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IMAGINING UVic is a summer camp intended to encourage young women to pursue STEM fields. The program will be held on the UVic campus, providing participants with the opportunity to immerse themselves in the university environment.

The program will run July 2 – 5, 2024, from 10:00am to 3:30pm each day. Interactive sessions lead by cutting-edge researchers will introduce participants to new and exciting math concepts which go beyond the BC curriculum. Participants will also have various opportunities to interact and form connections with professors and graduate students in mathematics.

Collaborative problem-solving will be emphasized throughout the week through activities such as our escape room. Participants will also have the opportunity to work together to present new mathematical concepts to the rest of the group. The camp will conclude with a fun team math challenge, designed to allow participants to apply the knowledge and skills they have learned throughout the week.

The program is open to girls (which we interpret in the broadest sense of the word) in the Victoria area who will be entering grade 11 or 12 in September 2024. We welcome non-binary, two-spirit, trans, and genderqueer individuals, as well as any other youth who identify with the program.

This program is by invitation only. In order to be considered, students must submit a personal application form, along with a reference form from a high school math or science teacher. Personal application and reference forms are both due by end of day on April 12, 2024. Late applications will only be considered if space is available.

Click here to apply now!

Abstract: For fixed n, consider the symmetric group S_n on the symbols 1,...,n and the set of *star* transpositions, the transpositions that contain the symbol n. A *star factorization* of a permutation b in S_n of length k is the writing of b as the product of k star transpositions. Goulden and Jackson (2009) showed that the number of such factorizations only depends on the conjugacy class of b and not on b itself, a remarkable fact given the special role the symbol n plays amongst star transpositions. We supply the first fully combinatorial proof of this fact that works for all lengths k, and our methods connect star factorizations to monotone factorizations. Star transpositions are connected to Jucys-Murphy elements, and we explain how our result can give expressions for the *transitive* image of certain symmetric functions evaluated at Jucys-Murphy elements.

This is joint work with Jesse Campion Loth (Heilbronn Institute and the University of Bristol).

The University of Victoria and Camosun College partner once again to bring Science Rendezvous to Victoria – May 11, 2024! For complete event details or to volunteer see the event website

The theory of combinatorial limits, which provides analytic tools to represent and study large discrete structures, resulted in new views on a wide range of topics in mathematics and computer science and also opened new connections between combinatorics and other areas of mathematics. In the talk, we will introduce basic concepts from the theory of combinatorial limits and apply its methods to several specific problems from extremal combinatorics and particularly from Ramsey theory.

Ramsey theory statements guarantee the existence of ordered substructures in large objects such as in the following classical statement proven by Ramsey in 1930: if N is sufficiently large, then for any partition of k-tuples of N points into finitely many classes, there exist n points such that all k-tuples formed by these n points belong to the same class. We will study quantitative versions of Ramsey type statements and present a solution of a 30-year-old problem on the existence of high chromatic graphs with small Ramsey multiplicity. In relation to general questions concerning the interplay of combinatorial limits and extremal combinatorics, we will present, among others, a counterexample to a conjecture of Lovász on finitely forcible optima of extremal combinatorics problems

The theory of combinatorial limits, which provides analytic tools to represent and study large discrete structures, resulted in new views on a wide range of topics in mathematics and computer science and also opened new connections between combinatorics and other areas of mathematics. In the talk, we will introduce basic concepts from the theory of combinatorial limits and apply its methods to several specific problems from extremal combinatorics and particularly from Ramsey theory.

Ramsey theory statements guarantee the existence of ordered substructures in large objects such as in the following classical statement proven by Ramsey in 1930: if N is sufficiently large, then for any partition of k-tuples of N points into finitely many classes, there exist n points such that all k-tuples formed by these n points belong to the same class. We will study quantitative versions of Ramsey type statements and present a solution of a 30-year-old problem on the existence of high chromatic graphs with small Ramsey multiplicity. In relation to general questions concerning the interplay of combinatorial limits and extremal combinatorics, we will present, among others, a counterexample to a conjecture of Lovász on finitely forcible optima of extremal combinatorics problems

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

MATHEW LEWIS
BSc (University of Victoria, 2021)

“Large-scale structures in two-dimensional Rayleigh–Bénard convection driven by fixed heat fluxes”

Department of Mathematics and Statistics
Friday, May 3, 2024
10:00 A.M.
David Strong Building
Room C114
https://uvic.zoom.us/j/82433823321

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

External Examiner:
Dr. Philipp Vieweg, Department of Applied Mathematics and Theoretical Physics, Cambridge University

Chair of Oral Examination:
Dr. Timothy Iles, Department of Pacific and Asian Studies, UVic

Abstract
Existence of large-scale structures in two-dimensional Rayleigh–Bénard convection is investigated in the case of boundaries that are no-slip and have fixed heat fluxes. Direct numerical simulations are carried out using the code Dedalus, which implements spectral methods. Simulations are carried out in a horizontally periodic domain, primarily with a horizontal period 20 times the layer height. The large-scale structure of interest is a pair of wide convection rolls. After finding one such two-roll state at fixed values of the Rayleigh number R and Prandtl number Pr, the parameters are varied slowly in time to find two-roll states elsewhere in the R-Pr plane. Loss of a two-roll state occurs by transition to a four-roll state, which is detected using several criteria. The R-Pr plane is divided into one region where we have found two-roll states that persist, and one region where we have not. Along part of the boundary between these regions the two-roll states are steady, suggesting that their break-up is a linear instability. Elsewhere in the R-Pr plane the boundary is hard to locate precisely because the two-roll states are unsteady and can display metastable behaviour. The two-roll regime is found only when R is sufficiently small and Pr is sufficiently large, and these two-roll states are further classified as steady or unsteady. Contrasting our findings with simulations in the literature that have different boundary conditions and/or are three dimensional, we find that existence or nonexistence of large-scale structures is substantially affected by both thermal and velocity boundary conditions and by dimension. A simple model with one fitting parameter is found to capture the middle region of a wide roll at various parameter values, and partial results are presented towards using this model to understand the region in the R-Pr plane at which two-roll states are found.

The Final Oral Examination
for the Degree of

Master of Science
(Department of Mathematics and Statistics)

Xinning Liu BS.C (North China University of Technology (NCUT), 2022)

“On the Prediction Accuracy of Models Chosen by Constrained Minimum Criterion”

Thursday, April 18, 2024
4:00 P.M.
Virtual Defence: Zoom link

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

Chair of Oral Examination:
Dr. Xuekui Zhang, Department of Mathematics and Statistics, UVic

Abstract: We will discuss a recent result joint with In-Jee Jeong and Theo Drivas. We prove that twisting in Hamiltonian flows on annular domains, which can be quantified by the differential winding of particles around the center of the domain, is stable to general perturbations. In fact, we prove the all-time stability of the lifted dynamics in an L2 sense (though single particle paths are generically unstable). These stability facts are used to establish several results related to the long-time behavior of inviscid fluid flows.

Abstract: Response optimization is a process of identifying the input variable settings that optimize the response. Multistratum design arises naturally in industrial experiments due to the inconvenient and impractical completely randomization. Accounting for the model uncertainty, we apply the Bayesian model averaging method and predictive approach to investigate the optimization problem for data with multi-stratum structure. With the posterior probabilities of models as weights, we consider the weighted average of the predictive densities of the response over all potential models. The goal of the optimization is to identify the values of the factors that result in a maximum probability of a response in a given range. The method is illustrated with two examples.

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

SKYE DORE-HALL

MSc (University of Victoria, 2021)
BSc Hons. (Kwantlen Polytechnic University, 2018)

“Comparisons of Ramp Functions and Michaelis-Menten Functions in Biochemical Dynamical Systems”

Department of Mathematics and Statistics
Friday, April 12, 2024
9:00 A.M.
Clearihue Building Room B021

Join Zoom Meeting
https://uvic.zoom.us/j/87408735591

Supervisory Committee:
Dr. Roderick Edwards, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Junling Ma, Department of Mathematics and Statistics, UVic (Member)
Dr. Stephanie Willerth, Department of Mechanical Engineering, UVic (Outside Member)

External Examiner:
Prof. Jean-Luc Gouzé, French National Institute for Research in Digital Science and Technology

Chair of Oral Examination:
Dr. Tao Wang, Department of Economics, UVic

Abstract
Analysis of nonlinear dynamical systems, such as those modeled using Michaelis-Menten kinetics, can be difficult. Thus, it is natural to consider whether such systems can be simplified in a way that facilitates analysis while preserving qualitative behaviour. Previously, we showed that when the Michaelis-Menten terms in a model of plant metabolism are replaced by piecewise linear approximations called ramp functions, the qualitative behaviour of the model is maintained. We then defined a limited class of systems containing ramp functions called biochemical ramp systems and studied their properties, including the existence and stability of equilibria and global flow.

Here, we expand on our previous work by reforming the definition of a biochemical ramp system to describe a wider class of systems. We study the properties of several types of biochemical ramp systems that were previously not covered by the definition, and show that their qualitative behaviour is similar to that of their Michaelis-Menten counterparts. We then introduce concepts from chemical reaction network theory, such as the Deficiency Zero and Deficiency One Theorems, and explain how they are applicable to the analysis of biochemical ramp functions, but cannot be applied to the corresponding Michaelis-Menten systems. In the last chapter, we show that when ramp functions are used in systems that do not fall under the expanded definition of a biochemical ramp system, there can be qualitative differences in behaviour between these ramp systems and their Michaelis-Menten counterparts. We end with a look at periodic behaviour in ramp systems by studying a version of the Lotka-Volterra predator-prey model containing ramp functions.

