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The UVic Department of Mathematics and Statistics, in conjunction with the Association for Women in Mathematics and the Pacific Institute for the Mathematical Sciences, is pleased to announce an exciting new program for high school students. IMAGINING UVic (Inspiring Mathematical Growth and Intuition in Girls) is a summer camp and seminar series aimed at encouraging young women to pursue STEM fields. More information, including how to apply, can be found on our website: https://onlineacademiccommunity.uvic.ca/imagininguvic/

See announcement and abstract (PDF).

See announcement and abstract (PDF).

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

Xiaolin Huang
BSc (Washington University, 2019)

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

External Examiner
Dr. You Liang, Department of Mathematics, Toronto Metropolitan University

Chair of Oral Examination
Dr. Kirstin Lane, School of Exercise Science, Physical and Health Education, UVic

Abstract
COVID-19 has affected the lives of people worldwide. This thesis includes two studies on the response to COVID-19 using statistical methods. The first study explores the impact of lockdown timing on COVID-19 transmission across US counties. We used Functional Principal Component Analysis to extract COVID-19 transmission patterns from county-wise case counts, and used machine learning methods to identify risk factors, with the timing of lockdowns being the most significant. In particular, we found a critical time point for lockdowns, as lockdowns implemented after this time point were associated with significantly more cases and faster spread. The second study proposes an adaptive sample pooling strategy for efficient COVID-19 diagnostic testing. When testing a cohort, our strategy dynamically updates the prevalence estimate after each test, and uses the updated information to choose the optimal pool size for the subsequent test. Simulation studies showed that our strategy reduces the number of tests required to test a cohort compared to traditional pooling strategies.

The semi-random graph process can be thought of as a one player game. Starting with an empty graph on n vertices, in each round a random vertex u is presented to the player, who chooses a vertex v and adds the edge uv to the graph (hence 'semi-random'). The goal of the player is to construct a small fixed graph G as a subgraph of the semi-random graph in as few steps as possible. I will discuss this process, and in particular the asympotically tight bounds we have found on how many steps the player needs to win. This is joint work with Trent Marbach, Pawel Pralat and Andrzej Rucinski.

Speaker Biography: Natalie completed her PhD in 2020 at Queen Mary University of London under the supervision of Robert Johnson. Prior to this, she completed both her Bachelors and Masters degrees at the University of Cambridge. After finishing her PhD she spent a year at the University of Ryerson in Toronto with the Graphs at Ryerson research group. She has worked on various problems under the broad umbrella of probabilistic and extremal combinatorics, including automata, graph saturation, graph factorization and probabilistic zero-forcing (a model for infection or rumour spreading across networks). Since the start of 2022 she has been a postdoctoral fellow at the University of Victoria, working with Natasha Morrison and Jonathan Noel.

Read more about our PIMS PDFs on our Medium feature here.

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

Download poster (PDF).

Join us for a free screening of Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani in celebration of Women in Math Day!

Examine the life and mathematical work of Maryam Mirzakhani, an Iranian immigrant to the United States who became a superstar in her field. In 2014, prior to her untimely death at the age of 40, she became both the first woman and the first Iranian to be awarded the Fields Medal, the most prestigious award in mathematics, often equated in stature with the Nobel Prize.

University of Victoria will host a one-day mini-conference focussing on recent developments in probability theory. The goal of this endeavour is to bring together probabilisits in the northwest pacific area and provide a platform for possible future collaborations.

For more information, list of speakers and abstracts see the conference website.

Reviewers

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

External Examiner
Dr. Quyuan Lin, Department of Mathematics, University of California Santa Barbara

Chair of Oral Examination
Dr. Raad Nashmi, Department of Biology, UVic

Abstract
This thesis is devoted to the motion of the incompressible and inviscid ow which is axisymmetric and swirl-free in a cylinder, where the hydrostatic approximation is made in the axial direction. It addresses the problem of local existence and uniqueness in the spaces of analytic functions for the Cauchy problem for the inviscid primitive equations, also called the hydrostatic incompressible Euler equations, on a cylinder, under some extra conditions. Following the method introduced by Kukavica-Temam-Vicol-Ziane in Int. J. Differ. Equ. 250 (2011) , we use the suitable extension of the Cauchy-Kowalewski theorem to construct locally in time, unique and real-analytic solution, and find the explicit rate of decay of the radius of real- analiticity. Furthermore, this thesis discusses the problem of finite-time blowup of the solution of the system of equations. Following a part of the method introduced by Wong in Proc Am Math Soc. 143 (2015), we prove that the first derivative of the radial velocity blows up in time, using primary functional analysis tools for a certain class of initial data. Taking the solution frozen at r = 0, we can apply an a priori estimate on the second derivative of the pressure term, to derive a Ricatti type inequality.

