Introduction to Complex Differential Geometry -- Lecture 1 -- Intuition and Definition of Manifolds

I recently completed my Ph.D. under the supervision of Ben Andrews at the Australian National University and Gang Tian at Beijing and Princeton University. My Ph.D. thesis was in the subject of complex differential geometry, the interplay between complex analysis, algebraic geometry, and differential geometry. My Ph.D. thesis was written as with monograph/textbook-style, which was previously absent from the literature.

As an effort to prepare the manuscript for textbook publishing, to improve my own understanding, explore more of the manuscript, and to improve the manuscript, I am planning to give a series of lectures with my Ph.D. thesis as the textbook. The material will be considerably advanced, since the content not only covers the forefront of many aspects of Hermitian geometry, but treats a large number of links between areas that have not appeared in the literature explicitly.

In saying this, the reader is only expected to have an understanding of a first course in point-set topology and complex analysis (of a single complex variable). In saying this, there is an implicit assumption of differential geometry, and some algebraic geometry (e.g., what is contained in Hartshorne) would not hurt. But throughout the book, reminders are given for what I deem not to be `well-known' (to a general mathematical audience).

I was originally planning to give this series of lectures at the University of Queensland, where I am currently a postdoc, but do not believe the numbers will be there, given the difficulty of the material. As a consequence, I have decided to record these lectures and upload them directly to my channel. I will edit the videos to some capacity, making it very clear what part of the notes I am referring to, and plan to record at least one one hour lecture per week.

In the lectures, I will mention where the reader may find supplementary resources for additional detail, since I acknowledge that it can take some time to learn some of the concepts present within the course of lectures I intend to give.

My Ph.D. thesis is available here: https://www.researchgate.net/publicat...

Textbook recommendations for the content of the present video:
Munkres, J. A., Topology, Second Edition.
Shabat, B. V., Introduction to Complex Analysis Part II, Functions of Several Variables.
Lee, J. M., Introduction to Smooth manifolds.
Jost, J., Riemannian Geometry and Geometric Analysis.

💪🙏 Support the channel by signing up to a free trial of Skillshare using the affiliate link https://www.skillshare.com/r/profile/... If you would like free access to the manim course without signing up to Skillshare, send me an email and I'll send you a free link to the course :)

If you would like to become a member of the channel, the link is here:    / @kylebroder  

Different levels of membership offer different perks, ranging from shout outs at the end of videos, to one-on-one discussions over zoom. The funds from memberships will be used to reinvest in the channel, improving production quality and quantity.