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Abstract: In this talk, I will discuss several Turán-type results for point-line incidences. It will be based on joint works with Istvan Tomon and Mozhgan Mirzaei.

This is part of the Round the World Relay in Combinatorics.

Abstract:
Russo-Seymour-Welsh's theory is a key concept in two-dimensional statistical physics models. A version of Russo--Seymour--Welsh type assumption for random walks was exploited by Berestycki, Laslier and Ray to study the decorrelation of uniform spanning trees (UST) in a planar graph. This ultimately led to a universality result about convergence of the dimer model. I will discuss an extension of this decorrelation theorem to USTs on random planar graphs. The major example we consider is the infinite cluster of supercritical bond percolation in the square lattice. This is joint work with my supervisor Gourab Ray.

Please contact the organizer if you need the Zoom link.

Join Let's Talk Science and two UVic professors for a chat about the world of Mathematics and Statistics!

This event is open to all students in Grades 9-12 in Canada.

Monday, May 3, 10am - 11am PDT: Dr. Laura Cowen, statistical ecologist.

Wednesday, May 5, 11am - 12pm PDT: Dr. David Goluskin, applied mathematician.

Register here: https://letstalkscience-parlonssciences-ca.zoom.us/webinar/register/WN_T-fHEgI8RuanHLrHibbtFw

Join Let's Talk Science and two UVic professors for a chat about the world of Mathematics and Statistics!

This event is open to all students in Grades 9-12 in Canada.

Monday, May 3, 10am - 11am PDT: Dr. Laura Cowen, statistical ecologist.

Wednesday, May 5, 11am - 12pm PDT: Dr. David Goluskin, applied mathematician.

Register here: https://letstalkscience-parlonssciences-ca.zoom.us/webinar/register/WN_T-fHEgI8RuanHLrHibbtFw

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

Songwan Joun
BSc (Kyungpook National University, 2017)

“Multi-SNP prediction model for lung function decline”

Friday, April 16, 2021
11:00 A.M.
Conducted Virtually

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

Chair of Oral Examination:
Dr. Ke Xu, Economics, University of Victoria

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

KAI YZENBRANDT
BSc (University of Victoria, 2018)

“Minimax D-optimal Designs for Regression Models with Heteroscedastic Errors”

Department of Mathematics and Statistics

Wednesday, April 14, 2021 1:30 P.M. Conducted Virtually

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

External Examiner:
Dr. Xiaodai Dong, Department of Electrical and Computer Engineering, UVic

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

Abstract
Minimax D-optimal designs for regression models with heteroscedastic errors are studied and constructed. These designs are robust against possible misspecification of the error variance in the model. We propose a flexible assumption for the error variance and use a minimax approach to define robust designs. As usual it is hard to find robust designs analytically, since the associated design problem is not a convex optimization problem. However, the minimax D- optimal design problem has an objective function as a difference of two convex functions. An effective algorithm is developed to compute minimax D-optimal designs under the least squares estimator and generalized least squares estimator. The algorithm can be applied to construct minimax D-optimal designs for any linear or nonlinear regression model with heteroscedastic errors. In addition, several theoretical results are obtained for the minimax D-optimal designs.

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

XINQI XIE
BSc (University of Victoria, 2018)

“Within-host Dynamics of HIV/AIDS”

Department of Mathematics and Statistics
Tuesday, April 13, 2021 10:00 A.M.
Conducted Virtually

Supervisory Committee:
Dr. Pauline van den Driessche, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Junling Ma, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:
Prof. Libin Rong, Department of Mathematics, University of Florida

Chair of Oral Examination:
Dr. Nathan Lachowsky, School of Public Health and Social Policy, UVic

This thesis first investigates within-host HIV models for the acute stage. These models incorporate the immune responses and helper T cells produced from the activation of naive CD4 T cells. Because both naive CD4 T cells and helper T cells are susceptible classes, backward bifurcation and bistability may occur. We start with a simple model that ignores the CD8 T cell dynamics, then extend it to include this dynamics. We also extend our model to consider the latent infection of naive CD4 T cells. Backward bifurcation occurs in all these models. We numerically investigate the stability of viral equilibria, and show the bistability caused by backward bifurcation. Increasing the in flow of CTLs prevents the backward bifurcation. With a large homeostatic source of healthy naive CD4 T cells, the disease is easier to establish when the basic reproduction number is less than one. Reducing the reproduction number below one is not sufficient to control the infection of HIV. Secondly, this thesis investigates the development of AIDS caused by viral diversity, as proposed by Wodarz et al. using a model that does not include the details of immune responses. We extend their model to include density dependence, and show that the viral load increases with viral diversity. To study if this result still holds with more realistic HIV dynamics, we incorporate viral diversity into our first model. We conclude theoretically that the total viral load is positively correlated with the number of viral strains, and viral diversity can drive the development of AIDS. We also find that the total CD4 T cell count does not always decrease with viral diversity. Thus further investigation is needed to fully understand the development of AIDS.

Abstract: In an earlier work in collaboration with C. Liverani, we studied stochastic properties of slow-fast partially hyperbolic local diffeomorphisms of the 2-torus. We showed a Local Central Limit Theorem which was instrumental in our proof of existence of finitely many SRB measures and exponential decay of correlation towards SRB measures, provided that the system is mostly contracting (e.g. every SRB measure has negative center Lyapunov exponents). In this talk I will present a work in progress with K. Fernando (U. Toronto), which deals with mostly expanding systems. Such systems have paradoxical features (such as statistical sinks with positive Lyapunov exponents). Once again using the LCLT, we prove that in this situation we also have finitely many SRB measures and exponential decay of correlations with bounds similar to the mostly contracting case.

Please contact the organizer if you need the Zoom link.

We determine a complete existence theory for signed graph designs. We extend previous work on the existence of graph designs -- specifically working on designs based on complete bipartite graphs and their complements -- to finish the existence theory. We also introduce the problem for multigraphs, and describe how the concept of vertex potentials can be used to help us understand the more general problem.

Please contact the organizer if you need the Zoom link.

