MATH 5720-0

Brief

The course builds upon Differential Geometry I and covers various types of natural geometric structures on manifolds.

Description

Course keywords: Riemannian, complex, symplectic, Kahler, hyper-Kahler, Calabi-Yau and generalised geometry Prerequisites: Having taken Differential Geometry I (on manifolds, tangent spaces, vector fields, differential forms, orientability and integration, Stokes' theorem, de Rham cohomology, homotopy invariance, Poincare duality) with a grade of at least B- or have equivalent knowledge. Contents: In this course, building upon the first semester, we introduce various types of natural geometric structures on manifolds, including Riemannian, complex, symplectic, Kähler, hyper-Kähler, Calabi-Yau and related structures, eventually unified by a relatively recent approach called generalised geometry. The language of instruction is English. References: A.L.Besse, Einstein manifolds, Springer S.S.Chern, Complex manifolds without potential theory, 2nd ed., Springer B.Dubrovin, A.Fomenko & S.Novikov, Modern geometry, Vols.I-III, Springer M. Gualtieri, Generalized complex geometry, PhD thesis, Oxford S.Halperin, W.H.Greub & J.Vanstone, Connections, curvature, and cohomology, Vols.I-III, Academic Press N.Hitchin, Generalized geometry – an introduction, in: Handbook of pseudo- Riemannian geometry and supersymmetry, European Math Soc. D.Huybrechts, Complex geometry, an introduction, Springer S.Kobayashi & K.Nomizu, Foundations of differential geometry, I,II, Wiley & Sons C.H.Taubes, Differential geometry: bundles, connections and curvature, Oxford Evaluation: About four or five problem sets, the last of which can be regarded as a take-home final exam. Due to shortage of manpower, only two exercises will be graded in each problem set. Students' scores depend only on the graded exercises. So it is entirely possible that a student solved most problems except the two to be graded and receive a zero score. New problem sets are handed out during lectures or sent electronically shortly after the lectures, and are due in class two weeks later before the lecture starts. The two exercises to be graded will be announced shortly after the homework is collected. Late hand-in of homework is not accepted. Normally, homework is graded and handed back to students in class a week later. If there are disputes on the correctness of the grading, students need to address the issue within a week of the hand back of the homework. The problem sets are quite challenging and the use of AI is not expected to be of help.

Timetable

Scores

Average Percentage 91

Std. Deviation 4.17

Remarks

本課程為 16 週課程

Restrictions

Compulsory

Elective

Cross Discipline

First Specialization

Second Specialization