Past OxPDE Short Courses

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.

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.

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.

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.

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.

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.

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.

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.

Sessions will be as follows:

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

Wednesday 8th, 2:00pm-3:30pm

Sessions will be as follows:

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

Wednesday 8th, 2:00pm-3:30pm

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].

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].

Sessions will take place as follows:

17th May 14:00 -15:00

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

Sessions will take place as follows:

17th May 14:00 -15:00

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

Sessions will take place as follows:

17th May 14:00 -15:00

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

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].

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].

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].

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].

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

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

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

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

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

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

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

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

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