In these two talks, we will develop a C*-algebra analogue to Masur's criterion, which asserts that a finite genus minimal flat surface whose orbit under Teichmuller flow has an accumulation point in Teichmuller space is uniquely ergodic. When the space of flat surfaces is parameterized suitably, each flat surface has a canonical transversal to its translation flow, which is an interval exchange mapping. The Teichmuller flow passes down to the canonical transversal to a "renormalization procedure" known as Rauzy-Veech induction on interval exchange mappings. This procedure turns out to be the key tool in proving Masur's criterion. Renormalization procedures have been used (both implicitly and explicitly) to solve other interesting problems like the C^{\infty} conjugation problem for circle diffeos by M. Herman, and to explain the existence of (hairy) copies of the Mandelbrot set inside itself by A. Douady and J.H. Hubbard. Our approach will be to define the notion of a renormalization procedure for a family of C*-algebras, and state Masur's criterion in these terms, instead of looking for an analogue of Teichmuller flow.

In the first talk we will cover most of the necessary pre-requisites which will include C(X)-algebras, groupoid correspondences, C*-correspondences, and their induced maps on traces, considered first by M. Laca and S. Neshveyev. We will then define the notion of a renormalization procedure in terms of these concepts. Along the way, we will work through the example of the "space of all AF-algebras" and its renormalization procedure.

In the second talk we will prove the analogue of Masur's criterion, which will state that given a renormalization procedure on a space (family) of C*-algebras, a C*-algebra in this space has either one or no trace if its orbit under the renormalization procedure has an accumulation point of "finite type" that is a simple C*-algebra. We obtain as a corollary R. Trevino's unique ergodicity result for infinite genus flat surfaces constructed out of bi-infinite Bratteli diagrams. In the time remaining we will construct the renormalization procedure for the irrational rotation algebras, and sketch why it satisfies a strong form of renormalization, and obtain as a corollary the elementary fact that an irrational rotation algebra has unique trace.

Please contact the organizer if you need the Zoom link.

Abstract: The Grothendieck L_p problem is defined as an optimization problem that maximizes the quadratic form of a Gaussian matrix over the unit L_p ball. The p=2 case corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble, while p=\infty this problem is known as the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In this talk, I will describe the limit of this optimization problem for general p and discuss some results on the behavior of the near optimizers along with some open problems. This is based on a joint work with Arnab Sen.

Please email the organizer if you need the Zoom link.

Starting in 2021, PIMS has inaugurated a high-level network-wide colloquium series. Distinguished speakers will give talks across the full PIMS network with one talk per month during the academic term. The 2021 speaker series is part of the PIMS 25th Anniversary showcase.

Abstract: In The Hitchhiker’s Guide to the Galaxy, by Douglas Adams, the number 42 was revealed to be the “Answer to the Ultimate Question of Life, the Universe, and Everything”. But he didn’t say what the question was! I will reveal that here. In fact it is a simple geometry question, which then turns out to be related to the mathematics underlying string theory.

John Baez is a leader in the area of mathematical physics at the interface between quantum field theory and category theory, and has broad interests in mathematics, and science more generally. He created one of the earliest blogs "This week's finds in Mathematical Physics" (before the term blog existed!)

Baez did his PhD at MIT, and was a Gibbs Instructor at Yale before moving to University of California, Riverside in 1988.

Time:
All network wide colloquia take place at 1:30pm Pacific Time

Registration:
To attend this event please register here. Kindly note that this talk will be recorded.

Given a graph $G$, the {\it strong clique number} of $G$, denoted $\SC(G)$, is the maximum size of a set $S$ of edges such that every pair of edges in $S$ has distance at most $2$ in the line graph of $G$. As a relaxation of the renowned Erd\H{o}s--Ne\v set\v ril conjecture regarding the strong chromatic index, Faudree et al. suggested investigating the strong clique number, and conjectured a quadratic upper bound in terms of the maximum degree.

Recently, Cames van Batenburg, Kang, and Pirot (2020) conjectured a linear upper bound in terms of the maximum degree for graphs without even cycles. Namely, if $G$ is a $C_{2k}$-free graph with $\Delta(G)\geq \max\{4, 2k-2\}$, then $\SC(G)\leq (2k-1)\Delta(G)-{2k-1\choose 2}$, and if $G$ is a $C_{2k}$-free bipartite graph, then $\SC(G)\leq k\Delta(G)-(k-1)$. We prove the second conjecture in a stronger form. Regarding the first conjecture, we prove an upper bound that is off by the constant term.

Please contact the organizer if you need the Zoom link.

In these two talks, we will develop a C*-algebra analogue to Masur's criterion, which asserts that a finite genus minimal flat surface whose orbit under Teichmuller flow has an accumulation point in Teichmuller space is uniquely ergodic. When the space of flat surfaces is parameterized suitably, each flat surface has a canonical transversal to its translation flow, which is an interval exchange mapping. The Teichmuller flow passes down to the canonical transversal to a "renormalization procedure" known as Rauzy-Veech induction on interval exchange mappings. This procedure turns out to be the key tool in proving Masur's criterion. Renormalization procedures have been used (both implicitly and explicitly) to solve other interesting problems like the C^{\infty} conjugation problem for circle diffeos by M. Herman, and to explain the existence of (hairy) copies of the Mandelbrot set inside itself by A. Douady and J.H. Hubbard. Our approach will be to define the notion of a renormalization procedure for a family of C*-algebras, and state Masur's criterion in these terms, instead of looking for an analogue of Teichmuller flow.

In the first talk we will cover most of the necessary pre-requisites which will include C(X)-algebras, groupoid correspondences, C*-correspondences, and their induced maps on traces, considered first by M. Laca and S. Neshveyev. We will then define the notion of a renormalization procedure in terms of these concepts. Along the way, we will work through the example of the "space of all AF-algebras" and its renormalization procedure.

