Scientific

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.

Class: 
Scientific
Subject: 
Mathematics

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).

Class: 
Scientific
Subject: 
Mathematics
Number Theory

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.

Class: 
Scientific
Subject: 
Mathematics
Mathematical Biology

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.

Class: 
Scientific
Subject: 
Mathematics
Mathematical Biology

Spectral Geometry of Liouville Quantum Gravity 3

Speaker: 
Nathanael Berestycki
Date: 
Fri, Jun 27, 2025
Location: 
PIMS, University of British Columbia
Conference: 
2025 PIMS-CRM Summer School in Probability
Abstract: 

I will discuss Liouville Brownian motion, the canonical diffusion in the random geometry defined by Liouville quantum gravity (LQG). In particular I will present some recent results on the spectral geometry of LQG, showing that the eigenvalues satisfy a Weyl law. We will also discuss a number of striking conjectures which aim to relate LQG to a phenomenon known as "quantum chaos", which will also be explained.

Class: 
Scientific
Subject: 
Mathematics
Probability

Heat kernel estimates and Harnack inequalities 16

Speaker: 
Mathav Murugan
Date: 
Fri, Jun 27, 2025
Location: 
PIMS, University of British Columbia
Conference: 
2025 PIMS-CRM Summer School in Probability
Abstract: 

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalentlydescribed using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

Class: 
Scientific
Subject: 
Mathematics
Probability

Dimension dependence of critical phenomena in percolation 16

Speaker: 
Tom Hutchcroft
Date: 
Fri, Jun 27, 2025
Location: 
PIMS, University of British Columbia
Conference: 
2025 PIMS-CRM Summer School in Probability
Abstract: 

In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter p and study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability pc(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

No prior knowledge of percolation will be assumed.

Class: 
Scientific
Subject: 
Mathematics
Probability

Spectral Geometry of Liouville Quantum Gravity 2

Speaker: 
Nathanael Berestycki
Date: 
Thu, Jun 26, 2025
Location: 
PIMS, University of British Columbia
Conference: 
2025 PIMS-CRM Summer School in Probability
Abstract: 

I will discuss Liouville Brownian motion, the canonical diffusion in the random geometry defined by Liouville quantum gravity (LQG). In particular I will present some recent results on the spectral geometry of LQG, showing that the eigenvalues satisfy a Weyl law. We will also discuss a number of striking conjectures which aim to relate LQG to a phenomenon known as "quantum chaos", which will also be explained.

Class: 
Scientific
Subject: 
Mathematics
Probability

Dimension dependence of critical phenomena in percolation 15

Speaker: 
Tom Hutchcroft
Date: 
Thu, Jun 26, 2025
Location: 
PIMS, University of British Columbia
Conference: 
2025 PIMS-CRM Summer School in Probability
Abstract: 

In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter p and study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability pc(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

No prior knowledge of percolation will be assumed.

Class: 
Scientific
Subject: 
Mathematics
Probability

Heat kernel estimates and Harnack inequalities 15

Speaker: 
Mathav Murugan
Date: 
Thu, Jun 26, 2025
Location: 
PIMS, University of British Columbia
Conference: 
2025 PIMS-CRM Summer School in Probability
Abstract: 

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalentlydescribed using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

Class: 
Scientific
Subject: 
Mathematics
Probability

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