Nazim Acar

BSc (Uludağ Universitesi, 1998)

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

Reviewers

Supervisory Committee
Dr. Christopher Bose, Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Ahmed Sourour, Mathematics and Statistics, University of Victoria (Member)

External Examiner
Dr. Shafiqul Islam, School of Mathematical and Computational Sciences, University of Prince Edward Island

Chair of Oral Examination
Dr. Richard Keeler, Department of Physics, UVic

Abstract
A Pythagorean Triple (PT) is a triple of positive integers (a, b, c), that satisfies a² + b² = c². By requiring two of the entries being relatively prime, (a, b, c) becomes a Primitive Pythagorean Triple (PPT). This removes trivially equivalent PTs. Following up on the unpublished paper by D. Romik [1] we develop a sequence of mappings and show how each PPT has a unique path starting from one of the two initial nodes (3, 4, 5), (4, 3, 5). We explain a way of generating the PPTs through paper folding. Using a various techniques from dynamics we show how these mappings can be carried over to their conjugates in the first unit arc x² + y² = z², x, y ≥ 0 and the unit interval [0, 1]. Under these mappings and through the conjugacies we show that the PPTs, the pair of rational points on the first unit arc and the rational numbers on the unit interval correspond to each other with the forward orbits exhibiting similar behavior. We identify infinite, σ-finite invariant measures for one-dimensional systems. With the help of the developed conjugacies we extend the dynamics of the PPTs to the continued fraction expansion of the real numbers in the unit interval and show a connection to the Euclidean algorithm. We show that the dynamical system is conservative and ergodic.

Zoom link.

Abstract: Modern biomedical studies often collect multi-view data, that is, multiple types of data measured on the same set of objects. A typical approach to the joint analysis of two high-dimensional data views/sets is to decompose each data matrix into three parts: a low-rank common-source matrix that captures the shared information across data views, a low-rank distinctive-source matrix that characterizes the individual information within each single data view, and an additive noise matrix. Existing decomposition methods often focus on the orthogonality between the common-source and distinctive-source matrices, but inadequately consider the more necessary orthogonal relationship between the two distinctive-source matrices. The latter guarantees that no more shared information is extractable from the distinctive-source matrices. We propose a novel decomposition method that defines the common-source and distinctive-source matrices from the L2 space of random variables rather than the conventionally used Euclidean space, with a careful construction of the orthogonal relationship between distinctive-source matrices. The proposed estimators of common-source and distinctive-source matrices are shown to be asymptotically consistent and have reasonably better performance than some state-of-the-art methods in both simulated data and the real data analysis.

Please contact pims@uvic.ca for the Zoom link.

Large sets in Euclidean space should have large projections in most directions. Projection theorems in geometric measure theory make this intuition precise, by quantifying the words “large” and “most”.

How large can a planar set be if it contains a circle of every radius? This is the quintessential example of a curvilinear Kakeya problem, central to many areas of harmonic analysis and incidence geometry.

What do projections have to do with circles?

The talk will survey a few landmark results in these areas and point to a newly discovered connection between the two.

Abstract: Achievement games are a class of combinatorial games in which two players take turns selecting points from a set (the board), with the goal of being the first to occupy one of the previously designated "winning" subsets. In this talk, we will consider the avoidance variant, in which the first player to occupy such a set loses the game. As a strategy-stealing argument can be used to show that an achievement game cannot be a second-player win, one might expect that the avoidance variant cannot be a first-player win. However, it turns out that we can find transitive avoidance games that are first-player wins for all board sizes which are not primes or powers of two. This talk is based on the paper "Transitive Avoidance Games" by J. Robert Johnson, Imre Leader, and Mark Walters.

As studied in ​many examples, higher order correction added to the Newtonian potential often provides more realistic and accurate quasi-homogeneous models in astrophysics. Important examples include the Schwarzschild and the Manev potentials. The quasi-homogeneous N-body problem aims to study the interaction between N point particles under a prescribed potential. The classical (Newtonian) Hill’s lunar problem aims to improve the solution accuracy of the lunar motion obtained by solving the two-body (Earth-Moon) system. Hill's lunar equation under the Newtonian or homogeneous potentials has been derived from the Hamiltonian of the three-body problem in a uniform rotating coordinate system with angular speed $\omega$, by using symplectic scaling and heuristic arguments on various physical quantities. In this talk, we first introduce a new variational method characterizing relative equilibria with minimal energy. This enables us to classify the dynamic in terms of global existence and singularity for all possible ranges of the parameters. Then we derive Hill’s lunar problem with quasi-homogenous potential, and finally, we implement the same ideas to demonstrate the existence of ``black hole effect" for a certain range of the parameters: below and at some energy threshold, invariant sets (in the phase space) with non-zero Lebesgue measure that either contain global solutions or solutions with singularity are constructed.

Zoom link.

Abstract: I will present our recent results on estimating the Lyapunov exponents of weakly-damped, weakly-dissipated stochastic differential equations. Our primary tool is a new, mildly-quantitative version of Furstenberg’s criterion.

