Mathematics

A Nonsmooth Approach to Einstein's Theory of Gravity

Speaker: 
Robert McCann
Date: 
Tue, Aug 1, 2023
Location: 
PIMS, University of British Columbia
Abstract: 

While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under curvature and dimension bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while Ricci curvature is reformulated using entropic convexity along geodesics of probability measures.

This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. We highlight how the null energy condition of Penrose admits a nonsmooth formulation as a variable lower bound on timelike Ricci curvature.

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Paths and Pathways

Speaker: 
Jayadev Athreya
Date: 
Fri, May 19, 2023
Location: 
Zoom
Online
SFU Harbour Center
Conference: 
Changing the Culture 2023
Abstract: 

We talk about how some simple sounding problems about straight line paths on surfaces require many different kinds of mathematical thinking to solve, focusing on the example of understanding straight line paths on Platonic solids. We'll use this to start a discussion of how we can emphasize teaching different ways of thinking, and why geometry is an important resource for students. There will be lots and lots of fun pictures and hopefully interesting and provocative ideas!

The following resources are referenced during this talk:

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Linearised Optimal Transport Distances

Speaker: 
Matthew Thorpe
Date: 
Thu, May 4, 2023
Location: 
PIMS, University of Washington
Zoom
Conference: 
Kantorovich Initiative Seminar
Abstract: 

Optimal transport is a powerful tool for measuring the distances between signals and images. A common choice is to use the Wasserstein distance where one is required to treat the signal as a probability measure. This places restrictive conditions on the signals and although ad-hoc renormalisation can be applied to sets of unnormalised measures this can often dampen features of the signal. The second disadvantage is that despite recent advances, computing optimal transport distances for large sets is still difficult. In this talk I will extend the linearisation of optimal transport distances to the Hellinger–Kantorovich distance, which can be applied between any pair of non-negative measures, and the TLp distance, a version of optimal transport applicable to functions. Linearisation provides an embedding into a Euclidean space where the Euclidean distance in the embedded space is approximately the optimal transport distance in the original space. This method, in particular, allows for the application of off-the-shelf data analysis tools such as principal component analysis as well as reducing the number of optimal transport calculations from $O(n^2)$ to $O(n)$ in a data set of size n.

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Multivariate Symmetry: Distribution-Free Testing via Optimal Transport

Speaker: 
Bodhisattva Sen
Date: 
Thu, Apr 13, 2023
Location: 
PIMS, University of Washington
Zoom
Online
Conference: 
Kantorovich Initiative Seminar
Abstract: 

The sign test (Arbuthnott, 1710) and the Wilcoxon signed-rank test (Wilcoxon, 1945) are among the first examples of a nonparametric test. These procedures — based on signs, (absolute) ranks and signed-ranks — yield distribution-free tests for symmetry in one-dimension. However, multivariate distribution-free generalizations of these tests are not known in the literature. In this talk we propose a novel framework — based on the theory of optimal transport — which leads to distribution-free generalized multivariate signs, ranks and signed-ranks, and, as a consequence to analogues of the sign and Wilcoxon signed-rank tests that share many of the appealing properties of their one-dimensional counterparts. In particular, the proposed tests are exactly distribution-free in finite samples, with an asymptotic normal distribution, and adapt to various notions of multivariate symmetry such as central symmetry, sign symmetry, and spherical symmetry. We study the consistency of the proposed tests and their behaviors under local alternatives, and show that the proposed generalized Wilcoxon signed-rank test is particularly powerful against location shift alternatives. We show that in a large class of models, our generalized Wilcoxon signed-rank test suffers from no loss in (asymptotic) efficiency, when compared to the Hotelling’s T^2 test, despite being nonparametric and exactly distribution-free. These ideas can also be used to construct distribution-free confidence sets for the location parameter for multivariate distributions.

This is joint work with Zhen Huang at Columbia University.

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Discussion on the history of the Prime Number Theorem and bounds of $\psi(x)$

Speaker: 
Sreerupa Bhattacharjee
Date: 
Fri, May 12, 2023
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
A Day to Celebrate Women and Diversity in Mathematics
Abstract: 

In this talk I will be discussing the history of the Prime Number Theorem following the works of Legendre, Gauss, Riemann, Hadamard, and de la Vallée Poussin, followed by a survey on explicit bounds for $\psi(x)$ beginning with the work of Rosser in 1941. I will go over various improvements over the years including the works of Rosser and Schoenfeld, Dusart, Faber-Kadiri, and Büthe. I will finally briefly discuss my work on the survey of a paper by Büthe and its significance.

