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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.

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.

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.

Abstract: Reproductive value is the relative expected number of off-springs produced by an individual in its remaining lifetime. It is also an invariant function for population processes with birth and death rates independent of the time except in cases when they are periodic. In this study, we prove that the limiting ratio of the survival probability of two branching processes starting from distinct state-time positions is equal to the relative reproductive value. Moreover, we developed a method to obtain the reproductive value for continuous time general branching population models with state-dependent rates and a renewal state. We are using size-biased birth rates for constructing the spinal representation of the branching process, which establishes the relation between survival probability and the Martingale of the process. Further, we provide sufficient conditions for a successful coupling to compare the survival probabilities of two branching population models.

Abstract: I highlight what I consider to be the two major contributions of topology to economic theory: the application of algebraic topology to social choice theory and the application of differential topology to general equilibrium theory. Interesting, but not surprisingly, both of these were directly influenced by the Bourbaki group.

Coffee and muffins at 2:15pm

Abstract: How many lines are spanned by a set of planar points?. If the points are collinear, then the answer is clearly "one". If they are not collinear, however, several different answers exist when sets are finite and "how many" is measured by cardinality. I will discuss a bit of the history of this problem and present a recent extension to the discretized and continuum settings, obtained in collaboration with T. Orponen and H. Wang. No specialized background will be assumed.

Pablo is a leader in the area of fractal geometry. He is a recent speaker at the International Congress of Mathematicians; and a Canada Research Chair at UBC.

Abstract: An $(m,n)$-generalized bent function is a function from $\mathbb{Z}_2^n$ to $\mathbb{Z}_m$ so that its associated Fourier transformations have constant absolute value. It is known that an $(m,n)$-generalized bent function exists whenever one of the following holds:

1. both $m$ and $n$ are even.
2. $4 \mid m$.

Abstract: We address the question: when does a purely infinite, simple, separable classifiable C*-algebra have a groupoid model? The answer was given up to stable isomorphism by Jack Spielberg and later in the unital case by L.O. Clark, J. Fletcher and A. an Huef. We will describe an alternate approach. This first requires a construction of Deaconu and Renault which I will describe. We further need a construction of mine into dynamical systems which I call 'binary factors' and finally an idea of orbit-breaking for Deaconu-Renault groupoids. This is joint work with Robin Deeley and Karen Strung.

Abstract: This talk is devoted to the Relativistic Vlasov-Maxwell system in space dimension three. We prove the local smooth solvability for weak topologies. This result is derived from a representation formula informing about the momentum spread, showing that the domain of influence in momentum is controlled by mild information in the sense that derivatives are not (or less than before) implied. In doing so, we develop a Radon Fourier analysis on the RVM system, which leads to the study of a class of singular weighted integrals. The end of my talk, I show how this method is applied in order to construct smooth solutions to the RVM system in the regime of hot, dense and strongly magnetized plasmas. This is a joint work with C. Cheverrey.

Abstract: Quantifying the relationship between daily changes in air pollution and short-term effects on mortality and hospitalisations is a complex task with many steps. Air pollution data contains missing values and outliers. Uncertainty in exposures should be reflected in uncertainty in effect sizes. Inference must account for the multitude of factors unrelated to air pollution that can influence mortality. Exposure-response effects are non-linear. Analyses across multiple cities and regions must take into account possible city-level variation in effects.

This talk will describe ongoing work undertaken in under contract from Health Canada to estimate health effects from pollution in 50 Canadian cities. A key feature of the project is the case-crossover model, a form of partial likelihood which adjusts for most changes in time using control days.

Abstract: I will present our recent methodology for integrating functional data into deep neural networks. The model is defined for scalar responses with multiple functional and scalar covariates. A by-product of the method is a set of dynamic functional weights that can be visualized during the optimization process. This visualization leads to a greater interpretability of the relationship between the covariates and the response relative to conventional neural networks. The model is shown to perform well in a number of contexts including prediction of new data and recovery of the true underlying relationship between the functional covariate and scalar response; these results were confirmed through real data applications and simulation studies. An R package (FuncNN) has also been developed on top of Keras, a popular deep learning library—this allows for general use of the approach.