In the second talk we will prove the analogue of Masur's criterion, which will state that given a renormalization procedure on a space (family) of C*-algebras, a C*-algebra in this space has either one or no trace if its orbit under the renormalization procedure has an accumulation point of "finite type" that is a simple C*-algebra. We obtain as a corollary R. Trevino's unique ergodicity result for infinite genus flat surfaces constructed out of bi-infinite Bratteli diagrams. In the time remaining we will construct the renormalization procedure for the irrational rotation algebras, and sketch why it satisfies a strong form of renormalization, and obtain as a corollary the elementary fact that an irrational rotation algebra has unique trace.

Please contact the organizer if you need the Zoom link.

Abstract: Biochemical networks often take the form of systems of ODEs with Michaelis-Menten kinetics. A motif that arises in at least one such network, for the regulation of metabolism in plants, is the Precursor Shutoff Valve (PSV). Primary metabolism must be maintained at a steady (if low) level, while secondary metabolism (production of lignin to build wood in trees, for example) can be very highly active, when input is plentiful, but can be shut off completely when input is low. However, both pathways are fed by the same chain of input reactions. Two possible mechanisms by which a plant may differentially regulate the two output fluxes have recently been proposed. One is simply a difference in thresholds for the two output pathways; the other is the PSV, a motif in which a precursor of both pathways is required for a reaction in the secondary pathway only. The combination of the two mechanisms is the most effective. Analysis of such a system is facilitated by a piecewise-linear approximation of the Michaelis-Menten type interactions, comprising a linear region followed by a saturated (constant) region. It is likely that the PSV motif occurs in other contexts, but interestingly, it is not effective in steep-switching systems (like Glass networks). All of these results can be proven analytically.

Please contact the organizer if you need the Zoom link.

Cultures have their own identity; cultures interact. The medieval period contains within it widely varying cultures in Europe, India, and the middle East. The subject that eventually became modern mathematics did not live in a geographical cocoon during this period; it owes a great deal to several cultures. The journey of mathematics through Islam, for almost a millennium, changed it utterly. The shaping of algebra, the number system, arithmetic, geometry, optics, and mathematical astronomy had a major, yet unseen impact on how we think today. Yet, to understand the accomplishments of the medieval Islamic scientists, we must approach them on their terms. We shall explore some of the roots of modern mathematics, but also try to view the mathematical sciences in medieval Islam with eyes open to their vision --- not ours.​

Please register at https://uvic.zoom.us/meeting/register/tZcld--qrTkqGtYkNmAHVMTgPPkDoYeRV8P6.

Abstract: Leplaideur and Watbled recently applied the tools from thermodynamic formalism to the Curie-Weiss mean-field theory with a new twist: they used a variant of the pressure where the energy functional is quadratic. Inspired by this result, Buzzi and Leplaideur initiated an effort to broaden the thermodynamical approach by considering an arbitrary function of free energies. In this talk we discuss the concepts of the pressure and equilibrium states in such nonlinear settings. We establish a nonlinear variational principle and characterize the nonlinear equilibrium measures as classical equilibrium states for some multiples of the potential.​

Please contact the organizer for the Zoom link if you need it.

Mathematics on the Indian subcontinent has been flourishing for over two and a half millennia, and this culture of inquiry has produced insights and techniques that are central to many of our mathematical practices today, such as the base ten decimal place value system and trigonometry. Indeed, many of their technical procedures, such as infinite series expansions for various mathematical relations predated those that were developed with the advent of the Calculus in Europe, but notably in contrasting intellectual circumstances with distinctly different epistemic priorities. However, while many histories of mathematics have centered on the so-called "western miracle" in their analysis of the ignition and flourishing of modern science, they have done so at the expense of other non-European traditions. This talk will highlight some of the significant mathematical achievements of India, and explore the work that remains to be done integrating them into more standard histories of mathematics.​

Register for the Zoom meeting at https://uvic.zoom.us/j/84541878195?pwd=alY0K1k0d1hSdzJ2d2FUYkUxY05UUT09

The study of quasirandom combinatorial structures is a fertile research area whose origins can be traced to the seminal work R\"{o}dl, Thomason, and Chung, Graham and Wilson from the 80s on graph quasirandomness. Proving a conjecture of Garbe et al., we show that a sequence of Latin squares is quasirandom if and only if every $2\times 3$ pattern has density $1/720+o(1)$. Moreover, we show this characterisation is best possible in the sense that a weakening to $2\times 2$ and $1\times k$ patterns is not sufficient to guarantee quasirandomness. Our work draws from the newly developed limit theory of Latin squares introduced by Garbe et al. (2020+), paralleling the successful limit theories for dense graphs and permutations. The talk is self-contained, assuming no prior knowledge of combinatorial limits.

Based on joint work with Daniel Kr\'al', Ander Lamaison, and Samuel Mohr.

Please contact the organizer if you need the Zoom link.

Abstract: It is a well-established fact that when one regards the tail-equivalence relation on a Bratteli diagram as a principal groupoid, the resulting C*-algebra is an AF-algebra. By taking a quotient of the unit space (the path space), we produce a "factor groupoid" which yields a C*-subalgebra of the AF-algebra. If the quotient operation is carried out in a particular way, the relative K-theory of this inclusion may be computed using recent excision results. I will begin the talk by describing this quotient space construction and presenting some examples. If time permits I will outline the analysis of the K-theory of the resulting factor groupoid C*-algebras.

Please contact the organizer if you need the Zoom link.

I will discuss some results on extreme eigenvalue distributions of adjacency matrices of random $d$-regular graphs, which are believed to be universal following the Tracy-Widom distribution. In the first part of the talk, I will present the results on random $d$-regular graphs, where $d$ grows with the size $N$ of the graph. We confirm that on the regime $N^{2/9}<< d<< N^{1/3}$ the extremal eigenvalues after proper rescaling are concentrated at scale $N^{-2/3}$ and their fluctuations are governed by the Tracy-Widom statistics. Thus, in the same regime of $d$, about fifty-two percent of all $d$-regular graphs have the second-largest eigenvalue strictly less than $2\sqrt{d-1}$. In the second part of the talk, I will focus on random $d$-regular graphs with fixed $d≥3$, and give a new proof of Alon's second eigenvalue conjecture that with high probability, the second eigenvalue of a random $d$-regular graph is bounded by $2\sqrt{d-1}+o(1)$. Here, we can show that the error term is polynomially small in the size of the graph. These are based on joint work with Roland Bauerschmidt, Antti Knowles and Horng-Tzer Yau.