Zoom link.

This is our 5th talk of the PIMS Data Science Seminar Series. PIMS requests all seminar participants to complete the demographics form online at https://ubc.ca1.qualtrics.com/jfe/form/SV_6QcNr2rQcIlQGyy

Abstract
"Double dipping" is the practice of using the same data to fit and validate a model. Problems typically arise when standard statistical procedures are applied in settings involving double dipping. To avoid the challenges surrounding double dipping, a natural approach is to fit a model on one dataset, and then validate the model on another independent dataset. When we only have access to one dataset, we typically accomplish this via sample splitting. Unfortunately, in some problems, sample splitting is unattractive or impossible. In this talk, we are motivated by unsupervised problems that arise in the analysis of single cell RNA sequencing data, where sample splitting does not allow us to avoid double dipping. We first propose Poisson thinning, which splits a single observation drawn from a Poisson distribution into two independent pseudo-observations. We show that Poisson count splitting allows us to avoid double dipping in unsupervised settings. We next generalize the Poisson thinning framework to a variety of distributions, and refer to this general framework as "data thinning". Data thinning is applicable far beyond the context of single-cell RNA sequencing data, and is particularly useful for problems where sample splitting is unattractive or impossible.

Speaker bio
Website Link: https://www.lucylgao.com/

Abstract: A technique for efficiently describing surfaces was developed to solve the knot recognition problem. This method, using “normal surfaces”, was introduced by Kneser and applied to topological algorithms by Haken. In this talk we will show how normal surfaces can be used to solve a key problem in computer graphics: How to triangulate a surface so that no triangle has an angle that is close to zero. This is joint work with M. Trnkova.

Join Zoom Meeting https://uvic.zoom.us/j/89487222886

Ecological populations experience booms and busts, with abundances changing by orders of magnitude within just a few short years. Some of these fluctuations may be explained by predator-prey cycles or deterministic chaos, two phenomena that hold immense practical and historical significance in the field of ecology. Here, I estimate the prevalence of cycles and chaos, using classic time series models, new tools from nonlinear systems analysis, and a giant database of ecological time series. Previous assessments find a 6% incidence of cyclical patterns, whereas my robust model-based procedure indicates that 8-25% of populations are cyclical. Chaos appears to be rare (with a likely prevalence of 0%), yet uncertainty remains high (0-18%). In an attempt to control false positive rates, I devised a surrogate data hypothesis test; however, the test suffers from low power, indicating that even exceptionally long time series cannot be reliably classified. Through a sensitivity analysis, I show that high noise levels, moderate observation errors, and weak density dependence are the primary obstacles to detecting ecological chaos.

Zoom link.

This is our 6th talk of the PIMS Data Science Seminar Series. PIMS requests all seminar participants to complete the demographics form online at https://ubc.ca1.qualtrics.com/jfe/form/SV_6QcNr2rQcIlQGyy

Using representations of functional data can be more convenient and beneficial in subsequent statistical models than direct observations. These representations, in a lower-dimensional space, extract and compress information from individual curves. The existing representation learning approaches in functional data analysis usually use linear mapping in parallel to those from multivariate analysis, e.g., functional principal component analysis (FPCA). However, functions, as infinite-dimensional objects, sometimes have nonlinear structures that cannot be uncovered by linear mapping. Linear methods will be more overwhelmed by multivariate functional data. For that matter, this paper proposes a functional nonlinear learning (FunNoL) method to sufficiently represent multivariate functional data in a lower-dimensional feature space. Furthermore, we merge a classification model for enriching the ability of representations in predicting curve labels. Hence, representations from FunNoL can be used for both curve reconstruction and classification. Additionally, we have endowed the proposed model with the ability to address the missing observation problem as well as to further denoise observations. The resulting representations are robust to observations that are locally disturbed by uncontrollable random noises. We apply the proposed FunNoL method to several real data sets and show that FunNoL can achieve better classifications than FPCA, especially in the multivariate functional data setting. Simulation studies have shown that FunNoL provides satisfactory curve classification and reconstruction regardless of data sparsity.

In 2021, Mohar introduced the game of Cops and Robber on geodesic spaces. The game captures the behavior of the Cops and Robber game played on graphs and that of continuous pursuit-evasion games. Analogous to one of the main open problems for the Cops and Robber game on graphs, Mohar conjectured that the cop number of a geodesic surface of genus $g$ is at most $O(\sqrt{g})$. Surprisingly, this upper bound can be significantly improved on surfaces of constant curvature which will be the main focus of this talk.

             It turns out that the cop number of compact spherical and Euclidean surfaces is at most $2$. Even more surprisingly, the cop number of compact hyperbolic surfaces is also at most $2$, independently of their genus. We will also consider the strong cop number of these surfaces and present several generalizations to higher-dimensions.

Joint work with Bojan Mohar and Alexandra Wesolek.

Abstract: One of the foundational results about C*-algebras is Gelfand's theorem establishing a duality between commutative unital C*-algebras and compact Hausdorff spaces. I will discuss the question of how effectively computable this duality is; specifically, is there an algorithm that takes as input a description of a commutative unital C*-algebra and produces as output a description of the spectrum of A? In the first talk I will provide all the necessary background from computability theory to be able to make this question precise, and also describe previous work on computable dualities. In the second talk I will outline a proof that the Gelfand duality is computable.

These talks are based on joint work with P. Burton, A. Fox, I. Goldbring, M. Harrison-Trainor, T. McNicholl, A. Melnikov, and T. Thewmorakot.

Abstract: Feynman’s path integral expresses the probability amplitude of a quantum mechanical system as a “sum of trajectories” of the classical system. Since this integral has not been given a satisfactory mathematical meaning, a widely accepted treatment is M. Kac’s method which starts with the idea of rotating “real time” to “imaginary time.” The corresponding path integrals are stochastic, given by exponential functionals of Brownian motion.

This talk will introduce a Feynman–Kac-type formula given by a non-exponential multiplicative functional of a non-Gaussian process. The formula represents the many-body delta-Bose gas in two dimensions, extending technically the two-body case obtained earlier. To contrast the classical and new, for this seminar, a significant part of the talk will discuss the classical Feynman–Kac formula.

Given two graphs H and G, the homomorphism density t(H,G) represents the likelihood that a random mapping from V(H) to V(G) is a homomorphism. Sidorenko Conjecture states that for any bipartite graph H, t(H,G) is greater or equal to t(K_2,G)^{e(H)}.

Introducing a binary relation H \geq T if and only if t(H,G)^{e(T)} \geq t(T,G)^{e(H)} for all graphs G, we establish a partial order on the set of non-empty connected graphs. Employing a technique by Kopparty and Rossman, which involves the use of entropy to define a linear program, we derive several necessary and sufficient conditions for two trees T, F satisfy T\geq F. Furthermore, we show how important results and open problems in extremal graph theory can be reframed using this binary relation.

Joint work with Natalie Behage, Gabriel Crudele, and Jonathan Noel.

Abstract: One of the foundational results about C*-algebras is Gelfand's theorem establishing a duality between commutative unital C*-algebras and compact Hausdorff spaces. I will discuss the question of how effectively computable this duality is; specifically, is there an algorithm that takes as input a description of a commutative unital C*-algebra and produces as output a description of the spectrum of A? In the first talk I will provide all the necessary background from computability theory to be able to make this question precise, and also describe previous work on computable dualities. In the second talk I will outline a proof that the Gelfand duality is computable.

These talks are based on joint work with P. Burton, A. Fox, I. Goldbring, M. Harrison-Trainor, T. McNicholl, A. Melnikov, and T. Thewmorakot.

Abstract: The Lorentz gas was originally introduced as a model for the movement of electrons in metals. It consists of a massless point particle (electron) moving through Euclidean space bouncing off a given set of scatterers $\mathcal{S}$ (atoms of the metal) with elastic collisions at the boundaries $\partial \mathcal{S}$. If the set of scatterers is periodic in space, then the quotient system, which is compact, is known as the Sinai billiard. There is a great body of work devoted to Sinai billiards and in many ways their dynamics is well understood. In contrast, very little is known about the behavior of the Lorentz gases with aperiodic configurations of scatterers which model quasicrystals and other low-complexity aperiodic sets. This case is the focus of our joint work with Rodrigo Treviño. We establish some dynamical properties which are common for the periodic and quasiperiodic billiard. We also point out some significant differences between the two. The novelty of our approach is the use of tiling spaces to obtain a compact model of the aperiodic Lorentz gas on the plane.

Zoom link: https://uvic.zoom.us/j/89487222886.

The majority of attempts to enumerate the homeless population rely on point-in-time or shelter counts, which can be costly and inaccurate. As an alternative, we use electronic health data from the Vancouver Island Health Authority, British Columbia, Canada from 2013 to 2022 to identify adults contending with homelessness based on their self-reported housing status. We estimate the annual population size of this population using a flexible open-population capture-recapture model that takes into account 1) the age and gender structure of the population, including aging across detection occasions, 2) annual recruitment into the population, 3) behavioural-response, and 4) apparent survival in the population, including emigration and incorporating known deaths. With this model, we demonstrate how to perform model selection for the inclusion of covariates. We then compare our estimates of annual population size with reported point-in-time counts of homeless populations on Vancouver Island over the same time period, and find that using data extracts from electronic health records gives comparable estimates. We find similarly comparable results using only a subset of interaction data, when using only ER interactions, suggesting that even if cross-continuum data is not available, reasonable estimates of population size can still be found using our method.

