Study of analysis 'in the large' on (possibly infinite-dimensional) smooth manifolds — where local chart-by-chart calculus must be glued consistently and global invariants emerge. Core themes: (i) Morse theory (Morse 1925, Bott 1954,…
global-analysis
Morse theory: critical-point indices, handle-attachment, Morse inequalities
A smooth f : M → R on a compact manifold is Morse iff every critical point (df = 0) is non-degenerate (Hessian invertible). At a…
Banach manifold: charts into Banach spaces + Fréchet-derivative calculus
An infinite-dimensional generalisation of a smooth manifold: charts take values in a fixed Banach space E (rather than R^n), with smooth…
Jet bundle J^k(M,N): truncated Taylor fibre + prolongation functor
Ehresmann's 1951 jet-bundle construction. For M, N smooth manifolds the k-jet j^k f(x) of a smooth map f : M → N at x ∈ M is the…
Paraboloid f=x²−y²: grad(0)=0; Hessian diag(2,−2); det −4; Morse index 1
Sympy-exact Morse-critical-point witness for the canonical saddle f(x, y) = x² − y² on R². Step 1 — gradient: ∇f(x, y) = (2x, −2y),…
Height on S²: perfect Morse, c=(1,0,1), Σ(−1)^i c_i = χ(S²) = 2
Sympy-exact arithmetic witness of the Morse inequalities equality case on S². The height function f(x, y, z) = z restricted to the round…
2-jet of x³: j²(0)=(0,0,0); j²(1)=(1,3,6); j²(2)=(8,12,12)
Sympy-exact verification of the 2-jet prolongation operator for the canonical cubic f(x) = x³. Derivatives: f'(x) = 3x², f''(x) = 6x. …
Morse theory (critical points)
Morse 1934: smooth function f: M → ℝ on compact manifold has critical points c_i with Morse indices λ_i; Euler characteristic χ(M) = Σ…
Floer homology
Floer 1988: infinite-dim Morse theory of action functional on loop-space. Lagrangian intersection (Hamiltonian-isotopy invariant),…
Atiyah-Singer index theorem
Atiyah-Singer 1963 (Abel 2004): index of elliptic-differential-operator D on compact manifold M equals topological-index computed from…
Yang-Mills instantons
Belavin-Polyakov-Schwarz-Tyupkin 1975: anti-self-dual SU(2) connection on ℝ⁴ with topological charge k. ADHM construction parametrises…
Seiberg-Witten invariants
Seiberg-Witten 1994: 4-manifold invariants from monopole-equations replacing instantons. Computationally simpler than Donaldson;…
Gauge theory & connections
Principal G-bundle P → M with connection 1-form A; curvature F = dA + A∧A. Yang-Mills action ∫|F|² extremised by Maxwell (abelian) or…
Morse theory (Morse 1925)
Morse 1925: smooth f: M -> R with non-degenerate critical points -> CW-complex structure on M with one i-cell per index-i critical point;…
Nash-Moser implicit function theorem
Nash 1956 + Moser 1961: implicit-function theorem in tame Frechet spaces with smoothing operators; basis of Nash 1954 isometric embedding +…
Uhlenbeck compactness (1982)
Uhlenbeck 1982: bounded-energy Yang-Mills connections converge weakly + bubble-tree compactification; foundation of gauge-theoretic moduli…
Ricci flow (Hamilton 1982 / Perelman)
Hamilton 1982 dg/dt = -2 Ric; Perelman 2002-2003 entropy-monotonicity + neck-pinch surgery resolves Poincare-conjecture + geometrization…
Seiberg-Witten equations (1994)
Seiberg-Witten 1994: simpler 4-manifold invariants than Donaldson; D_A psi = 0 + F_A^+ = sigma(psi); SW-basic-classes recover + improve…
Atiyah-Singer applications
Atiyah-Singer 1963 (Abel 2004): ind(D) = integral_M ch(sigma_D) td(TM); recovers Riemann-Roch / Hirzebruch / Gauss-Bonnet / spin-Dirac;…
Morse theory (1925)
M Morse 1925 critical-points-of-functionals; modern Morse-Smale + Floer-homology + symplectic-topology.
Smale (1961)
S Smale 1961 (Fields 1966) generalized-Poincaré + h-cobordism; modern surgery-theory + classification high-dim-manifolds.
Atiyah-Singer index (1963)
M Atiyah-I Singer 1963 (Atiyah Fields 1966) index-theorem; modern foundation of K-theory + topological-invariants in physics.
Ricci flow (Perelman 2003)
G Perelman 2003 Ricci-flow-with-surgery proof of Poincaré-conjecture + geometrization (Fields 2006 declined).
Thurston (1976)
W Thurston 1976 (Fields 1982) 8-geometries 3-manifold; modern Mostow-rigidity + hyperbolic + Klein lemma + WP-foliation.
Yamabe problem (1960)
H Yamabe 1960 + Schoen 1984 Yamabe-problem; modern conformal-geometry + scalar-curvature + Schoen-Yau positive-mass.