Mathematics

Topological Data Analysis of Collective Behavior

Speaker: 
Dhananjay Bhaskar
Date: 
Wed, Sep 29, 2021
Location: 
PIMS, University of British Columbia
Online
Zoom
Conference: 
Mathematical Biology Seminar
Abstract: 

Active matter systems, ranging from liquid crystals to populations of cells and animals, exhibit complex collective behavior characterized by pattern formation and dynamic phase transitions. However, quantitative classification is challenging for heterogeneous populations of varying size, and typically requires manual supervision. In this talk, I will demonstrate that a combination of topological data analysis (TDA) and machine learning can uniquely identify the spatial arrangement of agents by keeping track of clusters, loops, and voids at multiple scales. To validate the approach, I will present 3 case studies: (1) data-driven modeling and analysis of epithelial-mesenchymal transition (EMT) in mammary epithelia, (2) unsupervised classification of cell sorting, and self-assembly patterns in co-cultures, and (3) parameter recovery from animal swarming trajectories.

Class: 

Unsolved Problems in Number Theory

Speaker: 
Ben Green
Date: 
Wed, Sep 29, 2021
Location: 
PIMS, University of Calgary
Zoom
Online
Conference: 
Louise and Richard K. Guy Lecture Series
Abstract: 

Richard Guy's book "Unsolved Problems in Number Theory" was one of the first mathematical books I owned. I will discuss a selection of my favorite problems from the book, together with some of the progress that has been made on them in the 30 years since I acquired my copy.

Speaker Biography

Ben Green was born and grew up in Bristol, England. He was educated at Trinity College, Cambridge and has been the Waynflete Professor of Pure Mathematics at Oxford since 2013.

About the Series

The Richard & Louise Guy Lecture Series, presented from Louise Guy to Richard in recognition of his love of mathematics and his desire to share his passion with the world, celebrates the joy of discovery and wonder in mathematics for everyone.

Class: 

Branes, Quivers, and BPS Algebras 4 of 4

Speaker: 
Miroslav Rapčák
Date: 
Thu, Aug 26, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

This series of lectures covers a brief introduction into some algebro-geometric techniques used in the construction of BPS algebras. The starting point of our construction is a physical picture of D0-branes bound to D-branes of higher dimension. Using methods of the derived category of coherent sheaves, we are going to derive a framed quiver with potential describing supersymmetric quantum mechanics capturing the low-energy behavior of such D0-branes. For a large class of quivers, we are going to identify the space of BPS states with different melted-crystal configurations. Finally, by employing correspondences, we are going to construct an action of a BPS algebra known as the affine Yangian on the space of BPS states. The action of the affine Yangian factors through the action of various vertex operator algebras, Cherednik algebras, and more. This construction leads to an enormously rich interplay between physics, geometry and representation theory.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Branes, Quivers, and BPS Algebras 3 of 4

Speaker: 
Miroslav Rapčák
Date: 
Wed, Aug 25, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

This series of lectures covers a brief introduction into some algebro-geometric techniques used in the construction of BPS algebras. The starting point of our construction is a physical picture of D0-branes bound to D-branes of higher dimension. Using methods of the derived category of coherent sheaves, we are going to derive a framed quiver with potential describing supersymmetric quantum mechanics capturing the low-energy behavior of such D0-branes. For a large class of quivers, we are going to identify the space of BPS states with different melted-crystal configurations. Finally, by employing correspondences, we are going to construct an action of a BPS algebra known as the affine Yangian on the space of BPS states. The action of the affine Yangian factors through the action of various vertex operator algebras, Cherednik algebras, and more. This construction leads to an enormously rich interplay between physics, geometry and representation theory.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Branes, Quivers, and BPS Algebras 2 of 4

Speaker: 
Miroslav Rapčák
Date: 
Tue, Aug 24, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

This series of lectures covers a brief introduction into some algebro-geometric techniques used in the construction of BPS algebras. The starting point of our construction is a physical picture of D0-branes bound to D-branes of higher dimension. Using methods of the derived category of coherent sheaves, we are going to derive a framed quiver with potential describing supersymmetric quantum mechanics capturing the low-energy behavior of such D0-branes. For a large class of quivers, we are going to identify the space of BPS states with different melted-crystal configurations. Finally, by employing correspondences, we are going to construct an action of a BPS algebra known as the affine Yangian on the space of BPS states. The action of the affine Yangian factors through the action of various vertex operator algebras, Cherednik algebras, and more. This construction leads to an enormously rich interplay between physics, geometry and representation theory.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Branes, Quivers, and BPS Algebras 1 of 4

Speaker: 
Miroslav Rapčák
Date: 
Mon, Aug 23, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

This series of lectures covers a brief introduction into some algebro-geometric techniques used in the construction of BPS algebras. The starting point of our construction is a physical picture of D0-branes bound to D-branes of higher dimension. Using methods of the derived category of coherent sheaves, we are going to derive a framed quiver with potential describing supersymmetric quantum mechanics capturing the low-energy behavior of such D0-branes. For a large class of quivers, we are going to identify the space of BPS states with different melted-crystal configurations. Finally, by employing correspondences, we are going to construct an action of a BPS algebra known as the affine Yangian on the space of BPS states. The action of the affine Yangian factors through the action of various vertex operator algebras, Cherednik algebras, and more. This construction leads to an enormously rich interplay between physics, geometry and representation theory.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Geometry of N=2 Supersymmetry 4 of 4

Speaker: 
Andy Neitzke
Date: 
Thu, Aug 26, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

Coulomb branches of N=2 supersymmetric field theories in four dimensions support a rich geometry. My aim in these lectures will be to explain some aspects of this geometry, and its relation to the physics of the N=2 theories themselves.