Abstract: If a tricky devil challenged you to complete a Sudoku puzzle of their design, what restrictions could you place on their design to guarantee it was impossible for them to thwart your attempts? More specifically, if you could dictate exactly how many times that devil could use each symbol or pre-fill any one row, column, or box, could you safely bet that their deviously designed puzzle was still completable? This notion of seeking out guaranteed completability conditions for Sudoku drives the discussion as we delve into various graph and matrix representations of the Sudoku, explore relaxed notions of completability such as fractional completability, and use these tools to approach an answer. Due to the ongoing nature of this research, this talk is designed to serve as an introduction to the machinery of the argument being employed but does not conclude with a definitive answer.

Abstract: We address the question: when does a purely infinite, simple, separable classifiable C*-algebra have a groupoid model? The answer was given up to stable isomorphism by Jack Spielberg and later in the unital case by L.O. Clark, J. Fletcher and A. an Huef. We will describe an alternate approach. This first requires a construction of Deaconu and Renault which I will describe. We further need a construction of mine into dynamical systems which I call 'binary factors' and finally an idea of orbit-breaking for Deaconu-Renault groupoids. This is joint work with Robin Deeley and Karen Strung.

The Pacific Institute for the Mathematical Sciences will host an evening of

Math Mania

This "Girls in Math" event will take place
6:30-8:00 p.m. Tuesday, March 7, 2023 at
St. Margaret's School
1080 Lucas Ave.
Victoria, B.C.

"Fun" methods to teach math and computer science concepts to children (and adults!) by games and art will be demonstrated. Lots of hands-on activities!

• A Sorting Network
• An Impossible Balancing Act
• Bubbles
• Mathematical Puzzles
• The Guessing Game
• Sudoku
• The Penny Game
• The Set Game
• and lots more!

For more information contact:

PIMS UVic Site Office, Department of Mathematics and Statistics, University of Victoria
at (250) 472-4271
email: pims@uvic.ca

Zoom link.

joint work with E. Lesigne, A. Quas, J. Rosenblatt.

Abstract: Let S= (s_1<s_2<\dots) be a strictly increasing sequence of positive integers. We say S is good if for every real \alpha the sequence

\left(\frac1N\sum_{n\le N}e^{2\pi is_n\alpha}\right)_N

of complex numbers is convergent. Equivalently, the sequence S is good if for every real \alpha the sequence (s_n\alpha) has asymptotic distribution modulo 1. We are interested in finding out what the limit probability measure

\mu_{S,\alpha}=\lim_N \frac1N\sum_{n\le N}\delta_{s_n\alpha}

can be. It turns out that for an irrational \alpha the limit measure must be continuous. So now the main question is can it be any continuous measure? An affirmative answer would also affirm Furstenberg's x2x3 conjecture. What we can prove is that the limit measure can be any absolutely continuous measure.

Abstract: We propose to estimate modal regression with spatial data observed over a rectangular domain by assuming that the conditional mode of the response variable given covariates follows a nonparametric regression structure. We study the newly developed spatial modal regression utilizing the local linear approximation augmented with shrinking bandwidths. The asymptotic normal distributions of the resultant modal estimators are established and the explicit formulas for their asymptotic biases and variances are derived under mild regularity assumptions. The selection of optimal bandwidths in theory and practice are discussed. We also show that the targeted spatial modal regression can be utilized as an alternative to a nonparametric spatial robust regression when the data are symmetrically distributed. The asymptotic distributions for such modal-based robust estimators are derived with appropriate choices of bandwidths, which demonstrate that the suggested estimators can achieve the full asymptotic efficiency of the mean estimators when there are no outliers and the error distribution is normal. Monte Carlo simulations and a real data analysis on soil data are presented to illustrate the good finite sample performance of the estimators. We in the end generalize the propounded spatial modal regression to an additive sum of the form to avoid the curse of dimensionality and develop a kernel-based backfitting algorithm for estimating. We substantiate that the proposed modal estimator of each additive component is asymptotically normal and converges at the univariate nonparametric modal optimal rate.

Abstract: Persistent Homology (PH) is an active research area in Algebraic Topology, especially in Topological Data Analysis, that concerns capturing structural information of data. It arises in critical applications such as machine learning, neuroscience, cosmology, medicine and biology among others. One of the primary challenges in TDA is distinguishing between signal (meaningful structures) and noise (arising from local randomness or other inaccuracies). A key tool of PH is the persistence diagram, which captures the birth and death of various features in a filtration of a dataset. Characterizing the distribution of these diagrams is an open problem in the field. This talk will introduce the open problem, address some fundamental results, and discuss our results so far on a given approach to the problem.

Abstract: Suppose we have a set P of n points in the plane, and another set S of n^{1/3} points in convex position. What can we say about the structure of this arrangement if P contains many translates of the set S? I will discuss a recent result showing that if P contains around n translates of S, then the translation vectors must come from a generalized arithmetic progression of low dimension.

The motivation for this problem comes from incidence geometry, where many constructions for strictly convex curves having many incidences with pointsets follow this general outline. In particular, I will discuss applications to the Erdos unit distance conjecture. This is joint work with Jozsef Solymosi and Ethan White.

Abstract: I will discuss four applications of statistics to COVID-19 research. 1) The disease COVID-19 is caused by infection by the SARS-CoV-2 virus. However, variation in human genetics can modulate the severity of the disease, or susceptibility to infection. HostSeq is a resource of ~10,000 DNA sequences of COVID-19+ Canadians. I will discuss statistical challenges and polygenic risk scores with HostSeq data. 2) Due to asymptomatic infection, and underreporting, the burden of COVID-19 is greater than indicated by confirmed case counts. I will discuss methods to estimate the true burden of COVID-19, and to combine confirmed case counts with serology surveys to provide more accurate estimates of prevalence. 3) I will discuss a statistical operationalization of SIR models to determine the lag between changes in non-pharmaceutical interventions and statistically significant changes in confirmed case count trajectories. 4) I will discuss evidence for neurological manifestation of COVID-19 in brain imaging. These applications are joint work with MAGPIE, UVic, SickKids and Oxford.

Abstract: I will describe the large-scale behavior of a random growth model (Internal DLA) on random planar maps which approximate a random fractal surface embedded in the plane (Liouville quantum gravity, LQG). No prior knowledge of these objects will be assumed. Joint work with Ewain Gwynne.

Math Challengers is a math competition for grades 8-10 students. Please see https://www.mathchallengers.ca/events.htm to sign up for the in-person version of the event at Camosun College.

Abstract: The concordance group of virtual knots is known to be nonabelian but still somewhat mysterious. Using the Gordon-Litherland form, we introduce new invariants of virtual knots defined in terms of non-orientable spanning surfaces for the knot. The associated mock Seifert matrices lead to a new algebraic concordance group, which is abelian but not finitely generated. This talk is based on joint work with Homayun Karimi.

Note that you need to register in advance at this web site.

The past twenty-five years have heralded an unparalleled increase in understanding of cancer. At the same time, mathematical modelling has emerged as a natural tool for unravelling the complex processes that contribute to the initiation and progression of tumours, for testing hypotheses about experimental and clinical observations, and assisting with the development of new approaches for improving its treatment. In this talk I will reflect on how increased access to experimental data is stimulating the application of new theoretical approaches for studying tumour growth. I will focus on two case studies which illustrate how mathematical approaches can be used to characterise and quantify tumour vascular networks, and to understand how microstructural features of these networks affect tumour blood flow.

Abstract: It is well known that a maximal planar graph of order at least 3 is 3-colourable if and only if it is Eulerian. It is also known that if a maximal planar graph of order at least 3 has exactly two vertices of odd degree, then these vertices are nonadjacent. I will show that these two results are equivalent.

The talk should be accessible to undergraduate students with a basic knowledge of graph theory.

Abstract: We consider a family of Pimsner algebras associated to correspondences constructed from self-similar group actions and we investigate the equilibrium states (the KMS states) for the natural time evolution on these algebras. For all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given in a very concrete way by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz-Pimsner algebra; if the self-similar group is contracting, then the Cuntz-Pimsner algebra has only one KMS state. We apply these results to a number of examples, including self-similar actions associated to integer dilation matrices, and the canonical self-similar actions of the basilica group and the Grigorchuk group.

Abstract: DNA methylation is an epigenetic mark intrinsically involved in regulating the activity of DNA, and methylation levels are known to change with age, exposures, and disease status. DNA methylation can be measured with a sequencing technique that gives methylated and unmethylated counts at each targeted position in the genome, but the data are very noisy. I will describe an over-dispersed quasi-binomial model with functional smoothing to model DNA methylation patterns in small genomic regions, and how these patterns depend on covariates. Results will be illustrated with an analysis of DNA methylation and a biomarker strongly associated with rheumatoid arthritis.

Math Challengers is a math competition for grades 8-10 students. Please see https://www.mathchallengers.ca/events.htm to sign up for the online version of the event.

Abstract: Farrell and Jones proved that the automorphism groups of compact hyperbolic manifolds in dimensions n≥1 do not have the homotopy-type of finite CW complexes. Here "automorphism group" means groups of diffeomorphisms, PL-homeomorphisms, and homeomorphisms respectively, i.e. we are not discussing homotopy-equivalences or isometries. Farrell and Jones primarily used the machinery of "Higher Simple Homotopy Theory", which reduces the problem of computing homotopy groups of automorphism groups (in a range) to that of the group of automorphisms of S^1 x D^{n-1}, which was worked-out by Hatcher and Wagoner, plus a good amount of K-theory. With David Gabai we found an alternate route to such theorems, that avoids Higher Simple Homotopy Theory entirely. For us the results occur in dimensions n≥4. I will describe how these kinds of arguments work.

This event is part of the PIMS sponsored Seminar Series: Mathematics of Ethical Decision-making Systems

Download Poster.

Speakers:

• Midori Ogasawara, Assistant Professor, UVic Sociology
• Patricia Cochran, Associate Professor, UVic Law
• Freya Kodar, Associate Professor, UVic Law
• Jentery Sayers, Associate Professor, UVic English

AGENDA:

• 3:00 pm Dr. Midori Ogasawara will speak on the social consequences of surveillance and data collection.
• 3:30 pm Break for refreshments
• 4:00 pm Dr. Patricia Cochran and Dr. Freya Kodar
  Title: Technology, access to justice and narratives of law in an unequal society In this presentation, we will share some information about our ongoing research collaborations about the relationship between access to justice and technologically-mediated decision-making (including algorithmic credit-scoring and the online Civil Resolution Tribunal). We will discuss some of the critical questions we think require interdisciplinary attention given the growing significance of technology in legal processes, and the challenges that such processes pose for important legal ideas such as equality.
• 4:30 pm Dr. Jentery Sayers
  Title: Modelling Games as Activity Systems to Evaluate the Generative Contradictions between Play and Work I’ll talk about the articulation of genre studies with cultural-historical activity theory to consider conflicting motives in video games. I’m hoping to focus on contradictions related to labour (e.g., between dev teams and players; between types of players), metagaming (e.g., between social norms and player activities that may be considered cheating), and genre (between storytelling and mechanics, or “ludonarrative dissonance”). I’ll ground the talk in the model of activity theory rather than the particulars of these contradictions and conflicts.
• 5:00 -5:30 pm Panel Discussion

**For those unable to attend in person, there will be a virtual component - e-mail Kristina at pims@uvic.ca for this ‘virtual attendance’ information.**

Abstract: Products of permutations are used to model problems in fields ranging from representation theory and algebraic geometry to topological graph theory. I will show how probability can be used to study such problems, with a focus on how graphs may be embedded on different surfaces.

Abstract: We introduce a model of a controlled random process. In this model, the vertices of a hypergraph are ordered randomly and then revealed, one by one, to an algorithm. The algorithm must decide, immediately and irrevocably, whether to keep each observed vertex. Given the total number of observed vertices ("time"), the algorithm's goal is to succeed - asymptotically almost surely - in completing a hyperedge by keeping ("purchasing") the smallest possible number of vertices.

We analyse this model in the context of random graph processes, where the corresponding hypergraph defines a natural graph property, such as minimum degree, connectivity, Hamiltonicity and the containment of fixed-size subgraphs.

Joint work with Alan Frieze and Michael Krivelevich.

Abstract: In single-cell RNA-sequencing studies, researchers often model the variation between cells with a latent variable, such as cell type or pseudotime, and investigate associations between the genes and the latent variable. As the latent variable is unobserved, a two-step procedure seems natural: first estimate the latent variable, then test the genes for association with the estimated latent variable. However, if the same data are used for both of these steps, then standard methods for computing p-values in the second step will fail to control the type I error rate.

In my talk, I will introduce two different approaches to this problem. First, I will apply ideas from selective inference to develop a valid test for a difference in means between clusters obtained from the hierarchical clustering algorithm. Then, I will introduce count splitting: a flexible framework that enables valid inference after latent variable estimation in count-valued data, for virtually any latent variable estimation technique and inference approach.

This talk is based on joint work with Jacob Bien (University of Southern California), Daniela Witten and Anna Neufeld (University of Washington), as well as Alexis Battle and Joshua Popp (Johns Hopkins University).

