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Abstract: The boxicity of a graph G is a key graph parameter introduced by Roberts in 1969. It represents the minimum dimension d such that G can be realized as the intersection graph of a family of axis-parallel boxes in R^d. Boxicity is an important measure of structural complexity in various networks, including ecological and social systems. It also exhibits strong connections with other graph parameters such as treewidth, chromatic number, and vertex-cover number.

Determining the boxicity of a graph is computationally challenging—it is NP-hard to decide if a graph has boxicity at most 2. Consequently, much research has focused on developing efficient algorithms for specific graph classes or establishing upper bounds under particular conditions.

In this talk, we introduce a simple, general approach to determining the boxicity of a graph by studying its interval-order subgraphs. Our method leads to several new bounds, including those for Kneser graphs (confirming a conjecture by Caoduro and Lichev [Discrete Mathematics, Vol. 346, 5, 2023]), line graphs, complements of line graphs, and complements of block graphs. Notably, our approach also yields polynomial-time algorithms. In particular, we present an algorithm to compute the boxicity of complements of block graphs—marking the first non-trivial family for which boxicity can be computed in polynomial time.

This is joint work with Amna Adnan, Matthew Barclay, Josh Childs, Will Evans, Tao Gaede, and András Sebő.

Abstract: I will speak about Toeplitz-Pimsner algebras of product systems of Hilbert bimodules over group-embeddable monoids. Given a product system $\E$ over a submonoid of a group $G$, with coefficient algebra $A$, we show that, under technical assumptions, the fixed-point algebra $\Toepr(\E)^G$ of the gauge action of $G$ is nuclear if and only if $A$ is nuclear. Assuming in addition that $G$ is amenable, we conclude that the Toeplitz algebra is nuclear if and only if $A$ is nuclear.

I will also discuss exactness results and applications to product system over abelian monoids, $ax+b$-monoids of integral domains and Baumslag--Solitar monoids $BS^+(m,n)$ that admit an amenable embedding, which we provide when $m$ and $n$ are relatively prime. This is recent joint work with Elias Katsoulis and Camila Sehnem.

Optimal regression design problems on discrete design spaces can be written as convex optimization problems. When the number of points in the discrete design space is not very large, several numerical algorithms can find optimal designs effectively. However, when the number of points is huge, say 10,000 or more, it is challenging to find optimal designs. There are also issues to discretize irregularly shaped design spaces. We develop an effective iterative procedure to compute approximate optimal designs on discrete design spaces with a huge number of points. This procedure includes several new ideas for computing optimal designs: (1) new strategy to discretizing design spaces, (2) new rule to updating design spaces in the iteration, (3) new step to clustering support points in optimal designs. It is easy to use and is very fast. It can be applied for any regression model and many convex optimality criteria.

Click https://uvic.zoom.us/j/83154536954 to start or join a scheduled Zoom meeting.

Abstract: A spanning tree of a finite connected graph G is a connected subgraph of G that includes every vertex and contains no cycles. In this talk we will consider uniformly drawn spanning trees of high-dimensional graphs, and explain why, under appropriate rescaling, they converge in distribution as metric-measure spaces to Aldous’ Brownian CRT. This extends an earlier result of Peres and Revelle (2004) who previously showed a form of finite-dimensional convergence. If time permits, we may also discuss scaling limits of random spanning trees with non-uniform laws. Based on joint works with Asaf Nachmias and Matan Shalev.

Join Zoom Meeting: https://uvic.zoom.us/s/84818303141

Moving-habitat models provide insight into the mechanisms that support local persistence of species within their suitable habitats which shift polewards or upwards in altitude as the planet warms. A species’ suitable habitat is described through the species’ net growth; where the specie’s net growth is positive, the habitat is suitable. Beyond temperature, topographical changes in space present obstacles and barriers that change the shape of the suitable habitat. With conservation in mind, we ask, when does a species persist and can we identify when a species is at high risk of extinction. Reaction-diffusion equations provide one framework to study how the density of a population changes in space and time over these climate-driven moving-habitats. In their simplest form, they provide fundamental insight into persistence ability, studied through asymptotic analysis assuming a constant shift of the suitable habitat. To capture the temporal and spatial complexities of these moving habitats, we introduce two numerical methods, the first for spatially one-dimensional systems, the second for spatially two-dimensional systems. Our methods can capture complex movements of the suitable habitat, like deformation, accelerating shifting speeds, and jumps in density across the interface of suitable and unsuitable habitat resulting from habitat dependent dispersal rates and habitat bias. We analyse our numerical methods and demonstrate their strength in application, particularly focusing on time variable suitable-habitat sizes.

Abstract: Counting lattice paths with unit up and right steps beginning at the origin that are somehow constrained by a boundary is an old problem. When the boundary is the line y=x the celebrated Chung-Feller theorem states the number of paths having k flaws (steps above the boundary) is independent of k. The classic Dyck paths are those with 0 flaws, and the Chung-Feller theorem can be used to give a simple proof of their count.

