**Clifford Algebras and Spinors**

by**Corentin Le Fur**

**Abstract:** Spinors are complex mathematical entities and play an important role in mathematical and theoretical physics. In their mathematical form, they were introduced by Cartan in 1913. The word « Spinor » seems to have been used for the first time by Ehrenfest and Wolfgang Pauli inaugurated its use in mathematical physics in 1927.

This course has two main goals : on the one hand, to build in a mathematical and rigorous way the notion of spinor and, on the other hand, inspire a conceptual and philosophical analysis of this notion.

We will begin by the study of Clifford Algebras defined by quadratic forms and will examine a few particular cases. Then, we will look at orthogonal and spin groups which will lead us to the concept of spinors space. Fundamental algebraic structures (as monoid, group, ring, algebra, field…) are considered as known.

Clifford Algebras and Spinors |
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Session | Date | Room |
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Clifford Algebras and Spinors IV | 06/03/2013 – 10:30 | Gris, 734A |

Clifford Algebras and Spinors III | 27/02/2013 – 10:30 | Gris, 734A |

Clifford Algebras and Spinors II | 20/02/2013 – 10:30 | Gris, 734A |

Clifford Algebras and Spinors I | 13/02/2013 – 10:30 | Gris, 734A |

**An Introduction to Hopf Algebras**

by**Julien Page**

**Abstract:** Hopf Algebras define a very rich algebraic structure, playing a crucial role in many fields of mathematics and mathematical physics, specially in non-commutative geometry. We will show how this notion solves a duality default of the category of finite groups by extending the Pontryagin duality of the category of abelian finite groups. Next, we will show how this algebraic structure generalizes the structure of a group and its concomitant notion : symmetry. (As an algebraic structure, it is a generalization by the existence of a generalized notion of an inverse, by its “actions” on mathematical objects, by the “integrals” it helps to define, etc). Finally, we will make use of this structure to analyze the notion of duality, focusing mainly in the (crucial for physics) duality between space of states and algebra of observables.

We will suppose that the following mathematical notions are already familiar to the audience: group, ring, field, module, vector space, algebra, tensorial product, category, functor.

An Introduction to Hopf Algebras |
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Session | Date | Room |
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An Introduction to Hopf Algebras IV | 13/02/2013 – 10:30 | Gris, 734A |

An Introduction to Hopf Algebras III | 06/02/2013 – 10:30 | Gris, 734A |

An Introduction to Hopf Algebras II | 30/01/2013 – 10:30 | Gris, 734A |

An Introduction to Hopf Algebras I | 23/01/2013 – 10:30 | Rothko, 412B |

**The Theory of Cartan Connections**

by**Gabriel Catren**

**Abstract:** While Ehresmann connections are the geometric object used to represent the “gauge fields” of the Yang-Mills theory (giving a geometric description of the electromagnetic and nuclear interactions), the Cartan connections enable a reformulation (and a generalization) of General Relativity in such a way that the fundamental variable of the theory will no longer be the space-time metric but will be a connection instead. This reformulation gives new perspectives to the program trying to understand General Relativity as a “gauge theory”, i.e. as a theory describing a dynamical connection on a fiber bundle over space-time. We will show in which precise manner General Relativity (and its generalization: the Einstein-Cartan theory) is the result of the “locality” of the affine group acting transitively on the vacuum solution of the theory (Poincaré, de Sitter or anti-de Sitter groups)

The Theory of Cartan Connections |
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Session | Date | Room |
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The Theory of Cartan Connections III | 19/12/2012 – 10:30 | Kupka, 512A |

The Theory of Cartan Connections II | 19/12/2012 – 10:30 | Gris, 734A |

The Theory of Cartan Connections I | 12/12/2012 – 10:30 | Gris, 734A |