in coding theory - minimizing the value of the inverse norm sums - will be presented after that.

Machine learning has been an active field of research since the mid 20th century. Over the last decade, however, it’s mindshare has exploded in academia and in industry, often crossing over into popular and political arenas. This explosion was catalyzed by advances in computing (GPUs) and data engineering (massive, curated data sets) catching up to an already mature theory, leading to the latter’s rapid validation. Investment by industry and government followed, leading to a virtuous cycle of theoretical advancement, software and hardware development, and compelling applications. In fleshing out this story, I’ll attempt to balance articulation of the powerful, high-level paradigms that powered machine learning’s rise and fuel optimism for its future with providing enough technical, low-level detail to satisfy a mathematically sophisticated audience.

Information obtained from statistical infectious disease transmission models can be used to inform the development of containment strategies. Inference procedures such as Bayesian Markov chain Monte Carlo are typically used to estimate parameters of such models, but are often computationally expensive. However, in an emerging epidemic, stakeholders must move quickly to contain spread. Here, we explore machine learning methods for carrying out fast inference via supervised classification. We consider the issues of regularization, model choice and parameter estimation. This is done within the context of spatial models, applied to both diseases of agriculture and the COVID-19 epidemic. We also consider how accurate such methods are in comparison with naïve, and much slower, MCMC approaches.

Quantifying and clustering the medical student learning curve for ECG rhythm strip interpretation

Obtaining competency in medical skills such as interpretation of electrocardiograms requires repeated practice and feedback. The diagnostic accuracy of the first-year medical students was measured throughout opportunistic training sessions. We quantified a learning curve that estimates the number of sessions and practice time required to achieve a pre-determined diagnostic accuracy level on the exam using Gompertz and Michaelis-Menten equations. Our discussion is continued with the 3-parameter logistic function and Wickelgren’s (1977) formula in the context of a nonlinear mixed-effect model to describe learning efficiency over training sessions. Finally, the K-means algorithm is used for clustering the students into fast and slow learners to discuss the speed-accuracy tradeoff. We believe that enhanced statistical literacy allows clinicians and educators to apply the findings to medical education practice.

Least-squares spectral and wavelet analyses and their applications in geodesy and geophysics

Fourier transform is one of the equations that changed our world. It has applications in many areas, such as signal processing (time series analysis), differential and partial differential equations, option pricing, cryptography, and many more. The discrete Fourier transform, and wavelet transform are not defined for unequally spaced data/time series, nor they can consider the measurement errors that may be provided in the form of a covariance matrix. In earth and environmental applications, researchers often deal with data/time series that are nonstationary and unequally sampled.

Therefore, optimization methods that can process such series effectively and reliably are extremely demanding. The least-squares spectral and wavelet analyses have shown promise in analyzing any types of data/time series that can be unequally spaced and non-stationary without the need for any pre-processing of the data, such as interpolation, gap filling, and de-spiking. In this talk, I briefly walk through the theoretical parts of these methods, and then I will demonstrate some of their applications in seismology, environmental monitoring, and geodesy.

In this talk I will present a new stochastic multi-particle model for global optimization of non-convex functions on the sphere. This model belongs to the class of Consensus-Based Optimization (CBO) methods. In fact, particles move over the sphere driven by a drift towards an instantaneous consensus point, computed as a combination of the particle locations weighted by the cost function according to Laplace’s principle. The consensus point represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached, then the stochastic component vanishes. To quantify the performances of the new approach, we show that the algorithm is able to perform essentially as good as ad hoc state of the art methods in challenging problems in signal processing and machine learning.

A topological insulator is a material that behaves as an insulator in its bulk, i.e. its interior, but whose surface contains conducting states, meaning that electrons can only move along the surface of the material. The existence and characteristics of such conducting states on the surface are intricately connected to certain topological properties of the matrix functions that describe the bulk. This connection is known as the bulk-boundary correspondence. In this talk, I will first provide an overview of the conventional approaches to establishing the bulk-boundary correspondence, while arguing that these approaches focus only on proving the existence of the conducting states on the surface. I will then present some results that connect other properties of these states, such as their stability and sensitivity to disorder in the material, to the properties of the bulk. These results are derived by leveraging two factorizations of matrix polynomials, namely the Smith normal form and the Wiener-Hopf factorization.