What is...the Zariski topology?
HTML-код
- Опубликовано: 3 окт 2024
- Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...the Zariski topology? Or: Varieties are closed, by definition.
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.dtubbenhaue...
Website with exercises.
www.dtubbenhaue...
Thumbnail.
Picture from the first video slides.
Classical algebraic geometry.
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...)
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
Modern algebraic geometry.
en.wikipedia.o...
en.wikipedia.o...)
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
Modern algebraic geometry version 2.
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
Applications of (algebraic) geometry.
math.stackexch...
Pictures used.
en.wikipedia.o...
en.wikipedia.o...
Picture created using reference.wolf...
study.com/cima...
Pictures created using reference.wolf...
Picture created using reference.wolf...
Some books I am using (I sometimes steal some pictures from there).
agag-gathmann....
www.cambridge....
bertini.nd.edu...
mathoverflow.n...
Computer talk.
magma.maths.us...
reference.wolf...
#algebraicgeometry
#geometry
#mathematics
A quick comment on this idea of the Zariski topology not being a "real" topology. Qiachu Yuan has a fantastic pair of answers on Math StackExchange that are worth reading if someone finds this bothersome. (The question "Why Zariski topology?" by @Dubious is most relevant, and links to the other.) In brief, Yuan's thesis is that a topology in the point-set sense is better understood as a *logical* construction than a *spatial* one. The geometry-type topology is accidental in the sense that (for instance) metrics naturally ask the question "Is this there a single point that this set is close to?", and it happens that this is a question that the logical notion of a topology is equipped to understand.
Clickbait title: Is Your Topologist LYING to You?? (They DO NOT Study Topologies!)
(In fairness, most topologists I know wouldn't claim to- they study topological spaces :P)
Great, thanks for the comment. Here is the link:
math.stackexchange.com/questions/161884/why-zariski-topology
And I am the lying topologist: I do not study topologies, I like the Euclidean metric 🤣
(Sorry for the roundabout citation; just wasn't sure how you felt about links in the comments :P)
@@enstucky Haha, all good. Academic links are fine 😀
Thank you!
Welcome ☺
4:29 Your rules for a topology threw me a bit. Is this some sort of characterization as a fixed-point of a closure operator? You use V, is that V for vector space?
Wikipedia's axiomatizations of Kuratowski closure are really well-written and paint a quite pretty picture. It's nice for people who like to think of these things in terms of processes rather than things that only exist somewhere out there in Plato's mind.
@@eternaldoorman5228 Thanks for the question 😀
V = variety and the P and Q are the corresponding sets of polynomials. The varieties are supposed to be the closed sets of our topology, and the characterization you see is ‘‘What a topology needs to satisfy in terms of closed sets’’:
proofwiki.org/wiki/Topology_Defined_by_Closed_Sets
(I ignored (3), but that is trivial.)
I hope that makes some sense!