Abstract: A graph is $P_4$-sparse if every set of five vertices induces at most one $P_4$. $P_4$-sparse graphs generalize cographs and admit a tree representation. We consider the problem of deciding whether a $P_4$-sparse graph can be partitioned into two forests (or equivalently, has vertex arboricity 2). We give a characterization in terms of minimal obstructions for $P_4$-sparse graphs of vertex arboricity 2. We also show how these minimal obstructions can be extended for constructing $P_4$-sparse graphs of higher vertex arboricity.

Zoom link.

Zoom link.

Abstract: Co-evolving network models, wherein dynamics such as random walks on the network influence the evolution of the network structure, which in turn influences the dynamics, are of interest in a range of domains. While much of the literature in this area is currently supported by numerics, providing evidence for fascinating conjectures and phase transitions, proving rigorous results has been quite challenging. We propose a general class of co-evolving tree network models driven by local exploration, started from a single vertex called the root. New vertices attach to the current network via randomly sampling a vertex and then exploring the graph for a random number of steps in the direction of the root, connecting to the terminal vertex. Specific choices of the exploration step distribution lead to the well-studied affine preferential attachment and uniform attachment models, as well as less well understood dynamic network models with global attachment functionals such as PageRank scores. We obtain local weak limits for such networks and use them to derive asymptotics for the limiting empirical degree and PageRank distribution. We also quantify asymptotics for the degree and PageRank of fixed vertices, including the root, and the height of the network. Two distinct regimes are seen to emerge, based on the expected exploration distance of incoming vertices, which we call the `fringe' and `non-fringe' regimes. These regimes are shown to exhibit different qualitative and quantitative properties. In particular, networks in the non-fringe regime undergo `condensation' where the root degree grows at the same rate as the network size. Networks in the fringe regime do not exhibit condensation. A non-trivial phase transition phenomenon is also displayed for the PageRank distribution, which connects to the well known power-law hypothesis. In the process, we develop a general set of techniques involving local limits, infinite-dimensional urn models, related multitype branching processes and corresponding Perron-Frobenius theory, branching random walks, and in particular relating tail exponents of various functionals to the scaling exponents of quasi-stationary distributions of associated random walks. These techniques are expected to shed light on a variety of other co-evolving network models.

Abstract: In the study of a discrete dynamical system defined by polynomials, we hope as a starting point to understand the growth of the degrees of the iterates of the map. This growth is measured by the dynamical degree, an invariant which controls the topological, arithmetic, and algebraic complexity of the system. I will discuss the history of this question and the recent surprising construction, joint with Bell, Diller, and Jonsson, of a transcendental dynamical degree for an invertible map of this type, and how our work fits into the general phenomenon of power series taking transcendental values at algebraic inputs.

Speaker Biography:
Holly Krieger is a leader in the area of arithmetic dynamics. She received a Ph.D. from the University of Illinois at Chicago, and was a postdoc at MIT before starting her present position in Cambridge. She was the Australian Mathematical Society’s Mahler Lecturer in 2019, and received a Whitehead Prize from the London Mathematical Society in 2020.

About the series: 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.

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

Registration:
To attend this event please register here to receive the meeting link. Kindly note that this talk will be recorded and posted on the PIMS resource page www.mathtube.org

Zoom link.

Abstract: Thomassen famously showed that every planar graph is 5-choosable, and that every planar graph of girth at least five is 3-choosable. These theorems are best possible for uniform list assignments: Voigt gave a construction of a planar graph that is not 4-choosable, and of a planar graph of girth four that is not 3-choosable. In this talk, I will introduce the concept of a local girth list assignment: a list assignment wherein the list size of each vertex depends not on the girth of the graph, but only on the length of the shortest cycle in which the vertex is contained. I will present a local choosability theorem that unifies the two theorems of Thomassen mentioned above. This is joint work with Luke Postle.

Join us for coffee at 5:00 pm in the Bob Wright Centre Lobby, talk at 5:30 pm

What do thunderstorms look like on the inside?

Were they any different 30 to 50 thousand years ago?

How might they change in the next 100 years as global temperatures continue to rise?

The presentation will start with how a thunderstorm looks in 3-D using radar technology and lightning mapping arrays. We will then travel tens of thousands of years into the past using chemistry analysis of cave stalactites in Texas to see how storms behaved as the climate underwent large shifts in temperature driven by glacial variability. I will end the talk with predictions of how lightning frequency may change over North America by the end of the century using numerical models run on supercomputers, and the potential impacts to humans and ecosystems.

Download poster (PDF)

Abstract: It was known from work of an Huef, Laca, and Raeburn that the right Toeplitz system for the ax+b semigroup over the naturals exhibits more than one equilibrium state at it's critical temperature, but otherwise the structure of the supercritical equilibria was unknown. In recent work with Marcelo Laca, we obtained a classification of these supercritical equilibria. Using the algebraic structure of the diagonal, we rederived a result of Afsar, Larsen, and Neshveyev which provides a correspondence between the KMS beta states and a class of measures on the circle. We provide explicit formulas for these measures and their Fourier coefficients, and describe the phase transition using properties of the Lerch zeta function.

