Mon, 22 Nov 2021

16:00 - 17:00
L3

Gibbs measures in infinite dimensions - Some new results on a classical topic

HENDRIK WEBER
(University of Bath)
Abstract

Gibbs measures on spaces of functions or distributions play an important role in various contexts in mathematical physics.  They can, for example, be viewed as continuous counterparts of classical spin models such as the Ising model, they are an important stepping stone in the rigorous construction of Quantum Field Theories, and they are invariant under the 
flow of certain dispersive PDEs, permitting to develop a solution theory with random initial data, well below the deterministic regularity threshold. 

These measures have been constructed and studied, at least since the 60s, but over the last few years there has been renewed interest, partially due to new methods in stochastic analysis, including Hairer’s theory of regularity structures and Gubinelli-Imkeller-Perkowski’s theory of paracontrolled distributions. 

In this talk I will present two independent but complementary results that can be obtained with these new techniques. I will first show how to obtain estimates on samples from of the Euclidean $\phi^4_3$ measure, based on SPDE methods. In the second part, I will discuss a method to show the emergence of phase transitions in the $\phi^4_3$ theory.


 

Tue, 31 May 2022

10:00 - 12:00
L3

Regularity Theory of Spaces with Lower Ricci Curvature Bounds

Daniele Semola
(Oxford University)
Further Information

Aimed at: people interested on Geometric Analysis, Geometric Measure Theory and regularity theory in Partial Differential Equations.

Prerequisites: Riemannian and Differential Geometry, Metric spaces, basic knowledge of Partial Differential Equations.


Outline of the course:

  • Lecture 1:
    • Quick introduction to non-smooth spaces with lower Ricci curvature bounds [1, 23, 20, 17];
    • Basic properties of spaces with lower Ricci bounds: Bishop-Gromov inequality and doubling metric measure spaces, Bochner’s inequality, splitting theorem [19, 22];
    • Convergence and stability: Gromov-Hausdorff convergence, Gromov pre-compactness theorem, stability and tangent cones [19, 22];
  • Lecture 2:
    • Functional form of the splitting theorem via splitting maps;
    • δ-splitting maps and almost splitting theorem [5, 7];
    • Definition of metric measure cone, stability of RCD property for cones [16];
    • Functional form of the volume cone implies metric cone [12];
    • Almost volume cone implies almost metric cone via stability.
  • Lecture 3:
    • Maximal function type arguments;
    • Existence of Euclidean tangents;
    • Rectifiability and uniqueness of tangents at regular points [18];
    • Volume convergence [9, 13];
    • Tangent cones are metric cones on noncollapsed spaces [5, 6, 13].
  • Lecture 4:
    • Euclidean volume rigidity [9, 6, 13];
    • ε-regularity and classical Reifenberg theorem [6, 15, 7];
    • Harmonic functions on metric measure cones, frequency and separation of variables [7];
    • Transformation theorem for splitting maps [7];
    • Proof of canonical Reifenberg theorem via harmonic splitting maps [7].
  • Lecture 5:
    • Regular and singular sets [6, 13];
    • Stratification of singular sets [6, 13];
    • Examples of singular behaviours [10, 11];
    • Hausdorff dimension bounds via Federer’s dimension reduction [6, 13];
    • Quantitative stratification of singular sets [8];
    • An introduction to quantitative differentiation [3];
    • Cone splitting principle [8];
    • Quantitative singular sets and Minkowski content bounds [8].
  • Lecture 6:
    • The aim of this lecture is to give an introduction to the most recent developments of the regularity theory for non collapsed Ricci limit spaces. We will introduce the notion of neck region in this context and then outline how they have been used to prove rectifiability of singular sets in any codimension for non collapsed Ricci limit spaces by Cheeger-Jiang-Naber [7].
Abstract

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.


In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.


The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

Tue, 24 May 2022

10:00 - 12:00
L3

Regularity Theory of Spaces with Lower Ricci Curvature Bounds

Daniele Semola
(Oxford University)
Further Information

Aimed at: people interested on Geometric Analysis, Geometric Measure Theory and regularity theory in Partial Differential Equations.

Prerequisites: Riemannian and Differential Geometry, Metric spaces, basic knowledge of Partial Differential Equations.