More recently, we have discovered a similar, though more complicated, result to the Chung-Feller theorem for paths constrained by a line of rational slope. In this talk, I will explain these new results and also explain other recent results on counting lattice paths.

My aim is to have a broadly accessible talk, and I will assume very little prior knowledge in this area.

This is joint work with F. Firoozi and J. Jedwab.

Join us for this free Café Scientifique event and explore how mathematics can help us answer emerging environmental problems with UVic math and biology professor Mark Lewis.

Emerging environmental problems are complex and multifaceted. Normally we have lots of data, but we must put the pieces together if we are going to find solutions. Mathematics provides a way to piece the puzzle together. This talk will show how mathematics has played a central role in answering questions ranging from polar bear survival to the impacts of forests pests under climate change.

Café Scientifique is an initiative from UVic’s Faculty of Science that bridges the research-to-public gap with casual community lectures inspired by the early coffeehouses of 18th century France. These free evening cafés provide an opportunity to learn about exciting new discoveries in modern science and meet experts in the field. Each event begins with a 45 minute talk followed by a moderated question and answer session.

Doors open at 5:30 p.m. Event starts at 7 p.m. This talk will also be livestreamed on the Hermann’s Jazz Club YouTube channel.

The Colloquium Series was inaugurated in 2021 for distinguished speakers to give talks across the full PIMS network with one talk per month during the main academic term.

Register here.

Abstract: For centuries, mathematicians have relied on computers to perform calculations, to suggest conjectures, and as components of mathematical proofs. In the light of more modern tools such as interactive theorem provers, machine learning algorithms, and generative AI, we are beginning to see machines used in more creative and substantive ways in our work. In this talk we survey some historical and recent developments, and speculate on the future roles of machine assistance in mathematics.

Abstract: The directed landscape is a random directed metric on the plane that arises as a scaling limit of a classical metric models in the KPZ universality class. In this talk, we will discuss a functional large deviation principle (LDP) for the entire random metric. Applying the contraction principle, our result yields an LDP for the geodesics in the directed landscape. If time permits, we will also mention certain interesting features of the rate function for the geodesic LDP. Based on a joint work with Duncan Dauvergne and Balint Virag.

Abstract: We analyse the metastable behaviour of the disordered Curie-Weiss-Potts (DCWP) model subject to a Glauber dynamics. The model is a randomly disordered version of the mean-field q-spin Potts model (CWP), where the interaction coefficients between spins are general independent random variables. This model comprises also, e.g., the CWP model on inhomogeneous dense random graphs. We are interested in a comparison of the metastable behaviour of the CWP and the DCWP models, for fixed temperature and the infinite volume limit. We prove the CWP model is metastable and through this prove metastability for the DCWP model.

Then we identify the ratio of the (random) mean time the DCWP model takes to reach the stable phase when it starts from a certain probability distribution on the metastable state (called the last-exit biased distribution) and the (non-random) corresponding quantity for the CWP model. In particular, we obtain the asymptotic tail behaviour and the moments of the ratio of the metastable hitting times of the disordered and annealed systems. Our proof is based on a combination of the potential theoretic approach to metastability and concentration of measure techniques, the later adapted to our particular setup.

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

WANMENG WANG

BSc (University of Manitoba, 2022)

AdaptVarLM: A Linear Regression Model for Covariate-Dependent Non-Constant Error Variance

Department of Mathematics and Statistics

Friday, August 16, 2024

11:00 A.M. Virtual Defence

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

External Examiner:
Dr. Ke Xu, Department of Economics, UVic

Chair of Oral Examination:
Dr. Amanda Bates, Department of Biology, UVic
Dr. Robin G. Hicks, Dean, Faculty of Graduate Studies

Abstract
In biological research, traditional multiple regression models assume homoscedasticity- constant variance of error terms-an assumption that is difficult to maintain in complex biological data. This thesis introduces AdaptVarLM, a novel linear regression model specialized in dealing with non-constant error variance dependent on one covariate. AdaptVarLM integrates an auxiliary linear relationship between the logarithmic variance of the error term and a specific explanatory variable, and uses maximum likelihood estimation (MLE) in the iterative updating process to improve the parameter estimation accuracy. By modelling non-constant error variance, AdaptVarLM outperforms the traditional regression model in capturing the complex variability inherent in biological data. Applying to the study of Alzheimer's disease, AdaptVarLM detects genetically linked genes associated with the disease and error variance. The results of analyzing both bulk and single-cell data validate the effectiveness of AdaptVarLM in detecting significant genes.

