Mathematics

Periods and Quantization

Speaker: 
Brent Pym
Date: 
Fri, Oct 24, 2025
Location: 
PIMS, University of Regina
Online
Conference: 
University of Regina PIMS Distinguished Lecture
Abstract: 

A number is called a "period" if it can be expressed as the volume of a region in Euclidean space, defined by polynomial inequalities with rational coefficients. Many famous constants, such as π, log(2) and special values of the Riemann zeta function, are periods. Consequently, periods play an important role in many parts of mathematics and science. For example, they arise naturally when relating the mathematics of classical and quantum mechanics (Poisson geometry and noncommutative algebra, respectively), via a procedure known as "deformation quantization". It turns out that algebraic geometry endows periods with a wealth of rich and surprising structure, such as a "motivic Galois group" of symmetries, which constrains their properties and facilitates their calculation. I will give an introduction to this circle of ideas, emphasizing their role in recent developments in deformation quantization.

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Mathematical Models of Tobacco Use Dynamics: Products, Flavors, and Networks

Speaker: 
Clinton H. Durney
Date: 
Wed, Oct 8, 2025
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
UBC Math Biology Seminar Series
Abstract: 

Mathematical biology offers powerful tools to tackle pressing problems at the interface of health and public policy. In this talk, I will share two vignettes demonstrating how mathematical and simulation modelling can be applied to tobacco regulatory science. The first uses a Markov state transition framework to capture population-level dynamics of two tobacco products, each with a flavour option. This structure highlights the challenges of modelling high-dimensional systems, parameter inference from sparse data, and representing policy interventions as modifications to initiation, cessation, and product switching rates. The second vignette focuses on social network modelling, where adolescent tobacco use is primarily shaped by peer influence and network structure. In this setting, stochastic processes and graph-based models describe how behaviours propagate and stabilise within adolescent populations. Together, these examples illustrate how applied mathematics can bridge data and policy in public health.

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Algebraic models for functor calculus

Speaker: 
Niall Taggart
Date: 
Tue, Oct 7, 2025
Location: 
PIMS, University of British Columbia
Online
Zoom
Abstract: 

There is a striking and useful analogy between equivariant homotopy theory and functor calculus. In the equivariant setting, Greenlees conjectured that the category of rational G-spectra has an algebraic model - meaning it is equivalent to the derived category of an abelian category with desirable finiteness properties. This talk will examine the functor calculus counterpart of this conjecture in (potentially) more than one flavour of functor calculus. (Joint work with D. Barnes and M. Kedziorek.)

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Vanishing Sums of Roots of Unity: from Integer Tilings to Projections of Fractal Sets

Speaker: 
Caleb Marshall
Date: 
Wed, Oct 8, 2025
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

A vanishing sum of roots of unity (VSRU) is a finite list $z_1,\ldots,z_K$ of $N$-th complex roots of unity whose sum is zero. While there are many simple examples—including the famous "beautiful equation" of Euler, $e^{i \pi} + 1 = 0$—such sums become extremely complex as the parameter $N$ attains more complex prime power divisors (and we will see several classical examples illustrating this idea, as well as new examples from my work).

One fruitful line of inquiry is to seek a quantitative relationship between the prime divisors of $N$, their associated exponents, and the cardinality parameter $K$. A theorem of T.Y. Lam and K.H. Leung from the early '90's states: $K$ must always be (at least) as large as the smallest prime dividing $N$. This generalizes the well known observation that that sum of all $p$-th roots of unity (where $p$ is any prime number) must vanish; and, one notices that Euler's equation is one example of this fact.

In this talk, we will discuss two significant strengthenings of this result (one due to myself and I. Łaba, another due to myself, G. Kiss, I. Łaba and G. Somlai), which are derived from complexity measurements for polynomials with integer coefficients which have many cyclotomic polynomial divisors. As applications, we give connections in two other areas of mathematics. The first is in the study of integer tilings: additive decompositions of the integers $Z = A+B$ as a sum set, where each integer is represented uniquely. The second application is to the Favard length problem in fractal geometry, which asks for bounds upon the average length of the projections of certain dynamically-defined fractals onto lines.

This talk is based upon my individual work, as well as my joint work with I. Łaba, as well as my joint work with G. Kiss, I. Łaba and G. Somlai. All are welcome, and the first 15-20 minutes will include introductory ideas and examples for all results discussed in the latter portion of the talk.