I will first describe various constructions of N=2 theories and the corresponding Coulomb branches. In this story the main geometry visible is that of a complex integrable system, fibered over the Coulomb branch; one nice class of examples is associated to the Hitchin integrable system (moduli space of Higgs bundles over a Riemann surface). Fundamental objects in the N=2 theory (local operators, line operators, surface operators) all have geometric counterparts in the integrable system, as I will explain.

Next I will discuss a deformation of the story, which arises in physics from the Nekrasov-Shatashvili Omega-background. In this deformation, the Coulomb branch is replaced by a closely related space; for instance, the base of the Hitchin integrable system is replaced by a space parameterizing opers over the Riemann surface. One can use this deformation to give a concrete picture of the space of opers; in so doing one meets Stokes phenomena which are governed by the BPS indices (Donaldson-Thomas invariants) of the N=2 theory. This turns out to be closely related to the "exact WKB method" in analysis of ODEs. It is also connected to Riemann-Hilbert problems of a sort recently investigated by Bridgeland, as I will describe if time permits.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Geometry of N=2 Supersymmetry 3 of 4

Speaker: 
Andy Neitzke
Date: 
Wed, Aug 25, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

Coulomb branches of N=2 supersymmetric field theories in four dimensions support a rich geometry. My aim in these lectures will be to explain some aspects of this geometry, and its relation to the physics of the N=2 theories themselves.

I will first describe various constructions of N=2 theories and the corresponding Coulomb branches. In this story the main geometry visible is that of a complex integrable system, fibered over the Coulomb branch; one nice class of examples is associated to the Hitchin integrable system (moduli space of Higgs bundles over a Riemann surface). Fundamental objects in the N=2 theory (local operators, line operators, surface operators) all have geometric counterparts in the integrable system, as I will explain.

Next I will discuss a deformation of the story, which arises in physics from the Nekrasov-Shatashvili Omega-background. In this deformation, the Coulomb branch is replaced by a closely related space; for instance, the base of the Hitchin integrable system is replaced by a space parameterizing opers over the Riemann surface. One can use this deformation to give a concrete picture of the space of opers; in so doing one meets Stokes phenomena which are governed by the BPS indices (Donaldson-Thomas invariants) of the N=2 theory. This turns out to be closely related to the "exact WKB method" in analysis of ODEs. It is also connected to Riemann-Hilbert problems of a sort recently investigated by Bridgeland, as I will describe if time permits.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Geometry of N=2 Supersymmetry 2 of 4

Speaker: 
Andy Neitzke
Date: 
Tue, Aug 24, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

Coulomb branches of N=2 supersymmetric field theories in four dimensions support a rich geometry. My aim in these lectures will be to explain some aspects of this geometry, and its relation to the physics of the N=2 theories themselves.

I will first describe various constructions of N=2 theories and the corresponding Coulomb branches. In this story the main geometry visible is that of a complex integrable system, fibered over the Coulomb branch; one nice class of examples is associated to the Hitchin integrable system (moduli space of Higgs bundles over a Riemann surface). Fundamental objects in the N=2 theory (local operators, line operators, surface operators) all have geometric counterparts in the integrable system, as I will explain.

Next I will discuss a deformation of the story, which arises in physics from the Nekrasov-Shatashvili Omega-background. In this deformation, the Coulomb branch is replaced by a closely related space; for instance, the base of the Hitchin integrable system is replaced by a space parameterizing opers over the Riemann surface. One can use this deformation to give a concrete picture of the space of opers; in so doing one meets Stokes phenomena which are governed by the BPS indices (Donaldson-Thomas invariants) of the N=2 theory. This turns out to be closely related to the "exact WKB method" in analysis of ODEs. It is also connected to Riemann-Hilbert problems of a sort recently investigated by Bridgeland, as I will describe if time permits.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Geometry of N=2 Supersymmetry 1 of 4

Speaker: 
Andy Neitzke
Date: 
Mon, Aug 23, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

Coulomb branches of N=2 supersymmetric field theories in four dimensions support a rich geometry. My aim in these lectures will be to explain some aspects of this geometry, and its relation to the physics of the N=2 theories themselves.

I will first describe various constructions of N=2 theories and the corresponding Coulomb branches. In this story the main geometry visible is that of a complex integrable system, fibered over the Coulomb branch; one nice class of examples is associated to the Hitchin integrable system (moduli space of Higgs bundles over a Riemann surface). Fundamental objects in the N=2 theory (local operators, line operators, surface operators) all have geometric counterparts in the integrable system, as I will explain.

Next I will discuss a deformation of the story, which arises in physics from the Nekrasov-Shatashvili Omega-background. In this deformation, the Coulomb branch is replaced by a closely related space; for instance, the base of the Hitchin integrable system is replaced by a space parameterizing opers over the Riemann surface. One can use this deformation to give a concrete picture of the space of opers; in so doing one meets Stokes phenomena which are governed by the BPS indices (Donaldson-Thomas invariants) of the N=2 theory. This turns out to be closely related to the "exact WKB method" in analysis of ODEs. It is also connected to Riemann-Hilbert problems of a sort recently investigated by Bridgeland, as I will describe if time permits.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

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