MAPSS

14 Jul 2024, 18:35 → 19 Jul 2024, 14:00 Europe/Zurich

Hotel Les Sources

Donald Ray Youmans (University Heidelberg), Elise Raphael, Nikita Nikolaev (University of Birmingham)

All materials (lecture notes and exercises) can be found at the bottom of this page.

The Mathematical Physics Summer School for masters students and beginning PhD students is organized by SwissMAP and offers introductory lectures to different aspects of mathematical physics.

Registrations are now closed.

Confirmed speakers:

• Donald Huber-Youmans (University of Heidelberg)
• Nikita Nikolaev (University of Birmingham)
• Chiara Saffirio (University of Basel)
• Alexander Thomas (University of Heidelberg)
• Fridrich Valach (University of Hertfordshire)
• Olga Chekeres (Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University)

 

Program

This summer school will contain 8 mini-courses on the following topics:

• Symplectic geometry
• Symplectic reduction (and its applications in physics)
• Differential geometry
• General relativity
• Algebraic topology
• Supergeometry (and its applications in physics)
• Singular ODEs
• Superconductivity

 

Abstracts are visible directly on the schedule.

 

 

 

General relativity - lecture notes .pdf

Riemannian geometry_exercises.pdf

Supergeometry - advanced exercises v2.pdf

Supergeometry - Lecture notes.pdf

Symplectic geom - advanced exercises- v3.pdf

Symplectic Geometry - Lecture notes.pdf

Symplectic reduction - Exercises v2.pdf

What is superconductivity?.pdf

Sunday, 14 July

Dinner

Monday, 15 July

1 Symplectic Geometry

This course is an introduction to the foundations of symplectic geometry. We will discuss a motivating example—Hamiltonian mechanics—before defining what it means to be symplectic. Afterwards we will study some consequences of the general definitions. Importantly, we will show that locally, all symplectic spaces look the same: there are no local symplectic invariants! This is a consequence of Darboux’s theorem.
Key words: Hamiltonian mechanics, symplectic manifolds, Darboux theorem

Speaker: Donald Huber-Youmans (Heidelberg University)

Coffee Break

2 Symplectic Geometry

This course is an introduction to the foundations of symplectic geometry. We will discuss a motivating example—Hamiltonian mechanics—before defining what it means to be symplectic. Afterwards we will study some consequences of the general definitions. Importantly, we will show that locally, all symplectic spaces look the same: there are no local symplectic invariants! This is a consequence of Darboux’s theorem.
Key words: Hamiltonian mechanics, symplectic manifolds, Darboux theorem

Speaker: Donald Huber-Youmans (Heidelberg University)

3 Symplectic Geometry

This course is an introduction to the foundations of symplectic geometry. We will discuss a motivating example—Hamiltonian mechanics—before defining what it means to be symplectic. Afterwards we will study some consequences of the general definitions. Importantly, we will show that locally, all symplectic spaces look the same: there are no local symplectic invariants! This is a consequence of Darboux’s theorem.
Key words: Hamiltonian mechanics, symplectic manifolds, Darboux theorem

Speaker: Donald Huber-Youmans (Heidelberg University)

Lunch

Coffee Break

4 Symplectic reduction and its application in physics

After the introductory course in symplectic geometry, we analyze symplectic manifolds which are symmetric under the action of a Lie group, leading in particular to the symplectic quotient construction. The important concept is the notion of the moment map, generalizing the concept of momentum and angular momentum in classical mechanics, and capturing all preserved quantities coming from Noethers theorem. The symplectic quotient, or Marsden-Weinstein quotient, allows then to define reduced phase spaces. We will see many examples both physically and mathematically motivated.

Key words: Hamiltonian actions, moment maps, symplectic quotient, Noether theorem, Poisson manifolds

Speaker: Alexander Thomas (Heidelberg University)

5 Symplectic reduction and its application in physics

After the introductory course in symplectic geometry, we analyze symplectic manifolds which are symmetric under the action of a Lie group, leading in particular to the symplectic quotient construction. The important concept is the notion of the moment map, generalizing the concept of momentum and angular momentum in classical mechanics, and capturing all preserved quantities coming from Noethers theorem. The symplectic quotient, or Marsden-Weinstein quotient, allows then to define reduced phase spaces. We will see many examples both physically and mathematically motivated.

Key words: Hamiltonian actions, moment maps, symplectic quotient, Noether theorem, Poisson manifolds

Speaker: Alexander Thomas (Heidelberg University)

6 Symplectic reduction and its application in physics

After the introductory course in symplectic geometry, we analyze symplectic manifolds which are symmetric under the action of a Lie group, leading in particular to the symplectic quotient construction. The important concept is the notion of the moment map, generalizing the concept of momentum and angular momentum in classical mechanics, and capturing all preserved quantities coming from Noethers theorem. The symplectic quotient, or Marsden-Weinstein quotient, allows then to define reduced phase spaces. We will see many examples both physically and mathematically motivated.

