The 7m length by 2m diameter cylindrical Søderberg electrode is the secret behind a yearly production capacity of 215,000 tonnes of silicon for Elkem ASA, the third largest silicon producer in the world. This electrode operates continuously thanks to its raw material: carbon paste, whose viscosity depends very sensitively on the temperature.
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.
Fri, 05 Nov 2021
16:00
16:00
N3.12
Holographic Duals of Argyres-Douglas Theories
Federico Bonetti
(Oxford University)
Further Information
This seminar will only be in person.
Abstract
Superconformal field theories (SCFTs) of Argyres-Dougles type are inherently strongly coupled and provide a window onto remarkable non-perturbative phenomena (such as mutually non-local massless dyons and relevant Coulomb branch operators of fractional dimension). I am going to discuss the first explicit proposal for the holographic duals of a class of SCFTs of Argyres-Douglas type. The theories under examination are realised by a stack of M5-branes wrapped on a sphere with one irregular puncture and one regular puncture. In the dual 11d supergravity solutions, the irregular puncture is realised as an internal M5-brane source.
Fri, 19 Nov 2021
16:00
16:00
N4.01
Symmetries and Completeness in EFT and Gravity
Jake McNamara
(Harvard)
Further Information
It is also possible to join online via Zoom.
Abstract
We discuss the formal relationship between the absence of global symmetries and completeness, both in effective field theory and in quantum gravity. In effective field theory, we must broaden our notion of symmetry to include non-invertible topological operators. However, in gravity, the story is simplified as the result of charged gravitational solitons.