Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.
This time.
What is...the Hodge conjecture? Or: Topology and algebraic geometry.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Slides.
http://www.dtubbenhauer.com/youtube.html
TeX files for the presentation.
https://github.com/dtubbenhauer/My-TeX-files
Thumbnail.
https://en.wikipedia.org/wiki/Orientability#/media/File:Torus.png
https://en.wikipedia.org/wiki/Kummer_surface#/media/File:Kummer_surface.png
Main discussion.
https://en.wikipedia.org/wiki/Hodge_conjecture
https://mathoverflow.net/questions/54197/why-is-the-hodge-conjecture-so-important
https://burttotaro.wordpress.com/2012/03/18/why-believe-the-hodge-conjecture/
https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/voisinhodge.pdf
https://www.claymath.org/millennium/hodge-conjecture/
https://www.redalyc.org/journal/5117/511766757039/html/
https://www.theguardian.com/science/blog/2011/mar/01/million-dollars-maths-hodge-conjecture
https://en.wikipedia.org/wiki/Millennium_Prize_Problems
Background material.
https://en.wikipedia.org/wiki/Projective_variety
https://en.wikipedia.org/wiki/Complex_algebraic_variety
https://en.wikipedia.org/wiki/Homology_(mathematics)
https://en.wikipedia.org/wiki/Hodge_theory
https://en.wikipedia.org/wiki/Hodge_theory#Hodge_theory_for_complex_projective_varieties
https://en.wikipedia.org/wiki/Hodge_structure
https://en.wikipedia.org/wiki/De_Rham_cohomology
https://en.wikipedia.org/wiki/Weil_cohomology_theory
https://en.wikipedia.org/wiki/Poincar%C3%A9_duality
https://en.wikipedia.org/wiki/Fundamental_class
https://en.wikipedia.org/wiki/Algebraic_cycle
Computer talk.
https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/de_rham_cohomology.html
https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/characteristic_cohomology_class.html
Pictures used.
https://en.wikipedia.org/wiki/Complex_algebraic_variety#/media/File:RiemannKugel.svg
Picture from https://www.youtube.com/watch?v=INCSbaTRNTs
https://en.wikipedia.org/wiki/Kummer_surface#/media/File:Kummer_surface.png
https://en.wikipedia.org/wiki/Complex_algebraic_variety#/media/File:RiemannKugel.svg
https://www.researchgate.net/publication/257072438/figure/fig1/AS:601614754791432@1520447543566/Evidence-based-medicine-pyramid-The-levels-of-evidence-are-appropriately-represented-by.png
YouTube and co.
claymath.msri.org/hodgeconjecture.mov
https://www.youtube.com/watch?app=desktop&v=gIi92JSZ9J4
https://www.youtube.com/watch?app=desktop&v=SGesX6bkTOg
https://www.youtube.com/watch?v=UNNAoCy3yrk
#algebra
#geometry
#mathematics