The primitive equation is an important model for large scale fluid model including oceans and atmosphere. While solutions to the viscous model enjoy global regularity, inviscid solutions may develop singularities in finite time. In this talk, I will review the methods to show blowup, and share more recent progress on the qualitative properties of the singularity formation. Most notably, I will provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic in time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameters family of non-smooth profiles, and is stable only by smoother perturbations. This is a joint work with C. Collot and Q. Lin.

Zoom link. ​​

Zoom link.

Abstract:
We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs and Z–invariant weights. Specifically, we show that in the massive scaling limit, (i. e. as the mesh size tends to zero at the same rate as the inverse temperature goes to the critical one) the two-point spin correlations converges to a rotationally invariant function, which is universal among isoradial graphs and independant of the local geometry. We also give a simple proof of the fact that the infinite-volume sub-critical magnetization is independent of the site and the local geometry of the lattice. Finally, we provide a geometrical interpretation of the correlation length using the formalism of s-embeddings introduced recently by Chelkak. Based on a joint work (arXiv:2104.12858) with Dmitry Chelkak (ENS), Konstantin Izyurov (Helsinki).

ABSTRACT: Convective storms are highly intermittent and intense, making their occurrence and strength difficult to predict. This is especially true for climate models, which have grid resolutions much coarser (e.g., 100 km) than the scale of a storm's microphysical and dynamical processes (< 1 km). Physically-based parameterizations struggle to account for this scale mismatch, causing large model errors in rain and lightning. This talk will explore some avenues of using statistical techniques (such as generalized linear and log-Gaussian Cox process models) and machine learning methods (such as random forests and neural networks) that are trained by satellite observations of thunderstorms to see how well they can improve upon existing physical parameterizations in producing accurate rain and lightning characteristics given a set of large-scale environmental conditions.

Please contact the organizer for the Zoom link.

Abstract: Monitoring marked individuals is a common strategy in studies of wild animals (referred to as mark-recapture or capture-recapture experiments) and hard to track human populations (referred to as multi-list methods or multiple-systems estimation). A standard assumption of these techniques is that individuals can be identified uniquely and without error, but this can be violated in many ways. In some cases, it may not be possible to identify individuals uniquely because of the study design or the choice of marks. Other times, errors may occur so that individuals are incorrectly identified. I will discuss work with my collaborators over the past 10 years developing methods to account for problems that arise when are only individuals are only partially identified. I will present theoretical aspects of this research, including an introduction to the latent multinomial model and algebraic statistics, and also describe applications to studies of species ranging from the golden mantella (an endangered frog endemic to Madagascar measuring only 20 mm) to the whale shark (the largest known species of fish measuring up to 19 m).​​

Download poster (PDF).

Abstract:
One of the $\geq 10^{10^{10}}$ things Erd\H{o}s did in 1982 was ask the following question: for which values $k$ do there exist a set of $k$ points on the plane, no three on a line and no four on a circle, such that for every $i \in \{1,\ldots,k-1\}$, there exists a pairwise distance between its points that occurs exactly $i$ times. A set satisfying the above property is called Erd\H{o}s-deep. Using this problem as a launch point, I will introduce a related problem about finding families of $s$ Erd\H{o}s-deep sets in $\mathbb{Z}_n$ that, as a family, also satisfy a version of the Erd\H{o}s-deep property. After building the context for this related problem, I will present an overview of our argument proving the classification of the case when $s=2$. I will then briefly mention a construction for square $s$ that leverages a nice binomial identity. Time permitting: A significant non-mathematical motivation for classifying Erd\H{o}s-deep families is that they have a nice musical interpretation as systems of rhythms, so if there is time, I hope to share some thoughts on that (and possibly a brief and humble .midi example!). Based on joint work with Peter Dukes.

Zoom link.

Abstract: ​​ I will talk about the C*-algebra generated by the right regular representation of the semigroup of affine transformations of the natural numbers. This `right Toeplitz algebra' has three distinguished ideals that give three boundary quotients, and I will describe their structure and discuss their representations. From this we can see that the right Toeplitz algebra is very different from the left Toeplitz algebra studied by Raeburn and myself, although Cuntz-Echterhoff-Li (2013) have shown they have the same K-theory. When we compute KMS states, we see that the low temperature equilibrium states of the right Toeplitz algebra are also parametrized by measures on the​ circle, just as Raeburn and I showed for the left one. So right and left Toeplitz algebras also share the same crystalline phases. This is joint work with Astrid an Huef and Iain Raeburn [NZJM 2021].

Please contact the organizer for the Zoom link.

Abstract: In this talk, we will discuss the relation between two types of two-dimensional lattice models: on one hand, we will consider the spin models with an O(2)-invariant interaction, such as the famous XY and Villain models. On the other, we study integer-valued height function models, where the interaction depends on the discrete gradient. We show that delocalization of a height function model implies that an associated O(2)-invariant spin model has a power-law decay phase. Motivated by this observation, we also extend the recent work of Lammers to show that a certain class of integer-valued height functions delocalize for all doubly periodic graphs (in particular, on the square lattice). Together, these results give a new perspective on the celebrated Berezinksii-Kosterlitz-Thouless phase transition for two-dimensional O(2)-invariant lattice models. This is joint work with Michael Aizenman, Ron Peled, and Jacob Shapiro.

