Scientific

Size-Ramsey numbers of powers of hypergraph trees and long subdivisions

Speaker: 
Liana Yepremyan
Date: 
Thu, Mar 11, 2021
Location: 
Zoom
PIMS, University of Victoria
Conference: 
PIMS-UVic Discrete Math Seminar
Abstract: 

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.

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Mathematics, Colonization and Empire

Speaker: 
Tom Archibald, SFU
Date: 
Thu, Mar 4, 2021
Location: 
Online
University of Victoria, Victoria, Canada
PIMS, University of Victoria
Conference: 
PIMS-UVic Math Colloquium
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.

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Fusion rings and their categorifications

Speaker: 
Andrew Schopieray
Date: 
Wed, Feb 24, 2021
Location: 
Zoom
PIMS, University of Alberta
Conference: 
Emergent Research: The PIMS Postdoctoral Fellow Seminar
Abstract: 

Fusion rings are a special class of associative unital rings with nonnegative integer structure constants and a notion of duality. For example, the group ring of a finite group is a fusion ring. We study fusion rings mainly because they arise as Grothendieck rings of categories associated to Hopf algebras, semisimple Lie algebras, vertex operator algebras, etc. In turn, these categories have application to topological quantum field theory, invariants of knots and links, and quantum computation, to name a few. In this talk we will discuss the brief history of the classification of categorifiable fusion rings and how number-theoretic properties of fusion rings dictate the existence of, or properties of, their categorifications.

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Stochastic Optimal Transport, Control Theory, and PDEs

Speaker: 
Aaron Palmer
Date: 
Thu, Feb 25, 2021
Location: 
Zoom
PIMS, University of British Columbia
Conference: 
Kantorovich Initiative Seminar
Abstract: 

The optimal transport problem provides a fundamental and quantitative way to measure the distance between probability distributions. Recently, it has been successfully used to analyze the evolutionary dynamics in physics and biology. Motivated by questions of pricing in financial mathematics and control of distributed agents, stochastic variants of optimal transport have been developed. Over the past few years, my postdoc supervisors at the University of British Columbia (Nassif Ghoussoub and Young-Heon Kim) and I have developed a robust method to analyze these problems using convex duality, stochastic optimal control theory, and partial differential equation analysis.

This talk will focus on these variants of optimal transport, their applications, and our methods of analysis. Particular attention will be paid to the connections with mean field games and to a new direction of research that incorporates the practical limitation of partial information.

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Large orbit closures of translation surfaces are strata or loci of double covers V

Speaker: 
Paul Apisa
Alex Wright
Date: 
Thu, Feb 18, 2021
Location: 
Zoom
Abstract: 

Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.
In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.
We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.
The five lectures will introduce novel and broadly applicable techniques, organized as follows:
An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
Completion of the proof of the main theorem.

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A Snapshot of Early 20th Century Women Mathematicians

Speaker: 
Kieka Mynhardt
Date: 
Thu, Feb 4, 2021
Location: 
Zoom
Abstract: 

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.​

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Large orbit closures of translation surfaces are strata or loci of double covers III

Speaker: 
Paul Apisa
Alex Wright
Date: 
Thu, Feb 4, 2021
Location: 
Zoom
Abstract: 

Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.

In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.

We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.

The five lectures will introduce novel and broadly applicable techniques, organized as follows:

An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
Completion of the proof of the main theorem.

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Subject: 

Large orbit closures of translation surfaces are strata or loci of double covers IV

Speaker: 
Paul Apisa
Alex Wright
Date: 
Thu, Feb 11, 2021
Location: 
Zoom
Conference: 
Pacific Dynamics Seminar
Abstract: 

Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.
In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.
We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.
The five lectures will introduce novel and broadly applicable techniques, organized as follows:
An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
Completion of the proof of the main theorem.

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New lower bounds for van der Waerden numbers

Speaker: 
Ben Green
Date: 
Thu, Feb 11, 2021
Location: 
Zoom
Conference: 
PIMS Network Wide Colloquium
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 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((logN)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

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Graph Density Inequalities, Sums of Squares and Tropicalization

Speaker: 
Annie Raymond
Date: 
Thu, Feb 11, 2021
Location: 
Zoom
PIMS, University of Victoria
Conference: 
PIMS-UVic Discrete Math Seminar
Abstract: 

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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