What is...algebraic geometry?

Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...algebraic geometry? Or: Polynomials, and more.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
In this course I try to cover my favorite topics in algebraic geometry, from classical ideas such as algebraic varieties, to modern ideas such as schemes, to really modern ideas such as tropical varieties. I give a very biased collection of topics, and not nearly all that can be said will be addressed. Sorry for that.
Slides.
www.dtubbenhauer.com/youtube.html
Website with exercises.
www.dtubbenhauer.com/lecture-a...
Thumbnail.
Picture from the first video slides.
Classical algebraic geometry.
en.wikipedia.org/wiki/Algebra...
en.wikipedia.org/wiki/Commuta...
en.wikipedia.org/wiki/Multiva...
en.wikipedia.org/wiki/Algebra...
en.wikipedia.org/wiki/Affine_...
en.wikipedia.org/wiki/Project...
en.wikipedia.org/wiki/Quasi-p...
en.wikipedia.org/wiki/Line_(g...)
en.wikipedia.org/wiki/Circle
en.wikipedia.org/wiki/Parabola
en.wikipedia.org/wiki/Ellipse
en.wikipedia.org/wiki/Hyperbola
en.wikipedia.org/wiki/Cubic_p...
en.wikipedia.org/wiki/Ellipti...
Modern algebraic geometry.
en.wikipedia.org/wiki/Algebra...
en.wikipedia.org/wiki/Scheme_...)
en.wikipedia.org/wiki/Formal_...
en.wikipedia.org/wiki/Ind-scheme
en.wikipedia.org/wiki/Algebra...
en.wikipedia.org/wiki/Algebra...
Modern algebraic geometry version 2.
en.wikipedia.org/wiki/Algebra...
en.wikipedia.org/wiki/Gr%C3%B...
en.wikipedia.org/wiki/Cylindr...
en.wikipedia.org/wiki/Tropica...
Applications of (algebraic) geometry.
math.stackexchange.com/questi...
Pictures used.
Pictures created using reference.wolfram.com/languag...
A variation of pictures from • What is...algebraic to...
en.wikipedia.org/wiki/Conic_s...
mathinstitutes.org/uploads/20...
pbelmans.ncag.info/atlas/mumf...
Picture from people.bath.ac.uk/cel34/docs/...
www.researchgate.net/publicat...
www.researchgate.net/publicat...
hackaday.com/wp-content/uploa...
Some books I am using (I sometimes steal some pictures from there).
agag-gathmann.math.rptu.de/cl...
www.cambridge.org/core/books/...
bertini.nd.edu/book.html
mathoverflow.net/questions/24...
Computer talk.
magma.maths.usyd.edu.au/magma...
reference.wolfram.com/languag...
#algebraicgeometry
#geometry
#mathematics

[@yankeed4793]

Nice intro to paint a picture of the scope of what you want to cover and motivate what it’s all going to be about. Not sure I can keep up, but I can always count on you to have enough in your lectures to keep even the lay person entertained, and having fun is what it’s all about. Math is hard, that’s why it’s rewarding when that “aha” moment hits and the fog lifts enough to see the next goal post. Always appreciate your particular perspectives on these complex topics instead of just reciting some textbook material. I understand the “textbook” axiom/theorem/proof approach is important for rigor, but can be intimidating for a lot of us. Keep having fun!

[@VisualMath]

AG is quite abstract, but I will give it my best to make it as accessible as possible. That is better for me as well 😁 If you try to follow as long as its entertaining, then I am happy 😀

[@armchairLibrarian]

Thrilled to see that you are creating a series on AG! Your clarity and engaging teaching style are immensely appreciated! I’m personally looking forward to stacks!

[@VisualMath]

Thanks, I hope you will enjoy AG! I am looking forward for stacks as well. I always wanted to learn that properly anyway...😂

[@jshores4318]

This is going to be an awsome series! Very excited!

[@VisualMath]

I will try my best; feedback is very welcome 👍

[@vladimirbadalyan1195]

So excited!!! Always wanted to dabble into AlgGeo, but never found an accessible resource. Hopefully I can understand this since your explanations are so clear

[@VisualMath]

I am excited, too 😀 Let me know how AG goes for you; I hope the video will be helpful.

[@Naverb]

I am extremely excited for this series. I quite enjoy your presentation style, and think it could finally help elucidate for me how the various flavours of algebraic geometry relate

[@VisualMath]

I would be extremely happy if the series is helpful for you. Any comments and remarks along the way are very welcome 😀

[@conoroneill8067]

Thanks so much for this series! I've wanted to self study Algebraic Geometry for a long time, but when trying to figure out how, the standard advice seems to be "Read Hartshorne", which I couldn't understand at all when I tried to read. Hopefully this will be a little bit more approachable.