Abstract: Asymptomatic and paucisymptomatic presentations of COVID-19 along with restrictive testing protocols result in undetected COVID-19 cases. Estimating undetected cases is crucial to understand the true severity of the outbreak. We introduce a new hierarchical disease dynamics model based on the N-mixtures hidden population framework. The new models make use of three sets of disease count data per region: reported cases, recoveries, and deaths. Treating the first two as under-counted through binomial thinning, we model the true population state at each time point by partitioning the diseased population into the categories active, recovered, and died. Both domestic spread and imported cases are considered. These models are applied to estimate the level of under-reporting of COVID-19 in the Northern Health Authority region of British Columbia, Canada during thirty weeks of the provincial recovery plan. Parameter covariates are easily implemented, and used to improve model estimates. When accounting for changes in weekly testing volumes, we found under-reporting rates varying from 60.2% to 84.2%.

Zoom link.

Abstract: Is it possible to partition a disk in the plane into finitely many pieces and re-assemble those pieces (without distortion) to yield a partition of the square? This question was asked by Tarski back in 1925 and answered by Laczkovich some 65 years later. Spoiler alert: the answer is "yes."

Laczkovich's proof uses the Axiom of Choice in a strong way and so the pieces of his partition can be very hard to imagine. In 2017, two new proofs emerged which achieve pieces that are Lebesgue measurable or even Borel; the latter result is fully "constructive." We improve on these results by constructively achieving pieces which have (a) lower Borel complexity and (b) "small" boundaries. A benefit of the second condition is that, in contrast to the previous results, the pieces of our partition can, in some sense, be "visualized." The proof uses three concepts that are familiar to any graph-theorist: matchings, network flows and Euler circuits. Based on joint work with András Máthé and Oleg Pikhurko.

The talk will be live, in-person in Clearihue C115, and also available (livestreamed) on Zoom via the link below. The speaker will be in the seminar room.

Zoom link.

Abstract: We prove that the class of crossed product C*-algebras arising from the action of the multiplicative group of a number field on its rings of finite adeles is rigid in the following explicit sense: Any *-isomorphism between full corners of such C*-algebras gives rise to an isomorphism between the underlying number fields that is constructed from the *-isomorphism of corners. I will discuss this result, an application, and I will indicate the idea of proof if time permits. This talk is based on joint work with Takuya Takeishi (Kyoto Institute of Technology).

Zoom link.

Abstract: Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). While it is well-known that the PEs with full viscosity are globally well-posed in Sobolev spaces, the study of well-posedness of the inviscid PEs is more challenging. In this talk, I will first review some results on the deterministic inviscid PEs. Then I will discuss the differences and challenges in the stochastic case and show the existence of local martingale solutions and pathwise uniqueness. Finally I will discuss some interesting open problems. This is a joint work with Ruimeng Hu.

Zoom link.

Abstract: A process Y is a factor of a process X if it can be written as Y=F(X) for some function F which commutes with translations. The factor is finitary if Y_0 is almost surely determined by some finite portion of the input X. Given a process Y, the question of whether Y is a (finitary) factor of an IID process is fundamental in ergodic theory and has received much attention in probability as well. As it turns out, contrary to the prevailing belief, some classical results about factors do not have finitary counterparts, as was recently shown by Gabor. We will present a complementary result that any process Y which is a finitary factor of an IID process furthermore admits an entropy-efficient finitary coding by an IID process. Here entropy-efficient means that the IID process has entropy arbitrarily close to that of Y. As an application we give an affirmative answer to an old question of van den Berg and Steif about the critical Ising model. Joint work with Tom Meyerovitch.

E-mail pims@uvic.ca to request Zoom meeting ID

Abstract: Estimating the COVID-19 infection fatality rate (IFR) has proven to be challenging, since data on deaths and data on the number of infections are subject to various biases. I will describe some joint work with Harlan Campbell and others on both methodological and applied aspects of meeting this challenge, in a meta-analytic framework of combining data from different populations. I will start with the easier case when the infection data are obtained via random sampling. Then I will discuss drawing in additional infection data obtained in decidedly non- random manner.

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Abstract: We consider a locally compact Hausdorff etale groupoid G constructed from a family of k commuting local homemorphisms acting on a compact Hausdorff space X. We characterize the continuous one-cocycles in the groupoid G taking on real values in terms of k-tuples of continuous real-valued on functions on X satisfying certain canonical identities. Under appropriate conditions, we construct a continuous one-parameter automorphism group acting on the C*-algebra associated to G coming from a continuous real-valued one-cocycle on G. The question of the existence of KMS states on the groupoid C*-algebra associated to these automorphism groups is addressed. The work discussed is joint with C. Farsi, L. Huang, and A. Kumjian.

