An affine variety can be embedded into proj space in many ways. Do they all give the same Betti tables? Or are the Betti tables somehow related? So much I've found starts w projective varieties / graded rings. Are there good keywords to search for to understand 👆?

#AlgebraicGeometry #algebra

Anyone know a good source for multivariate Hensel lifting w/ more equations than variables?

I wrote up some brief notes for myself on it a while back, and I'd post them, but I'd feel silly doing so if there's a good source out there already.

#Algebra #AlgebraicGeometry #NumberTheory

Is there a term for when a class of equations has sol'ns over an extension field iff it has sol'ns over the ground field?

Examples:
1. Linear eqns
2. Eqns saying that two fin. dim. rep'ns of a fin. generated algebra are equivalent

Non-ex: eqns saying two 3-tensors are isomorphic

#Algebra #AlgebraicGeometry

Question for #AlgebraicGeometry #AlgebraicTopology folk:

Suppose I have a Zariski-constructible set S defined over the reals R (or even over the integers Z). Is there a nice relationship between the reduced singular (co)homology of its real points and its complex points?

Why do we call it #AlgebraicGeometry ? There is no metric !

Except sometime there is a metric, and its called, guess what:
*Metric* Algebraic Geometry.

They've taken us for fools!

Algebraic geometry friends! If I have a projective variety given as the vanishing set of a bunch of explicit homogeneous quadratics, and I want to determine the dimension of the biggest linear subspace it contains, is there a standard way to go about that? #math #AlgebraicGeometry

I've just posted a set of lecture notes and slides about the monster/Semple tower construction. They are from a 4-lecture series I delivered with Susan Colley in 2017.
We surveyed the construction and use of the monster/Semple tower in three distinct areas of mathematics:

1. Compactifying curvilinear data (#AlgebraicGeometry)

2. Studying Goursat distributions (#DifferentialGeometry)

3. Analyzing a truck with trailers (#Mechanics and #ControlTheory)

https://u.osu.edu/kennedy.28/2022/11/24/lecture-notes-aspects-of-the-monster-tower-construction/

#Stacks #AlgebraicGeometry #mathematics https://www.quantamagazine.org/wei-ho-is-drawn-to-algebra-geometry-and-the-human-side-of-math-20221122/

just found out u can make lists on here. need to figure how out how to use this feature effectively so I can have my #Sapphic list, an #AlgebraicGeometry list, #BirdPhotography list, #Bunny list, etc

#Mastodon #Lists

While preparing for #juliacon :julia: 2023, #throwback to last year and one of the #projects I presented.

GeometricTheoremProver.jl: simple package for automated theorem proving in Euclidean geometry, powered by #julialang amazing metaprogramming and Ritt-Wu method from #algebraicgeometry.

Still a lot of room for improvement and development, hopefully over time...

https://github.com/lucaferranti/GeometricTheoremProver.jl

Dang, this #AlgebraicGeometry text has a very poetic title, "The Rising Sea", which refers to an oneiric metaphor of Grothendieck about how a nut of knowledge is slowly cracked by the slowness of an incoming ocean tide.

It also cites Morpheus from The Matrix in its preface, haha.

Now I really want to read this. I like poetical mathematics. Kinda like Ada Lovelace.

http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf