• home
• contact us
• events

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

MACKENZIE CARR

BSc Hons (Acadia University, 2018)

Enumerating Digitally Convex Sets in Graphs

Department of Mathematics and Statistics

Wednesday, June 24, 2020 10:00 A.M
Remote Defence

Supervisory Committee: Dr. Kieka Mynhardt, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Ortrud Oellermann, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
External Examiner: Dr. Danielle Cox, Department of Mathematics, Mount Saint Vincent University
Chair of Oral Examination: Dr. Catherine Costigan, Department of Psychology, UVic

Abstract

Given a finite set V , a convexity, C , is a collection of subsets of V that contains both the empty set and the set V and is closed under intersections. The elements of C are called convex sets. We can define several different convexities on the vertex set of a graph. In particular, the digital convexity, originally proposed as a tool for processing digital images, is defined as follows: a subset S _ V (G) is digitally convex if, for every v 2 V (G), we have N[v] _ N[S] implies v 2 S. Or, in other words, each vertex v that is not in the digitally convex set S needs to have a private neighbour in the graph with respect to S. In this thesis, we focus on the generation and enumeration of digitally convex sets in several classes of graphs. We establish upper bounds on the number of digitally convex sets of 2-trees, k-trees and simple clique 2-trees, as well as conjecturing a lower bound on the number of digitally convex sets of 2-trees and a generalization to k-trees. For other classes of graphs, including powers of cycles and paths, and Cartesian products of complete graphs and of paths, we enumerate the digitally convex sets using recurrence relations. Finally, we enumerate the digitally convex sets of block graphs in terms of the number of blocks in the graph, rather than in terms of the order of the graph.

SUSPENDED UNTIL FURTHER NOTICE

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

Math Mania

This Alternative Math Education event will take place
6:30-8:00 p.m. Thursday, May 14, 2020 at
Lakehill Elementary
1031 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@math.uvic.ca

bcCOVID-19 group Lecture:

Real time modelling of the COVID-19 epidemic - perspectives from British Columbia

May 14, 2020 01:00 PM in Vancouver

Register at : https://www.pims.math.ca/scientific-event/200514-pbglrtmcepbc

The COVID-19 global pandemic has led to unprecedented public interest in mathematical modelling as a tool to understand the dynamics of disease spread and predict the impact of public health interventions.

In this pair of talks, we will describe how mathematical models are being used, with particular reference to the British Columbia epidemic.

In the first talk, Prof. Caroline Colijn (Dept. of Mathematics and Statistics, Simon Fraser University) will outline the key features of the British Columbia data and focus on how modelling has allowed us to estimate the effectiveness of the provincial response.

In the second talk, Prof. Daniel Coombs (Dept. of Mathematics and Inst. of Applied Mathematics, University of British Columbia) will describe forward-looking modelling approaches that can provide some guidance as the province moves towards partial de-escalation of measures.

Each talk will be 30 mins in length and followed by a question and discussion period.

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

RONGZHENG XIE

BSc (Swansea University, 2014)

“A population approach to systems of Izhikevich neurons: can neuron interaction cause bursting?”

Department of Mathematics and Statistics

Thursday, April 23, 2020 8:30 A.M.
Remote Defence

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

External Examiner: Dr. Julien Modolo, Inserm -LTSI, Université de Rennes

Chair of Oral Examination: Dr. Caterina Valeo, Department of Mechanical Engineering, UVic

Abstract
In 2007, Modolo and colleagues derived a population density equation for a population of Izhekevich neurons. This population density equation can describe oscillations in the brain that occur in Parkinson’s disease. Numerical simulations of the population density equation showed bursting behaviour even though the individual neurons had parameters that put them in the tonic firing regime. The bursting comes from neuron interactions but the mechanism producing this behaviour was not clear. In this thesis we study numerical behaviour of the population density equation and then use a combination of analysis and numerical simulation to analyze the basic qualitative behaviour of the population model by means of a simplifying assumption: that the initial density is a Dirac function and all neurons are identical, including the number of inputs they receive, so they remain as a point mass over time. This leads to a new ODE model for the population. For the new ODE system, we define a Poincaré map and then to describe and analyze it under conditions on model parameters that are met by the typical values adopted by Modolo and colleagues. We show that there is a unique fixed point for this map and that under changes in a bifurcation parameter, the system transitions from fast tonic firing, through an interval where bursting occurs, the number of spikes decreasing as the bifurcation parameter increases, and finally to slow tonic firing.

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

JOSEPH ANTHONY HORAN

MSc (University of Victoria, 2015)
BMath (University of Waterloo, 2013)

“Spectral gap and asymptotics for a family of cocycles of Perron-Frobenius operators”

Department of Mathematics and Statistics

Friday, April 17, 2020 9:30 A.M.
Remote Defence

Supervisory Committee:
Dr. Christopher Bose, Department of Mathematics and Statistics, University of Victoria (CoSupervisor)
Dr. Anthony Quas, Department of Mathematics and Statistics, UVic (Co-Supervisor)
Dr. Sue Whitesides, Department of Computer Science, UVic (Outside Member)

External Examiner:
Dr. Jérôme Buzzi, Département de Mathématiques d’Orsay, Université Paris-Saclay

Chair of Oral Examination:
Dr. Elisabeth Gugl, Department of Economics, UVic

Abstract
At its core, a dynamical system is a set of things and rules for how they change. In the study of dynamical systems, we often ask questions about long-term or average phenomena: whether or not there is an equilibrium for the system, and if so, how quickly the system approaches that equilibrium. These questions are more challenging in the non-autonomous (or random) setting, where the rules change over time. The main goal of this dissertation is to develop new tools with which to study random dynamical systems, and demonstrate their application in a nontrivial context. We prove a new Perron-Frobenius theorem for cocycles of bounded linear operators which preserve and sometimes contract a cone in a Banach space; this new theorem provides an explicit upper bound for the second-largest Lyapunov exponent of the cocycle, which determines how quickly the system approaches its equilibrium-like state. Using this theorem and other tools (including a new Lasota-Yorke-type inequality for Perron-Frobenius operators for use with a family of maps), we show that a class of cocycles of piecewise linear maps has a Lyapunov spectral gap (hence answering the equilibrium question in the affirmative), and we moreover have an explicit lower bound on the spectral gap. We also prove asymptotics for a family of cocycles arising from a perturbation of a fixed map with two invariant densities; we obtain a linear upper bound for the second-largest Lyapunov exponent, and the bound is sharp, in the sense that there are members of this family of perturbations where the second-largest Lyapunov exponent is linear in the perturbation parameter. The sharpness example is studied through an in-depth determinant-free linear algebra computation for Markov operators.

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

MATTHEW PARKER
BSc (University of Victoria, 2016)

“N-mixture models with auxiliary populations and for large population abundances”

Department of Mathematics and Statistics

Friday, April 17, 2020 9:00 A.M.
Remote Defence

Supervisory Committee:
Dr. Laura Cowen, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Julie Zhou, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Simon Bonner, Department of Biology, Western University

Chair of Oral Examination:
Dr. Hossein Nassaji, Department of Linguistics, UVic

Abstract
The key results of this thesis are (1) an extension of N-mixture models to incorporate the additional layer of obfuscation brought by observing counts from a related auxiliary population (rather than the target population), (2) an extension of N-mixture models to allow for grouped counts, the purpose being two-fold: to extend the applicability of Nmixtures to larger population sizes, and to allow for the use of coarse counts in N-mixture models, (3) a new R package allowing the easy application of the new Nmixture models, (4) a new R package allowing for optimization of multi-parameter functions using arbitrary precision arithmetic, which was a necessary tool for optimization of the likelihood in large population abundance N-mixture models, as well as (5) simulation studies validating the new grouped count models and comparing them to the classic N-mixtures models.