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Modelling and calibrating the outbreak of an infectious disease in a small population

Speaker: 
JC Loredo-Osti
Date: 
Wed, Sep 24, 2025
Location: 
PIMS, University of British Columbia
Conference: 
UBC Math Biology Seminar Series
Abstract: 

The many ways to model an infectious disease go from simple predator-prey Lotka-Volterra compartmentalised models to highly dimensional models. These models are also commonly expressed as the solution to a system of deterministic differential equations. One issue with models that are highly parametrised, which makes them unsuitable for the early stages of an outbreak, is that estimation with a few data points may be impractical. In terms of sampling, small populations are peculiar, e.g., one may find very effective contact tracing along quite noisy data collection and management due to the lack of resources, and a scarcity of methodological developments crafted for those populations. In this presentation, I will argue that in small jurisdictions, stochastic branching and self-exciting processes or variations of basic compartmentalised models are more relevant because of the volatile nature of the disease dynamics, particularly at early stages of an outbreak. Then, we will focus on continuous-time Markov chain compartmentalised models and their parameter estimation through the likelihood. Finally, we comment on the connection of SIR-like models with Hawkes processes. For those unable to attend in person, you can join via Zoom using the link below.

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Complexity of Lagrangian submanifolds

Speaker: 
Octav Cornea
Date: 
Thu, Apr 17, 2025
Location: 
PIMS, University of Regina
Online
Conference: 
University of Regina PIMS Distinguished Lecture
Abstract: 

Lagrangian submanifolds are a central object of study in symplectic topology. Their rigidity properties have been uncovered via Floer theory since the early ’90’s. The talk will briefly review the subject, in particular how triangulated category structures naturally arise in this context through work of Donaldson, Kontsevich, Fukaya, and others. Further, will be discussed the more recent, natural role of persistence theory, in the sense common in data science. Finally, we will outline how complexity measurements based on persistence methods reflect topological and dynamical invariants, such as topological entropy.

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The question mark function, welding and complex dynamics

Speaker: 
Curtis McMullen
Date: 
Thu, Sep 25, 2025
Location: 
PIMS
Online
Conference: 
PIMS Network Wide Colloquium
Abstract: 

In this talk, titled "The Question Mark Function, Welding, and Complex Dynamics," Prof. McMullen will explore a fascinating interplay of ideas drawn from number theory, conformal geometry, and dynamical systems. While the abstract remains intentionally open-ended, the colloquium promises a thought-provoking look at mathematical structures that defy easy categorization.

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Short Proofs For Some Known Cohomological Results

Speaker: 
Abbas Maarefparvar
Date: 
Wed, Sep 24, 2025
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

In this talk, we first introduce the Brumer-Rosen-Zantema exact sequence (BRZ), a four-term sequence related to strongly ambiguous ideal classes in finite Galois extensions of number fields. Then, using BRZ, we obtain some known cohomological results in the literature concerning Hilbert's Theorem 94, the capitulation map, and the Principal Ideal Theorem. This is a joint work with Ali Rajaei (Tarbiat Modares University) and Ehsan Shahoseini (Institute for Research in Fundamental Sciences).

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Branching out from compartmental models to analyze genomic data: using phylogenies to learn about pathogen populations

Speaker: 
Alex Beams
Date: 
Wed, Sep 17, 2025
Location: 
PIMS, University of British Columbia
Conference: 
UBC Math Biology Seminar Series
Abstract: 

Phylogenetic trees are mathematical objects that encode information about ancestry relationships and are often used in the interpretation of genomic data. They have proved especially useful for advancing our understanding of pathogen populations that evolve on observable timescales, and the construction of phylogenies and our interpretations of them rely on mathematical models at every step. In this talk, we will discuss ongoing projects that focus on the bacterium that causes Tuberculosis. In the first project, we connect compartmental models of disease transmission to pathogen phylogenies in order to understand how epidemiological processes affect tree shape. In the second project, we aim to reconstruct movement patterns on phylogenies to inform the likely efficacy of geographically-targeted public health interventions. In both of these projects, mathematical models play an essential role in the interpretation of phylogenies, and that seems likely to be the case for any statistical inferences we hope to draw from genomic data for the foreseeable future.

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The Rainbow and the Brain

Speaker: 
Cindy Greenwood
Date: 
Wed, Sep 10, 2025
Location: 
PIMS, University of British Columbia
Zoom
Conference: 
UBC Math Biology Seminar Series
Abstract: 

The rainbow and the brain have in common that frequencies are produced. In both cases there is a function of frequency, f, called the power spectral density (PSD). In both cases invasive investigation spoils the investigated object. This talk will describe using noninvasive electroencephalography (EEG) to evaluate the PSD of the brain, via stochastic modelling of associated brain structure. We explore the popular question: does the human brain manifest the mysterious property called "1/f"? Is the PSD of the brain proportional to the function "f to the power -a", for some a > 0, and hence scale-free? What would that mean about the brain? Independent of these fascinating questions, the exponent, a, has many successful applications as a diagnostic of brain disorders and treatments.

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