Let X be a finite subset of \mathbb{Z}^d of cardinality at least 3. We say a subset of [n]^d is X-free if it does not contain a non-trivial scaled or translated copy of X. Let r_X(n) be the cardinality of the largest X-free subset of [n]^d. How big can r_X(n) be? The Multidimensional Szemerédi's Theorem of Furstenberg and Katznelson states that r_X(n) = o(n), though exact asymptotics are not known. A natural second question asks: how many X-free subsets of [n]^d are there? We show that for infinitely many n \in \mathbb{N}, the number of X-free subsets is 2^{O(r_X(n))}. This work generalizes previous work of Balogh, Liu, and Sharifzadeh, who considered when X is a k-term arithmetic progression, and Kim, who considered when X is a corner.

Abstract: One of the few cases where the Chow groups of quadrics over an arbitrary field are known is that of excellent quadrics, including the special case of Pfister quadrics. The computation of the Chow groups of Pfister quadrics also gives partial information about the Chow groups of Pfister neighbours, the forms stably birational to Pfister forms. By understanding how the Chow groups behave under field extensions, we can obtain important information about the quadric over the ground field, and by extending to an algebraic closure, we obtain interesting numerical invariants of quadratic forms such as the J-invariant. It appears the J-invariant can be useful in determining whether a form is a virtual Pfister neighbour (a form which over some extension becomes a Pfister neighbour), and we are currently interested in just how much information can be obtained from it.

Abstract: The Enskog-Vlasov equation extends the Enskog equation for the dense hard sphere fluid by accounting for the attractive forces between, and the finite volume of, the gas particles. Hence, it gives a van-der-Waals-like description of a non-ideal gas, including liquid-vapor phase change. Specifically, the equation describes the liquid phase, the vapor phase, and a diffusive transition region connecting both phases. Solutions of the Enskog-Vlasov equation exhibit all relevant phenomena occurring in the evaporation and condensation of rarefied or dense vapors.

Using Grad’s moment method we derived macroscopic transport equations—moment equations with 13 and 26 variables—from the Enskog-Vlasov equation, which describe liquid vapor and transition region in terms of a few macroscopic properties.

Focussing on 1-D heat and mass transfer problems, we compare moment solutions to DSMC solutions for transport across the interface, and the interplay between interface and Knudsen layers. Interface resistivities for jump interface conditions are determined from the simulations, which show marked differences to those found from classical kinetic theory, where dimensionless resistivities are constants. In contrast, the EV models give temperature dependent resistivities, some negative off-diagonal resistivities, and indicate non-linear behavior where resistivities depend on mass and heat fluxes through the interface.

For more information and registration: https://www.pims.math.ca/seminars/PIMSPDF.

ABSTRACT:
In this talk, I will describe how certain non-linear exponential tilts of the Discrete Gaussian Free Field yield invariant measures for a discretized PDE, the focusing Non Linear Schrödinger equation. I will briefly talk about how this tilted measure relates to self-intersections of the random walk on $\mathbb{Z}^{d}$. I will then describe joint work with Partha Dey and Kay Kirkpatrick on how this measure undergoes a phase transition in the thermodynamic limit with general $d\geq 3$ and non-linearity $p$. I will conclude with some new results with Gourab Ray, on the weak convergence of the random field sampled with respect to this measure to the massive discrete Gaussian Free Field.

SPEAKER BIO:
Kesav is a PIMS postdoctoral fellow at the University of Victoria who works with Gourab Ray. Kesav’s academic trajectory began as an undergraduate student in physics at the University of Delhi. After realizing that mathematical methods (specifically probability) excited him more, Kesav switched track and completed his masters in mathematics at the Indian Statistical Institute, Bangalore. As a PhD student at the University of Illinois, Urbana-Champaign, Kesav focused on problems in probability that have origins in statistical and mathematical physics. He continues to work in this field, specifically on phase transitions in lattice field theories and disordered models in statistical mechanics. He is thrilled to have gotten the opportunity to spend two years in the Pacific Northwest and explore its spectacular natural beauty.

Download poster (PDF).

Zoom link.

Abstract: We discuss ergodic properties of coded shift spaces. A coded shift is defined as a closure of all bi-infinite concatenations of words from a fixed countable generating set. It turns out that many well-known classes of shifts are coded including transitive subshifts of finite type, S-gap shifts, generalized gap shifts, transitive Sofic shifts, Beta shifts, and many more. We derive sufficient conditions for the uniqueness of measures of maximal entropy based on the partition of the coded shift into its sequential set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). Under these conditions we provide a simple explicit description of the measure. (Joint work with M. Schmoll and C. Wolf)

Join Zoom Meeting https://uvic.zoom.us/j/89487222886.

In the Arctic, specific microbial communities thrive in and around sea ice, leading to blooms, within, beneath, and at the edge of sea ice. The Arctic Ocean is strongly impacted by climate change, with important modifications of the sea ice environment set to continue during the 21st century, and as a result ice algae are poised to undergo significant changes. Conversely, these primary producers play a role in biogeochemical cycles and the Arctic climate. They produce sulfur compounds, such as dimethylsulfide (DMS) which once emitted to the atmosphere is an aerosol precursor, thereby impacting cloud formation. To examine the future of sea ice algae and the production of DMS at the Arctic scale, a regional ocean and sea ice biogeochemical model has been developed. The complexity of such models raises the issue of analyzing model simulations and understanding the impact of modeling choices. In this talk, I will present diagnostics that help to understand model output, both for the analysis of future projections but also to guide model development. To explore the climate-ocean interactions in the Arctic, diagnostics have been developed that allow to link environmental conditions to the growth and phenology of ice algae as well as the production of DMS. These diagnostics also reveal how the mathematical representation of biogeochemical processes impacts model projections.

Zoom link: https://uvic.zoom.us/j/84724365947?pwd=OTRzYml3a29oeTJkNW5ucjhUWmxpdz09

This is our 4th talk of the PIMS Data Science Seminar Series. PIMS requests all seminar participants to complete the demographics form online at https://ubc.ca1.qualtrics.com/jfe/form/SV_6QcNr2rQcIlQGyy

Abstract: Single-cell omics data play a pivotal role in identifying cell-to-cell heterogeneity, understanding cell differentiation, unveiling cell population structures, and ultimately deciphering disease pathogenesis. Due to inherent high-dimensionality, sparsity, noise, and high correlation of single cell data, machine learning (ML) models, known for its assumption-free flexibility, scalability, and predictive power, have surged in analyzing single cell data to address these challenges. In this talk, I will present some of our recent work on ML-based approaches to accurately and efficiently encode single-cell gene expressions and chromatin accessibility. Our proposed method OCAT, One Cell At A Time, is a ML-based method that sparsely encodes single-cell gene expressions to integrate data from heterogeneous sources without highly variable gene selection or explicit batch effect correction (Wang et al., Genome Biology, 2022). We have demonstrated that OCAT efficiently integrates multiple heterogeneous scRNA-seq datasets and achieves the state-of-the-art performance in cell type clustering, especially in challenging scenarios of non-overlapping cell types. In addition, OCAT can efficaciously facilitate a variety of downstream analyses, such as differential gene analysis, trajectory inference, pseudo time inference and cell type inference. OCAT has proven its efficiency and accuracy in characterizing the transcriptomic difference between healthy and diseased kidney samples (McEvoy et al., Nature Communications, 2022). We have further developed OCAT2 that maps multiple complementary single-cell omics to the same domain through multi-modal diffusion mapping. We have demonstrated its accuracy and high computational efficiency on integrating real multi-omics datasets.

Abstract: Turán-type problems for hypergraphs have been an intriguing area of research. Despite significant efforts, the Turán density of F is known for only a few three-uniform hypergraphs F. This talk concerns Turán-type problems for 3-uniform tight cycles C_k, where the number of vertices k is not divisible by 3.

The Turán density of a hypergraph F is the maximum density of an n-vertex hypergraph that does not contain any member of F. Mubayi and Rödl gave an ``iterated blow-up'' construction showing that the Turán density of C_5 is at least 2sqrt{3}-3, and this bound is conjectured to be tight. Interestingly, their construction also excludes C_k for larger k not divisible by 3, indicating that it might be the extremal construction for these hypergraphs as well. Indeed, we have recently shown that the Turán density of C_k is 2sqrt{3}-3 for sufficiently large k, in a joint result with Shoham Letzter and Alexey Pokrovskiy.

Abstract: Given an almost periodic C*-algebraic dynamical system (A,\sigma) we define a new C*-algebra, the crystal of A, whose states parametrize the ground states of A, and which, in several classes of examples, has the same K-theory as A. This addresses a question of Connes about whether a quantum system can be cooled down until it becomes quasi-classical without losing its main equilibrium and topological features. Our work generalizes the two main classes of motivating examples: equilibrium states of Toeplitz-Pimsner algebras of correspondences (from work with Neshveyev) and of Toeplitz algebras of ax+b monoids, (from work with Raeburn and with Cuntz and Deninger) for which the K-theory was computed by Cuntz, Echterhoff, and Li. This is joint work with Sergey Neshveyev and Makoto Yamashita at Oslo.

