Specialized Topics in Geometry (ΓΕ8)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ8
Semester	2
Course Title	Special Topics in Geometry
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	In this lecture we discuss several topics concerning contemporary topics in Differential Geometry, e.g. symplectic and Kähler manifolds, theory of isometric immersions, minimal surfaces and geometric evolution equations.
General Competences	Work autonomously Work in teams Develop critical thinking skills

Syllabus

• Complex manifolds.
• Kählerian manifolds.
• Riemannian submersions and projective spaces.
• Homogeneous and symmetric spaces.
• Holonomy groups.
• The Bochner technique.
• Harmonic maps and harmonic forms.
• Minimal submanifolds.
• Convergence of Riemannian manifolds.
• Comparison theorems.
• Geometric flows.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Autonomous Study 78 Solution of Exercises - Homeworks 70.5 Course total 187.5
Student Performance Evaluation	Weakly homeworks, presentations.

Attached Bibliography

• B. Andrews and C. Hopper, The Ricci flow in Riemannian Geometry, Springer, 2011.
• T. Colding and W. Minicozzi, A course in minimal surfaces, Graduate Studies in Mathematics, Volume 121, 2011.
• M. Dajczer and R. Tojeiro, Submanifolds theory beyond an introduction, Springer, 2019.
• J. Jost, Riemannian Geometry and Geometric Analysis, 7th edition, Springer, 2017.
• P. Petersen, Riemannian Geometry, 3rd edition, Graduate Texts in Mathematics, 171, Springer, 2016.