Higher Math: Geometry and Topology

Riemannian and esp. hyperbolic geometry in Sarnak course... e.g. Laplacian of hyperbolic spaces used early.

Terry Tao 285G, Lecture 0: Riemannian manifolds and curvature... want to be less scared of this.

Tao: What is a gauge?

MO: one-parameter subgroup and geodesics on Lie group

Tao: geodesic flow and Euler-Arnold equation

Maximum number of Killing vectors (google it... mentioned in Dafermos course Fall 2017)

MO: sheaves and bundles in differential geometry

References from Mohan... John Pardon: algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. McDuff Salamon. Gromov Chris Wendl (nonsqueezing theorem notes)?

Special values of L-functions and hyperbolic manifolds. Zagier paper (Google it)?

Why are all knots trivial in 4D? (MSE?)

Jacobi identity intuitive explanation: MSE... intuitively, differentiate $g_th_txh_t^{-1}g_t^{-1}$ in two different ways?

Wick rotation in physics (Minkowski vs Euclidean space)... Wayne.

Sheaves of differentials

Nice to have different perspectives. See even Neukirch for "arithmetic Riemann-Roch theory" (including a notion of genus of number field, and so on).

Want to merge classical and modern perspectives on calculus, tangent/cotangent spaces, etc. See Liu and Vakil for examples, intuition, and pictures. At the beginning, Vakil's "Key fact 21.2.3." on generators and relations of module of differentials (see November 18, 2017 version) is quite nice, with a tight connection to the affine conormal exact sequence:

21.2.D. EXERCISE. Verify Key fact 21.2.3. (If you wish, use the affine conormal exact sequence, Theorem 21.2.12, to verify it; different people prefer to work through the theory in different orders. Just take care not to make circular arguments.)

Qing Liu Chapter 6, Proposition 1.8, relating differentials on towers of fiber spaces (i.e. "affine cotangent sequence" and "affine conormal sequence"). It's interesting that while differentials are (at least in some special cases of interest) easy to compute directly and explicitly by hand (at least intuitively, though checking the universal property may be boring), the proof goes more geometrically through derivations (Yoneda-esque dual perspective, using universal property of the module of relative differential forms). Reflect more deeply on this?

[EDIT after first reading: See also Vakil, "relative cotangent sequence, affine version" and "conormal exact sequence, affine version" for other discussion.]

Could perhaps be proved directly with more yoga, but the derivations sequence is much more transparent than the originals. I suppose it's not too honest to call derivations "purely geometric", but at least elements of the derivation sets are more or less tangent vectors, and the restriction and inducing of derivations can be understood geometrically in that way.

(Technical note: see the algebraic fact that let us conclude exactness from dual exactness: MSE: Dual sequence and its exactness; MSE: Verifying statement of exact sequences; MSE: Hom functors and exactness; MSE: Confusion about a proof: exactness of modules.)

Frobenius' theorem on vector fields (more precisely, on integrable subbundles of the tangent bundle)

A good high-level perspective I like is at MSE: Wanted: A purely algebraic proof of the Frobenius theorem on distributions.

Together with MSE: Any set of linearly independent and commuting vector fields CAN be realized (locally) as partial derivatives of a local coordinate, this gives a complete proof.