Spring 2025
Math 121: Introduction to Hodge Theory
E-mail: salim.tayou@dartmouth.edu,
Office: Kemeny Hall 341.
Schedule:
- Meeting times: Monday-Wednesday: 3:30 PM-5:20 PM. (3A)
- X-hour: Monday, block 3AX (5:30 - 6:20 p.m.).
- Room: 343 in Kemeny Hall.
- First/last meeting: Monday, March 31st/ Wednesday, June 4th, 2025.
- Office hours: See Canvas or by appointment, in Kemeny Hall 341.
Syllabus:
This course is an introduction to Hodge theory, which lies at the intersection of analysis, topology, and algebraic geometry. It is a fundamental framework for studying complex manifolds and a very active area of research. Topics covered include: complex manifolds, Kähler geometry, de Rham cohomology, Laplace operator, harmonic forms, Hodge structures, and applications to algebraic geometry, including a discussion of the famous Hodge conjecture. If time permits, we will talk about variations of Hodge structures and their period domains.
This course is intended for graduate students and advanced undergraduates.
Textbook:
- Claire Voisin, Hodge theory and complex algebraic geometry I.
- Philip Griffiths and Joseph Harris, Principles of algebraic geometry.
Prerequisites:
Differential Topology at the level of Math 104 and Complex Analysis at the level of Math 43.
Grading:
- Homework will be assigned. The solutions can either be scanned or typed and uploaded on Canvas. Late homework can only be accepted under special circumstances. Collaborative work on homework is accepted but you must write your own solution as well as the names of the collaborators, see policy on Canvas for collaborative work on homework.
- Students must also write a short report (no more than 5 pages) and give a short oral presentation (around 30 minutes) on a topic of their choice related to the class. A list of possible topics will be handed out at the beginning of the class.
- The Grade will be based on homework (30%), written report (40%), and oral presentation (30%).
Learning Outcomes:
By the end of this course, you should be able to:
- Understand of the basic structures of complex geometry: define terms, explain their significance, and apply them in context;
- Solve mathematical problems: utilize abstraction and think creatively; and
- Write clear mathematical proofs: recognize and construct mathematically rigorous arguments.