Mon, 07 Mar 2022

16:30 - 17:30
Virtual

Nonlinear wave equations, the weak null condition, and radiation fields

Joseph Keir
(Oxford University)
Abstract

Nonlinear wave equations are ubiquitous in physics, and in three spatial dimensions they can exhibit a wide range of interesting behaviour even in the small data regime, ranging from dispersion and scattering on the one hand, through to finite-time blowup on the other. The type of behaviour exhibited depends on the kinds of nonlinearities present in the equations. In this talk I will explore the boundary between "good" nonlinearities (leading to dispersion similar to the linear waves) and "bad" nonlinearities (leading to finite-time blowup). In particular, I will give an overview of a proof of global existence (for small initial data) for a wide class of nonlinear wave equations, including some which almost fail to exist globally, but in which the singularity in some sense takes an infinite time to form. I will also show how to construct other examples of nonlinear wave equations whose solutions exhibit very unusual asymptotic behaviour, while still admitting global small data solutions.

Thu, 17 Feb 2022

15:00 - 16:00
C2

Torsion points on varieties and the Pila-Zannier method - TALK POSTPONED UNTIL WEEK 5

Francesco Ballini
(Oxford University)
Abstract

In 2008 Pila and Zannier used a Theorem coming from Logic, proven by Pila and Wilkie, to give a new proof of the Manin-Mumford Conjecture, creating a new, powerful way to prove Theorems in Diophantine Geometry. The Pila-Wilkie Theorem gives an upper bound on the number of rational points on analytic varieties which are not algebraic; this bound usually contradicts a Galois-theoretic bound obtained by arithmetic considerations. We show how this technique can be applied to the following problem of Lang: given an irreducible polynomial f(x,y) in C[x,y], if for infinitely many pairs of roots of unity (a,b) we have f(a,b)=0, then f(x,y) is either of the form x^my^n-c or x^m-cy^n for c a root of unity.

Thu, 27 Jan 2022

15:00 - 16:00
Virtual

Ricci curvature lower bounds for metric measure spaces.

Dimitri Navarro
(Oxford University)
Abstract

In the '80s, Gromov proved that sequences of Riemannian manifold with a lower bound on the Ricci curvature and an upper bound on the dimension are precompact in the measured Gromov--Hausdorff topology (mGH for short). Since then, much attention has been given to the limits of such sequences, called Ricci limit spaces. A way to study these limits is to introduce a synthetic definition of Ricci curvature lower bounds and dimension upper bounds. A synthetic definition should not rely on an underlying smooth structure and should be stable when passing to the limit in the mGH topology. In this talk, I will briefly introduce CD spaces, which are a generalization of Ricci limit spaces.

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, 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.

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.

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