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

SHUAI YOU
BSc (University of British Columbia, 2019)

“Predictive modelling, Anomaly Detection, and Empirical Extraction of Variation Patterns within Longitudinal Data”

Department of Mathematics and Statistics

Wednesday, March 13, 2024
2:30 P.M.
Virtual Defence https://uvic.zoom.us/j/86576342841

Supervisory Committee:
Dr. Mary Lesperance, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Xuekui Zhang, Department of Mathematics and Statistics, UVic (Co-Supervisor)
Dr. Youlian Pan, Department of Mathematics and Statistics, UVic (Member)
Dr. Tao Lu, Department of Electrical and Computer Engineering, UVic (Outside Member)

External Examiner:
Dr. Depeng Jiang, Department of Community Health Sciences, University of Manitoba

Chair of Oral Examination:
Dr. Jamie Lawson, Department of Political Science, UVic

Abstract
This thesis represents a comprehensive exploration and enhancement of statistical methodologies, addressing complex challenges in transdisciplinary data analysis. It integrates novel techniques across various domains to bridge gaps in existing algorithms, focusing on advancing multitask prediction and anomaly detection. Significant contributions involve extending stacking algorithms for survival and longitudinal data prediction, a generalizable workflow, and an innovative unsupervised learning algorithm for real-time spectral anomaly identification and classification. This thesis emphasizes the collective contributions of these interconnected research findings, aiming to advance predictive modeling methodologies and anomaly detection across disciplines.

Abstract: In this talk, I will introduce my recent work on the asymptotic behaviors of the stochastic heat equation (SHE) and the Kardar-Parisi-Zhang (KPZ) equation for spatial dimensions d >== 3. The former describes the dynamics of non-equilibrium growth processes arising in statistical physics, and the latter is connected with the directed polymer model, which describes the evolution of a hydrophilic polymer chain wafting in water. As an application, the asymptotic structures of the partition function and the free energy of the continuous directed polymer will be clarified.

Abstract: Given graphs $G$ and $H$ and a positive integer $q$, an $(H,q)$-coloring of $G$ is an edge-coloring in which each copy of $H$ receives at least $q$ colors. Erdős and Shelah raised the question of determining the minimum number of colors, $f(G,H,q)$, which are required for an $(H,q)$-coloring of $G$. Determining $f(K_n,K_p,2)$ for all $n$ and $p$ is equivalent to determining the classical multicolor Ramsey numbers. Recently, Mubayi and Joos introduced the use of a new method for proving upper bounds on these generalized Ramsey numbers; by finding a “conflict-free" matching in an appropriate auxiliary hypergraph, they determined the values of $f(K_{n,n},C_4,3)$ and $f(K_n,K_4,5)$. In this talk, we will show how to generalize their approach to give bounds on the generalized Ramsey numbers for several families of graphs. This is joint work with Deepak Bal, Patrick Bennett, and Shira Zerbib.

Abstract: Given an almost periodic C*-algebraic dynamical system (A,\sigma) we define a new C*-algebra, the crystal of A, whose states parametrize the ground states of A, and which, in several classes of examples, has the same K-theory as A. This addresses a question of Connes about whether a quantum system can be cooled down until it becomes quasi-classical without losing its main equilibrium and topological features. Our work generalizes the two main classes of motivating examples: equilibrium states of Toeplitz-Pimsner algebras of correspondences (from work with Neshveyev) and of Toeplitz algebras of ax+b monoids, (from work with Raeburn and with Cuntz and Deninger) for which the K-theory was computed by Cuntz, Echterhoff, and Li. This is joint work with Sergey Neshveyev and Makoto Yamashita at Oslo.

Abstract: We consider the COVID-19 pandemic in two Canadian regions: the Yukon Territory and the BC Interior. Our goal is to understand the effect of control measures and opinion dynamics on the transmission of the disease. We first incorporate an averaged measure of nine restriction policies - called the ‘Stringency Index’ - into the famous Susceptible-Infected-Recovered Model. We fit this model to case data for YT and BC Interior. To account for the effect of opinion changes, we extend the model splitting the susceptible compartment into four sub-compartments based on their view on vaccination and non-pharmaceutical interventions (NPIs). We also include vaccination though the vaccines are assumed to be 100% effective (no waning immunity). We fit this larger model to case data for each region and simulate the model results along with the Stringency Index to analyze the effect of policies and opinion dynamics.

Zoom link.

Abstract: In this talk, I will present a recent work in which two co-authors and myself studied the behaviour of a local algorithm optimizing the weight of a graph. More precisely, the process starts with a given subgraph H of the complete graph with uniform weights and a maximal weight W, and inductively replaces a subgraph of H and of weight less than W by the minimum spanning tree on the corresponding set of vertices. Our main result shows that there is a sharp threshold for W regarding the asymptotic behaviour of this algorithm (i.e. with high probability): if W is less than 1, it is impossible to reach the global minimum spanning tree, whereas it is possible when W is larger than 1. Since this work introduces a new type of local algorithm, I will also present some related open problems. In particular, our results prove when such algorithms can reach the global minimum spanning tree and it is then only natural to ask how fast they can do so when possible. The answer to this question actually relates to efficiently packing sets of uniforms into a special type of partition and leads to a surprisingly difficult open question.

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Intraguild predation (IGP) consists of two (or more) consumers of the same shared resource exhibiting a predator-prey relation among themselves, and is a very present phenomena in terrestrial, freshwater and marine ecological systems. Theoretical works show that IGP allows for coexistence between two consumers of the same guild, as long as IG prey is a more effective consumer than IG predator, revealing an important mechanism for consumer coexistence in food chains. Here we explore biological invasions forming IGP communities, by either introducing IG prey or IG predator to established (single) Consumer-Resource populations in homogeneous and heterogeneous landscapes. We use reaction-diffusion equations as our modeling framework, and explore them through numerical simulations and homogenization techniques. In homogeneous landscapes, we find that asymptotic spreading speeds are linearly determinate and also that the formation of traveling wave solutions and dynamical stabilization regimes are possible. On heterogeneous landscapes, we find that coexistence regimes in highly heterogeneous landscapes can occur regardless of IG-Prey being the least effective consumer, or be hindered even when IG-Prey remains as the dominant competitor, depending on habitat preferences of each of the species involved. We provide some conclusions of the work and venues of future research.

Abstract:
We will introduce the research area of complex dynamics by studying the family of (complex) quadratic polynomials, $z->z^2+c$. Historically, mathematicians first studied the dynamics of {\em real} quadratic polynomials and then moved to the complex realm. Recently, there has been a resurgence of mathematical activity surrounding real quadratic polynomials. We will explore some well known features and some rather new features of real polynomials in contrast with those enjoyed by their complex counterparts.

Speaker Biography:
Sarah Koch is a an Arthur F. Thurnau Professor in the Department of Mathematics at the University of Michigan. Her research incorporates topology, algebraic geometry, complex analysis, and Teichmueller theory to better understand complex dynamical systems (in one and several variables) and their associated dynamical moduli spaces from both analytic and algebraic points of view.

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Abstract: In this talk we consider the oriented colouring problem for graphs with bounded Euler genus. That is we consider the smallest $k$ such that all oriented graphs embeddable on surfaces of Euler genus at most $g$ necessarily have an oriented homomorphism to a graph of order $k$. For convenience given a fixed $g$ and $k$, we let $\chi_o(g) = k$. We will discuss our proofs that $\Omega((\frac{g^2}{\log{g}})^{\frac{1}{3}}) \leq \chi_o(g) \leq (1+o(1))g^{6400}$, which improves the prior upper bound of order $2^{O(g^{\frac{1}{2}+o(1)})}$ and lower bound of order $\Omega(\sqrt{g})$, as well as exploring how our bounds might be improved in future work.

Joint work with Peter Bradshaw (University of Illinois Urbana Champaign), and Jingwei Xu (University of Illinois Urbana Champaign).

Abstract: Given an almost periodic C*-algebraic dynamical system (A,\sigma) we define a new C*-algebra, the crystal of A, whose states parametrize the ground states of A, and which, in several classes of examples, has the same K-theory as A. This addresses a question of Connes about whether a quantum system can be cooled down until it becomes quasi-classical without losing its main equilibrium and topological features. Our work generalizes the two main classes of motivating examples: equilibrium states of Toeplitz-Pimsner algebras of correspondences (from work with Neshveyev) and of Toeplitz algebras of ax+b monoids, (from work with Raeburn and with Cuntz and Deninger) for which the K-theory was computed by Cuntz, Echterhoff, and Li. This is joint work with Sergey Neshveyev and Makoto Yamashita at Oslo.

Abstract: The epidemic curve is generally exponential during the early stage of an epidemic, presenting a challenge in identifying parameters of mathematical models that incorporate multiple control measures. This presents a major hurdle in disentangling and evaluating the effectiveness of contact tracing and other non-pharmaceutical public health interventions (NPIs) separately. In this presentation, we show how to use a novel contact tracing model and a simulation study to determine the dataset required for such an assessment. Our results show that the daily counts of new cases, cases diagnosed via contact tracing, and symptom onsets are necessary for this evaluation. We apply our method to the early stage of the Covid-19 pandemic in Ontario, Canada.

