Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This introductory course, the first in a two-quarter sequence, develops the basic theory of the subject, beginning with seminal theorems—the Hilbert Basis Theorem and Hilbert’s Nullstellensatz—that establish the dual relationship between so-called varieties—both affine and projective—and certain ideals of the polynomial ring in some number of variables. Topics covered in this first quarter include: algebraic sets, projective spaces, Zariski topology, coordinate rings, the Grassmannian, irreducibility and dimension, morphisms, sheaves, and prevarieties. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology. Prerequisites: Some exposure to advanced mathematical methods, particularly those pertaining to ring theory, fields extensions, and point-set topology.
Yes math fans, as previously hinted at in prior conversations, we’ll be taking a deep dive into the overlap of algebra and geometry. Be sure to line up expeditiously as registration for the class won’t happen until July 31, 2017.
While it’s not yet confirmed, some sources have indicated that this may be the first part of a two quarter sequence on the topic. As soon as we have more details, we’ll post them here first. As of this writing, there is no officially announced textbook for the course, but we’ve got some initial guesses and the best are as follows (roughly in decreasing order):
- Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics) 4th ed. by David A. Cox, John Little, and Donal O’Shea
- Algebraic Geometry: An Introduction (Universitext) by Daniel Perrin
- An Invitation to Algebraic Geometry (Universitext) by Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen, William Traves
- Algebraic Geometry (Dover Books on Mathematics) by Solomon Lefschetz (Less likely based on level and age, but Dr. Miller does love inexpensive Dover editions)
For those who are new to Dr. Miller’s awesome lectures, I’ve written some hints and tips on what to expect.
Most of his classes range from about 20-30 people, many of them lifelong regulars. (Yes, there are dozens of people like me who will take almost everything he teaches–he’s that good. This class, my 22nd, will be the start of my second decade of math with him.)
Congratulations on your new math class, and welcome to the “family”! Beginners Welcome! Invariably the handful of new students every year eventually figure the logistics…
Syndicated copies:
Dr. Miller emailed me yesterday to confirm that the textbook for his Fall UCLA Extension course Introduction to Algebraic Geometry will be Elementary Algebraic Geometry by Klaus Hulek (AMS, 2003) ISBN: 0-8218-2952-1.
Sadly, I totally blew the prediction of which text he’d use. I was so far off that this book wasn’t even on my list to review! I must be slipping…
Syndicated copies:
I actually ordered the book. But I’m spending a lot of time studying machine learning! So I’m not sure, but I’d like to.
First class was great. Come on down!
I originally made this compilation on May 31, 2016 to share with some friends and never got around to posting it. Now that I’m actually in the midst of a class on the topic, I thought I’d dust it off and finally publish it for those who are interested.
If you’re aware of things I’ve missed, or which have appeared since, please do let me know in the comments.
A List of video lectures for Algebraic Geometry
Harpreet Bedi (YouTube) 68 lectures (Note: His website also has some other good lectures on Galois Theory and Algebraic Topology)
Miles Reed(How to Download Miles Reid’s Algebraic Geometry videos)
Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity (NPTEL)
Algebraic geometry for physicists by Ugo Bruzzo
Lectures on Algebraic Geometry by L. Goettsche (ICTP)
2007/2008 Lectures
2008/2009 Lectures
2009/2010 Lectures
2010/2011 Lectures
Talks given at the AMS Summer Institute in Algebraic Geometry (2015)
Classical Algebraic Geometry Today (MSRI Workshop 2009)
Lectures by Harris, Hartshorne, Maclagan, and Beelen at ELGA2011
Some other places with additional (sometimes overlapping resources), particularly for more advanced/less introductory lectures:
Video Lectures for Algebraic Geometry (MathOverflow)
Sites to Learn Algebraic Geometry (MathOverflow)
Video lectures of Algebraic Geometry-Hartshorne-Shafarevich (MathOverflow)
Syndicated copies: