Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.

This time.
What are...algebraic varieties? Or: Zeros!

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.

Error.
There is an error on the first slide: the circle equation should be with R^2. See that comments below (thanks for spotting!).

Typo.
The equation for the variety with the selfintersection is y^2=x(x^2+1).

Slides.
http://www.dtubbenhauer.com/youtube.html

Website with exercises.
http://www.dtubbenhauer.com/lecturea...

Thumbnail.
Picture from the first video slides.

Classical algebraic geometry.
https://en.wikipedia.org/wiki/Algebra...
https://en.wikipedia.org/wiki/Commuta...
https://en.wikipedia.org/wiki/Multiva...
https://en.wikipedia.org/wiki/Algebra...
https://en.wikipedia.org/wiki/Affine_...
https://en.wikipedia.org/wiki/Project...
https://en.wikipedia.org/wiki/Quasip...
https://en.wikipedia.org/wiki/Line_(g...)
https://en.wikipedia.org/wiki/Circle
https://en.wikipedia.org/wiki/Parabola
https://en.wikipedia.org/wiki/Ellipse
https://en.wikipedia.org/wiki/Hyperbola
https://en.wikipedia.org/wiki/Cubic_p...
https://en.wikipedia.org/wiki/Ellipti...

Modern algebraic geometry.
https://en.wikipedia.org/wiki/Algebra...
https://en.wikipedia.org/wiki/Scheme_...)
https://en.wikipedia.org/wiki/Formal_...
https://en.wikipedia.org/wiki/Indscheme
https://en.wikipedia.org/wiki/Algebra...
https://en.wikipedia.org/wiki/Algebra...

Modern algebraic geometry version 2.
https://en.wikipedia.org/wiki/Algebra...
https://en.wikipedia.org/wiki/Gr%C3%B...
https://en.wikipedia.org/wiki/Cylindr...
https://en.wikipedia.org/wiki/Tropica...

Applications of (algebraic) geometry.
https://math.stackexchange.com/questi...

Pictures used.
https://www.chilimath.com/wpcontent/...
https://upload.wikimedia.org/wikipedi...
https://upload.wikimedia.org/wikipedi...
Picture created using https://reference.wolfram.com/languag...
https://d33wubrfki0l68.cloudfront.net...
https://upload.wikimedia.org/wikipedi...
https://en.wikipedia.org/wiki/Affine_...
https://upload.wikimedia.org/wikipedi...

Some books I am using (I sometimes steal some pictures from there).
https://agaggathmann.math.rptu.de/cl...
https://www.cambridge.org/core/books/...
https://bertini.nd.edu/book.html
https://mathoverflow.net/questions/24...

Computer talk.
https://magma.maths.usyd.edu.au/magma...
https://reference.wolfram.com/languag...

#algebraicgeometry
#geometry
#mathematics