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On the quality of the ABC-solutions

Speaker: 
Solaleh Bolvardizadeh
Date: 
Fri, May 12, 2023
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
A Day to Celebrate Women and Diversity in Mathematics
Abstract: 

Let the triple $(a, b, c)$ of integers be such that $\gcd{\left(a, b, c\right)} = 1$ and $a + b = c$. We call such a triple an ABC-solution. The quality of an ABC-solution $(a, b, c)$ is defined as
$$
q(a, b, c) = \frac{\max\left\{\log |a|, \log |b|, \log |c|\right\}}{\log \operatorname{rad}\left({|abc|}\right)}
$$
where $\operatorname{rad} (|abc|)$ is the product of distinct prime factors of $|abc|$. The ABC-conjecture states that given $\epsilon >0$ the number of the ABC-solutions $(a, b, c)$ with $q(a, b, c)\geq 1 + \epsilon$ is finite. In the first part of this talk, under the ABC-conjecture, we explore the quality of certain families of the ABC-solutions formed by terms in Lucas and associated Lucas sequences. We also introduce, unconditionally, a new family of ABC-solutions that has quality $> 1$. In the remainder of the talk, we provide a result on a conjecture of Erdõs on the solutions of the Brocard-Ramanujan equation
$$
n! + 1 = m^2
$$
by assuming an explicit version of the ABC-conjecture proposed by Baker.

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Twelve on the twelfth

Speaker: 
Kristine Bauer
Date: 
Fri, May 12, 2023
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
A Day to Celebrate Women and Diversity in Mathematics
Abstract: 

The under representation of women, especially women of color, has been persistently well documented (see for example the data dashboard on www.womendomath.org). One reason that this is a problem is that it can be difficult for women to identify role models - this can in turn make it harder for women to envision their own success. During my career, I found it very helpful to learn the stories of women in STEM and to draw on aspects of their success to try to invent my own path. In this talk, I will retell twelve stories of women in STEM that influenced me. I can’t promise that the stories will be historically accurate, but I will try to say what I learned from the stories as I heard them and what lessons I hope others might take from them as well.

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Mathematics for Humanity

Speaker: 
Laleh Behjat
Date: 
Fri, May 12, 2023
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
A Day to Celebrate Women and Diversity in Mathematics
Abstract: 

The world is going through major changes with the climate crisis and the dual revolutions of artificial intelligence and bio-technology. These changes require us to rethink our systems and build new ones. In this talk, I will discuss why mathematics as the common language for sciences is the most important factor in building this new world and how diverse perspectives can help us solve these problems better.

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Closure of bulk spectral gap for topological insulators with general edges

Speaker: 
Xiaowen Zhu
Date: 
Wed, Jun 7, 2023
Location: 
PIMS
Online
Conference: 
Emergent Research: The PIMS Postdoctoral Fellow Seminar
Abstract: 

Topological insulators are materials that exhibit unique physical properties due to their non-trivial topological order. One of the most notable consequences of this order is the presence of protected edge states as well as closure of bulk spectral gaps, which is known as the bulk-edge correspondence.

In this talk, I will discuss the mathematical description of topological insulators and their related spectral properties. The presentation will begin with an overview of Floquet theory, Bloch bundles, and the Chern number. We will then examine the bulk-edge
correspondence in topological insulators before delving into our research on closure of bulk spectral gaps for topological insulators with general edges. This talk is based on a joint work with Alexis Drouot.

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Quantum symmetries of finite dimensional algebras

Speaker: 
Amrei Oswald
Date: 
Wed, May 31, 2023
Location: 
PIMS
online
Conference: 
Emergent Research: The PIMS Postdoctoral Fellow Seminar
Abstract: 

The classical notion of symmetry can be formalized by actions of groups. Quantum symmetry is a generalization of the notion of symmetry to the quantum setting, where symmetries can no longer be completely described by the actions of groups. In this setting, quantum symmetries are given by Hopf actions of quantum groups on algebras. I will start with background on quantum groups and Hopf actions and then give examples of quantum symmetries of quiver path algebras. Path algebras can be described in terms of directed graphs and play an important role in the representation theory of finite-dimensional algebras. While quantum symmetries are not straightforward to visualize, path algebras give us a nice tool for doing so. Then, I will discuss a tensor categorical perspective for understanding quantum symmetry and how this perspective can be applied to quantum symmetries of path algebras and finite-dimensional algebras.

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