Finite-time Singularity in the Euler equation

Please contact the organizer if you need the Zoom link.

Abstract: When the acting group is the integers, classical results in symbolic dynamics guarantee that shifts of finite type (SFTs) and sofic shifts of positive entropy must contain rich families of subsystems. In this talk, I will discuss recent (ongoing) work with Robert Bland and Ronnie Pavlov on the subsystems of SFTs and sofic shifts on arbitrary countable amenable groups. In particular, we prove that for any countable amenable group G, any G-SFT X with positive entropy h(X) > 0 contains a family of SFT subsystems whose entropies are dense in the interval [0, h(X)]. We also establish analogous results for G-sofic shifts. Our results recover those of Desai in the case G = Z^d, and our proofs make use of recent work on exact tilings of amenable groups by Downarowicz, Huczek, and Zhang.

Please contact the organizer if you need the Zoom link.

The Arab mathematician al-Khwarizmi is usually said to be the 'father of algebra', or otherwise that 'the Arabs invented algebra'. There is probably nothing in the previous sentence that is true (except the 'usually'). It turns out that the traditional story is just intellectually, mathematically, and culturally lazy. A little bit of thinking about the original texts, the mathematics, and a little bit of historical context leads to a much more problematic, culturally rich, and technically subtle story. We still don't know the whole story – there is lots of room for further research, if you have the languages – and a lot of room for thinking about past mathematics (and by symmetry present day mathematics) as existing in a rich, complex social and intellectual matrix, and not just as a succession of correct theorems. The story might even involve the Sogdians, and you have never heard of them!​

Register in advance for this meeting: https://uvic.zoom.us/meeting/register/tZcrf-mtqTwpG9T547KsKzp1p733IrTo8w5v

After registering, you will receive a confirmation email containing information about joining the meeting.​

Abstract: The eternal domination number of a graph is the minimum number of guards necessary to defend the graph against any sequence of attacks on its vertices. It is well known that the clique covering number of a graph is an upper bound on its eternal domination number. In this talk, we will show that the smallest order of a graph having eternal domination number strictly less than its clique covering number is 10 and discuss related results. This is joint work with Gary MacGillivray and Kieka Mynhardt.

Please contact the organizer if you need the Zoom link.

Abstract: We first review several dilation results for contractive semigroup representations and their connection to Nica's covariance conditions. We then consider the dilation theory for certain type of semigroup dynamical systems that encode the right LCM structure of the semigroup. This leads to a generalized Naimark dilation theorem and Stinespring's dilation theorem. As an application, we prove a dilation result for contractive representations of the boundary quotient. This is a joint work with Marcelo Laca.

Please contact the organizer if you need the Zoom link.

We will discuss recent results on singularity formation in the Euler equation and related models.

Please contact the organizer if you need the Zoom link.

Abstract: In this talk, we will discuss equidistribution of the orbits of the horocycle subgroup acting on homogeneous spaces (e.g. SL2(R)/SL2(Z)). This roughly means that orbits spend "the expected" amount of time in every set, or that it "fills out" the space. In the case that the space has finite volume, this follows from Ratner's famous equidistribution theorem. It was recently proved by Mohammadi and Oh for certain infinite volume cases as well. Such results have seen many applications, especially in number theory. However, in applications, one often needs to know more than just equidistribution: specifically, at what rate does the orbit equidistribute? We call a statement that includes a quantitative error term effective. In this talk, I will present an effective equidistribution theorem, with polynomial rate, for the horocycle subgroup acting on certain infinite volume quotients (called geometrically finite), and if time permits, an application showing a fractal distribution of certain vectors in R^n under the action of a geometrically finite subgroup of SO(n,1) (or e.g. SL2(R)). No prior knowledge of homogeneous dynamics will be assumed. This is joint work Nattalie Tamam.

Please contact the organizer if you need the Zoom link.

Starting in 2021, PIMS has inaugurated a high-level network-wide colloquium series. Distinguished speakers will give talks across the full PIMS network with one talk per month during the academic term. The 2021 speaker series is part of the PIMS 25th Anniversary showcase.

The asymmetric exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice. While it was initially introduced by Macdonald-Gibbs-Pipkin to provide a model for translation in protein synthesis, the stationary distribution of the ASEP and its variants has surprising connections to combinatorics. I will explain how the study of the ASEP on a ring leads to new formulas for Macdonald polynomials, a remarkable family of multivariate polynomials which generalize Schur polynomials. In a different direction, the inhomogeneous ASEP on a ring is closely connected to Schubert polynomials, which represent classes of Schubert varieties in the flag variety. This talk is based on joint work with Corteel-Mandelshtam, and joint work with Donghyun Kim.

Speaker Biography: Lauren Williams is the Dwight Parker Robinson Professor of Mathematics at Harvard University and the Sally Starling Seaver Professor at the Radcliffe Institute. She is a leader in the field of algebraic combinatorics, with research programs on the positive Grassmannian, and combinatorial models in statistical mechanics. Before moving to Harvard, she was a professor at Berkeley, where she held numerous prestigious prizes and grants including a Sloan fellowship and an NSF Career grant. In a recent Quanta article on her work, she was described as fearless, for her willingness to take mathematical leaps.

Time:
All network wide colloquia take place at 1:30pm Pacific Time

Registration:
To attend this event please register here. Kindly note that this talk will be recorded.

The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and R\"odl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers that are linear in the number of vertices. This extends recent results about the linearity of size-Ramsey numbers of powers of bounded degree trees and of long subdivisions of bounded degree graphs.

This is joint work with Shoham Letzter and Alexey Pokrovskiy.

Please contact the organizer if you need the Zoom link.

