Reply to A (very) gentle comment on Algebraic Geometry for the faint-hearted | Ilyas Khan

Replied to A (very) gentle comment on Algebraic Geometry for the faint-hearted by Ilyas KhanIlyas Khan (LinkedIn)
This short article is the result of various conversations over the course of the past year or so that arose on the back of two articles/blog pieces that I have previously written about Category Theory (here and here). One of my objectives with such articles, whether they be on aspects of quantum computing or about aspects of maths, is to try and de-mystify as much of the associated jargon as possible, and bring some of the stunning beauty and wonder of the subject to as wide an audience as possible. Whilst it is clearly not possible to become an expert overnight, and it is certainly not my objective to try and provide more than an introduction (hopefully stimulating further research and study), I remain convinced that with a little effort, non-specialists and even self confessed math-phobes can grasp some of the core concepts. In the case of my articles on Category Theory, I felt that even if I could generate one small gasp of excited comprehension where there was previously only confusion, then the articles were worth writing.

I just finished a course on Algebraic Geometry through UCLA Extension, which was geared toward non-traditional math students and professionals, and wish I had known about Smith’s textbook when I’d started. I did spend some time with Cox, Little, and O’Shea’s Ideals, Varieties, and Algorithms which is a pretty good introduction to the area, but written a bit more for computer scientists and engineers in mind rather than the pure mathematician, which might recommend it more toward your audience here as well. It’s certainly more accessible than Hartshorne for the faint-of-heart.

I’ve enjoyed your prior articles on category theory which have spurred me to delve deeper into the area. For others who are interested, I thought I’d also mention that physicist and information theorist John Carlos Baez at UCR has recently started an applied category theory online course which I suspect is a bit more accessible than most of the higher graduate level texts and courses currently out. For more details, I’d suggest starting here: https://johncarlosbaez.wordpress.com/2018/03/26/seven-sketches-in-compositionality/

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👓 Mathematicians Explore Mirror Link Between Two Geometric Worlds | Quanta Magazine

Read Mathematicians Explore Mirror Link Between Two Geometric Worlds by Kevin Hartnett (Quanta Magazine)
Decades after physicists happened upon a stunning mathematical coincidence, researchers are getting close to understanding the link between two seemingly unrelated geometric universes.

An interesting story in that physicists found the connection first and mathematicians are tying the two areas together after the fact. More often it’s the case that mathematicians come up with the theory and then physicists are applying it to something. I’m not sure I like some of the naming conventions laid out, but it’ll be another decade or two after it’s all settled before things have more logical sounding names. I’m a bit curious if any category theorists are playing around in either of these areas.

After having spent the last couple of months working through some of the “rigidity” (not the best descriptor in the article as it shows some inherent bias in my opinion) of algebraic geometry, now I’m feeling like symplectic geometry could be fun.

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RSVP to MATH X 451.43 Introduction to Algebraic Geometry: The Sequel | UCLA Extension

RSVPed Attending MATH X 451.43 Introduction to Algebraic Geometry: The Sequel
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 course is the second in a two-quarter introductory sequence that develops the basic theory of this classical mathematical field. Whereas the fall-quarter course focused more on the subject’s algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include Bézout’s Theorem, rational varieties, cubic curves and surfaces (including the remarkable 27-line theorem), and the connection between varieties and manifolds. 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.

I’m definitely attending the Winter Quarter!

MATH X 451.43 Introduction to Algebraic Geometry: The Sequel | UCLA Extension

Bookmarked MATH X 451.43 Introduction to Algebraic Geometry: The Sequel (UCLA Extension)
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 course is the second in a two-quarter introductory sequence that develops the basic theory of this classical mathematical field. Whereas the fall-quarter course focused more on the subject’s algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include Bézout’s Theorem, rational varieties, cubic curves and surfaces (including the remarkable 27-line theorem), and the connection between varieties and manifolds. 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.

Alright math nerds, it’s that time again! Be sure to register for Mike Miller’s excellent follow-on course for Algebraic Geometry.

Don’t forget to use the coupon code EARLY to save 10% with an early registration–time is limited!

