Mathematical Institute

Mon, 29 Nov 2021
12:45

Scattering amplitudes and tropical Grassmannians

Omer Gurdogan

(University of Southampton)

Abstract

The analytic structure of scattering amplitudes exhibit striking
properties that are not at all evident from the first principles of
Quantum Field Theory. These are often rich and powerful enough to be
considered as their defining features, and this makes the problem of
finding a set of universal rules a compelling one. I will review the
recently mounting evidence for the relevance of tropical Grassmannians
in this respect, including implications on symbol alphabets and
adjacency conditions

The First Term

So, the first term at university. And, more specifically, the first mathematical term at Oxford. What's in store? Well, our students' mathematical experience in their first term (and beyond) comprises two parts: lectures and tutorials. How do they work?

Wed, 03 Nov 2021

14:00 - 15:00

Virtual

Scattering Amplitudes and Cluster Algebras

Anders Schreiber

Abstract

In this talk we will study scattering amplitudes N=4 super-Yang-Mills theory. In this theory, scattering amplitudes are known to be functions of cluster variables of Gr(4,n) and certain algebraic functions of cluster variables. We will give an overview of how this cluster algebraic structure manifests, and will exploit it in an algorithm for computing symbol alphabets by solving matrix equations of the form C.Z = 0 associated with plabic graphs. These matrix equations associate functions on Gr(m,n) to parameterizations of certain cells of Gr_+ (k,n) indexed by plabic graphs. We are able to reproduce all known algebraic functions of cluster variables appearing in known symbol alphabets. We further show that it is possible to obtain all rational symbol letters (in fact all cluster variables) by solving C.Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of Gr_+ (n-4,n).

Tue, 16 Nov 2021
14:00

The singularity probability of a random symmetric matrix is exponentially small

Matthew Jenssen

Abstract

Let $A$ be drawn uniformly at random from the set of all $n \times n$ symmetric matrices with entries in $\{-1,1\}$. We show that $A$ is singular with probability at most $e^{-cn}$ for some absolute constant $c>0$, thereby resolving a well-known conjecture. This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.

Tue, 02 Nov 2021
14:00

A nonabelian Brunn-Minkowski inequality

Yifan Jing

(Oxford)

Abstract

Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions in the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. If time allows I will also discuss some applications of this result. (Joint with Chieu-Minh Tran and Ruixiang Zhang)

Modelling cylinder addition in Søderberg electrodes

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

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

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

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.

Tue, 26 Apr 2022

10:00 - 12:00

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

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.

Tue, 17 May 2022

10:00 - 12:00

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

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.

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