Outline of the course:

  • Lecture 1:
    • Quick introduction to non-smooth spaces with lower Ricci curvature bounds [1, 23, 20, 17];
    • Basic properties of spaces with lower Ricci bounds: Bishop-Gromov inequality and doubling metric measure spaces, Bochner’s inequality, splitting theorem [19, 22];
    • Convergence and stability: Gromov-Hausdorff convergence, Gromov pre-compactness theorem, stability and tangent cones [19, 22];
  • Lecture 2:
    • Functional form of the splitting theorem via splitting maps;
    • δ-splitting maps and almost splitting theorem [5, 7];
    • Definition of metric measure cone, stability of RCD property for cones [16];
    • Functional form of the volume cone implies metric cone [12];
    • Almost volume cone implies almost metric cone via stability.
  • Lecture 3:
    • Maximal function type arguments;
    • Existence of Euclidean tangents;
    • Rectifiability and uniqueness of tangents at regular points [18];
    • Volume convergence [9, 13];
    • Tangent cones are metric cones on noncollapsed spaces [5, 6, 13].
  • Lecture 4:
    • Euclidean volume rigidity [9, 6, 13];
    • ε-regularity and classical Reifenberg theorem [6, 15, 7];
    • Harmonic functions on metric measure cones, frequency and separation of variables [7];
    • Transformation theorem for splitting maps [7];
    • Proof of canonical Reifenberg theorem via harmonic splitting maps [7].
  • Lecture 5:
    • Regular and singular sets [6, 13];
    • Stratification of singular sets [6, 13];
    • Examples of singular behaviours [10, 11];
    • Hausdorff dimension bounds via Federer’s dimension reduction [6, 13];
    • Quantitative stratification of singular sets [8];
    • An introduction to quantitative differentiation [3];
    • Cone splitting principle [8];
    • Quantitative singular sets and Minkowski content bounds [8].
  • Lecture 6:
    • The aim of this lecture is to give an introduction to the most recent developments of the regularity theory for non collapsed Ricci limit spaces. We will introduce the notion of neck region in this context and then outline how they have been used to prove rectifiability of singular sets in any codimension for non collapsed Ricci limit spaces by Cheeger-Jiang-Naber [7].
Abstract

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.


In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.


The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

Tue, 26 Apr 2022

10:00 - 12:00
L3

Regularity Theory of Spaces with Lower Ricci Curvature Bounds

Daniele Semola
(Oxford University)
Further Information

Aimed at: people interested on Geometric Analysis, Geometric Measure Theory and regularity theory in Partial Differential Equations.

Prerequisites: Riemannian and Differential Geometry, Metric spaces, basic knowledge of Partial Differential Equations.


Outline of the course:

  • Lecture 1:
    • Quick introduction to non-smooth spaces with lower Ricci curvature bounds [1, 23, 20, 17];
    • Basic properties of spaces with lower Ricci bounds: Bishop-Gromov inequality and doubling metric measure spaces, Bochner’s inequality, splitting theorem [19, 22];
    • Convergence and stability: Gromov-Hausdorff convergence, Gromov pre-compactness theorem, stability and tangent cones [19, 22];
  • Lecture 2:
    • Functional form of the splitting theorem via splitting maps;
    • δ-splitting maps and almost splitting theorem [5, 7];
    • Definition of metric measure cone, stability of RCD property for cones [16];
    • Functional form of the volume cone implies metric cone [12];
    • Almost volume cone implies almost metric cone via stability.
  • Lecture 3:
    • Maximal function type arguments;
    • Existence of Euclidean tangents;
    • Rectifiability and uniqueness of tangents at regular points [18];
    • Volume convergence [9, 13];
    • Tangent cones are metric cones on noncollapsed spaces [5, 6, 13].
  • Lecture 4:
    • Euclidean volume rigidity [9, 6, 13];
    • ε-regularity and classical Reifenberg theorem [6, 15, 7];
    • Harmonic functions on metric measure cones, frequency and separation of variables [7];
    • Transformation theorem for splitting maps [7];
    • Proof of canonical Reifenberg theorem via harmonic splitting maps [7].
  • Lecture 5:
    • Regular and singular sets [6, 13];
    • Stratification of singular sets [6, 13];
    • Examples of singular behaviours [10, 11];
    • Hausdorff dimension bounds via Federer’s dimension reduction [6, 13];
    • Quantitative stratification of singular sets [8];
    • An introduction to quantitative differentiation [3];
    • Cone splitting principle [8];
    • Quantitative singular sets and Minkowski content bounds [8].
  • Lecture 6:
    • The aim of this lecture is to give an introduction to the most recent developments of the regularity theory for non collapsed Ricci limit spaces. We will introduce the notion of neck region in this context and then outline how they have been used to prove rectifiability of singular sets in any codimension for non collapsed Ricci limit spaces by Cheeger-Jiang-Naber [7].
Abstract

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.


