Forthcoming events in this series


Tue, 30 May 2023
15:30
C4

Multivalued Dir-Minimizing Functions

Dr Immanuel Ben Porat
((Oxford University))
Further Information

The course will serve as an introduction to the theory of multivalued Dir-minimizing functions, which can be viewed as harmonic functions which attain multiple values at each point.

Aimed at Postgraduate students interested in geometric measure theory and its link with elliptic PDEs, a solid knowledge of functional analysis and Sobolev spaces, acquaintance with variational
methods in PDEs and some basic geometric measure theory are recommended.

Sessions led by Dr Immanuel Ben Porat will take place on

09 May 2023 15:30 - 17:30 C4

16 May 2023 15:30 - 17:30 C4

23 May 2023 15:30 - 17:30 C4

30 May 2023 15:30 - 17:30 C4

Should you be interested in taking part in the course, please send an email to @email.

Abstract

COURSE_PROPOSAL (12)_2.pdf

The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers. 

Tue, 30 May 2023

10:00 - 12:00
C2

Nonlinear Fokker-Planck equations modelling large networks of neurons

Dr Pierre Roux
((Oxford University))
Further Information
Sessions led by Dr Pierre Roux will take place on

30 May 2023 10:00 - 12:00 C2

6 June 2023 15:00 - 17:00 C2

8 June 2023 10:00 - 12:00 C2

13 June 2023 15:00 - 17:00 C2

Participants should have a good knowledge of Functional Analysis; basic knowledge about PDEs and distributions; and notions in probability. Should you be interested in taking part in the course, please send an email to @email.

Abstract

PhD_course_Roux.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.

Tue, 23 May 2023
15:30
C4

Multivalued Dir-Minimizing Functions

Dr Immanuel Ben Porat
((Oxford University))
Further Information

The course will serve as an introduction to the theory of multivalued Dir-minimizing functions, which can be viewed as harmonic functions which attain multiple values at each point.

Aimed at Postgraduate students interested in geometric measure theory and its link with elliptic PDEs, a solid knowledge of functional analysis and Sobolev spaces, acquaintance with variational
methods in PDEs and some basic geometric measure theory are recommended.

Sessions led by Dr Immanuel Ben Porat will take place on

09 May 2023 15:30 - 17:30 C4

16 May 2023 15:30 - 17:30 C4

23 May 2023 15:30 - 17:30 C4

30 May 2023 15:30 - 17:30 C4

Should you be interested in taking part in the course, please send an email to @email.

Abstract

COURSE_PROPOSAL (12)_1.pdf

The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers. 

Tue, 16 May 2023
15:30
C4

Multivalued Dir-Minimizing Functions

Dr Immanuel Ben Porat
((Oxford University))
Further Information

The course will serve as an introduction to the theory of multivalued Dir-minimizing functions, which can be viewed as harmonic functions which attain multiple values at each point.

Aimed at Postgraduate students interested in geometric measure theory and its link with elliptic PDEs, a solid knowledge of functional analysis and Sobolev spaces, acquaintance with variational
methods in PDEs and some basic geometric measure theory are recommended.

Sessions led by  Dr Immanuel Ben Porat will take place on

09 May 2023 15:30 - 17:30 C4

16 May 2023 15:30 - 17:30 C4

23 May 2023 15:30 - 17:30 C4

30 May 2023 15:30 - 17:30 C4

Should you be interested in taking part in the course, please send an email to @email.

Abstract

COURSE_PROPOSAL (12)_0.pdf

The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers. 

Tue, 09 May 2023
15:30
C4

Multivalued Dir-Minimizing Functions

Dr Immanuel Ben Porat
(University of Oxford)
Further Information

The course will serve as an introduction to the theory of multivalued Dir-minimizing functions, which can be viewed as harmonic functions which attain multiple values at each point.

Aimed at Postgraduate students interested in geometric measure theory and its link with elliptic PDEs, a solid knowledge of functional analysis and Sobolev spaces, acquaintance with variational
methods in PDEs, and some basic geometric measure theory are recommended.

Sessions led by  Dr Immanuel Ben Porat will take place on

09 May 2023 15:30 - 17:30 C4

16 May 2023 15:30 - 17:30 C4

23 May 2023 15:30 - 17:30 C4

30 May 2023 15:30 - 17:30 C4

Should you be interested in taking part in the course, please send an email to @email.

Abstract

COURSE_PROPOSAL (12).pdf

The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers. 

Wed, 22 Mar 2023

10:00 - 12:00
L6

Gradient flows in metric spaces: overview and recent advances

Dr Antonio Esposito
(University of Oxford)
Further Information

Sessions led by Dr Antonio Esposito will take place on

14 March 2023 10:00 - 12:00 L4

16 March 2023 10:00 - 12:00 L4

21 March 2023 10:00 - 12:00 L6

22 March 2023 10:00 - 12:00 L6

Should you be interested in taking part in the course, please send an email to @email.