Abstract. Achievement games are a class of combinatorial games in which two players take turns selecting points from a set (the board), with the goal of being the first to occupy one of the previously designated "winning" subsets. In this talk, we will consider the avoidance variant, in which the first player to occupy such a set loses the game. As a strategy-stealing argument can be used to show that an achievement game cannot be a second-player win, one might expect that the avoidance variant cannot be a first-player win. However, it turns out that we can find transitive avoidance games that are first-player wins for all board sizes which are not primes or powers of two. This talk is based on the paper "Transitive Avoidance Games" by J. Robert Johnson, Imre Leader, and Mark Walters.

Judith Packer is a professor of Mathematics at the University of Colorado Boulder. Her research interests include operator algebras, K-theory and wavelets.

Zoom link.

Abstract: We describe several aspects of an analytic/geometric framework for the three-dimensional Navier-Stokes regularity problem, which is directly inspired by the morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of three-dimensional turbulence. Among these, we present our proof that the scaling gap in the 3D Navier-Stokes regularity problem can be reduced by an algebraic factor within an appropriate functional setting incorporating the intermittency of the spatial regions of high vorticity.

Zoom link.

Abstract: In this talk, I will describe some progress towards the construction of the 3D Yang-Mills (YM) measure. In particular, I will introduce a state space of “distributional gauge orbits” which may possibly support the 3D YM measure. Then, I will describe a result which says that assuming that 3D YM theories exhibit short distance behavior similar to the 3D Gaussian free field (which is the expected behavior), then the 3D YM measure may be constructed as a probability measure on the state space. The underlying technical details involve analyzing the YM heat flow (which is a certain PDE) with random distributional initial data. This is based on joint work with Sourav Chatterjee.

Mathematics can help Art Historians and Art Conservators in studying and understanding art works, their manufacture process and their state of conservation. The presentation will review several instances of such collaborations, explaining the role of mathematics in each instance, and illustrating the approach with extensive documentation of the art works.

Speaker Biography:
Ingrid Daubechies is a Belgian Physicist and Mathematician, one of the leaders in the area of wavelets, a part of applied harmonic analysis. Wavelets are widely used in data compression and image encoding. Indeed, a wavelet pioneered by Daubechies is the basis of the standard for digital cinema. Ingrid Daubechies has held positions at the Free University in Brussels, Princeton University, and is currently James B. Duke Professor at Duke University. She is a Member of the National Academy of Sciences and of the National Academy of Engineering and a Fellow of the American Association for the Advancement of Science. Ingrid Daubechies has received many awards including the Leroy P. Steele Prize for Seminal Contribution to Research of the American Mathematical Society.

About the PIMS Network-Wide Colloquium Series:
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.

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

Registration:
To attend this event please register here to receive the meeting link. Kindly note that this talk will be recorded and posted on the PIMS resource page www.mathtube.org.

Abstract: Multivariate count data are commonly encountered through high-throughput sequencing technologies in bioinformatics. Although the Poisson and negative binomial distributions are routinely used to model these count data, its multivariate extension is computationally expensive, thus restricting their use to small dimensional datasets. Hence, independence between genes is assumed in most cases and this fails to take into account the correlation between genes. Fitting such univariate models for multivariate analysis is not only biologically inappropriate, misspecifying the correlation (covariance) structure can result in poor fit to the data. Recently, we developed mixtures of multivariate Poisson lognormal (MPLN) models to analyze these multivariate count measurements. In the MPLN model, the observed counts, conditional on the latent variable, are modeled using a Poisson distribution and the latent variable comes from a multivariate Gaussian distribution. Due to this hierarchical structure, the MPLN model can account for over-dispersion and allows for correlation between the variables. We show that the univariate version of MPLN provides a similar fit to the widely used negative binomial distribution in terms of capturing the mean-variance trends of the RNA-seq data. Moreover, we developed a computationally efficient framework for parameter estimation for MPLN models that utilizes variational Gaussian approximation and opened the possibilities for extending these models for large datasets. Some recent and on-going extensions of MPLN will be briefly discussed.

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The list homomorphism problem for a fixed signed graph $(H,\pi)$ takes as input a signed graph $(G, \sigma)$, equipped with lists $L(v) \subseteq V(H)$, $v \in V(G)$, of allowed images, and asks if there is a homomorphism $\varphi: (G, \sigma) \to (H, \pi)$ classify the computational complexity of this problem when $H$ is a tree (irreflexive, reflexive, and general). The polynomial targets exhibit interesting structures. The tools developed are useful for general targets, and the patterns discovered for trees suggest nice families (such as the bi-arc graphs which characterize the problem for graphs) may classify the polynomial cases in general.