Abstract
The 15-puzzle which acts as an inspiration for this talk is a simple game consisting of 15 square tiles numbered 1–15 in a frame that is 4 tiles high and 4 tiles wide, leaving one unoccupied tile position. Tiles in the same row or column of the open position can be moved by sliding them horizontally or vertically. Like the popular Rubik's cube, the sliding puzzle is often ‘jumbled up‘ and the goal of the puzzle is to place the tiles in numerical order from left to right and top to bottom with the empty tile in the bottom right corner. In the late 1880’s a famous American puzzler named Sam Loyd offered $1000 to anyone who could provide him with a series of moves that would swap only the tiles labelled 14 and 15 while leaving all other positions in their standard numerical order. In this talk we will discuss which mathematical principles guaranteed that Sam Loyd would keep his money and discuss how the mathematician Richard Wilson used graph theory ‘inception’ to generalize this notion to all sliding puzzles, determining exactly which solutions were possible and which could never be obtained.

Abstract: C*-algebras are often said to resemble `noncommutative spaces.' One way of making this idea precise makes use of the idea of Morita Equivalence. This is an introductory talk on Morita equivalence and its role in K-theory, representation theory (of groups), and group actions.

Zoom link.

Abstract: The Perceptron model was proposed as early as the 1950's as a toy model of a one-layer neural network. The basic model consists of a set of solutions (either the Hamming cube or the sphere of dimension n) and a set of constraints given by independent n-dimensional Gaussian vectors. The constraints are that the inner product of a solution vector with each constraint vector scaled by sqrt{n} must lie in some interval on the real line. Probabilistic questions about the model include the satisfiability threshold (or the "storage capacity") and questions about the typical structure of the solution space. Algorithmic questions include the tractability of finding a solution (the learning problem in the neural network interpretation). I will describe the model, the main problems, and recent progress.

Zoom link

Abstract: 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 an AIDS dataset with new findings.

The Helly number of a set in the plane is the smallest N such that the following is true. If any N members of a finite family of convex sets contains a point of S, then there is a point of S which is contained in all members of the family. An exponential lattice with base x consists of points whose coordinates are positive integer powers of x. We prove lower and upper bounds on Helly numbers of exponential lattices in terms of x, and we determine their values exactly in some cases. We also consider asymmetric exponential lattices, and characterise those that have finite Helly numbers.

Joint work with Gergely Ambrus, Martin Balko, Attila Jung and Márton Naszódi.

Zoom link.

Abstract: It is common in probability theory and statistics to study distributional convergences of sums of random variables conditioned on another such sum. In this talk I will present a novel approach using Stein’s method for exchangeable pairs that allows to derive a conditional central limit theorem of the form $(X_n|Y_n = k)$ with explicit rate of convergence as well as its extensions to a multidimensional setting. We will apply these results to particular models including pattern counts in a random binary sequence and subgraph counts in Erdös-Rényi random graph. This talk is based on joint work with Partha S. Dey.

Abstract:Gene co-expression network analysis is widely used in microarray and RNA sequencing data analysis. It groups genes based on their co-expression network. And genes within a group infer similarity in function or coregulation in the pathway.

In literature, the approaches to group genes are mainly unsupervised, which may introduce instability and variation across different datasets. Inspired by ensemble learning, we propose a novel approach that ensemble supervised and unsupervised learning techniques and simultaneously works on two tasks, gene module identification and phenotype prediction, during the data analysis process. The identified gene modules from this approach could suggest more candidate genes to the original pathway, and those genes are potential biomarkers for pathway-related diseases. In addition, the novel approach also improves the prediction accuracy for phenotypes. The algorithm can be used as a general prediction algorithm. And, as it is specially designed to handle large samples, it is suitable for handling single-cell data with many cells. We showcased the use of the algorithm in single-cell cell-type auto-annotation.

Abstract: I'll show an algorithm for 3 colouring triangle free graphs embedded on the Klein bottle that runs in polynomial time if most of the faces are 4 faces. Joint work with Jean-Sebastian Sereni and Zdenek Dvorak.