Abstract: The first talk will be an introductory survey of the history of C*-algebras generated by isometries on Hilbert space. I will begin by recounting classical theorems of Coburn, Douglas, and Cuntz from the 1960’s and 1970’s and discuss their proofs. Douglas’ and Cuntz’s approaches already indicate, in an implicit way, that semigroup crossed products play a central role in the theory. This was not formalized until the 1990’s when Stacey, Murphy, and then Raeburn and I developed an explicit approach focusing on a covariance condition that works well for the quasi-lattice semigroups introduced by Nica. I will elaborate a bit on this approach and show how it works on selected examples. I will finish by giving a brief introduction to Xin Li's generalization from the 2010’s of covariance using constructible ideals, and by discussing some examples from number theory. This talk should be accessible to those with a modicum of exposure to operators on Hilbert space.

The second talk will be about recent joint work with C. Sehnem on a universal Toeplitz algebra for submonoids of groups defined via generators and relations. Our C*-algebra coincides with Xin Li's when the semigroup satisfies his independence condition but behaves as expected also when independence fails; in particular, it is isomorphic to the left regular Toeplitz algebra in situations where weak-containment holds. I will give a characterization of faithful representations and uniqueness theorems for these universal Toeplitz algebras that are new even for right LCM monoids. Time permitting I will also discuss how Sehnem’s covariance algebra of a product system leads to a full boundary quotient of the universal Toeplitz algebra, generalizing the boundary relations of quasi-lattice orders introduced by Crisp and myself in the 2000’s.

Please contact the organizer if you need the Zoom link.

Abstract: Universality in disordered systems has always played a central role in the direction of research in Probability and Mathematical Physics, a classical example being the Gaussian universality class (the central limit theorem). In this talk, I will describe a different universality class for random growth models, called the KPZ universality class. Since Kardar, Parisi and Zhang introduced the KPZ equation in their seminal paper in 1986, the equation has made appearances everywhere from bacterial growth, fire front, coffee stain to the top edge of a randomized game of Tetris; and this field has become a subject of intense research interest in Mathematics and Physics for the last 15 to 20 years. The random growth processes that are expected to have the same scaling and asymptotic fluctuations as the KPZ equation and converge to the universal limiting object called the KPZ fixed point, are said to lie in the KPZ universality class, though this KPZ universality conjecture has been rigorously proved for only a handful of models till now. Here, I will talk about some recent results on universal geometric properties of the KPZ fixed point and the underlying landscape and show that the KPZ equation and exclusion processes converge to the KPZ fixed point under the 1:2:3 scaling, establishing the KPZ universality conjecture for these models, which were long-standing open problems in this field. The talk is based on joint works with Jeremy Quastel, Balint Virag and Duncan Dauvergne.

Please contact the organizer if you need the Zoom link.

Abstract: Mathematics has been an important tool in various colonizing enterprises; and in the last 2 centuries the colonizing enterprise has often involved teaching mathematics to the new subjects of the imperial or colonial regime. In this rather informal discussion we will look at mathematics and mathematicians as instruments in this process, using examples from various time periods and places.

Register for the Zoom meeting at https://uvic.zoom.us/j/85116042781?pwd=UVRjNi9IYThuWmV0OTBabUw4Q1gvdz09

Given a collection of points S in a metric space and a probability p in (0,1), one can make a random graph by independently joining any pair of points in S that are less than 1 apart by an edge with probability p; and no edge with probability 1-p. Points 1 or more apart are never joined by edges. In a surprising paper, Bonato and Janssen showed that if the metric space is the real line and S is a randomly chosen countable dense set, then the random graphs obtained by this process are almost surely isomorphic. Furthermore, somewhat like the Rado graph, the isomorphism class of the graph obtained does not depend on the particular p or the particular S (with probability 1). They call the resulting graph the Geometric Rado graph on R. Bonato and Janssen also showed that this Rado property fails in two dimensions.

I will discuss Bonato and Janssen's results, and then describe some joint work, extending the results to larger metric spaces. One of the motivating questions is "which features of the metric space and/or the vertex set S are revealed by the isomorphism class of the graph?"

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Abstract: The first talk will be an introductory survey of the history of C*-algebras generated by isometries on Hilbert space. I will begin by recounting classical theorems of Coburn, Douglas, and Cuntz from the 1960’s and 1970’s and discuss their proofs. Douglas’ and Cuntz’s approaches already indicate, in an implicit way, that semigroup crossed products play a central role in the theory. This was not formalized until the 1990’s when Stacey, Murphy, and then Raeburn and I developed an explicit approach focusing on a covariance condition that works well for the quasi-lattice semigroups introduced by Nica. I will elaborate a bit on this approach and show how it works on selected examples. I will finish by giving a brief introduction to Xin Li's generalization from the 2010’s of covariance using constructible ideals, and by discussing some examples from number theory. This talk should be accessible to those with a modicum of exposure to operators on Hilbert space.

The second talk will be about recent joint work with C. Sehnem on a universal Toeplitz algebra for submonoids of groups defined via generators and relations. Our C*-algebra coincides with Xin Li's when the semigroup satisfies his independence condition but behaves as expected also when independence fails; in particular, it is isomorphic to the left regular Toeplitz algebra in situations where weak-containment holds. I will give a characterization of faithful representations and uniqueness theorems for these universal Toeplitz algebras that are new even for right LCM monoids. Time permitting I will also discuss how Sehnem’s covariance algebra of a product system leads to a full boundary quotient of the universal Toeplitz algebra, generalizing the boundary relations of quasi-lattice orders introduced by Crisp and myself in the 2000’s.

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Abstract: This talk is about a series of papers with Quoc-Hung Nguyen, devoted to the study of solutions with critical regularity for the two-dimensional Muskat equation. I will describe our main result, which states that the Cauchy problem is well-posed on the endpoint Sobolev space of L^2 functions with three-half derivative in L^2 (locally in time for large data, and globally for small enough data). This result is optimal with respect to the scaling of the equation.

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Abstract: Let (X,sigma) be a subshift. A flow equivalence of two dynamical systems is an orientation-preserving homeomorphism of the suspensions of the systems. The mapping class group of a subshift is the group of self-flow equivalences up to isotopy. We compute the mapping class group for various classes of minimal low complexity subshifts.