In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.


The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

Tue, 17 May 2022

10:00 - 12:00
L3

Regularity Theory of Spaces with Lower Ricci Curvature Bounds

Daniele Semola
(Oxford University)
Further Information

Aimed at: people interested on Geometric Analysis, Geometric Measure Theory and regularity theory in Partial Differential Equations.

Prerequisites: Riemannian and Differential Geometry, Metric spaces, basic knowledge of Partial Differential Equations.


Outline of the course:

  • Lecture 1:
    • Quick introduction to non-smooth spaces with lower Ricci curvature bounds [1, 23, 20, 17];
    • Basic properties of spaces with lower Ricci bounds: Bishop-Gromov inequality and doubling metric measure spaces, Bochner’s inequality, splitting theorem [19, 22];
    • Convergence and stability: Gromov-Hausdorff convergence, Gromov pre-compactness theorem, stability and tangent cones [19, 22];
  • Lecture 2:
    • Functional form of the splitting theorem via splitting maps;
    • δ-splitting maps and almost splitting theorem [5, 7];
    • Definition of metric measure cone, stability of RCD property for cones [16];
    • Functional form of the volume cone implies metric cone [12];
    • Almost volume cone implies almost metric cone via stability.
  • Lecture 3:
    • Maximal function type arguments;
    • Existence of Euclidean tangents;
    • Rectifiability and uniqueness of tangents at regular points [18];
    • Volume convergence [9, 13];
    • Tangent cones are metric cones on noncollapsed spaces [5, 6, 13].
  • Lecture 4:
    • Euclidean volume rigidity [9, 6, 13];
    • ε-regularity and classical Reifenberg theorem [6, 15, 7];
    • Harmonic functions on metric measure cones, frequency and separation of variables [7];
    • Transformation theorem for splitting maps [7];
    • Proof of canonical Reifenberg theorem via harmonic splitting maps [7].
  • Lecture 5:
    • Regular and singular sets [6, 13];
    • Stratification of singular sets [6, 13];
    • Examples of singular behaviours [10, 11];
    • Hausdorff dimension bounds via Federer’s dimension reduction [6, 13];
    • Quantitative stratification of singular sets [8];
    • An introduction to quantitative differentiation [3];
    • Cone splitting principle [8];
    • Quantitative singular sets and Minkowski content bounds [8].
  • Lecture 6:
    • The aim of this lecture is to give an introduction to the most recent developments of the regularity theory for non collapsed Ricci limit spaces. We will introduce the notion of neck region in this context and then outline how they have been used to prove rectifiability of singular sets in any codimension for non collapsed Ricci limit spaces by Cheeger-Jiang-Naber [7].
Abstract

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.


In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.


The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

Tue, 10 May 2022

10:00 - 12:00
L3

Regularity Theory of Spaces with Lower Ricci Curvature Bounds

Daniele Semola
(Oxford University)
Further Information

Aimed at: people interested on Geometric Analysis, Geometric Measure Theory and regularity theory in Partial Differential Equations.

Prerequisites: Riemannian and Differential Geometry, Metric spaces, basic knowledge of Partial Differential Equations.


Outline of the course:

  • Lecture 1:
    • Quick introduction to non-smooth spaces with lower Ricci curvature bounds [1, 23, 20, 17];
    • Basic properties of spaces with lower Ricci bounds: Bishop-Gromov inequality and doubling metric measure spaces, Bochner’s inequality, splitting theorem [19, 22];
    • Convergence and stability: Gromov-Hausdorff convergence, Gromov pre-compactness theorem, stability and tangent cones [19, 22];
  • Lecture 2:
    • Functional form of the splitting theorem via splitting maps;
    • δ-splitting maps and almost splitting theorem [5, 7];
    • Definition of metric measure cone, stability of RCD property for cones [16];
    • Functional form of the volume cone implies metric cone [12];
    • Almost volume cone implies almost metric cone via stability.
  • Lecture 3:
    • Maximal function type arguments;
    • Existence of Euclidean tangents;
    • Rectifiability and uniqueness of tangents at regular points [18];
    • Volume convergence [9, 13];
    • Tangent cones are metric cones on noncollapsed spaces [5, 6, 13].
  • Lecture 4:
    • Euclidean volume rigidity [9, 6, 13];
    • ε-regularity and classical Reifenberg theorem [6, 15, 7];
    • Harmonic functions on metric measure cones, frequency and separation of variables [7];
    • Transformation theorem for splitting maps [7];
    • Proof of canonical Reifenberg theorem via harmonic splitting maps [7].
  • Lecture 5:
    • Regular and singular sets [6, 13];
    • Stratification of singular sets [6, 13];
    • Examples of singular behaviours [10, 11];
    • Hausdorff dimension bounds via Federer’s dimension reduction [6, 13];
    • Quantitative stratification of singular sets [8];
    • An introduction to quantitative differentiation [3];
    • Cone splitting principle [8];
    • Quantitative singular sets and Minkowski content bounds [8].
  • Lecture 6:
    • The aim of this lecture is to give an introduction to the most recent developments of the regularity theory for non collapsed Ricci limit spaces. We will introduce the notion of neck region in this context and then outline how they have been used to prove rectifiability of singular sets in any codimension for non collapsed Ricci limit spaces by Cheeger-Jiang-Naber [7].
Abstract

