Divisors and Linear Systems
Cartier and Weil divisors, the sheaf O_X(D), linear systems, and ampleness
Recent posts
Posts sorted by last_modified_at (most recently updated first).
Classifying Spaces
Classification of principal G-bundles and construction of the classifying space BG
The Divergence Theorem and Stokes' Theorem
Divergence theorem, Stokes’ theorem, irrotational and conservative fields, unification of integral theorems
Green's Theorem
Green’s theorem, area formulas, circulation and flux forms, simply connected domains and conservative fields
Surface Integrals and Flux
Parametric surfaces, normal vectors, surface area, scalar surface integrals, flux
Line Integrals
Scalar and vector line integrals, work, fundamental theorem, and conservative fields
Vector Fields
Vector fields, gradient and conservative fields, divergence and curl, differential identities
Curves and Vector-Valued Functions
Vector-valued functions, parametric curves, velocity and tangent, arc length, curvature and acceleration decomposition
Multiple Integrals
Multiple integrals, Fubini’s theorem, change of variables and the Jacobian determinant
Functions of Several Variables and Partial Derivatives
Partial derivatives, gradient, differentiability, multivariable chain rule, extrema
Minimal Polynomial
Cayley-Hamilton theorem and minimal polynomial
Quadratic Forms and Sylvester's Law of Inertia
Classification of real symmetric bilinear forms
Singular Value Decomposition
Orthogonal decomposition of arbitrary real matrices
The Fundamental Theorem of Calculus
Fundamental theorem, existence of primitives, Leibniz rule, termwise integration of power series
Improper Integrals
Infinite and singular integrals, comparison test, absolute convergence
Integration
Antiderivatives, Riemann sums, definite integrals, properties, and the mean value theorem
Taylor's Theorem
Taylor polynomials, Lagrange remainder, Maclaurin series, approximation and limits
Complex Spectral Theorem
unitary diagonalization of normal operators
Spectral Theorem
Orthogonal diagonalization of self-adjoint operators
Mean Value Theorem
Mean value theorem and its applications: monotonicity, extrema, convexity, L’Hospital’s rule, optimization
Complex Inner Product Spaces
Hermitian inner products over complex numbers
Differentiation
Termwise differentiation of power series, derivatives of elementary functions, product, quotient, and chain rules
Power Series
Power series, radius of convergence, elementary function expansions, and analytic functions
Infinite Series
Partial sums and convergence, geometric and p-series, convergence tests, absolute and conditional convergence
Limits of Sequences
Convergence, limit laws, standard limits, e, and monotone convergence
Quasi-coherent Sheaves
Sheaf of O_X-modules, equivalence on affine schemes, and quasi-coherence
Sheaf Cohomology of Schemes
Cohomology of quasi-coherent sheaves, Serre vanishing, the cohomological criterion for ampleness, and Hilbert polynomials
Kähler Differentials and Cotangent Sheaves
Kähler differentials, cotangent sheaf, tangent sheaf, Euler sequence, and canonical sheaf
Division Rings
Division rings, quaternions, Wedderburn’s little theorem, and Schur’s lemma
Central Idempotents and Ring Decomposition
Central idempotents, ring product decomposition, and the Chinese remainder theorem
Invertible Elements and Zero Divisors
Units, regular elements, and when regular elements become units in finite commutative rings
Continuous Functions
Definition of continuity and properties of continuous functions: extreme and intermediate value theorems
Quotient Spaces
Quotient spaces of vector spaces by subspaces
Limits of Functions
Defining limits of functions via ε-δ and proving limit laws and the squeeze theorem
Richardson Varieties and Peterson Varieties
Richardson varieties as transversal intersections of opposite Schubert varieties, and Peterson varieties from regular nilpotents: definitions, dimensions, in...
Bruhat Decomposition
Cell decomposition of homogeneous spaces, parabolic subgroups, and Schubert varieties on Grassmannians
Differentiation and Derivatives
Definition of derivative, differentiability and continuity, derivatives and higher-order derivatives
Givental J-function and Mirror Theorem
Fundamental solution of quantum differential equation and I=J theorem
Dubrovin Connection
A flat connection on a Frobenius manifold with a spectral parameter
Frobenius Manifolds
Frobenius manifolds and the WDVV equation
Mirror Symmetry: An Overview
Historical background and the Hori–Vafa mirror
Fano Varieties
Reflexive polytopes and the corresponding Gorenstein Fano toric varieties
Torus-Invariant Divisors and Line Bundles
Torus-invariant divisors and line bundles arising from the rays of a fan
Definition of Toric Varieties
General toric varieties obtained by gluing affine toric varieties from a fan
Chow Groups
Chow groups and the cycle class map
Intersection Product
The intersection product on Chow groups
Kodaira Vanishing Theorem
The Kodaira vanishing theorem and its applications
From Varieties to Schemes
From varieties to schemes
The Riemann–Roch Theorem for Surfaces
Intersection theory on surfaces and its applications
The Riemann–Roch Theorem for Curves
The Riemann–Roch theorem for curves