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Cops and Robber is a well-studied pursuit-evasion game played on graphs. In this talk, I’ll discuss a variation of the game with a new capture condition. Instead of a win for the cop side resulting from at least one of the cops occupying the same vertex as the robber as in the original game, the cops in this surrounding version win by occupying each of the robber's neighbouring vertices.

The surrounding copnumber is analogous to the copnumber. We present a variety of results for this parameter, including exact values for several classes of graphs as well as more general bounds. Classes of interest include graph products, graphs arising from combinatorial designs, and generalized Petersen graphs.

This is joint work with A. Burgess, R. Cameron, P. Danziger, S. Finbow, C. Jones, and D. Pike.

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See abstract (PDF).

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Abstract: We establish a central limit theorem for the maximal Lyapunov exponent of typical cocycles (in the sense of Bonatti and Viana) over irreducible subshifts of finite type with respect to the unique equilibrium state for a Hölder potential. We also establish other related results such as the analytic dependence of the top Lyapunov exponent on the underlying equilibrium state and a large deviation principle. The transfer operator and its spectral properties play key roles in establishing these limit laws. This is joint work with Kiho Park.

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Abstract: Various types of calculi can be defined for manifolds equipped with specific geometric structures; for instance, the recent calculi defined by van Erp-Yunken for manifolds with a Lie filtration, the b-calculus of Melrose for manifolds with boundary, and, more generally, the calculus on manifolds with corners studied by Monthubert. These calculi are typically defined as distributions on a (Lie) groupoid which are conormal along the object space. The groupoids associated with the aforementioned calculi are usually defined by starting with the pair groupoid and then performing a deformation or blow-up procedure. Recently, a theory of "weighted manifolds" has been developed by Meinrenken and Loizides as a way to understand the work of Haj-Higson and van Erp-Yunken on filtered manifolds (a general. In this talk I will briefly explain the theory of weighted manifolds, and explain how a weighting of a manifold naturally defines a class of conormal distributions using a "weighted deformation" process. As an example, I will explain how a Lie filtration on a manifold M defines a weighting of the pair groupoid M×M and how the calculus associated to this weighting is the calculus of van Erp-Yunken. If time permits, I will explain the usefulness of the deformation process to index theory. This is based on work in progress with E. Meinrenken.

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Time:
All network wide colloquia take place at 1:30pm Pacific Time

Registration:
To attend this event please register here. Kindly note that this talk will be recorded.

Abstract:
Colour {1,..,N} red and blue, in such a manner that no 3 of the blue elements are in arithmetic progression. How long an arithmetic progression of red elements must there be? It had been speculated based on numerical evidence that there must always be a red progression of length about sqrt{N}. I will describe a construction which shows that this is not the case - in fact, there is a colouring with no red progression of length more than about exp ((log N)^{3/4}), and in particular less than any fixed power of N.

I will give a general overview of this kind of problem (which can be formulated in terms of finding lower bounds for so-called van der Waerden numbers), and an overview of the construction and some of the ingredients which enter into the proof. The collection of techniques brought to bear on the problem is quite extensive and includes tools from diophantine approximation, additive number theory and, at one point, random matrix theory.

Speaker Biography:
Ben Green works in additive combinatorics and related areas, such as harmonic analysis and approximate algebraic structure. The underlying relationship is that ideas rooted in harmonic analysis are used to give a quantitative description of randomness versus structure. Among many important results, the results with Terence Tao stating that the prime numbers contain arbitrarily long arithmetic progressions is a landmark. His work has revolutionized the field and Dr. Green has been awarded many prizes, including the Salem Prize and the Sylvester Medal of the Royal Society. Dr. Green is a Fellow of the Royal Society.

Ben Green is the Waynflete Professor of pure mathematics at the University of Oxford. Previously he was a Professor at the University of Cambridge. The paper with Terence Tao on arithmetic progressions in the primes was written when Ben Green was a PIMS postdoctoral fellow at the University of British Columbia.

About the PIMS Network-Wide Colloquia:
Starting in 2021, PIMS has inaugurated a high-level network-wide colloquium series. Distinguished speakers will give talks across the full PIMS network with one talk per month during the academic term. The 2021 speaker series is part of the PIMS 25th Anniversary showcase.

Establishing inequalities among graph densities is a central pursuit in extremal graph theory. One way to certify the nonnegativity of a graph density expression is to write it as a sum of squares or as a rational sum of squares. In this talk, we will explore how one does so and we will then identify simple conditions under which a graph density expression cannot be a sum of squares or a rational sum of squares. Tropicalization will play a key role for the latter, and will turn out to be an interesting object in itself. This is joint work with Greg Blekherman, Mohit Singh, and Rekha Thomas.

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Abstract: In this second talk, we introduce a new numerical method to solve partial differential equations, especially elliptic ones, in unbounded domains, such as whole space, half-space or external domains. The method, due to the speaker, does not require any domain truncation. ​After a rigorous analysis of its convergence, we present some numerical results obtained with this method on various one-dimensional, ​two-dimensional and three-dimensional problems.

Please contact the organizer if you need the Zoom link.

Abstract: What is the chance that among a group of n friends, there are s friends all of whom have the same birthday? This is the celebrated birthday problem which can be formulated as the existence of a monochromatic s-clique (s-matching birthdays) in the complete graph K_n, where every vertex of K_n is uniformly colored with 365 colors (corresponding to birthdays). More generally, for a connected graph H, let T(H, G_n) be the number of monochromatic copies of H in a uniformly random coloring of the vertices of the graph G_n with c_n colors. In this talk, characterization theorems for the limiting distribution of this quantity, and related random multilinear polynomials, will be derived. Applications of these results in testing high-dimensional discrete distributions, motif estimation in large networks, and the discrete logarithm problem will also be discussed.

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Over the millennia, from Theano (born c. 546 B.C.), the wife of the Greek mathematician Pythagoras and herself a mathematician, to Maryam Mirzakhani (May 1977 – July 2017), who in 2014 became the first woman to win the Fields medal, the most prestigious award in mathematics, there have been many brilliant female mathematicians working in all areas of math. I will mention a few who were active in the late 19th and the first half of the 20th centuries, and discuss the work and impact of one of them in greater depth.​

Register in advance for this meeting: https://uvic.zoom.us/meeting/register/tZcrf-Ghpj0pHNTYW8T5Ofcnm-Fwed0LcXDd

After registering, you will receive a confirmation email containing information about joining the meeting.​

A connected matching is a matching contained in a connected component. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs.