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

CHRIS BRUCE BSc (University of Victoria, 2014)

“C*-algebras from actions of congruence monoids”

Department of Mathematics and Statistics

Thursday, April 16, 2020 10:00 A.M.
Remote Defence

Supervisory Committee:
Dr. Marcelo Laca, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Ian Putnam, Department of Mathematics and Statistics, UVic (Member)
Dr. Michel Lefebvre, Department of Physics and Astronomy, UVic (Outside Member)

External Examiner:
Dr. Judith A. Packer, Department of Mathematics, University of Colorado at Boulder

Chair of Oral Examination:
Dr. David Berg, Department of Chemistry, UVic

We initiate the study of a new class of semigroup C*-algebras arising from number theoretic considerations; namely, we generalize the construction of Cuntz, Deninger, and Laca by considering the left regular C*-algebras of ax+b-semigroups from actions of congruence monoids on rings of algebraic integers in number fields. Our motivation for considering actions of congruence monoids comes from class field theory and work on Bost–Connes type systems. We give two presentations and a groupoid model for these algebras, and establish a faithfulness criterion for their representations. We then explicitly compute the primitive ideal space, give a semigroup crossed product description of the boundary quotient, and prove that the construction is functorial in the appropriate sense. These C*-algebras carry canonical time evolutions, so that our construction also produces a new class of C*-dynamical systems. We classify the KMS (equilibrium) states for this canonical time evolution, and show that there are several phase transitions whose complexity depends on properties of a generalized ideal class group. We compute the type of all high temperature KMS states, and consider several related C*-dynamical systems.

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

JESSE JOHNSON
BSc (University of Victoria, 2010)
New and Existing Results on Circular Words
Department of Mathematics and Statistics

Wednesday, April 15, 2020 10:30 A.M
Remote Defence

Supervisory Committee: Dr. James Currie, Department of Mathematics and Statistics, University of Victoria
(Co-Supervisor) Dr. Gary MacGillivray, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner: Dr. Jacobus Swarts, Department of Mathematics, Vancouver Island University

Chair of Oral Examination: Dr. Gina Harrison, Department of Educational Psychology and Leadership Studies, UVic

Abstract
A brief history on circular words, also known as necklaces, is given, from their early conception to present results. We introduce the concept of level word, that being a word containing a nearly equal amount of each letter. We show the existence of a level circular word on three letters avoiding squares and having for all lengths greater than 18. This is accomplished through a modification of Shur’s construction of square-free circular words. A word on two letters is called a Frankel-Simpson word if the only squares it contains are 00, 11, and 0101. Using the result mentioned above and several computer searches, we show the existence of circular Frankel Simpson words for every length greater than 84.

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

Sofiia ROMANOVSKA, BSC

Evaluating citizen science Golden Eagle migration data collected by the Rocky Mountain Eagle Research Foundation, Mount Lorette, Alberta 1993-2018

April 9, 2020 10:00am

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

Chair of Oral Examination:
Dr. Michelle Miranda, Department of Mathematics and Statistics, UVic

This event has been postponed but you can watch the film online by registering below.

The Committee of Women in Mathematics of International Mathematical Union is celebrating International Women Mathematician day on May 12. They have an agreement with ZALA films for a free screening of their beautiful film “Secrets of the Surface” devoted to Maryam Mirzakani. Please register here https://may12.womeninmaths.org/screening-secrets and you will receive a link by email for a free screening at the date you chose, in the period April 1 to May 15.

POSTPONED TO NEW DATE TBA.

The UVic AWM Student Chapter and the Department of Math and Stats EDI Committee are hosting a FREE screening of the film Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani. Check out the trailer here: http://www.zalafilms.com/secrets/

When: Friday April 3rd from 4:30-5:45pm.
Where: DSB C103 (https://goo.gl/maps/V9LgUccPYEdH532P7)

Light snacks will be provided by SIGMAS on behalf of the AWM Student Chapter.

Bring your friends! Anyone is welcome to attend.

***** POSTPONED WITH NEW DATE TBA *****

Abstract: Phase I clinical trials explore the toxicity of new therapeutic agents. Phase II trials seek to establish some sign of efficacy. But practical applications vary creating a suite of statistical expressions for these trials’ goals and objectives. There is now a robust statistical literature on these topics. We review the development of adaptive dose selection procedures in early phase clinical trials. We will show how a suite of modern approaches developed out of classical stochastic approximation, random walks and optimal design theory. Critical effects that design choices have on analysis procedures are described. We focus on the distinguishing concepts and goals behind the various approaches.

Download poster (PDF).

