Symplectic Geometry

• Donald Huber-Youmans (Heidelberg University)

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

Symplectic Geometry

• Donald Huber-Youmans (Heidelberg University)

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

Symplectic Geometry

• Donald Huber-Youmans (Heidelberg University)

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

Symplectic reduction and its application in physics

• Alexander Thomas (Heidelberg University)

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

Symplectic reduction and its application in physics

• Alexander Thomas (Heidelberg University)

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

Symplectic reduction and its application in physics

• Alexander Thomas (Heidelberg University)

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

Algebraic Topology

• Nikita Nikolaev (University of Birmingham)

Algebraic Topology

• Nikita Nikolaev (University of Birmingham)

Algebraic Topology

• Nikita Nikolaev (University of Birmingham)

Supergeometry - oddities of the square

• Donald Huber-Youmans

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

Supergeometry - oddities of the square

• Donald Huber-Youmans

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

Supergeometry - oddities of the square

• Donald Huber-Youmans

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

Differential Geometry

• Olga Chekeres

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.

Differential Geometry

• Olga Chekeres

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.

Differential Geometry

• Olga Chekeres

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.

General Relativity

• Fridrich Valach (University of Hertfordshire)

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

General Relativity

• Fridrich Valach (University of Hertfordshire)

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

General Relativity

• Fridrich Valach (University of Hertfordshire)

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

Singular ODes

• Nikita Nikolaev (University of Birmingham)

Singular ODes

• Nikita Nikolaev (University of Birmingham)

Singular ODes

• Nikita Nikolaev (University of Birmingham)