Zoom link.

Abstract: We consider a class of disordered mean-field combinatorial optimization problems, focusing on the Gibbs measure, where the inverse temperature does not vary with the size of the graph and the edge weights are sampled from a general distribution. We prove Central Limit Theorems for the log-partition function, the weight of a typical configuration, and the Gibbs average in both quenched and annealed forms. We also derive quenched Poisson convergence for the size of the intersection of two independent samples, yielding replica symmetry of the model. Applications cover popular models from the literature, such as the Minimal Matching Problem, Traveling Salesman Problem, and Minimal Spanning Tree Problem, on a sequence of deterministic and random dense block graphs of increasing size. Joint work with Grigory Terlov.

Bayesian hierarchical models have gained popularity for analyzing complex ecological data gathered through contemporary animal tracking technologies, including acoustic telemetry. Their widespread use is attributed to several advantages such as remarkable flexibility, the capacity to integrate prior knowledge, and enhanced precision. Acoustic telemetry systems are increasingly employed to investigate fish movement patterns, habitat utilization, and demographic parameter estimation. Nonetheless, the data produced by omnidirectional acoustic telemetry studies are intricate, characterized by multiple sources of variability. In this presentation, we explore several applications of Bayesian hierarchical modeling for acoustic telemetry data, focusing on estimating individual-level Walleye movement paths in Lake Winnipeg and examining survival probability variations of Arctic Char in the Cambridge Bay area. Additionally, we confront the practical challenges encountered in modeling such data and underscore the benefits of the Bayesian approach compared to traditional methods.

See announcement and abstract.

Zoom link.

PIMS requests all seminar participants to complete the demographics form online at https://ubc.ca1.qualtrics.com/jfe/form/SV_6QcNr2rQcIlQGyy

Abstract:
Measuring the serial dependence across time is critical in model identification and diagnosis in time series (TS) analysis. In classic TS analysis, the autocorrelation function is perhaps the most widely used method to examine the temporal relationship of the scalar or vector-valued observations. In functional TS (FTS), which refers to TS of functional data, their dependence is best summarised by an autocovariance operator. Evaluating the size and information contained in such an object can be difficult. Existing methods are relatively constrained and unable to capture certain characteristics contained in the FTS objects, such as the "direction" of dependence. We develop a new method to address this problem by projecting lagged pairs unit sphere and computing the angle between them, which we refer to as spherical autocorrelation. We establish the asymptotic properties of the empirical spherical autocorrelation, and we study its use in an application to European electricity data.

In recognition of Rod Edwards' remarkable contributions to the field of Mathematical Modeling, we are inviting Rod's current and former students, postdocs, close collaborators, friends and colleagues to attend this event, scheduled to take place on campus on February 17th and 18th, 2024.

See the Program for more details

In Graph Theory, the well-known problems of packing and independence are generalized by broadcast packing and broadcast independence. As an analogy, placing cell towers (of various powers) in a network so that the signals do not interfere is a broadcast packing problem. Placing cell towers in a network where the signals can interfere at any point except the towers is a broadcast independence problem. In this talk, I will present explicit formulas for the broadcast independence and packing numbers of perfect binary and k-ary trees, along with the proof techniques used to determine these formulas. Furthermore, I will present a linear time algorithm for computing the broadcast independence number of spiders. This research was motivated by the question “Can we determine the broadcast independence number of other subclasses of the class of trees? In particular, what about k-ary trees?”, which was posed by Ahmane et al. in their paper On the broadcast independence number of caterpillars.

This is joint work with Dr. Richard Brewster.

Abstract: "(Strong) Morita equivalence of C*-algebras was introduced by M. Rieffel as a generalization of Morita equivalence of rings, and is an extremely useful tool in studying K-theory, ideals and representations of C*-algebras. It was generalized by Combes ('84) to 'Morita equivalence of C*-algebras with a group action' (i.e. equivariant Morita equivalence). One of Combes results is that, if C*-dynamical systems are Morita equivalent, then their (full/reduced) crossed-product C*-algebras are Morita equivalent.

In this talk, I will introduce another generalization to 'Morita equivalence of C*-correspondences.' I show that if two C*-correspondences are Morita equivalent, then the Toeplitz algebras are Morita equivalent. This will start with some background on C*-correspondences and Toeplitz algebras, a proof-sketch of the main result, and some examples."

Abstract: In a linear population model that has a unique “largest” eigenvalue, the corresponding left and right (Perron) eigenvectors determine the long-term relative prevalence and reproductive value of different types of individuals, as described by the Perron-Frobenius theorem and generalizations. It is therefore of interest to study how the Perron vectors depend on the generator of the model. Even when the generator is a finite-dimensional matrix, there are several approaches to the corresponding perturbation theory. We explore an approach that hinges on stochasticization (re-weighting the space of types to make the generator stochastic) and interprets formulas in terms of the corresponding Markov chain. The resulting expressions have a simple form that can also be obtained by differentiating the renewal-theoretic formula for the Perron vectors. The theory appears well-suited to the study of infection spread that persists in a population at a relatively low prevalence over an extended period of time, via a fast-slow decomposition with the fast/slow variables corresponding to infected/non-infected compartments, respectively. This is joint work with MSc student Tareque Hossain.

Abstract: For a population model that encodes parent-child relations, an ordered representation is a partial or complete labelling of individuals, in order of their descendants’ long-term success in some sense, with respect to which the ancestral structure is more tractable. The two most common types are the lookdown and the spinal decomposition(s), used respectively to study exchangeable models and Markov branching processes. We study the lookdown for an exchangeable model with a fixed, arbitrary sequence of natural numbers, describing population size over time. We give a simple and intuitive construction of the lookdown via the complementary notions of forward and backward neutrality. We discuss its connection to the spinal decomposition in the setting of Galton-Watson trees. We then use the lookdown to give sufficient conditions on the population sequence for the existence of a unique infinite line of descent. For a related but slightly weaker property, takeover, the necessary and sufficient conditions are more easily expressed: infinite time passes on the coalescent time scale. The latter property is also related to the following question of identifiability: under what conditions can some or all of the lookdown labelling be determined by the unlabelled lineages? A reasonably good answer can be obtained by comparing extinction times and relative sizes of lineages.

Announcement (PDF)

Zoom link.

We consider the SARS-CoV-2 pandemic, between March 2020 and February 2021, in three Canadian regions: Greater Vancouver, Greater Toronto, and Calgary. This period was dominated by the wild-type strain of SARS-CoV-2 and occurred prior to widespread vaccination roll-out. To assess the effectiveness of age-specific control measures on viral transmission, we construct a Susceptible-Exposed-Infectious-Recovered (SEIR) deterministic model with two age groups: youth (0-19 years), and adults (20+ years). We fit this model to SARS-CoV-2 case data over sequential time frames that capture the changes in public health control measures in each region. The age-specific control parameters of the fitted model are then adjusted to simulate alternative policy scenarios and assess their relative potential effectiveness in reducing the Fall/Winter epidemic peaks.

Abstract: Let q=1 mod 4 be a prime power and let F_q be the finite field of q elements. The Paley graph of order q is the graph with vertex set F_q, such that two vertices are adjacent if and only if their difference is a square in F_q. Paley graphs play an important role in many branches of combinatorics and number theory. Among many exciting questions related to Paley graphs, estimating their clique number is of importance. In this talk, I will report recent progress on the lower bounds and upper bounds on the clique number of Paley graphs and Paley-like graphs. Joint work with Seoyoung Kim, Jozsef Solymosi, and Semin Yoo.

Abstract: "(Strong) Morita equivalence of C*-algebras was introduced by M. Rieffel as a generalization of Morita equivalence of rings, and is an extremely useful tool in studying K-theory, ideals and representations of C*-algebras. It was generalized by Combes ('84) to 'Morita equivalence of C*-algebras with a group action' (i.e. equivariant Morita equivalence). One of Combes results is that, if C*-dynamical systems are Morita equivalent, then their (full/reduced) crossed-product C*-algebras are Morita equivalent.

In this talk, I will introduce another generalization to 'Morita equivalence of C*-correspondences.' I show that if two C*-correspondences are Morita equivalent, then the Toeplitz algebras are Morita equivalent. This will start with some background on C*-correspondences and Toeplitz algebras, a proof-sketch of the main result, and some examples."

Zoom link.

Abstract: We will discuss local well-posedness and ill-posedness results of some active scalar equations, including 2D incompressible Euler equations and the SQG equation. We will discuss how one can obtain instantaneous growth of solutions using a perturbation of a steady initial data as well as making use of unboundedness of the Riesz transform in $L^\infty$. We will then discuss the first result of an instantaneous gap loss of Sobolev regularity for 2D Euler. Namely, we will describe a construction of initial vorticity for the 2D Euler equations that belongs to the Sobolev space $H^\beta$, $\beta \in (0,1)$ which gives rise to a unique global-in-time solution that instantaneously leaves not only $H^\beta$, but also $H^{\beta'}$ for every $\beta' >(2-\beta )\beta /(2-\beta^2)$. This is joint work with Diego Cordoba and Luis Martinez-Zoroa.