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Video lectures for Algebraic Geometry

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

Some other places with additional (sometimes overlapping resources), particularly for more advanced/less introductory lectures:

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👓 Vladimir Voevodsky, Fields Medalist, Dies at 51 | IAS

Read Vladimir Voevodsky, Fields Medalist, Dies at 51 (Institute for Advanced Study)
The Institute for Advanced Study is deeply saddened by the passing of Vladimir Voevodsky, Professor in the School of Mathematics. Voevodsky, a truly extraordinary and original mathematician, made many contributions to the field of mathematics, earning him numerous honors and awards, including the Fields Medal. Celebrated for tackling the most difficult problems in abstract algebraic geometry, Voevodsky focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and algebraic topology. He made one of the most outstanding advances in algebraic geometry in the past few decades by developing new cohomology theories for algebraic varieties. Among the consequences of his work are the solutions of the Milnor and Bloch-Kato Conjectures. More recently he became interested in type-theoretic formalizations of mathematics and automated proof verification. He was working on new foundations of mathematics based on homotopy-theoretic semantics of Martin-Löf type theories. His new "Univalence Axiom" has had a dramatic impact in both mathematics and computer science.

 

Sad to hear of Dr. Voevodsky’s passing just as I was starting into my studies of algebraic geometry…

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Algebraic Geometry Lecture 1

For those who are still on the fence about taking Algebraic Geometry this quarter (or the follow on course next quarter), here’s a downloadable copy of the written notes with linked audio that will allow you to sample the class:

Algebraic Geometry-Lecture 1 notes [.pdf file with embedded and linked audio]

I’ve previously written some notes about how to best access and use these types of notes in the past. Of particular note, one must download the .pdf file and open in a recent version of Adobe Acrobat to take advantage of the linked/embedded audio file. (Trust me, it’s worth doing as it will be like you were there with the 20 of us who showed up last night!)

For those who prefer just the audio files separately, they can be listened to here, or downloaded.

Lecture 1 – Part 1

Lecture 1 – Part 2

Again, the recommended text is Elementary Algebraic Geometry by Klaus Hulek (AMS, 2003) ISBN: 0-8218-2952-1.

For those new to Dr. Miller’s classes, I’ve written up some hints/tips about them in the past as well.

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🔖 Elementary Algebraic Geometry by Klaus Hulek

Bookmarked Elementary Algebraic Geometry (Student Mathematical Library, Vol. 20) (American Mathematical Society)
This is a genuine introduction to algebraic geometry. The author makes no assumption that readers know more than can be expected of a good undergraduate. He introduces fundamental concepts in a way that enables students to move on to a more advanced book or course that relies more heavily on commutative algebra. The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory. The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary. It is also an excellent text for those working in neighboring fields (algebraic topology, algebra, Lie groups, etc.) who need to know the basics of algebraic geometry.

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…

Elementary Algebraic Geometry by Kaus Hulek (AMS, 2003)

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Introduction to Algebraic Geometry | UCLA Extension in Fall 2017

Bookmarked MATH X 451.42 Introduction to Algebraic Geometry (UCLA Extension)
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.

Dr. Michael Miller has announced the topic for his Fall math class at UCLA Extension: Algebraic Geometry!!

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):

  1. 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
  2. Algebraic Geometry: An Introduction (Universitext) by Daniel Perrin
  3. An Invitation to Algebraic Geometry (Universitext) by Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen, William Traves
  4. 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.)

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Shinichi Mochizuki and the impenetrable proof of the abc conjecture

Read The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof of the ABC Conjecture (Nature News & Comment)
A Japanese mathematician claims to have solved one of the most important problems in his field. The trouble is, hardly anyone can work out whether he's right.

The biggest mystery in mathematics

This article in Nature is just wonderful. Everyone will find it interesting, but those in the Algebraic Number Theory class this fall will be particularly interested in the topic – by the way, it’s not too late to join the class. After spending some time over the summer looking at Category Theory, I’m tempted to tackle Mochizuki’s proof as I’m intrigued at new methods in mathematical thinking (and explaining.)

The abc conjecture refers to numerical expressions of the type a + b = c. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities a, b and c. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers — those that cannot be further factored out into smaller whole numbers: for example, 15 = 3 × 5 or 84 = 2 × 2 × 3 × 7. In principle, the prime factors of a and b have no connection to those of their sum, c. But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c.

Thanks to Rama for bringing this to my attention!

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Axiom of Choice? “Would you rather be deaf or blind?”

Sir Michael Francis Atiyah, OMFRSFRSEFAA, a British mathematician
in Mathematics in the 20th Century

 

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