Abstract

This course will serve as an introduction to the theory of gradient flows with an emphasis on the recent advances in metric spaces. More precisely, we will start with an overview of gradient flows from the Euclidean theory to its generalisation to metric spaces, in particular Wasserstein spaces. This also includes a short introduction to the Optimal Transport theory, with a focus on specific concepts and tools useful subsequently. We will then analyse the time-discretisation scheme à la Jordan--Kinderlehrer-Otto (JKO), also known as minimising movement, and discuss the role of convexity in proving stability, uniqueness, and long-time behaviour for the PDE under study. Finally, we will comment on recent advances, e.g., in the study of PDEs on graphs and/or particle approximation of diffusion equations.

PhD_course_Esposito_1.pdf

Tue, 21 Mar 2023

10:00 - 12:00
L6

Gradient flows in metric spaces: overview and recent advances

Dr Antonio Esposito
(University of Oxford)
Further Information

Sessions led by Dr Antonio Esposito will take place on

14 March 2023 10:00 - 12:00 L4

16 March 2023 10:00 - 12:00 L4

21 March 2023 10:00 - 12:00 L6

22 March 2023 10:00 - 12:00 L6

Should you be interested in taking part in the course, please send an email to @email.

Abstract

This course will serve as an introduction to the theory of gradient flows with an emphasis on the recent advances in metric spaces. More precisely, we will start with an overview of gradient flows from the Euclidean theory to its generalisation to metric spaces, in particular Wasserstein spaces. This also includes a short introduction to the Optimal Transport theory, with a focus on specific concepts and tools useful subsequently. We will then analyse the time-discretisation scheme à la Jordan--Kinderlehrer-Otto (JKO), also known as minimising movement, and discuss the role of convexity in proving stability, uniqueness, and long-time behaviour for the PDE under study. Finally, we will comment on recent advances, e.g., in the study of PDEs on graphs and/or particle approximation of diffusion equations.

PhD_course_Esposito_0.pdf

Thu, 16 Mar 2023

10:00 - 12:00
L4

Gradient flows in metric spaces: overview and recent advances

Dr Antonio Esposito
(Unviersity of Oxford)
Further Information

Sessions led by Dr Antonio Esposito will take place on

14 March 2023 10:00 - 12:00 L4

16 March 2023 10:00 - 12:00 L4

21 March 2023 10:00 - 12:00 L6

22 March 2023 10:00 - 12:00 L6

Should you be interested in taking part in the course, please send an email to @email.

Abstract

PhD_course_Esposito.pdfThis course will serve as an introduction to the theory of gradient flows with an emphasis on the recent advances in metric spaces. More precisely, we will start with an overview of gradient flows from the Euclidean theory to its generalisation to metric spaces, in particular Wasserstein spaces. This also includes a short introduction to the Optimal Transport theory, with a focus on specific concepts and tools useful subsequently. We will then analyse the time-discretisation scheme à la Jordan--Kinderlehrer-Otto (JKO), also known as minimising movement, and discuss the role of convexity in proving stability, uniqueness, and long-time behaviour for the PDE under study. Finally, we will comment on recent advances, e.g., in the study of PDEs on graphs and/or particle approximation of diffusion equations.

Tue, 14 Mar 2023

10:00 - 12:00
L4

Gradient flows in metric spaces: overview and recent advances

Dr Antonio Esposito
(University of Oxford)
Further Information

Sessions led by Dr Antonio Esposito will take place on

14 March 2023 10:00 - 12:00 L4

16 March 2023 10:00 - 12:00 L4

21 March 2023 10:00 - 12:00 L6

22 March 2023 10:00 - 12:00 L6

Should you be interested in taking part in the course, please send an email to @email.

Abstract

This DPhil short course will serve as an introduction to the theory of gradient flows with an emphasis on the recent advances in metric spaces. More precisely, we will start with an overview of gradient flows from the Euclidean theory to its generalisation to metric spaces, in particular Wasserstein spaces. This also includes a short introduction to the Optimal Transport theory, with a focus on specific concepts and tools useful subsequently. We will then analyse the time-discretisation scheme à la Jordan--Kinderlehrer-Otto (JKO), also known as minimising movement, and discuss the role of convexity in proving stability, uniqueness, and long-time behaviour for the PDE under study. Finally, we will comment on recent advances, e.g., in the study of PDEs on graphs and/or particle approximation of diffusion equations.

Wed, 08 Jun 2022

14:00 - 16:00
L3

Shock Reflection and free boundary problems

Professor Mikhail Feldman
(University of Wisconsin-Madison)
Further Information

Sessions will be as follows:

Tuesday 7th, 2:00pm-4:00pm

Wednesday 8th, 2:00pm-3:30pm

Abstract

We will discuss shock reflection phenomena, mathematical formulation of shock reflection problem, structures of  shock reflection configurations, and von Neumann conjectures on transition between regular and Mach reflections. Then we will describe the results on existence and properties of regular reflection solutions for potential flow equation. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear  elliptic equation in self-similar coordinates, where the reflected shock is the free boundary, and ellipticity degenerates near a part of a fixed boundary. We will discuss the techniques and methods used in the study of such free boundary problems.