This is joint work with Jan Bok, Tom\'{a}s Feder, Pavol Hell, and Nikola Jedli\v{c}kov\'{a}

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Abstract: For the 3D incompressible Euler equation, a distinguished solution is the flow leading to rigid rotation about an axis u(x,y,z)=(-y,x,0). We show that this solution is stable under small axisymmetric perturbations, and that the perturbations exist globally and scatter linearly back to the equilibrium through a dispersive effect induced by the linearized Euler-Coriolis equations. This is joint work with Y. Guo and K. Widmayer.

Deep reinforcement learning (RL) has been widely used in a variety of challenging tasks, from game playing to robot navigation. However, sample inefficiency and slow convergence rate, i.e. the required number of interactions with the environment and training time is impractically high, remain challenging problems in RL. To address these issues, we propose a general acceleration method for deep RL algorithms built on Anderson mixing, which is an effective approach to accelerating the iterates of the fixed point problems. Specifically, we provide deeper insights into the acceleration schemes in policy iteration by establishing a connection between Anderson mixing and quasi-Newton methods and proving that Anderson mixing increases the convergence radius of policy iteration schemes by an extra contraction factor. We further propose a stabilization strategy by introducing a stable regularization term in Anderson mixing and a differentiable, non-expansive MellowMax operator that can allow both faster convergence and more stable behavior. The effectiveness of our proposed method is evaluated on a variety of Atari games. Experiment results show that our proposed method enhances the convergence, stability, and performance of state-of-the-art deep RL algorithms.

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Abstract: For negatively curved Riemannian manifolds, various natural geometric quantities grow exponentially quickly: the volume of a ball in the universal cover; the number of "distinguishable" geodesics of a given length; the number of closed geodesics with length below a given threshold. Margulis gave very precise asymptotic estimates in this setting. After surveying the general background and history of Margulis-type results, I will describe joint work with Gerhard Knieper and Khadim War in which we obtain Margulis asymptotics for surfaces without conjugate points.

Abstract: The semi-random graph process can be thought of as a one player game. Starting with an empty graph on n vertices, in each round a random vertex u is presented to the player, who chooses a vertex v and adds the edge uv to the graph. Given a graph property, the objective of the player is to force the graph to satisfy this property in as few rounds as possible.

We will consider the property of constructing a fixed graph G as a subgraph of the semi-random graph. Ben-Eliezer, Gishboliner, Hefetz and Krivelevich proved that the player can asymptotically almost surely construct G given n^{1 – 1/d}w rounds, where w is any function tending to infinity with n and d is the degeneracy of the graph G. We have proved a matching lower bound. I will talk about this result, and also discuss a generalisation of our approach to semi-random hypergraphs. I will finish with some open questions.

This is joint work with Trent Marbach, Pawel Pralat and Andrzej Rucinski.

Abstract: I will report on progress on the problem of constructing an example of what Connes' calls a `twisted' (finitely summable) spectral triple over the crossed product of a (classical) hyperbolic group acting on its boundary. This particular triple is important because it represents the Dirac class of the action, and is the analogue in Type III of Connes' Dolbeault spectral triple over the irrational torus. (The Dirac class equals in this case the K-homology class of the boundary extension of the group, as well.) My description uses Patterson-Sullivan theory fairly extensively.

Abstract: The 2d incompressible Euler equation is globally well posed for smooth initial data. However the long term dynamics of general solutions is difficult to understand due to the lack of global relaxation mechanisms. Numerical simulations and physical experiments show that vortices (steady solutions with radial vorticity functions) play an important role in the global dynamics, through a process called vortex symmetrization of small perturbations. In this talk, I will discuss some recent progress on this problem, including a full nonlinear symmetrization result near a special point vortex and precise linearized symmetrization result near general vortices. Difficulties of full nonlinear vortex symmetrization around general vortices will also be discussed.​

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Abstract: The Oseledets multiplicative ergodic theorem plays a fundamental role in smooth ergodic theory. Here we present a generalization where instead of multiplying matrices along an orbit, we compose linear operators. Theorems of this type play a role in the study of delay differential equations, and have applications in oceanographic measurements.

Abstract: Given a graph G, a decomposition of G is a partition of its edges. A graph is (d, h)-decomposable if its edge set can be partitioned into a d-degenerate graph and a graph with maximum degree at most h. In this talk, we present results related to the (d,h)-decomposition problem of planar graphs.

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Abstract: I will report on progress on the problem of constructing an example of what Connes' calls a `twisted' (finitely summable) spectral triple over the crossed product of a (classical) hyperbolic group acting on its boundary. This particular triple is important because it represents the Dirac class of the action, and is the analogue in Type III of Connes' Dolbeault spectral triple over the irrational torus. (The Dirac class equals in this case the K-homology class of the boundary extension of the group, as well.)

My description uses Patterson-Sullivan theory fairly extensively.

Abstract: We consider the classical problem of water waves on the surface of an ideal fluid in 2D and in particular, we study the growth of perturbations to nearly limiting Stokes waves at infinite depth. The stability of Stokes waves is determined by the linearization of the equations of motion of fluid around a Stokes wave.

The eigenvalue problem is solved numerically via shift and invert method.