Abstract: Lifting theorems have played a key role in many recent breakthrough results in diverse areas of theoretical cs and mathematics via simple yet profound connections between these areas and communication complexity. A typical lifting theorem translates the query complexity (a very simple model of computation) of a Boolean function to the communication complexity (a more powerful model) of an associated function obtained by a central operation of Boolean functions known as block-composition by composing with another function known as inner function. Most lifting theorems work for any Boolean function f and depend upon the pseudo-random properties of the inner function g known as the gadget. The main parameter of efficiency in lifting theorems is the input size of the inner function g. Obtaining lifting theorems for constant-sized gadgets would give us breakthrough results and a nearly complete understanding of lifting phenomena; current techniques are far from achieving this goal. The main barrier is the existence of “nice” pseudo-random properties for well-known gadgets when their input length is relatively small compared to the outer function. Recent results have shown that understanding the pseudo-random properties is inherently connected to interesting questions in combinatorics (like the sunflower lemma) and in Boolean function analysis. In this talk, we will go through a high level overview of lifting theorems and the necessary condition that enable lifting phenomenon. We will also see some applications of lifting theorems in diverse areas of theoretical cs and mathematics, achieved via surprising yet simple connections. We will also go over recent advances in understanding the pseudo-random properties that drive the lifting phenomena and important barriers and open problems related to this.

Zoom link.

Areas in which B cells, T cells and macrophages assemble in close spatial proximity to each other, known as lympho-myeloid aggregates (LMAs), are prognostic for patients with High Grade Serous Ovarian Cancer (HGSC), yet there is still much to be discovered about their vital role in anti-tumour immunity. This includes quantitative descriptors of their cellular makeup; the specifics of the interactions and collaborations within these communities; and the immunomodulatory and immunostimulatory factors that promote and prevent their formation. We are investigating spatial distributions of the immune cells that comprise LMAs, their gene expression profiles, and their receptor (ie. T-Cell receptor - TCR, and B-Cell receptor - BCR) sequences via modification of the 10X genomics Visium technology for this purpose. We are identifying LMAs, their subtypes, and the specific role these subtypes pose in anti-tumour immunity using kernel density analysis, clustering, and spatial point process statistics. We are also leveraging our data to identify pairs of collaborating T and B cells. Specifically, we used (i) cross-K functions and simulation envelopes to evaluate spatial dependence, and (ii) number of pixels shared to flag putative sites of direct contact between B and T cells. Overall, we aim to understand how location, phenotype and antigen specificity of tumor-infiltrating T-Cells and B-Cells contribute to antitumor immunity. This information is essential for understanding the clinical benefits of immune cells and LMAs, and maximizing their functionality for therapeutic purposes.

Since suggested by Tur\'an in 1941, determining the Tur\'an density of hypergraphs has been a notoriously difficult problem at the center of extremal combinatorics. Subsequently, several natural variants of this problem have been suggested, most prominently the uniform Tur\'an density by Erd\H{o}s and S\'os and the codegree Tur\'an density by Mubayi and Zhao. Roughly speaking, the Tur\'an density is the threshold of the edge density above which large hypergraphs are guaranteed to contain a copy of a fixed hypergraph~$F$. Similarly, the uniform Tur\'an density and the codegree Tur\'an density are the thresholds of the local density and the minimum codegree, respectively, above which large hypergraphs are guaranteed to contain a copy of~$F$. In this talk, we will discuss recent results which determine several variants of the Tur\'an density in new instances and make progress towards a problem of Erd\H{o}s and S\'os. Further, we will present our recent result with Piga which states that there are hypergraphs with arbitrarily small codegree Tur\'an density. This is in contrast to the behaviour of the classical Tur\'an density and the uniform Tur\'an density due to results by Erd\H{o}s and by Reiher, R\"odl, and Schacht, respectively.

Based on joint works with Chen, Conlon, Piga, and Sales.

Abstract: I will start with a short review of dimension groups (no background needed) and show how, in the case of a minimal action of the group of integers on a Cantor set, how they arise in a natural way (helped, in part, by the associated crossed product C*-algebra). This further leads to a complete model for such systems. In the second talk, I will discuss how these ideas can be extended, in a surprising way, to minimal actions of the group Z^2. This is joint work with Thierry Giordano (Ottawa) and Christian Skau (Oslo).

Abstract:
In a recent article by Guglielmi and Hairer (SIADS 2015), an analysis in the $\varepsilon\to 0$ limit was proposed of regularized discontinuous ODEs in codimension-2 switching domains (intersections of discontinuity surfaces). This was obtained by studying a certain 2-dimensional system describing the so-called hidden dynamics. In particular, the existence of a unique limit solution was not proved in all cases, a few of which were labeled as ambiguous, and it was not clear whether or not the ambiguity could be resolved. We now show that it cannot be resolved in general. We show that three types of non-uniqueness or ambiguity can occur.

Firstly, we show that the limit solution can depend on the form of the regularization function.

Secondly, we show that behaviour of the hidden dynamics in the structurally ambiguous cases can depend on parameters, with bifurcations between different macroscopic outcomes. Thus the structure does not directly determine the behaviour.

Finally, we investigate the extreme sensitivity of solutions to initial conditions or parameters in the transition from codimension-2 domains to codimension-3 when there is a limit cycle in the hidden dynamics.

Zoom link.

Abstract: The word complexity function p(n) of a subshift X measures the number of n-letter words appearing in sequences in X, and X is said to have linear complexity if p(n)/n is bounded. It's been known since work of Ferenczi that linear word complexity highly constrains the dynamical behavior of a subshift. In recent work with Darren Creutz, we show that if X is a transitive subshift with limsup p(n)/n < 3/2, then X is measure-theoretically isomorphic to a compact abelian group rotation. On the other hand, limsup p(n)/n = 3/2 can occur even for X measurably weak mixing. Our proofs rely on a substitutive/S-adic decomposition for such subshifts. I'll give some background/history on linear complexity, discuss our results, and will describe several ways in which 3/2 turns out to be a key threshold (for limsup p(n)/n) for several different types of dynamical behavior.

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Climate change, global land use change and increasing intensive anthropogenic interventions make grasslands and forests to highly endangered ecosystems. Effective protection in the future requires a fundamental understanding of the dynamics of these ecosystems and their major drivers. Complementary to field experiments, ecological models enable us to analyze the long-term effects of changes as well as to gain a deeper understanding of the underlying ecological processes. I combined individual-based grassland model and network science with extensive field measurements to systematically explore the complex structures and dynamics of highly diverse ecological systems. The aim was to understand the community structure and dynamics emerging from individual plant interactions – in relation to plant traits, ecological processes, environmental and anthropogenic impacts, and the small-scale spatial distribution of plants.

Description: We consider metric tensors on undirected weighted graphs G, which allows us to treat P(G), the set of probability vectors on G, as a length space. On defines a divergence operator div_\mu(G) for mu in P(G), in such a way that we can use control vectors m to define paths s:[0,T] \to P(G), satisfying the system of ODEs: d\sigma/dt + div_G(m) + \hbar div_\sigma(\nabla_G log \sigma)=0. These paths serve as characteristics for Hamilton-Jacobi equations involving graph-individual noise operators. We propose a well posedness theory on P(G). (This talk is based on a joint work with C. Mou and A. Swiech)

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

Registration:
Participants register once on Zoom and can attend any of the Colloquium talks. Please remember to download the calendar information to save the dates on your calendar. PIMS will resend the confirmation from Zoom prior to the event date.

Evelyne will be giving her talk remotely (on Zoom). Local participants are still encouraged to come to COR A128 to view it on the projector.

Abstract: Suppose a graph G has list chromatic number k. It is easy to see that if L is a (k+1)-list assignment for G, then G admits two L-colourings f and g where f(v) =/= g(v) for every vertex v in the graph. But what if we want still more disjoint L-colourings? In this talk, I will discuss recent progress towards determining the list packing number of various classes of planar graphs: that is, the smallest number k such that if L is a k-list assignment for an arbitrary graph G in the class under study, then L can be decomposed into k disjoint L-colourings. All results I will discuss also hold in the correspondence colouring framework. Joint work with Daniel Cranston.

Abstract: I will start with a short review of dimension groups (no background needed) and show how, in the case of a minimal action of the group of integers on a Cantor set, how they arise in a natural way (helped, in part, by the associated crossed product C*-algebra). This further leads to a complete model for such systems. In the second talk, I will discuss how these ideas can be extended, in a surprising way, to minimal actions of the group Z^2. This is joint work with Thierry Giordano (Ottawa) and Christian Skau (Oslo).

Abstract: I will talk about the `minimal spanning arborescence', a directed version of the Minimal spanning tree. I will explain how this naturally leads to a new type of stochastic process which we call `loop contracting random walk'. I will show how this can be analyzed in the setting of trees. I will finish with some simulations and open questions. Joint work with Arnab Sen.

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Few statistical measures receive as much attention (implicitly or explicitly) as the arithmetic mean. The history for this particular measure is rooted in the historic development of quantitative science and the personalities that influenced it. Today, environmental measures, predictions, and evaluations of environmental and sustainable development interventions and policy are routinely done on mean responses, even when the "average" may not be the best way to understand a system or inform a decision. This talk will be a broad survey of ecological, evolutionary, and sustainable development contexts where measuring the "average" may mislead more than enlighten, and will serve as an exploration and an invitation to openly question and consider alternatives to the kinds of technical measures we take for granted when doing environmental research.