I will describe Łuczak's reduction, introduce the new reduction, and mention potential applications of the improved method.

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Abstract: Various types of calculi can be defined for manifolds equipped with specific geometric structures; for instance, the recent calculi defined by van Erp-Yunken for manifolds with a Lie filtration, the b-calculus of Melrose for manifolds with boundary, and, more generally, the calculus on manifolds with corners studied by Monthubert. These calculi are typically defined as distributions on a (Lie) groupoid which are conormal along the object space. The groupoids associated with the aforementioned calculi are usually defined by starting with the pair groupoid and then performing a deformation or blow-up procedure. Recently, a theory of "weighted manifolds" has been developed by Meinrenken and Loizides as a way to understand the work of Haj-Higson and van Erp-Yunken on filtered manifolds (a general. In this talk I will briefly explain the theory of weighted manifolds, and explain how a weighting of a manifold naturally defines a class of conormal distributions using a "weighted deformation" process. As an example, I will explain how a Lie filtration on a manifold M defines a weighting of the pair groupoid M×M and how the calculus associated to this weighting is the calculus of van Erp-Yunken. If time permits, I will explain the usefulness of the deformation process to index theory. This is based on work in progress with E. Meinrenken.

Please contact the organizer if you need the Zoom link.

Abstract: We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schrödinger equation with nonlinear exponent close to the mass critical case and for any space dimension. These profiles bifurcate from the ground state solitary wave. In this talk, we present the argument which relies on the matched asymptotics method and we derive an exponentially smallness condition on the Sobolev critical exponent as conjectured by Sulem and Sulem in 1997.

This is a joint work with Yvan Martel and Pierre Raphaël.

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Cancers form through a complex interplay of mutational, microenvironmental, and immunological factors that give rise to highly heterogeneous tumors. This heterogeneity can critically determine the outcome of anticancer therapies and shed light on the processes that drive cancer formation, but is frequently masked because traditional genetic and genomic approaches lack sufficient resolution to capture it. By integrating single cell RNA sequencing with high-dimensional flow cytometry, multiplexed histological analysis, and metabolomic profiling, our group has begun to resolve key aspects of the tumor microenvironment and the anticancer immune response. Here, I present new insights derived from these approaches, alongside the analytical challenges and opportunities inherent in these nascent technologies.

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Abstract: One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. We will explore a dynamical approach to this topic. After introducing a new dynamical framework for treating questions in multiplicative number theory, we will present an ergodic theorem which contains various classical number-theoretic results, such as the Prime Number Theorem, as special cases. This naturally leads to a formulation of an extended form of Sarnak's Mobius randomness conjecture, which deals with the disjointness of actions of (N,+) and (N,*). This talk is based on joint work with Vitaly Bergelson.

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Nati Linial raised the following basic problem in 2006: given a k-dimensional simplicial complex S, how many facets can a k-complex on n vertices have if it contains no topological copy of S? This is a beautiful and natural question, but results in low dimensions apart (k ≤ 2), very little was previously known. In this talk, I'll provide an answer in all dimensions and take the scenic route to the answer, surveying many natural problems about simplicial complexes along the way.

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Abstract: In this second talk, we introduce a new numerical method to solve partial differential equations, especially elliptic ones, in unbounded domains, such as whole space, half-space or external domains. The method, due to the speaker, does not require any domain truncation. ​After a rigorous analysis of its convergence, we present some numerical results obtained with this method on various one-dimensional, ​two-dimensional and three-dimensional problems.

Please contact the organizer if you need the Zoom link.

Abstract: In this talk I will describe recent joint work with Kevin McGoff. We consider topological dynamical systems (X,T) defined by a compact metrizable space X and action T of a countable amenable group G on X by homeomorphisms.

For two such systems (X,T) and (Y,S) and a factor map pi from X to Y, an intermediate factor is a topological dynamical system (Z,R) for which pi can be written as a composition of factor maps from X to Z and from Z to Y. Our main result establishes that if G is countable amenable and X and Y are zero-dimensional, then every number between h(X,S) and h(Y,T) can be realized as the entropy of an intermediate zero-dimensional factor.

I'll describe some connections to work of Shub, Weiss, and Lindenstrauss on "lowerability of entropy," and will give a vague outline of some parts of the proof.

Please contact the organizer is you need the Zoom link.

Description:
Starting in 2021, PIMS has inaugurated a high-level network-wide colloquium series. Distinguished speakers will give talks across the full PIMS network with one talk per month during the academic term. The 2021 speaker series is part of the PIMS 25th Anniversary showcase.

Abstract: Diffusion methods help understand and denoise data sets; when there is additional structure (as is often the case), one can use (and get additional benefit from) a fiber bundle model.

Time:
All network wide colloquia take place at 1:30pm Pacific Time

Registration:
To attend this event please register here. Kindly note that this talk will be recorded.

The $\ell$-deck of a graph $G$ is the multiset of all induced subgraphs of $G$ on $\ell$ vertices. The first major result on reconstruction was due to Kelly in 1957 who showed that trees can be reconstructed from their $(n-1)$-deck, and this was later improved to their $(n-2)$-deck by Giles in 1976. I will talk about recent work which shows that trees can be reconstructed from their $63n/64$ deck, where a constant fraction of the vertices have been removed. Before reconstructing the tree, we first want to recognise that the deck belongs to a tree, and I will also consider the problem of recognising if a graph is connected.

Joint work with Carla Groenland, Alex Scott and Jane Tan.