Key words: Hamiltonian actions, moment maps, symplectic quotient, Noether theorem, Poisson manifolds

Speaker: Alexander Thomas (Heidelberg University)

Dinner

Tuesday, 16 July

7 Algebraic Topology

Speaker: Nikita Nikolaev (University of Birmingham)

Coffee Break

8 Algebraic Topology

Speaker: Nikita Nikolaev (University of Birmingham)

9 Algebraic Topology

Speaker: Nikita Nikolaev (University of Birmingham)

Lunch

Coffee Break

10 Supergeometry - oddities of the square

This course is an excursion into the marvelous world of supergeometry which plays an important role in mathematics and physics. On one hand, it is a natural, albeit at first glance unintuitive, generalization of ordinary geometry. On the other hand it plays a pivotal role in the theory of supersymmetry. Naively, one can replace “super” by “Z/2Z”-graded, alongside introducing the Koszul sign-rule. This naive idea has far reaching consequences leading to the definition of supermanifolds and supersymmetry.
Key words: Super algebras, super manifolds, odd coordinates, fuzzy points, Supersymmetry, super quantum mechanics

Speaker: Donald Huber-Youmans

11 Supergeometry - oddities of the square

This course is an excursion into the marvelous world of supergeometry which plays an important role in mathematics and physics. On one hand, it is a natural, albeit at first glance unintuitive, generalization of ordinary geometry. On the other hand it plays a pivotal role in the theory of supersymmetry. Naively, one can replace “super” by “Z/2Z”-graded, alongside introducing the Koszul sign-rule. This naive idea has far reaching consequences leading to the definition of supermanifolds and supersymmetry.
Key words: Super algebras, super manifolds, odd coordinates, fuzzy points, Supersymmetry, super quantum mechanics

Speaker: Donald Huber-Youmans

12 Supergeometry - oddities of the square

This course is an excursion into the marvelous world of supergeometry which plays an important role in mathematics and physics. On one hand, it is a natural, albeit at first glance unintuitive, generalization of ordinary geometry. On the other hand it plays a pivotal role in the theory of supersymmetry. Naively, one can replace “super” by “Z/2Z”-graded, alongside introducing the Koszul sign-rule. This naive idea has far reaching consequences leading to the definition of supermanifolds and supersymmetry.
Key words: Super algebras, super manifolds, odd coordinates, fuzzy points, Supersymmetry, super quantum mechanics

Speaker: Donald Huber-Youmans

Dinner

Wednesday, 17 July

Lunch

Dinner

Thursday, 18 July

13 Differential Geometry

Differential geometry in 1, 2, 3 and more dimensions.
Imagine an n-dimensional Riemannian manifold and then set n=1, 2, 3, 4+.

Prerequisites: Come as you are. Understanding the notion of a differentiable manifold is assumed though.

Consequences: To embrace the mightiness of general relativity prepared thou shall be.

Key words: Riemannian and pseudo-Riemannian manifolds, metric tensor, connection, covariant derivative, curvature.

Speaker: Olga Chekeres

Coffee Break

14 Differential Geometry

Differential geometry in 1, 2, 3 and more dimensions.
Imagine an n-dimensional Riemannian manifold and then set n=1, 2, 3, 4+.

Prerequisites: Come as you are. Understanding the notion of a differentiable manifold is assumed though.

Consequences: To embrace the mightiness of general relativity prepared thou shall be.

Key words: Riemannian and pseudo-Riemannian manifolds, metric tensor, connection, covariant derivative, curvature.

Speaker: Olga Chekeres

15 Differential Geometry

Differential geometry in 1, 2, 3 and more dimensions.
Imagine an n-dimensional Riemannian manifold and then set n=1, 2, 3, 4+.

Prerequisites: Come as you are. Understanding the notion of a differentiable manifold is assumed though.

Consequences: To embrace the mightiness of general relativity prepared thou shall be.

Key words: Riemannian and pseudo-Riemannian manifolds, metric tensor, connection, covariant derivative, curvature.

Speaker: Olga Chekeres

Lunch

Coffee Break

16 General Relativity

Abstract: General relativity is a beautifully geometric and mathematically rigorous theory describing our universe on large scales, where gravity plays a crucial role. After an introduction to differential and Riemannian geometry we will look in more detail at the mathematical underpinnings of this theory and talk about geodesics, normal coordinates, Einstein equations, and other interesting topics.

Keywords: spacetime, metric, curvature, geodesics, normal coordinates, Einstein equations

Speaker: Fridrich Valach (University of Hertfordshire)

17 General Relativity

Abstract: General relativity is a beautifully geometric and mathematically rigorous theory describing our universe on large scales, where gravity plays a crucial role. After an introduction to differential and Riemannian geometry we will look in more detail at the mathematical underpinnings of this theory and talk about geodesics, normal coordinates, Einstein equations, and other interesting topics.

Keywords: spacetime, metric, curvature, geodesics, normal coordinates, Einstein equations

Speaker: Fridrich Valach (University of Hertfordshire)

18 General Relativity

Abstract: General relativity is a beautifully geometric and mathematically rigorous theory describing our universe on large scales, where gravity plays a crucial role. After an introduction to differential and Riemannian geometry we will look in more detail at the mathematical underpinnings of this theory and talk about geodesics, normal coordinates, Einstein equations, and other interesting topics.

Keywords: spacetime, metric, curvature, geodesics, normal coordinates, Einstein equations

Speaker: Fridrich Valach (University of Hertfordshire)

Dinner

Friday, 19 July

19 Singular ODes

Speaker: Nikita Nikolaev (University of Birmingham)

Coffee Break

20 Singular ODes

Speaker: Nikita Nikolaev (University of Birmingham)

21 Singular ODes

Speaker: Nikita Nikolaev (University of Birmingham)

Lunch