Ph.D. course in geometry and topology HT2021

Lecturer

Georgios Dimitroglou Rizell
office: Å14161.
e-mail: georgios.dimitroglou@math.uu.se.

Material

Examination

Schedule and preliminary plan

Nr.	Date	Topics	Lecture notes	References
01.	Thu. 16/9	Topological spaces, Manifolds, Homotopy.		[Hat], [Mil]
02.	Thu. 23/9	Homotopy groups and the fundamental group, computations		[Hat]
03.	Thu. 30/9	Dependence on basepoint, Seifert-Van Kampen		[Hat]
04.	Thu. 7/10	Tangent bundle, Fibre bundles		[Mil], [Hus]
05.	Mon. 11/10	Principal bundles, Homogeneous spaces		[Sha], [Hus]
06.	Thu. 14/10	Homotopy lifting property, Long exact sequence of homotopy groups		[Bre], [Hat], [Hus]
07.	Mon. 18/10	More about principal bundles		[Hus]
08.	Mon. 25/10	Examples of homology and homotopy spheres. Morse functions.		[Kos]
09.	Thu. 28/10	Morse functions, the Morse Lemma.		[Kos]
10.	Mon. 1/11	More about Morse functions. Classifiation of surfaces, Heegaard splittings.		[Kos]
11.	Thu. 4/11	Kirby diagrams, Surgeries, Morse homology		[Kos], [Jos]
12.	Mon. 8/11	The de Rham complex, LES of a pair		[Bot]
13.	Thu. 11/11	Barcode induced by a Morse function.
14.	Mon. 15/11	Introduction to knot theory. Smooth isotopies, submanifolds, and isotopy extension.		[Kos],[Lic]
15.	Thu. 18/11	Reidemeister moves, Wirtinger presentation.		[Lic]
16.	Thu. 25/11	Quandles and colourings, Linking number, Seifert surfaces.		[Lic]
17.	Mon. 29/11	Alexander polynomial and Skein relation		[Lic]
18.	Thu. 2/12	Jones polynomial, Connection, Parallel transport		[Son]
19.	Mon. 6/12	Flat connections, Lie bracket		[Son]
20.	Thu. 9/12	Adjoint representations, Connection one-form, Curvature two-form, Gauge transformations		[Son]

References

• [Bot]: Bott, R. and Tu, L. W., Differential Forms in Algebraic Topology, Gradiate texts in mathematics 82, Springer, New York, 1982.
• [Hus]: Husemoller. D., Fibre bundles, Gradiate texts in mathematics, Springer, New York, 1994.
• [Jos]: Jost. J., Riemannian Geometry and Geometric Analysis, Universitext, 2017, 7th ed. 2017.
• [Hat:]: Hatcher, A. Algebraic Topology.
• [Kos]: Kosinski, A. A. Differential Manifolds, Boston: Academic, cop. 1993.
• [Lic] Lickorish, W. B. R. Introduction to Knot Theory, Graduate Texts in Mathematics, Springer, New York, 2012.
• [Mil]: Milnor, J. W., Topology from the differentiable viewpoint, Princeton, N.J. ; Chichester : Princeton Univ. Press, 1997.
• [Sha] Sharpe, R. W. Differential geometry: Cartan's generalization of Klein's Erlangen program, Graduate texts in mathematics, Springer, New York, 1997.
• [Son]: Sontz, S. B.Principal Bundles, The classical Case, UTX, Springer, Electronic, 2015.