SYLLABUS for Introduction to Geometric Analysis -Spring 2026
TEXT: Introduciton to Geometric Analysis
INSTRUCTOR: C.J. Anna Sung
OFFICE: Math. Building 525 ; Tel: ext 62308; Email: cjsung@math.nthu.edu.tw
OFFICE HOURS: by appointment.
LECTURES: Friday 11:10-12:30 AM and 13:20-14:30
Contents
In this course, we will study Harmonic functions, Green's functions, and Spectral theorems on Riemannian Manifolds.
We will build a connection between geometry and PDE at the graduate level.
(a) Gradient Estimate.
(b) Existence of Green's function.
(c) Harmonic function and end.
(d) Manifold with positive curvature.
(e) Manifold with positive bottom spectrum.
(f) Manifold with weigthed Poincar\'{e} inequality.
(g) Applications.

References:
L. Evans: Partial Differential Equations.
Jost: Riemannian Geometry and Geometric Analysis.
Davies and Safrov: Spectral Theory and Geometry.
S. Axler,P. Bourdon, W. Ramey; Harmonic Function Theory.
Warner: Foundations of Differential Manifolds and Lie Groups.
Do Carmo: Riemannian Geometry.
P. Li: Lecture notes on Geometric Analysis.
R. Courant and D. Hilbert: Methods of Mathematical Physics. Vol. I and II.
E.B.Davies: Heat Kernels and Spectral Theory.

Course Policies:
1) Attendance and Participation: Regular attendance is expected. I encourage you to participate in discussions.
2) Coursework and Evaluation: Grades will be based on problem sets, possible presentations, and a final project (if applicable).
Detailed grading criteria will be announced in class.
3) Homework Policy: Homework must be submitted in class on the designated due date. While discussion with classmates is permitted,
each student must write up and fully understand their own solutions and be able to reproduce them independently.
Late homework will not be accepted. So that you know, all necessary steps must be completed to receive credit.
4) Examinations: Make-up examinations will not be given. A student who misses an exam without a legitimate excuse
will receive a grade of zero. Any request to be excused from a scheduled exam must be discussed
with the instructor at least one week in advance.
5) Academic Integrity: Please make sure that all submitted work is your own. You'll need to cite any referenced material.
6) Auditing: Enrollment is limited to officially registered students. This course is not open to auditors.
7) Classroom Conduct: Please turn off cellular phones during class.
*********Note: All policies are subject to change based on in-class announcements.*********