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

XIAOXIAO MA
MSc (Tianjin University, 2020)
BSc (Tianjin University, 2017)

On Bilevel Programs and Minimax Problems

Department of Mathematics and Statistics

Monday, August 12, 2024
9:00 A.M.
Virtual Defence

Supervisory Committee:
Dr. Jane Ye, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Julie Zhou, Department of Mathematics and Statistics, UVic (Member)
Dr. Yang Shi, Department of Mechanical Engineering, UVic (Outside Member)

External Examiner:
Dr. Patrick Mehlitz, Department of Mathematics and Computer Science, University of Marburg

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

Abstract
Second-order optimality conditions usually offer more precise insights into local optimality compared to their first-order counterparts. Concurrently, there has been a growing prevalence of bilevel programs and minimax problems in recent years. In our research, we intricately explore second-order optimality conditions within the realm of bilevel programs and minimax problems.

First, we provide a comprehensive exploration of second-order combined approaches for bilevel problems. Building on the well-known first-order combined approach, the research introduces novel techniques that incorporate lower-level second-order information to overcome the difficulty of the constraint qualification for bilevel problems. By characterizing lower-level optimal solutions using both first and second-order necessary optimality conditions, together with the value function constraint, we give some new single-level reformulations for bilevel problems for which the important partial calmness condition can be more likely to hold.

We then focus on the introduction and analysis of calm local minimax points, which is an appropriate local notion for nonconvex-nonconcave nonsmooth minimax problems. We study the properties of calm local minimax points, establishing their strong connections with existing optimality concepts. We provide a comprehensive exploration of first-order and second-order sufficient and necessary optimality conditions for calm local minimax points.

See announcement and abstract.

See announcement and abstract (PDF).

A proper conflict-free colouring of a graph is a (vertex-)colouring with no monochromatic edges such that for every nonisolated vertex v, the neighbourhood N(v) contains a vertex w coloured with a colour not appearing on N(v)-{w}. For a real number h, a colouring of a graph with no monochromatic edges is h-conflict-free if for every vertex v, N(v) contains at least min{deg(v), h} vertices coloured with a colour used only once in N(v). For a real number p, we define a p-peaceful colouring to be a colouring f with no monochromatic edges in which for every vertex v,

|{w in N(v) : there exists u in N(v)-{w} with f(u)=f(w)}| ≤ p.

We note that for a d-regular graph, a colouring is an h-conflict-free proper colouring precisely if it is a (d-h)-peaceful colouring. In contrast, if G is an irregular graph of maximum degree Delta then while a p-peaceful colouring and a (\Delta-p)-conflict-free colouring impose the same condition on maximum degree vertices, the peaceful colouring imposes weaker conditions on low degree vertices. We present some results on these three types of colourings. These are joint work with Chun-hung Liu.

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

KHASHAYAR NESHAT TAHERZADEH
MSc (Sharif University of Technology, 2019)
BSc (Azad University, 2016)

“Invariant conic optimization with basis-dependent cones: scaled diagonally dominant matrices and real *-algebra decomposition”

Department of Mathematics and Statistics
Wednesday, July 17, 2024 9:00 A.M.
Engineering and Computer Science Building Room 130

Supervisory Committee:
Dr. David Goluskin, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Heath Emerson, Department of Mathematics and Statistics, UVic (Member)
External Examiner:
Dr. Cordian Riener, Department of Mathematics and Statistics, University of Tromsø
Chair of Oral Examination:
Dr. Violeta Iosub, Department of Chemistry, UVic

Abstract
Symmetry reduction for a semidefinite program (SDP) with symmetries makes computational solution of the SDP easier by decomposing the semidefiniteness constraint into multiple smaller semidefineness constraints. This decomposition requires changing to a symmetry-adapted basis that block diagonalizes the matrix variable, but this does not change the optimum value of the SDP because the semidefinite cone is basis-independent. For other cones that are basis-dependent, if optimization problems over those cones have symmetries one can still change to a symmetry-adapted basis that block diagonalizes the matrix. However, this change of basis generally changes the constraint cone and can change the optimum. In this thesis we develop a framework for determining when symmetry reduction for basis-dependent conic optimization makes the optimum increase, decrease, or stay the same. The aim is to determine this using general features such as the symmetry group of the optimization problem, without having to solve the problem computationally. We then use our framework to prove various results of this type for scaled diagonally dominant programs (SDDPs), which are convex optimization problems over the cone of scaled diagonally dominant matrices. These results depend on the orbital structure of the underlying representation of invariant SDDPs. Using the regular representation, we demonstrate that analysis of SDDPs of any size can be confined to a smaller SDDP that is invariant under a particular representation. Our approach uses real *-algebra decomposition of equivariant maps, which is not needed for existing symmetry reduction of SDPs. Because polynomial optimization problems with sum-of-squares and sum-of-binomial-squares can be represented as SDPs and SDDPs, respectively, our results on SDDPs have implications for polynomial optimization. Using several polynomial optimization problems as examples, we give computational results that illustrate our theorems. For polynomial optimization subject to sum-of-binomial-squares, our examples included cases in which symmetry reduction causes the optimum to increase, decrease, or stay the same.

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!

See abstract and announcement (PDF).

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.