POSTPONED TO NEW DATE TBA

DEPARTMENT LANSDOWNE LECTURE

Abstract

Progress in dissecting the genetic determinants of complex human traits requires cost-efficient genomic technologies, biological knowledge accessible in open data resources, and computationally feasible statistical methods that reliably infer association in the context of high-dimensional multiple testing. It has been estimated that there are 8 million common single nucleotide polymorphisms (SNPs) with variants occurring at a frequency of at least 1% in a population. Well-developed array technologies, widely used in genome-wide association studies (GWAS) of complex traits, can genotype individuals at 1 million SNP positions chosen to capture common variation. Emerging next generation sequencing (NGS) technologies generate genotypes at the nucleotide level, including rare variants, and enable imputation of 8-9 million high-quality unmeasured SNPs from 1 million genotyped SNPs. At the same time, investigators are focussing on use of population biobanks with many traits available on each individual. High-dimensional GWAS, NGS, and trait data, together with variant, gene, and biological pathway annotation resources are increasingly available for data mining and meta-analysis, but present major analytic challenges including joint analysis of multiple data types, use of biological knowledge to inform statistical modelling, and choice of study design. In this talk, I will highlight some of these challenges in two settings: joint modelling of multiple traits in a longitudinal study of unrelated individuals; and discovery of rare mutations in families ascertained on disease status.

***** POSTPONED TO NEW DATE TBA *****

PUBLIC LANSDOWNE LECTURE

Abstract

Human genomic DNA is composed of approximately 3 billion nucleotides in a linear string, organized into 23 chromosome pairs. Genome-wide association study (GWAS) analysis takes a comprehensive agnostic screening approach to discover genetic variants that are associated with complex traits such as physiological measures or disease status. Typically, the analysis proceeds by examining association of a trait with each genetic variant, one at a time. This involves making several million statistical tests across the genome. Motivated by genetic architectures in which analysis of sets of variants within a genomic region can uncover associations missed by single-variant analysis, in this talk I will describe a two-stage strategy for region-level genetic association testing. In the first stage, genomic regions are identified by computationally partitioning each chromosome into blocks according to local correlation structure known as linkage disequilibrium. Within each of the blocks, statistical association analysis then proceeds in stage two using multiple-variant methods. I will discuss some of the analytic issues we face in applying this strategy in association analysis of quantitative traits and disease status using dense GWAS genotype data.

See announcement and abstract (PDF).

Abstract: I will discuss ongoing joint work with Takuya Takeishi (Kyoto) in which we compute partition functions and other C*-dynamical invariants for the (lift of) the gauge action on the Toeplitz algebra of a countable graph. One application of our results is a generalization of a recent reconstruction theorem by Brownlowe, Laca, Robertson, and Sims.

Abstract: We consider the unique measure of maximal entropy for proper 3-colorings of Z2, or equivalently, the so-called zero-slope Gibbs measure. We will prove that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translation- equivariant function of independent and identically distributed random variables placed on Z2.​ Joint with Y. Spinka (UBC).

Do you strive to complete the Sudoku puzzle in the paper? Or are you distracted by the geometric patterns of the tiled floor? Learn how the two are more related than it seems and how new research can help determine what makes a puzzle solvable or impossible in a presentation and discussion session. Then enjoy a hands-on exploration of the research by modelling and solving some puzzles for yourself.

This presentation/workshop is part of Ideafest

The Coast Combinatorics Conferences are low-key meetings that welcome 30-minute (or so) talks on any topic in discrete mathematics and/or theoretical computer science. They are informal and largely "self-serve", meaning that there is no registration fee, and no main speaker.

Everyone is welcome to attend the CCC. Please let us know if you are coming by sending email to Gary MacGillivray: gmacgill at UVic dot ca. It makes planning easier.

For more information see the conference website.

ABSTRACT:
Given a data set of (x,y) pairs, a common learning task is to fit a model predicting y (a label or dependent variable) conditioned on x. We consider the similar but much less-understood problem of modeling "higher-order" statistics of y’s distribution conditioned on x (such as the conditional variance). Such statistics are often challenging to estimate using traditional empirical risk minimization (ERM) approaches. We develop and theoretically analyze an ERM-like approach with multi-observation loss functions, an approach that we call multi-observation regression. This approach is rooted in property elicitation, and so the idea of property elicitation will be introduced first. We propose four multi-observation regression algorithms, two of which have statistical guarantees in settings allowing both slow and fast convergence rates, but which are outperformed empirically by the other two. Empirical results illustrate potential practicality of these algorithms in low dimensions and significant improvement over standard approaches in some settings. This is joint work with my collaborators Rafael Frongillo, Tom Morgan, and Bo Waggoner.