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Abstract: I will not assume that anyone knows what a translation (or flat) surface is. It is a type of dynamical system (more precisely, a foliation). I will start with a short introduction. Kathryn Lindsey and Rodrigo Trevino gave an interesting construction of translation surfaces from a Bratteli diagram. I will explain this (in different terms than they did). The first fundamental question is then: what is the relationship between the AF-algebra associated with the Bratteli diagram and the foliation C*-algebra of the flat surface? One appealing feature of the Lindsey-Trevino construction is that it reproduces all finite-genus translation surfaces, but also gives many infinite genus ones as well. The second question is how to use the relationship between the C*-algebras to extend known results from the finite-genus case, to these more exotic surfaces. I'll give a fairly complete answer to the first question and discuss some work in progress giving promising partial results on the second. This is joint work with Rodrigo Trevino (Maryland).

Abstract: In this first talk, we consider a family of Weighted Sobolev spaces usually used to solve PDEs in unbounded domains and we ask the following question: what is their image by Fourier transform? The answer to this question contains several surprises. One of these surprises is that the Fourier transform is an isometry of an infinite number of weighted spaces (which leads to a generalization of the Plancherel identity).

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Abstract: The Erdos-Hajnal conjecture states that for every graph H there exists c > 0 such that every n-vertex graph G either contains H as an induced subgraph, or has a clique or stable set of size at least n^c. I will talk about a proof of this conjecture for the case H = C5 (a five-cycle), and related results. The proof is based on an extension of a lemma about bipartite graphs due to Pach and Tomon.

This is joint work with Maria Chudnovsky, Alex Scott, and Paul Seymour.

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Abstract: The work deals with the studies of the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in R^d, d=4,5. The proof of the existence of solu​tions relies on a fixed point

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Abstract: In Bernoulli(p) bond percolation, each edge of a given graph is declared open with probability p. The set of open edges is a random subgraph. The arboreal gas is the probability measure that arises from conditioning the random subgraph to be a spanning forest, i.e., to contain no cycles. In the special case p=1/2 the arboreal gas is the uniform measure on spanning forests. What are the percolative properties of these forests? This turns out to be a surprisingly rich question, and I will discuss what is known and conjectured. I will also describe a magic formula for the connection probabilities of the arboreal gas. This formula, analogous to the magic formula for reinforced random walks, arises due to an important connection between the arboreal gas and spin systems with hyperbolic symmetry. Based on joint work with Roland Bauerschmidt, Nick Crawford, and Andrew Swan.

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Motivated by a famous inverse problem in matrix theory (the principal minor assignment problem), we consider the notion of a rank characteristic sequence associated with an $n \times n$ matrix.

From this initial work of Pauline and others, a given $n \times n$ matrix, its {\em principal rank characteristic sequence} (pr-sequence) is a sequence of length $n+1$ of 0s and 1s where, for $k=0,1,2,\ldots n$, a 1 in the $k$th position indicates the existence of a principal submatrix of rank $k$ and a 0 indicates the absence of such a submatrix. Subsequent research by Pauline and her collaborators developed a more versatile notion referred to as the {\em enhanced principal rank characteristic sequence} (epr-sequence) of an $n \times n$ matrix is a sequence $\ell_1, \ell_2,\ldots \ell_n$, where $\ell_k$ is is {\tt A}, {\tt S}, or {\tt N} according as all, some, or none of its principal minors of order $k$ are nonzero.

In this talk, I will survey numerous interesting facts and properties for both pr- and epr-sequences for symmetric matrices, including a discussion on forbidden sub-sequences. In addition, I will present three extensions: (1) epr-sequences for skew-symmetric matrices (research with Pauline and Dale during a visit to UVic); (2) qpr-sequences, which include both principal and almost-principal minors of symmetric matrices; and (3) apr-sequences, which study only almost-principal minors of symmetric matrices (both (2) and (3) represent recent research with X. Martinez-Rivera).

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See announcement and abstract (PDF file)

See announcement and abstract (PDF file)

See announcement and abstract (PDF file).

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

KURTIS MANKE
BSc (Thompson Rivers University, 2017)

“Generating function approach for the effective degree SIR Model”

Department of Mathematics and Statistics

Thursday, December 10, 2020 2:00 P.M. Conducted Remotely

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

External Examiner:
Dr. Hao Wang, Department of Mathematical and Statistical Sciences, University of Alberta

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

Abstract
The effective degree model has been applied to both SIR and SIS type diseases (those which confer permanent immunity and those which do not, respectively) with great success. The original model considers a large system of ODEs to keep track of the number of infected and susceptible neighbours of an individual. In this thesis, we use a generating function approach on the SIR effective degree model to transform the system of ODEs into a single PDE. This has the advantage of allowing the consideration of infinite networks. We derive existence and uniqueness of solutions to the PDE. Furthermore, we show that the linear stability of the PDE is governed by the same disease threshold derived by the ODE model, and we also show the nonlinear instability of the PDE agrees with the same disease threshold.

Abstract (PDF file)

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Abstract: The iteration of contractive functions on Euclidean space can produce geometrically interesting self-similar sets, or fractals. Many classic examples, such as the Siérpinski Gasket, can be constructed from affine maps which share the same matrix transformation, and differ only by shifts. Moreover, a groupoid C*-algebra construction has yet to be provided for many of the standard iterated function systems used in these classic examples. In Part 1, we will review the relevant ideas from iterated function systems, and discuss how this single-matrix property lets us define étale equivalence relations for this broad class of examples. In Part 2, we will discuss the associated groupoid C*-algebras, including ideals related to the open set condition, and some K-theory calculations for the Siérpinski Gasket.

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Abstract: In 2019, Adams, Ehlting, and Edwards developed a model of phenylalanine metabolism in plants that followed Michaelis-Menten kinetics. Two versions of the model were studied, one that included a series of reactions known as the Shikimate Ester Loop (SEL), and one that did not include the SEL. Analysis suggested that in the version with the SEL, phenylalanine use in protein synthesis is prioritized over use in secondary metabolic pathways when concentrations of shikimate, a phenylalanine precursor, are low, via a regulatory mechanism called the Precursor Shutoff Valve(PSV).

In this talk, we will simplify Adams and colleagues’ model by approximating the Michaelis-Menten terms with piecewise linear ramp functions. We will examine equilibria and stability in the simplified model, and show that the version with the SEL still suggests PSV-type regulation. Finally, we will briefly consider general systems with ramp functions.

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