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Geometry

代数几何

Classical Algebraic Geometry

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Videos

• Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity
  • Videos on YouTube
• NPTEL videos on algebraic geometry

Books

• Algebraic Curves: An Introduction to Algebraic Geometry, William Fulton - section 2.10 has exercises about exact sequences
• Rick Miranda. Algebraic Curves and Riemann Surfaces. An introduction and preparation for algebraic geometry coming from complex geometry.
• Learn: affine complex varieties
• Andrei Okounkov Bridging probability, representation theory and algebraic geometry.
• Discriminant of elliptic curve.

Elliptic integrals

• Note in the Wikipedia article on Particular values of the gamma function the connection between {$\Gamma(\frac{n}{24})$} and Complete elliptic integrals of the first kind
• How do prime ideals relate number theory and algebraic geometry?
• Presheaf - restrictions can be thought of as morphisms? associative?
• Daping Weng. A Categorical Introduction to Sheaves I noticed that we can think of presheaves in terms of category theory, or more specifically, in terms of topological spaces with open sets. So I am curious how the topologically defined sheaves might be thought more generally in terms of category theory. I think this paper answers that question.
• https://math.stackexchange.com/questions/3362833/examples-of-presheaves-which-are-not-sheaves
• https://math.stackexchange.com/questions/3291785/on-the-functorial-point-of-view-in-algebraic-geometry Schemes as functors. Moduli problems. Geometry of moduli spaces.
• Michael Atiyah - From Algebraic Geometry to Physics - a Personal Perspective, slides
• Algebraic geometry deals with crosssections (zeros) and considers values in various grids (natural numbers, integers, rationals, reals, complexes).
• Resolution of singularities. For each projective variety X, there is a birational morphism W->X where W is smooth and projective. (This brings to mind universal covering spaces, the unfolding of loops into paths.)
• Sheaf theory can be used to model perspectives. They impose a global context (question) (the type of data that we want such as a ring of continuous functions) upon the local domains (open sets) with regard to which the data (answer) is provided (the question is answered).
• How do we compose sheaves?
• https://en.wikipedia.org/wiki/Sheaf_(mathematics)
• Algebraic geometry studies crosssections (zeroes).

Masaki Kashiwara, Pierre Schapira. Categories and Sheaves. 2006

Sheaf Representation of Monoidal Categories

• Monoidal Categories MonCat. Presheaf F: L->MonCat and Sheafs. Generalizing Stone's Theorem.
• https://julesh.com/2018/08/16/lenses-for-philosophers/ David Jaz Myers