This work appeared in AISTATS 2019 (link)

Abstract: In this talk, we consider the initial-value problem for Schrödinger maps with helicity terms. We prove the local well-posedness in two different settings: the general case and the zero-homotopic case. In the former case, we establish local well-posedness by applying the energy method of McGahagan with improving her argument. In the later case, we show further properties for solutions, such as blow-up criterion, quantitative bounds and difference estimates. The key idea for the later result is the reduction of the problem to a kind of nonlinear Schrödinger equation with magnetic potentials, so-called the modified Schrödinger map equation. We obtain quantitative estimates for solutions via this equation.

Abstract: We consider a sequence of matrices that are associated to Markov dynamical systems and use determinant-free linear algebra techniques (as well as some algebra, complex analysis, and perhaps surprisingly, algebraic geometry) to rigorously estimate the eigenvalues of every matrix simultaneously without doing any calculations on the matrices themselves. These estimates give rise to mixing rates for the dynamical systems and provide an asymptotic sharpness result in the study of random compositions of the maps in these dynamical systems.

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

Math Mania

This Alternative Math Education event will take place
6:30-8:00 p.m. Thursday, Feb. 27, 2020 at
École Doncaster School
1525 Rowan Street
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@math.uvic.ca

Abstract: Informally, a combinatorial object is "quasirandom" if its global structure resembles that of a random object of the same type. In this talk, we will discuss two problems in which quasirandomness plays a key role. First, we provide several new characterizations of quasirandom permutations which have both practical and theoretical applications. We then turn our attention to an extremal problem on cycles in tournaments in which quasirandom structure appears in the tight examples. Our proofs involve a blend of combinatorial arguments and methods from semi-definite programming and spectral theory. This talk contains joint work with Timothy Chan, Andrzej Grzesik, Daniel Kral, Yanitsa Pehova, Maryam Sharifzadeh and Jan Volec.

Abstract: One of the remarkable features of the Bost-Connes system from the 90’s, which had been foreshadowed by the Riemann gas introduced by physicists B. Julia and D. Spector in the late 80’s is that the associated partition function is the Riemann zeta function. This feature has been a guiding principle in the formulation of Bost-Connes type systems for algebraic number fields, despite the fact that the partition function of a C*-dynamical system is not a precisely formulated concept in full generality, especially for systems that have no intrinsic Hamiltonian. It has been possible to avoid this imprecision because most of the specific systems studied thus far have extra structure and symmetries that admit a unique (albeit ad-hoc) natural candidate for partition function. However, some new examples have forced us to take a more careful look at this problem. After motivating the study as an abstract spectroscopic analysis, I will describe recent joint work with C. Bruce (Victoria) and T. Takeishi (Kyoto), in which we show that for type I factor KMS states, which correspond to pure equilibrium phases at low-temperature, there is a canonical family of partition functions, which can be used to identify number fields by intepreting them as the possible absorption spectra.

Full title: Improving Response Rates for Statistics Canada’s Household Surveys, and Identifying Vulnerable Children in BC at High Risk of Poor Outcomes

Abstract: In addition to the national census of the Canadian population and surveys of businesses, Statistics Canada conducts surveys of households. For valid statistical inference, an adult in the household must be selected randomly to respond to the survey. This random selection introduces challenges when the self-reported survey is online and the household receives an invitation by mail. In this talk, I will present a comparison of the standard household enumeration method and two alternative methods of respondent selection, including their response rates and respondent selection accuracy from a field test.

One of the six program areas of the BC Ministry of Children and Family Development is child protection services. When a child is unable to safely remain with their parents or extended family, the child is admitted into the care of the ministry and usually placed with a foster family. The ministry’s Modelling and Analysis team has explored the use of predictive modelling to help social workers identify children in foster care who are at high risk of poor outcomes in the future. In this seminar, I will discuss a study on identifying children at high risk of recurrence of maltreatment when they leave the care of a foster family.