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.


In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.


The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

Tue, 03 May 2022

10:00 - 12:00
L3

Regularity Theory of Spaces with Lower Ricci Curvature Bounds

Daniele Semola
(Oxford University)
Further Information

Aimed at: people interested on Geometric Analysis, Geometric Measure Theory and regularity theory in Partial Differential Equations.

Prerequisites: Riemannian and Differential Geometry, Metric spaces, basic knowledge of Partial Differential Equations.


Outline of the course:

  • Lecture 1:
    • Quick introduction to non-smooth spaces with lower Ricci curvature bounds [1, 23, 20, 17];
    • Basic properties of spaces with lower Ricci bounds: Bishop-Gromov inequality and doubling metric measure spaces, Bochner’s inequality, splitting theorem [19, 22];
    • Convergence and stability: Gromov-Hausdorff convergence, Gromov pre-compactness theorem, stability and tangent cones [19, 22];
  • Lecture 2:
    • Functional form of the splitting theorem via splitting maps;
    • δ-splitting maps and almost splitting theorem [5, 7];
    • Definition of metric measure cone, stability of RCD property for cones [16];
    • Functional form of the volume cone implies metric cone [12];
    • Almost volume cone implies almost metric cone via stability.
  • Lecture 3:
    • Maximal function type arguments;
    • Existence of Euclidean tangents;
    • Rectifiability and uniqueness of tangents at regular points [18];
    • Volume convergence [9, 13];
    • Tangent cones are metric cones on noncollapsed spaces [5, 6, 13].
  • Lecture 4:
    • Euclidean volume rigidity [9, 6, 13];
    • ε-regularity and classical Reifenberg theorem [6, 15, 7];
    • Harmonic functions on metric measure cones, frequency and separation of variables [7];
    • Transformation theorem for splitting maps [7];
    • Proof of canonical Reifenberg theorem via harmonic splitting maps [7].
  • Lecture 5:
    • Regular and singular sets [6, 13];
    • Stratification of singular sets [6, 13];
    • Examples of singular behaviours [10, 11];
    • Hausdorff dimension bounds via Federer’s dimension reduction [6, 13];
    • Quantitative stratification of singular sets [8];
    • An introduction to quantitative differentiation [3];
    • Cone splitting principle [8];
    • Quantitative singular sets and Minkowski content bounds [8].
  • Lecture 6:
    • The aim of this lecture is to give an introduction to the most recent developments of the regularity theory for non collapsed Ricci limit spaces. We will introduce the notion of neck region in this context and then outline how they have been used to prove rectifiability of singular sets in any codimension for non collapsed Ricci limit spaces by Cheeger-Jiang-Naber [7].
Abstract

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.


In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.


The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

Thu, 11 Nov 2021
14:00
L3

Higher Form Symmetries: Part 2

Dewi Gould
((Oxford University))
Further Information

Junior strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research areas. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Thu, 04 Nov 2021

16:00 - 17:00
L3

Blow-up in the supercooled Stefan problem with noise: unstable states and discontinuity of the temperature

ANDREAS SOJMARK
(University of Oxford)
Abstract

Following on from Christoph's talk last week, I will present a version of the supercooled Stefan problem with noise. I will start by discussing the physical intuition and then give a probabilistic representation of solutions. From there, I will identify a simple relationship between the initial heat profile and a single parameter for how the liquid solidifies, which, if violated, forces the temperature to develop a discontinuity in finite time with positive probability. On the other hand, when the relationship is satisfied, the temperature remains globally continuous with probability one. The work is part of a new preprint that should soon be available on arXiv.

 

Thu, 25 Nov 2021

16:00 - 17:00
L3

TBC

BEN HAMBLY
(University of Oxford)
Abstract

TBC

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