[@VisualMath]

"Read Hartshorne" never worked for me, so that will not be my explanation strategy 😂 I will be much more explicit; let me know how that works for you.

[@tuongnguyen9391]

Damn ! this is so exiting, as an engineer I hope the content of Cylindrical Algebraic Decomposition got cover !

[@VisualMath]

I am excited as well 😁 Not sure yet what will be covered in the later parts, but I right now CAD is not on my list, sorry for that 😰

[@Jaylooker]

There is the classical Red Book by Mumford and Algebraic Geometry by Hartshorne. I like Milne’s notes in general and he has notes on at least algebraic geometry and étale cohomology. The Stacks project can also be a helpful reference at times for a specific topic or mathematical object in algebraic geometry. There are some important theorems referenced in Grothendieck’s SGA and EGA but it’s difficult and requires prior knowledge on algebraic geometry to understand. Derived algebraic geometry is a somewhat new application of algebraic geometry. At least for me tropical geometry sounds interesting because it is related to clusters algebras and integrable system. It may interest you because it is related to matroids.

[@VisualMath]

Haha, we will see whether my flavor of AG is suited for you. I am quite combinatorial as you know ☺

[@Jaylooker]

@@VisualMath It sounds like schemes, affine spaces, projective spaces, and possibly moduli spaces like the parameterized example that classifies conic might be covered so yes. I think I could use more combinatorial ideas

[@VisualMath]

@@Jaylooker More combinatorics? Good, then I might be able to help 😂

[@spenxerbdp9809]

Greetings from Switzerland. Nice Serie

[@VisualMath]

Excellent, greetings back to Switzerland 😀 I hope you will enjoy AG!

[@spenxerbdp9809]

@@VisualMath I really just like commutative algebra I have not yet done much AG, look forward to it

[@VisualMath]

@@spenxerbdp9809 Some people say the two are the same 🤣 So you should be in a good shape!

[@M0rph1sm]

Very nice work on your videos Hartshorne is very dense 📚. Some Abstract algebra texts have an introduction to Alg-Geom ie Dummit and Foote 👍

[@VisualMath]

Indeed Hartshorne is dense. Or I am dense, or maybe both, haha. I hope you will like the videos! In any case, I hope its enjoyable.

[@vinbo2232]

Waiting for the robotics application...

[@VisualMath]

Me too ☺ We will get there eventually

[@UnforsakenXII]

nice

[@VisualMath]

Thanks for the feedback; let me know how it goes for you 😀

[@preetheshputhran2307]

your link with exercises returns a 404

[@VisualMath]

Oh, sorry. The plan is to create the website as the video series move one. In other words, yes, right now its empty, but I hope to fix that soonish 😅

[@Achrononmaster]

@5:20 oh man, what a downer. I really like your series and relaxed delivery, but Mathematica™? Seriously? That prices out a lot of poor kids (and myself). Can't you bend a little to redo interactives in SAGE or Maxima or similar. In Jupyter you can use Sympy and Galgebra (the pypi library, not the gui Geogebra, although the latter is useful too) combined with Plotly. You have to support free-libre software dude. So much of the world runs on free-libre, we all should give back by refusing proprietary software. (I do realize the irony of posting this on youtube.)

[@VisualMath]

Well, nobody is perfect 😅 and every subscription model (free or paid or in between like KZbin 😁) has advantages. Even Python has some advantages 🤣 Anyway, thanks for the additional references, those might indeed be useful for someone.

[@Sidionian]

Wow Awesome!!! 🎉🎉🎉🎉

[@VisualMath]

Finally 😂 If you (or anyone) has any suggestions along the way, please let me know!

[@Sidionian]

@@VisualMath Definitely Topos Theory. I believe Grothendieck and friends developed Topos Theory to deal with problems in Alg. Geom., but it later became a plausible contender for modern foundations of mathematics to be studied in its own right. Isham and Döring later used Topos Theory to provide a mathematical framework for the foundations of Quantum Theory and physics in general. Topos Theory has gained more attention from the Quantum Gravity community as a possible language to unify gravity with particle physics. In any case, I believe you should do at least 2 or 3 parts to Topos Theory, in order to do it some justice. Pre-requisites would be Category Theory of course...Thanks 🤓🙏

[@VisualMath]

@@SidionianTopos theory is not on the list so far. We will see whether it fits 🤔