Abstract: We study the κ-color cyclic particle system on the one-dimensional integer lattice, first introduced by Bramson and Griffeath. In their original article they show that almost surely, every site changes its color infinitely often if κ ∈ {3, 4} and only finitely many times if κ ≥ 5. In addition, they conjecture that for κ ∈ {3, 4} the system clusters, that is, for any pair of sites x, y, with probability tending to 1 as t → ∞, x and y have the same color at time t. Here we prove that conjecture.

(joint work with Hanbaek Lyu)

A permutation code of length n and minimum distance d is a subset of the symmetric group S_n in which any two distinct elements differ in at least d positions. For the case d = n-1, these objects are closely connected with mutually orthogonal n-by-n latin squares (MOLS). Indeed, existence of k MOLS of side n implies such a permutation code of size kn, built from the level sets of the squares. As a partial converse, we show how certain permutation codes admit a recursive construction similar to a standard one for MOLS. After some finite-field constructions and a number sieve, we obtain a permutation code of length n, minimum distance n-1, and size n^{1.0797} for sufficiently large n. This is joint work with Sergey Bereg and based on the short paper at https://arxiv.org/pdf/1902.04153.pdf.

Abstract: Computational topologists recently developed a method, called persistent homology to analyze data presented in terms of similarity or dissimilarity. Indeed, persistent homology studies the evolution of topological features in terms of a single index, and is able to capture higher order features beyond the usual clustering techniques. There are three descriptive statistics of persistent homology, namely barcode, persistence diagram and more recently, persistence landscape. Persistence landscape is useful for statistical inference as it belongs to a space of p-integrable functions, a separable Banach space. We apply tools in both computational topology and statistics to DNA sequences taken from Clostridioides difficile infected patients treated with an experimental fecal microbiota transplantation. Our statistical and topological data analysis are able to detect interesting patterns among patients and donors. It also provides visualization of DNA sequences in the form of clusters and loops.

In this talk, I will discuss stochastic surface growth dynamics within the anisotropic class of the Kardar–Parisi–Zhang equation. Fluctuations of the dynamics are conjectured to be Gaussian in the limit of large time as if an unusual nonlinear term in the equation does not exist. The proofs obtained recently consider iterated scaling limits of some particle systems. I will discuss some aspects of these results and explain the particular role of the classical central limit theorem for the problem.

Meet and greet with refreshments at 1:00 pm. Talk is at 1:30 pm

ABSTRACT: A large part of stochastic portfolio theory, as initiated by Robert Fernholz in the 1990s, is concerned with construction of practical equity portfolios that can beat the stock market index by active rule-based trading. The truly remarkable part of the theory is that it requires no probabilistic modeling on the future behavior of stock prices. There is a Monge-Kantorovich optimal transport problem that naturally arises in the construction of such portfolios. This transport problem is a multiplicative analog of the well-studied quadratic Kantorovich- Wasserstein transport with equally striking properties. We will see aspects of this transport problem from theoretical uses such as defining gradient flows in a non-metric setting to practical uses such as in determining the right frequency of trading. Interesting probability theory comes in as we consider entropic relaxation of this problem giving rise to multiplicative Schrodinger bridges.

Abstract: We continue the theme of the last two talks and present the results of the index pairing for the spectral triple introduced last week. We then use this to deduce the action of the b-twist (from Heath’s talk) on the K-theory of the irrational rotation algebra.

Abstract: In his early work, Schur constructed a powerful link between the symmetric group and the general linear group via commuting actions on a common vector space. Much more recently, there has been a cascade of developments surrounding diagram algebras that share commuting actions with other Lie groups, Lie algebras, and deformations thereof, with consequences in many fields, including representation theory, combinatorics, knot theory, and statistical mechanics. We will take a brief tour of some important examples of diagram algebras, and discuss how seeing them from Schur's perspective can reveal a wealth of information.

Abstract: Determinantal varieties are algebraic varieties defined by the simultaneous vanishing of minors of matrices. They are important in algebraic geometry because many naturally occurring algebraic varieties have determinantal structure.

After reviewing some of these ideas, I'll focus on two such families of algebraic varieties: (i) Schubert varieties in flag varieties and multiple flag varieties; and (ii) representation varieties of Dynkin quivers. The study of each family is motivated by questions in algebraic geometry and representation theory and has led to beautiful combinatorics. Furthermore, each family contains, as special cases, classes of determinantal varieties of independent interest.

I'll discuss an ongoing research program on unifying problems about the equivariant geometry and combinatorics of representation varieties of Dynkin quivers with the corresponding problems for Schubert varieties in multiple flag varieties (joint with R. Kinser), as well as some consequences on combinatorial formulas for degeneracy loci (joint with R. Kinser and A. Knutson).

Abstract: In this talk, we first present some elementary new proofs (using Cayley digraphs and spectral graph theory) for Waring's problem over finite fields, and explain how in the process of proving these results, we obtain an original result that provides an analogue of S{\'a}rk{\"o}zy's theorem in the finite field setting (showing that any subset $E$ of a finite field $\Bbb F_q$ for which $|E| > \frac{qk}{\sqrt{q - 1}}$ must contain at least two distinct elements whose difference is a $k^{\text{\tiny th}}$ power). Once we have our results for finite fields, we can apply some classical mathematics to extend our Waring's problem results to the context of general (not necessarily commutative) finite rings. In the second half of our talk, we talk about some sum-product formulas related to matrix rings over finite fields, which can again be proven using Cayley digraphs and spectral graph theory in an efficient way.

Abstract: Spectral triples provide a more geometric approach to K-homology, and as such their index pairing with K-theory usually requires more geometric tools. In this talk, we introduce these tools and provide explicit computations for the Dirac-Schrodinger operator associated to a smooth ergodic flow. If time permits, we will demonstrate how this relates to the 'b-twist' obtained by Emerson and Duwenig.

Abstract: In this talk I will discuss my approach to pattern forming phenomena in nonlinear systems. This work outlines a dynamical systems perspective to understand the existence, stability, dynamics, and bifurcations of nonlinear waves and coherent structures to mathematical equations. Particular focus is paid to understanding how symmetry of an equation can influence the solutions of differential equations and how breaking that symmetry changes the expected behaviour. I focus on both spiral waves and localized structures in this talk and demonstrate how using lattice dynamical systems can help us to better understand them from a mathematical point-of-view. This work shows how using lattice dynamical systems can be used to solve old problems from a new angle.

Abstract: Task-based functional magnetic resonance imaging (fMRI) studies are a powerful tool to understand human sensory, cognitive, and emotional processes. The fMRI measurement consist of an indirect and non-invasive measure of brain activity based on the Blood Oxygen Level Dependent (BOLD) contrast. A critical challenge is to account for the dependencies from the complex structure of the brain. In this talk, I propose an adaptive composite-hybrid basis approach to model task fMRI data. The proposed basis approach tackles the high-dimensionality inherent to the brain data while simultaneously accounting for nearby and distant spatial correlation in a multilevel basis procedure. The first level accounts for whitin-ROI spatial correlation by selecting a sparse set of bases for each ROI. The second level considers between-ROIs correlation and obtains a set of basis functions that represents the entire brain spatial structure. The third level of the approach models time dependencies in the BOLD time series by projecting the time course into a wavelet space and then assuming a long memory process that accounts for differences in each wavelet decomposition level. Moreover, residual connectivity between ROIs is incorporated into the estimation procedure and can provide further insights into fMRI tasks. The spatial strategy gives us a sparse representation of the brain improving many folds in computational speed and effectively providing full Bayesian inference at the voxel or ROI level.

Abstract:

Bootstrap percolation processes are a family of cellular automata that were originally introduced in 1979 by the physicists Chalupa, Leith and Reich as a model of ferromagnetism. Since then, it has been used throughout a variety of disciplines to model real world phenomena such as the spread of influence in a social network, information processing in neural networks or the spread of a computer virus.

The $r$-neighbour bootstrap percolation process on a graph $G$ starts with an initial set $A$ of \emph{infected} vertices and, at each step of the process, a \emph{healthy} vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of the graph eventually becomes infected, we say that $A$ \emph{percolates}.