Abstract: Verifying nonlinear stability of a laminar fluid flow against all perturbations is a classic challenge in fluid dynamics. All past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. This "energy method" cannot show global stability of any flow in which perturbation energy may grow transiently. For the many flows that allow transient energy growth but seem to be globally stable (e.g. pipe flow and other parallel shear flows at certain Reynolds numbers) there has been no way to mathematically verify global stability. After explaining why the energy method was the only way to verify global stability of fluid flows for over 100 years, I will describe a different approach that is broadly applicable but more technical. This approach, proposed in 2012 by Goulart and Chernyshenko, uses sum-of-squares polynomials to computationally construct non-quadratic Lyapunov functions that decrease monotonically for all flow perturbations. I will present a computational implementation of this approach for the example of 2D plane Couette flow, where we have verified global stability at Reynolds numbers above the energy stability threshold. This energy stability result for 2D Couette flow had not been improved upon since being found by Orr in 1907. The results I will present are the first verification of global stability – for any fluid flow – that surpasses the energy method. This is joint work with Federico Fuentes (Universidad Católica de Chile) and Sergei Chernyshenko (Imperial College London).

Abstract: Take an Ising model with very low temperature. What is the largest p such that the Ising model dominates Bernoulli percolation with parameter p ? We will show that the answer to this question depends drastically on the geometry of the graph. We also obtain similar results for for two Ising models at very low, but close temperatures. A process is a finitary factor of iid if it can be written as a measurable and equivariant function of an iid process. As an application of the domination results, we show that the very low temperature Ising model on a nonamenable graph is a finitary factor of iid. This is in stark contrast with the amenable setting, where it is known through a celebrated result of Van Den Berg and Steif that the low temperature Ising model is not a finitary factor of iid. Joint work with Yinon Spinka.

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

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

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

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

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

Abstract: The arboricity \Gamma(G) of an undirected graph G = (V,E) is the minimal number such that E can be partitioned into \Gamma(G) forests. Nash-Williams' formula states that \Gamma(G) = \ceil{ \gamma(G) }, where \gamma(G) is the maximum of |E_H|/(|V_H|-1) over all subgraphs (V_H, E_H) of G with |V_H| ≥ 2.

The Strong Nine Dragon Tree Conjecture states that if \gamma(G) ≤ k + d / (d+k+1) for natural numbers k, d, then there is a partition of the edge set of G into k+1 forests such that one forest has at most d edges in each connected component.

We settle the conjecture for d ≤ k + 1. For d ≤ 2(k+1), we cannot prove the conjecture, however we show that there exists a partition in which the connected components in one forest have at most d + \ceil{kd/(k+1)} - k edges.

As an application of this theorem, we show that every 5-edge-connected planar graph G has a 5/6-thin spanning tree, a spanning trees whose edges fill up at most 5/6 of every cut. This theorem is best possible, in the sense that we cannot replace 5-edge-connected with 4-edge-connected, even if we replace 5/6 with any positive real number less than 1. This strengthens a result of Merker and Postle which showed 6-edge-connected planar graphs have a 18/19-thin spanning tree. This is joint work with Benjamin Moore.

Abstract: Schrödinger operators with delta potentials are of longstanding interest for admitting closed analytic solutions, and they receive renewed attention for connections to the Kardar–Parisi–Zhang equation. Along with a review of this background, the talk will discuss a standard model of the operators in 2D and its counterpart in Feynman–Kac formulas.

We provide a brief motivational overview of recent developments of extensions of stochastics tools to deal with uncertainty. These are: Peng's nonlinear expectations and Ito's calculus without probabilities. We then describe a non-probabilistic version of a supermartingale theory closely motivated by financial considerations of no-arbitrage. The basic object replacing the classical filtered probability space is a structured trajectory set which allows the definition of conditional outer integrals as well as null sets. The conditional outer integrals are non linear functionals that allow to circumvent the linearity of the classical conditional expectations in proofs and definitions. Integrability notions emerge in our setting through non-classical conditional integral operators that lead to the special case of non-probabilistic martingales.

One can define non-probabilistic supermartingales and prove analogous of classical results like: Doob's optional sampling theorem, Dubin's upcrossing inequalities and Doob's a.e. convergence for non-negative supermartingales. All constructions and results have a hedging and superhedging interpretation and there is a direct way in which the new results generalize the classical case. Null sets appearing in the results have a financial interpretation and are handled in a more concrete way than in the classical theory.

See announcement and abstract (PDF)

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.

Zoom link.

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.

Zoom link.

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.

See announcement and abstract.

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.

Zoom link.

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)

Zoom link.

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.

Zoom link.

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.

Zoom link

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.

Zoom link

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.

Zoom link.

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 ***

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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.

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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.