 

Tue, 07 Jun 2022

14:00 - 16:00
N3.12

Shock Reflection and free boundary problems

Professor Mikhail Feldman
(University of Wisconsin-Madison)
Further Information

Sessions will be as follows:

Tuesday 7th, 2:00pm-4:00pm

Wednesday 8th, 2:00pm-3:30pm

Abstract

We will discuss shock reflection phenomena, mathematical formulation of shock reflection problem, structures of  shock reflection configurations, and von Neumann conjectures on transition between regular and Mach reflections. Then we will describe the results on existence and properties of regular reflection solutions for potential flow equation. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear  elliptic equation in self-similar coordinates, where the reflected shock is the free boundary, and ellipticity degenerates near a part of a fixed boundary. We will discuss the techniques and methods used in the study of such free boundary problems.

 

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.

Fri, 20 May 2022

10:30 - 12:00
L5

General Linear PDE with constant coefficients

Bogdan Raiță
(Scuola Normale Superiore di Pisa)
Further Information

Sessions will take place as follows:

17th May 14:00 -15:00

18th and 20th May 10:30 -12:00

Abstract

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

Wed, 18 May 2022

10:30 - 12:00
L5

General Linear PDE with constant coefficients

Bogdan Raiță
(Scuola Normale Superiore di Pisa)
Further Information

Sessions will take place as follows:

17th May 14:00 -15:00

18th and 20th May 10:30 -12:00

Abstract

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

Tue, 17 May 2022

14:00 - 15:00
L5

General Linear PDE with constant coefficients

Bogdan Raiță
(Scuola Normale Superiore di Pisa)
Further Information

Sessions will take place as follows:

17th May 14:00 -15:00

18th and 20th May 10:30 -12:00

Abstract

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

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.

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.

Wed, 02 Mar 2022

14:00 - 16:00
Virtual

Topics on Nonlinear Hyperbolic PDEs

Gui-Qiang G. Chen
(Oxford University)
Further Information

Dates/ Times (GMT): 2pm – 4pm Wednesdays 9th, 16th, 23rd Feb, and 2nd March

Course Length: 8 hrs total (4 x 2 hrs)

Abstract

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.

Wed, 02 Mar 2022

10:00 - 12:00
Virtual

Controllability of smooth and non smooth vector fields

Franco Rampazzo
(Università degli Studi di Padova)
Further Information

Dates and Times (GMT):

10am – 12pm Monday’s 2nd, 9th, 16th, 23rd March

8am – 10am Friday’s 4th, 11th, 18th, 25th March

Course Length: 16 hrs total (8 x 2 hrs)

Click here to enroll

Abstract

Courserequirements: Basicmathematicalanalysis.

Examination and grading: The exam will consist in the presentation of some previously as- signed article or book chapter (of course the student must show a good knowledge of those issues taught during the course which are connected with the presentation.).

SSD: MAT/05 Mathematical Analysis
Aim: to make students aware of smooth and non-smooth controllability results and of some

applications in various fields of Mathematics and of technology as well.

Course contents:

Vector fields are basic ingredients in many classical issues of Mathematical Analysis and its applications, including Dynamical Systems, Control Theory, and PDE’s. Loosely speaking, controllability is the study of the points that can be reached from a given initial point through concatenations of trajectories of vector fields belonging to a given family. Classical results will be stated and proved, using coordinates but also underlying possible chart-independent interpretation. We will also discuss the non smooth case, including some issues which involve Lie brackets of nonsmooth vector vector fields, a subject of relatively recent interest.

Bibliography: Lecture notes written by the teacher.

Wed, 23 Feb 2022

14:00 - 16:00
Virtual

Topics on Nonlinear Hyperbolic PDEs

Gui-Qiang G. Chen
(Oxford University)
Further Information

Dates/ Times (GMT): 2pm – 4pm Wednesdays 9th, 16th, 23rd Feb, and 2nd March

Course Length: 8 hrs total (4 x 2 hrs)

Abstract

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.

Wed, 16 Feb 2022

14:00 - 16:00
Virtual

Topics on Nonlinear Hyperbolic PDEs

Gui-Qiang G. Chen
(Oxford University)
Further Information

Dates/ Times (GMT): 2pm – 4pm Wednesdays 9th, 16th, 23rd Feb, and 2nd March

Course Length: 8 hrs total (4 x 2 hrs)

Abstract

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.

Wed, 09 Feb 2022

14:00 - 16:00
Virtual

Topics on Nonlinear Hyperbolic PDEs

Gui-Qiang G. Chen
(Oxford University)
Further Information

Dates/ Times (GMT): 2pm – 4pm Wednesdays 9th, 16th, 23rd Feb, and 2nd March

Course Length: 8 hrs total (4 x 2 hrs)

Abstract

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.