In this talk I will discuss the solution to a conjecture of Balogh and Bollob\'{a}s that concerns the $r$-neighbour bootstrap process on the $d$-dimensional hypercube. I will also discuss results on related processes, including the weak saturation process introduced by Bollob\'{a}s in 1968, in both the extremal and probabilistic settings.

This talk contains work joint with Gal Kronenberg, Taisa Martins, Jonathan Noel and Alex Scott.

PIMS - UVic Distinguished Lecture

Meet and Greet at 3:00, talk at 3:30

Understanding how environmental randomness affects evolution is of fundamental importance for biology. The presence of temporal or spatial randomness significantly affects the competition dynamics in populations and gives rise to some counterintuitive observations. In this talk, I will present some recent results on the evolutionary dynamics in systems where spatial and temporal randomness affects division and/or death parameters of cells. Of particular interest are the dynamics of non-selected mutants, whose rates come from the same distribution as those of wild type cells. Temporal and spatial types of randomness possess fundamentally different properties. Under temporal randomness, depending on the exact formulation of the update rules, minority mutants can be advantageous, disadvantageous, or neutral. In contrast to this, under spatial randomness, minority mutants are always advantageous. Applications to biomedical problems, including biofilms and cancer, are discussed.

Download poster (PDF).

Abstract: The annihilation and creation operators x + d/dx, x-d/dx which play an important role in Bott Periodicity in KK theory, also appear in connection with recent approaches to self-duality for the irrational torus. In this talk we describe how this happens.

I will describe why the trivial knot S²-->S⁴ has non-unique spanning discs up to isotopy. This comes from a chain of deductions that include a description of the low-dimensional homotopy-groups of embeddings of S¹ in S¹xSⁿ, a group structure on the reducing discs of S¹xDⁿ, and the action of Diff(S¹xSⁿ) on Emb(S¹, S¹xSⁿ).

Abstract: We develop a method for estimating population abundance for difficult to count populations. This is made possible using an easy to count auxiliary population with a known link to the target population. We apply the new methods to estimate the breeding population of a particular Ancient Murrelet seabird colony, using Ancient Murrelet chicks as an auxiliary population. We compare our results to those of the established survey methodology from the Canadian Wildlife Service (CWS), which similarly shows a decrease in population over 1995 to 2006. Our methods improve upon the CWS survey methods by allowing density differences across sampling sites, and by allowing estimates to be made between CWS survey years.

Abstract: The Lyapunov spectrum of Perron-Frobenius operator cocycles contains relevant information about dynamical properties of time-dependent (non-autonomous, random) dynamical systems. In this talk we characterize the Lyapunov spectrum of a class of analytic expanding maps of the circle, and discuss stability and instability properties of this spectrum under perturbations. (Joint work with Anthony Quas.)

Abstract: I will start by briefly recalling the construction and main properties of the ideal class group of a "number field". This is a finite group, whose order bears deep arithmetic information about the field. It has been known since the work of Dirichlet at the end of the Nineteenth Century that in some cases there are relationships among the orders of class groups in Galois extensions. Dirichlet's results have been put in a broader and conceptually clearer contest by Brauer's work on special values of Artin L functions at s=0 and I will discuss the interplay between analytic and representation-theoretic tools leading to these results. Finally, I will report on a joint work with L. Caputo focusing on the case of extensions whose Galois group is dihedral.

Abstract: Dynamical entropy is an important tool in classifying measure-preserving or topological dynamical systems up to measure or topological conjugacy. Classical dynamical entropy theory, of an action of a single transformation, has been studied since the 50s and 60s. Recently L. Bowen and Kerr-Li have introduced entropy theory for actions of sofic groups. Although a conjugacy invariant, sofic entropy in general appears to be less well-behaved than classical entropy. In particular, sofic entropy may depend on the choice of sofic approximation, although only degenerate examples have been known until now.

We present an example, inspired by hypergraph 2-colorings from statistical physics literature, of a mixing subshift of finite type with two different positive topological sofic entropies corresponding to different sofic approximations. The measure-theoretic case remains open. This is joint work with Lewis Bowen and Dylan Airey.

See announcement (PDF).