Abstract: In 1974 Erdős and Rothschild asked the following question: given integers k, s and a large n, what is the maximum number of s-edge-colourings of an n-vertex graph free of a monochromatic k-vertex clique? A follow up question is to determine which graph(s) attain this maximum. Recent work of Pikhurko, Staden and Yilma shows that in most cases this problem reduces to considering complete multipartite graphs and only counting a natural family of K_k-free "template" colourings of their edge set. Despite this reduction, the Erdős-Rothschild problem has only been solved for some small s and k. Various generalisations of this problem have been considered in the literature, where instead of forbidding monochromatic cliques, we forbid other, non-monochromatic, patterns on a clique. We consider the problem of maximising the number of s-edge-colourings without triangles which see exactly two colours, and show that for every s, the extremal graph is an r-partite Turán graphs where r is easy to compute for any given s. This is joint work with Pranshu Gupta, Emil Powierski and Katherine Staden.

Abstract: I will start with a short review of dimension groups (no background needed) and show how, in the case of a minimal action of the group of integers on a Cantor set, how they arise in a natural way (helped, in part, by the associated crossed product C*-algebra). This further leads to a complete model for such systems. In the second talk, I will discuss how these ideas can be extended, in a surprising way, to minimal actions of the group Z^2. This is joint work with Thierry Giordano (Ottawa) and Christian Skau (Oslo).

Abstract: We are interested in the design of forcing in the Navier–Stokes equation such that the resultant flow maximizes the heat transfer between two differentially heated walls for a given power supply budget. Previous work established that heat transport cannot scale faster than 1/3-power of the power supply. Recently, Tobasco and Doering (PRL'17) and Doering and Tobasco (CPAM'19) constructed self-similar two-dimensional steady branching flows, saturating this upper bound up to a logarithmic correction to scaling. We present a construction of three-dimensional ``branching pipe flows'' that eliminates the possibility of this logarithmic correction and for which the corresponding heat transport scales as a clean 1/3-power law in power supply. Our flows resemble previous numerical studies of the three-dimensional wall-to-wall problem by Motoki, Kawahara and Shimizu (JFM'18). However, using an unsteady branching flow construction, it appears that the 1/3 scaling is also optimal in two dimensions. After carefully examining these designs, we extract the underlying physical mechanism that makes the branching flows ``efficient,'' based on which we present a design of mechanical apparatus that, in principle, can achieve the best possible case scenario of heat transfer. We will further discuss some interesting implications of branching flows, for example, anomalous dissipation in turbulent flows and Rayleigh--B\'enard convection.

Abstract: We consider an arbitrary sequence of d*d matrices (A_n) of norm at most 1 and subject them to noiselike perturbations of size epsilon. We show that the perturbed system has simple Lyapunov spectrum (no repeated exponents) and discuss universal bounds on gaps between exponents as a function of d and epsilon.

Join Zoom meeting: https://uvic.zoom.us/j/89487222886.

We currently have limited understanding of dynamic food web consequences arising from temperature- and nutrient-driven algal blooms that typify many freshwater aquatic ecosystems. Such studies are needed to better understand and potentially mediate the impact of climate change and fertilizer run-off in agricultural landscapes. Based on controlled benchtop trials, we developed size-structured consumer-resource models to predict the impact of temperature increase and fertilizer input on growth and population dynamics of the green algal species (Chlorella vulgaris), cyanobacteria (Microcystis aeruginosa), and a common aquatic herbivore (Daphnia magna) and tested those predictions in replicated time series trials in freshwater mesocosms. As predicted by our temperature- and nutrient-driven consumer-resource models, algal and zooplankton populations exhibited massive blooms, followed by collapse to much lower densities, suggesting that physiologically-based size-structured models can help predict the impact of climate change and fertilizer-induced eutrophication on freshwater ecosystems. We also evaluated genotypic variation in Daphnia demographic response to cyanobacteria toxicity and the impact of genotypic variability on seasonal plankton dynamics.

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

FREDDIE MULLIN

BEd (University of Victoria, 2013)
BSc (University of Victoria, 2011)
BA (University of Victoria, 2011)

“Studies on Math Education and Trigraph Homomorphisms”

Department of Mathematics and Statistics
Thursday, December 14, 2023
3:30 P.M.
Virtual Defence

Supervisory Committee:
Dr. Gary MacGillivray, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Jane Butterfield, Department of Mathematics and Statistics, UVic (Co-Supervisor)
Dr. Richard Brewster, Department of Mathematics and Statistics, UVic (Member)

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

Chair of Oral Examination:
Dr. Marlea Clarke, Department of Political Science, UVic

Abstract
This thesis is comprised of two parts: (i) a study of homomorphisms of weak trigraphs and (ii) an analysis of the effectiveness of the University of Victoria (UVic) Department of Mathematics and Statistics’ Pretest. In the first part of the thesis, we study homomorphisms of weak trigraphs. Results analogous to those for graph homomorphisms are developed. In particular, we determine the complexity of deciding whether there is a weak trigraph homomorphism of a weak trigraph G to a weak trigraph H, the complexity of deciding whether a given weak trigraph has a weak trigraph homomorphism to a proper subgraph (the complexity of deciding whether it is not a core) and describe an efficient algorithm based on Consistency Checking that determines whether there is a weak trigraph homomorphism from a given cactus weak trigraph to a fixed weak trigraph H. In the second part of the thesis, we analyse the effectiveness of the UVic Pretest. First we compare the current online UVic Pretest with the past paper Pretest in terms of their respective effectiveness in identifying students who are ready for Calculus I. We also analyse the current online UVic Pretest in greater detail, to identify which precalculus skills are most likely to predict success on that test itself. Finally, we use odd ratios to categorize each question on the online UVic Pretest and identify questions that are particularly useful to the test.

Zoom link.

This is our 3rd talk of the PIMS Data Science Seminar Series. PIMS requests all seminar participants to complete the demographics form online.

Abstract:
In this data explosion era, machine learning has accelerated research in biology and medicine at an unprecedent speed and in multiple dimensions. The increased data volume and capacity for data aggregation and analytics power, along with decreasing costs of genome sequencing has spurred the growth in bioinformatics and need for novel tools to integrate the highly heterogenous data from multiple sources and of varying types, and extract meaningful patterns. The big data analytics and AI tools have already created significant impact in many fields of life sciences including medicine. However, the data complexity and multi-dimensionality have led to technical challenges in developing and validating AI solutions that generalize to diverse populations and imped the progress in their implementation in clinical practice due to imbalance in data distribution across population demography and data sparsity. This leads to the unconscious biases in the generated models and algorithms. In this talk, the speaker will discuss applications of AI in biology and medical research, advances and major challenges.

Short bio of the speaker:
Dr. Youlian Pan is an international expert in integrative pattern recognition from big data in Life Sciences. He has authored and co-authored over 80 refereed articles and created significant applications of data mining, machine learning, AI and bioinformatics in genomics, transcriptomics and systems biology with various medical applications, such as cancers, infectious diseases and neurodegenerative diseases. He also has extensive research interest in plants’ pathogenesis and embryogenesis, their interaction with environment, and biological oceanography specifically in marine pollution. Dr. Pan is a Senior Research Scientist at the National Research Council Canada and an Adjunct Professor at the University of Victoria and Brock University. He received his PhD in Biology and Master of Computer Science from Dalhousie University. He has served at various capacities in editorial board of six international journals, such as Journal of Computations & Modeling, Open Medical Informatics, and Frontiers in Genetics, Microbiology and Plant Sciences; and various national and international grant evaluation panels such as Natural Sciences and Engineering Research Council (NSERC) of Canada, National Science Foundation (NSF) of US.

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

J. ETHAN WILLIAMS
BSc (University of Victoria, 2021)

“Eternal Domination Problems”

Department of Mathematics and Statistics

Wednesday, December 13, 2023
10:30 A.M.
David Turpin Building
Room A203

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

External Examiner:
Dr. William Klostermeyer, School of Computing, University of North Florida

Chair of Oral Examination:
Dr. Matt Moffitt, Department of Chemistry, UVic

Abstract
Consider placing mobile guards on the vertices of a graph. The vertices are then attacked by an assailant, requiring you to move guards to the attacked vertices. What is the minimum number of guards you need in order to be able to defend against any sequence of attacks? This question is the basis for the eternal domination problem. In this thesis we investigate this problem and introduce new parameters related to it.

These new parameters arise from changing three of the assumptions made when defining the game. Specifically we assume that any number of guards can move when defending against an attack; only one attack needs to be defended against at a time; and that any number of guards can occupy a vertex. Changing these assumptions gives rise to the maneuver, invasion, and stacking numbers respectively. We investigate these parameters throughout this thesis, especially as they relate to trees.

Additionally, we tackle the related problem of eternal Roman domination, which is based on the topic which originally gave rise to the eternal domination problem. We establish a best possible upper bound for this parameter over all graphs. Finally, we present exponential time algorithms for solving all of these problems, as well as a host of other related problems.

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

Abstract: Zero forcing is a type of graph propagation based on the color-change rule: Given graph $G$, if each vertex of $G$ is colored either white or blue, and vertex $v$ is a blue vertex with only one white neighbor $w$, then change the color of $w$ to blue. In this talk we prove a conjecture formulated by the automated conjecturing program called \emph{TxGraffiti}. The conjecture states that in a claw-free graph, the vertex cover number of the graph is at least the zero forcing number of the graph. We also prove a relationships about the zero forcing and independence number of a connected subcubic graph.

