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Topics in spin geometry
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Topics in spin geometry
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Academic year 2016/2017
- Teacher
- Ioannis Chrysikos (Lecturer)
- Year
- 3° anno
- Type
- A scelta dello studente
- Credits/Recognition
- 6 CFU
- Course disciplinary sector (SSD)
- MAT/03 - geometria
- Delivery
- Tradizionale
- Language
- Inglese
- Prerequisites
- Basic knowledge of linear algebra, smooth manifolds and differential forms
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Sommario del corso
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Course objectives
The course introduces the students to Clifford algebras and Dirac operators, from a geometric point of view.
We will discuss the most basic facts related with the classification of Clifford algebras and we shall examine the geometry of Riemannian spin manifolds, focusing on Dirac and Twistor operators.
The eigenspinors of these differential operators play a fundamental role, both in theoretical physics and differential geometry. Our goal is to describe well-known methodologies for the examination of such special spinor fields, e.g. Killing spinors, with further aim the development of mathematical thinking in this so beautiful area of differential geometry and mathematical analysis.
The knowledge of this material can be essential for further motivation and research, related to more recent developments in the theory of Dirac operators induced by general metric connections. This currently is a subject of advanced research and since we want to approach some basic methods which appear in this area, we will often focus on examples in low dimensions.
From the students is required some basic knowledge of differential geometry.
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Results of learning outcomes
Clifford Algebras and Spin groups, Riemannian spin manifolds, Dirac and Twistor operatos, special spinors, parallel spinors, Killing spinors, twistor spinors, pure spinors, eigenvalue estimates.
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Course delivery
For the organisation of the course we shall provide notes, which basically follow known literature on the topic.
Exercises will be distributed and we will devote time in the class, to discuss their solutions.
The exam consists of written and oral exams. We shall propose exercises for solution, ask definitions and proofs of basic theorems and results.
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Learning assessment methods
Written and oral exam.
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Program
Docenti: Ioannis Chrysikos, Arman Taghavi-ChabertProgram:Clifford algebras and spin groups (introduction)Clifford algebras and spin groups(classification),Spin representations and the Clifford multiplication, examples.Spin structures on pseudo Riemannian manifolds, examples.Theory of connections on fibre bundles, the spin connection on the spinor bundleDirac operators (definition and basic properties)Further properties of the Dirac operator, the Schrödinger–Lichnerowicz formula,The Twistor operator and conformal invariance,Twistor and Killing spinors, parallel spinors on Riemannian spin manifolds,Spin Riemannian manifolds admitting real Killing spinors, the cone-construction of Bär,Pure spinors, CR geometries and foliations by null geodesicsConnections with skew-torsion, Dirac operator with torsion, generalized Killing spinors, spin structures on homogeneous spaces.Suggested readings and bibliography
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1) Simon Salamon, ''Riemannian Geometry and Holonomy Groups'', Reserach Notes in Mathematics Series, Longman Sc & Tech; Subsequent edition, 1989.
2) Helga Baum, Thomas Friedrich, Ralf Grunewald, Ines Kath, ''Twistors and Killing spinors on Riemannian manifolds''. Stuttgart etc. B.G. Teubner Verlagsgesellschaft, 1991.
3) Thomas Friedrich, ''Dirac Operators in Riemannian Geometry'', Graduate Studies in Mathematics, Vol 25, AMS, 2000.
4) Oussama Hijazi, ''Spectral properties of the Dirac properties and geometrical structures'', Proceedings of the Summer school on Geometric methods in Quantum Field Theory, Vila de Leyva, Colombia, July 12-30, (1999), World Scientific 2001.
5) Peter Petersen, ''Riemannian Geometry'', Graduate Texts in Mathematics, Vol 171, Springer, 2006 (Second Edition)
6) Nicolas Ginoux, ''The Dirac Spectrum'', Lecture Notes in Mathematics, Springer, 2009.
Bibliography for further reading.
1) Élie Cartan,
Michelsohn, ''Spin Geometry'', Princeton University Press, 1989.
3) Pertti Lounesto, ''Clifford Algebras and Spinors'', London Math. Soc., Lecture Note Series Vol. 286, Sec. Edition, Cambridge University Press, 2001.
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Note
Start date: February 15, 2017
11:00am -13:00 pm
Aula 5 (room 5)
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