We identify each positive eigenvalue with an unstable eigenmode in the equations of motion, and find that real positive eigenvalues of the linearized problem converge to a selfsimilar curve as a function of steepness. The power law is suggested for unstable eigenmodes in the immediate vicinity of the limiting Stokes wave. We will discuss the topic of Stokes waves in the finite depth fluid.

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Abstract: Given square matrices A_1, ..., A_d we can consider random products and the associated (top) Lyapunov exponent. We will consider two applications where the Lyapunov exponent plays an interesting role: firstly to barycentric subdivisions of triangles in the Euclidean plane; and secondly to random walks in the hyperbolic plane.

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Abstract: I will report on progress on the problem of constructing an example of what Connes' calls a `twisted' (finitely summable) spectral triple over the crossed product of a (classical) hyperbolic group acting on its boundary. This particular triple is important because it represents the Dirac class of the action, and is the analogue in Type III of Connes' Dolbeault spectral triple over the irrational torus. (The Dirac class equals in this case the K-homology class of the boundary extension of the group, as well.) My description uses Patterson-Sullivan theory fairly extensively.

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Abstract: In 1975, the physicists Imry and Ma predicted that the incorporation of a random field in low-dimensional spin systems leads to the rounding of first-order phase transitions. These predictions were rigorously established by Aizenman and Wehr in 1989 for general spin systems, and recently quantified in the case of the random field Ising model. While the Imry-Ma phenomenon has been mainly studied in the case of compact spin spaces, it has been observed that a similar effect occurs for other models of statistical physics such as random surfaces. In this talk, we will present how the qualitative properties of these models are affected by the additon of a random field and discuss some open questions.

The Monge transfer problem goes back to the 18th century. It consists in minimizing the transport cost of a material from a place to another (and chnaging the shape). Monge could not solve the problem and the next significant step was achieved 150 years later by Kantorovich who introduced the transport distance between two probability measures as well as the dual problem.

The Monge-Kantorovich distance is not easy to use for Partial Differential Equations and the method of doubling the variables is one of them. It is very intuitive in terms of stochastic processes and this provides us with a method for conservative PDEs as parabolic equations (possibly fractional), homogeneous Boltzman equation, scattering equation or porous medium equation...

Structured equations, as they appear in mathematical biology, is a particular class where the method can be used.

About the series: From 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.

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

Registration:
To attend this event please register here to receive the meeting link. Kindly note that this talk will be recorded and posted on the PIMS resource page www.mathtube.org

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Abstract: I will report on progress on the problem of constructing an example of what Connes' calls a `twisted' (finitely summable) spectral triple over the crossed product of a (classical) hyperbolic group acting on its boundary. This particular triple is important because it represents the Dirac class of the action, and is the analogue in Type III of Connes' Dolbeault spectral triple over the irrational torus. (The Dirac class equals in this case the K-homology class of the boundary extension of the group, as well.) My description uses Patterson-Sullivan theory fairly extensively. I will explain this theory for the hyperbolic plane and free groups, in the first talk, and define the `Dirac operator' in the second.

Abstract: We examined the relationship between the Bayes factor and the separation of credible intervals in between- and within-subjects designs under a range of effect and sample sizes. For the within-subject case, we considered five intervals: (a) the within-subject confidence interval of Loftus and Masson (1994), (b) the within-subject Bayesian interval developed by Nathoo, Kilshaw, and Masson (2018), whose derivation conditions on estimated random effects, (c and d) two modifications of (b) based on a proposal by Heck (2019) to allow for shrinkage and account for uncertainty in the estimation of random effects, and (e) the standard Bayesian highest-density interval. We observed a clear and consistent relationship between the Bayes factor and the separation of credible intervals. Remarkably, this relationship is well described by a simple exponential curve and is most precise in case (d). We provide a benchmark to evaluate the Bayes factor when the credible intervals for two means touch at their boundaries. In contrast, interval (e) is relatively wide due to between-subjects variability and tends to obscure effects when used in within-subject designs, rendering its relationship with the Bayes factor less clear. We also provide an R package ‘rmBayes’ to enable computation of each of the within-subject credible intervals investigated here using a number of possible prior distributions. Joint work with Farouk Nathoo and Michael Masson. Keywords: Bayes factor, credible intervals, within-subject designs, within-subject inference

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Abstract: We prove that skew systems with a sufficiently expanding base have "approximate" statistical properties similar to random ergodic Markov chains. For example, they exhibit approximate exponential decay of correlations, meaning that the exponential rate is observed modulo a controlled error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. The error in the approximation is shown to go to zero when the expansion of the base tends to infinity.

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Abstract: Using the idea of ground states in nonlinear dispersive equations (e.g. Klein-Gordon and Schr\"odinger equations), we can characterize the solutions of the N-body problem with strong force under some energy constraints.In this talk, I will explore this method to a restricted 3-body problem (Hill’s lunar problem), and talk about the dynamics of the solutions below, at, and (slightly) above the ground state energy threshold. This is a joint work with Slim Ibrahim.​

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