Register for Zoom link.

In 1962, Tutte proposed a simple method to produce a straight-line embedding of a planar graph in the plane, known as Tutte’s spring theorem. It leads to a surprisingly simple proof of a classical theorem proved by Bloch, Connelly, and Henderson in 1984, which states that the space of geodesic triangulations of a convex polygon is contractible. In this talk, I will introduce spaces of geodesic triangulations of surfaces, review Tutte’s spring theorem, and present this short proof. It time permits, I will briefly report the recent progress in identifying the homotopy types of spaces of geodesic triangulations of general surfaces.

Zoom link.

Abstract: The random interlacement point process (introduced by Sznitman, generalized by Teixeira) is a Poisson point process on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift on a transient graph G. We show that the random interlacement point process on any transient transitive graph G is a factor of i.i.d., i.e., it can be constructed from a family of i.i.d. random variables indexed by vertices of the graph via an equivariant measurable map. Our proof uses a variant of the soft local time method (introduced by Popov and Teixeira) to construct the interlacement point process as the almost sure limit of a sequence of finite-length variants of the model with increasing length. We also discuss a more direct method of proving that the interlacement point process is a factor of i.i.d. which works if and only if G is non-unimodular. Based on joint work with Márton Borbényi and Sándor Rokob.

See announcement and abstract (PDF)

Zoom link.

PIMS requests all seminar participants to complete the demographics form online at https://ubc.ca1.qualtrics.com/jfe/form/SV_6QcNr2rQcIlQGyy

Abstract: In this talk, we will share a new evolutionary but ensemble method, that enable us to track different regimes of behavior and identify when the changes occurred. As opposed to traditional methods that relies on statistical change tracking to detect intrusions, we use the performance, and the predictive power of evolving models to detect when and which models can or cannot describe the observations anymore. By doing so, we obtain a much fine-grain, more adaptive outlier detection algorithm that can reliably model data while being robust to its variations. The algorithm can be viewed as an evolutionary algorithm with growth and new generation capabilities, but it is special in the sense that it includes ensemble of models with performance measures and age decay corrections to evolve and compare models. For some special cases, the algorithm can be related to Bayesian Change Point techniques.

Bio: Belaid Moa received the B.Sc. degree in electrical engineering from École Hassania des Travaux Publics, Casablanca, Morocco, the M.Eng. degree in electronics and signal processing from École Nationale Supérieure d'électronique, d'électrotechnique, d'informatique, d'hydraulique et des Télécommunications, Toulouse, France, the DEA Diploma degree in Telecommunications and Networks from the Institute National Polytechnique de Toulouse, Toulouse, and the Ph.D. degree in computer science from the University of Victoria. He is currently an adjunct faculty with ECE Dept., and Advanced Research Computing Specialist with the Digital Research Alliance of Canada/BCDRI /University Systems, at the University of Victoria. He has authored or co-authored many research articles and conference proceedings in various journals and multi-disciplinary research areas

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

Hanan Abousaleh
BSc (University of Victoria, 2021)

Examining Committee

Supervisory Committee
Dr. Julie Zhou, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Michelle Miranda, Department of Mathematics and Statistics, UVic (Member)

External Examiner
Dr. Nilanjana Roy, Department of Economics, UVic

Chair of Oral Examination
Dr. Hong-Chuan Yang, Department of Electrical and Computer Engineering, UVic

Abstract
We focus on a class of optimization problems known as optimal design problems, where the goal is to select design points optimally with respect to some criterion of interest. For regression models, the optimality criterion is based on the statistical model itself and is often a function of the information matrix. We solve A-, D-, and EI-optimal design problems in this thesis. The CVX program in MATLAB is a modelling tool and solver for convex optimization problems. As with other numerical methods in the literature, formulating an optimal design problem in a CVX-compatible way requires a discrete design space. We develop a CVX-based algorithm to solve optimal design problems on large and irregular discrete spaces for multiple regression models. The algorithm uses innovative rules to add several design points at each iteration, and clusters nearby points together at the end of iteration. Furthermore, we provide useful guidelines for discretizing irregular regions. These are based on derived theoretical properties which relate optimal designs on continuous and discrete design spaces. Several numerical examples and their MATLAB codes are presented for A-, D-, and EI-optimal designs for both linear and generalized linear models. The optimal designs found via the CVX solver are better than those presented in the literature. In addition, our guidelines to discretizing design spaces improve the efficiency of optimal designs, especially over irregular regions. We find that our iterative procedure overcomes the bottlenecks of typical sequential and multiplicative algorithms.

Abstract: Eternal domination is a graph game where mobile guards are placed on vertices and then moved in order to defend against sequences of attacks. If only one guard is allowed to move in response to an attack, then it has been shown that there is no benefit to allowing multiple guards to occupy a vertex. It was conjectured that if all guards can move there would similarly be no benefit from multiple guards occupying a vertex. Finbow et al. showed this was untrue, and provided a construction demonstrating that there are graphs which need fewer guards when 2 guards are allowed on the same vertex. We extend these results to show that for any $k$ and $s$ there are $k$-connected graphs which need fewer guards when up to $s$ guards are allowed on any vertex.

This is joint work with Georgia Penner.

PCIC is pleased to announce an upcoming talk on Wednesday, November 22nd, titled, Clustering for Climate Science Insights, as part of our Pacific Climate Seminar Series.

This talk will be delivered by Dr. John R.J. Thompson, an Assistant Professor at the University of British Columbia (Okanagan campus) whose areas of expertise are nonparametric and applied statistics and machine learning. This talk will be held between 3 p.m. and 4 p.m. Pacific Time, via Zoom meetings. For more on this talk, including registration information and an abstract, see the talk’s page on our site.

Register via Zoom

Abstract: In this talk we will first discuss this soon to be 100 years old conjecture, which states that the set of primes for which an integer $a$ different from $-1$ or a perfect square is a primitive root admits an asymptotic density among all primes. In 1967 Hooley proved this conjecture under the Generalized Riemann Hypothesis.

After that, we will look into a generalization of this conjecture, where we don't restrain ourselves to look for primes for which $a$ is a primitive root but instead elements of an infinite subset of $\N$ for which $a$ is a generalized primitive root. In particular, we will take this infinite subset to be either $\N$ itself or integers with few prime factors.

Speaker Biography: Paul Péringuey is a PIMS Postdoctoral fellow at the University of British Columbia where he is working with Prof. Greg Martin, under the sponsorship of the PIMS Collaborative Research Group "L-Functions in Analytic Number Theory". He obtained his PhD in 2022 at Université de Lorraine under the supervision of Cécile Dartyge. His area of research is in Analytic Number Theory, Comparative Prime Number Theory, as well as Additive Combinatorics.

Poster: The poster for the talk is available.

Zoom link

Abstract. We study a mean-field spin glass model whose coupling distribution has a power-law tail with exponent \alpha \in (0, 2). This is known as Levy spin glasses in literature. Though it is a fully connected model, many of its important characteristics are driven by the presence of strong bonds that have a sparse structure. In this sense, the Levy model sits between the widely studied Sherrington-Kirkpatrick model (with Gaussian couplings) and diluted spin glass models, which are more realistic but harder to understand. In this talk, I will report a number of rigorous results on the Levy model. For example, when 1< \alpha < 2, in the high-temperature regime, we obtain the limit and the fluctuation of the free energy. Also, we can determine the behaviors of the site and bond overlaps at the high temperature. Furthermore, we establish a variational formula of the limiting free energy that holds at any temperature. Interestingly, when 0< \alpha < 1, the effect of the strong bonds becomes more pronounced, which significantly changes the behavior of the model. For example, the free energy requires a different normalization (N^{1/\alpha} vs N), and its limit has a simple description via a Poisson point process at any temperature.

This is a joint work with Wei-Kuo Chen and Heejune Kim.

Speaker Bio:
Kristin Lauter is an American mathematician and cryptographer whose research interest is broadly in application of number theory and algebraic geometry in cryptography. She is particularly known for her work in the area of elliptic curve cryptography. She was a researcher at Microsoft Research in Redmond, Washington, from 1999 - 2021, and the head of the Cryptography Group from 2008 - 2021; her group developed Microsoft SEAL. In April 2021, Lauter joined Facebook AI Research (FAIR) as the West Coast Head of Research Science. She became the President-Elect of the Association for Women in Mathematics in February 2014 and served as President from 2015 - 2017.

Registration:
Participants register once on Zoom and can attend any of the Colloquium talks. Please remember to download the calendar information to save the dates on your calendar. PIMS will resend the confirmation from Zoom prior to the event date.

Zoom link.

Abstract: For every $k\ge 2$ and $\Delta$, we prove that there exists a constant $C_{\Delta,k}$ such that the following holds. For every graph $H$ with $\chi(H)=k$ and every tree with at least $C_{\Delta,k}|H|$ vertices and maximum degree at most $\Delta$, the Ramsey number $R(T,H)$ is $(k-1)(|T|-1)+\sigma(H)$, where $\sigma(H)$ is the size of a smallest colour class across all proper $k$-colourings of $H$. This is tight up to the value of $C_{\Delta,k}$, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.