PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Geometry → Algebraic Geometry ↓


Affine and Projective Varieties


In algebraic geometry, one of the fundamental concepts is varieties, which are geometric expressions of algebraic equations. Affine and projective varieties serve as building blocks in understanding more complex spaces. Let's dive deeper into these concepts, starting with affine varieties and then exploring projective varieties.

Affine varieties

Affine variety is defined as the set of solutions of a system of polynomial equations. These solutions live in an affine space, denoted as (mathbb{A}^n), where n is a positive integer indicating the dimension of the space.

Consider a simple example in a two-dimensional affine space, (mathbb{A}^2) Suppose we have the polynomial equation:

f(x, y) = x^2 + y^2 - 1 = 0

The set of all points (x, y) that make the equation true is an affine variety. In this case, it represents a circle centered at the origin with radius 1.

Visual example: circle

In terms of algebraic geometry, it is important to note that affine varieties are more complicated than their visual counterparts. They are the solution sets of finite sets of polynomial equations. The more equations you have, the more limited the points on your variety will be.

If you introduce a second polynomial equation, for example:

g(x, y) = x + y - 0.5 = 0

Then, to find the intersection, you solve f(x, y) = 0 and g(x, y) = 0 simultaneously, resulting in a new variety by their intersection.

Visual example: intersection of a circle and line

In this example the affine variety represents the two points where the line intersects the circle.

Projectile varieties

Projective varieties take the concept of affine varieties to a more holistic level by considering solutions in a projective space, denoted by (mathbb{P}^n) Projective spaces take into account points at infinity, which affine spaces ignore.

To transition from the affine to the projective variety, the polynomial equations are rewritten in homogeneous coordinates. In homogeneous coordinates, a point ((x, y, z)) in projective space satisfies the equivalence relation:

(x, y, z) sim (kx, ky, kz) quad text{for all} quad k neq 0

This makes it possible to identify points lying on the same line passing through the origin.

Consider the polynomials from the affine example. In projective space, the circle x^2 + y^2 - 1 = 0 is extended to:

X^2 + Y^2 = Z^2

When Z = 0, the equation X^2 + Y^2 = 0 defines the points on the circle at infinity.

Visual example: circles in projective space

Z=0 plane

The advanced circle now considers more than just the finite portion appearing in (mathbb{A}^2); it also includes aspects that extend to infinity, providing a global perspective on the geometry.

Properties of affine and projective varieties

Affine and projective varieties have intrinsic properties that make them useful in the study of algebraic geometry:

  • Irreducibility: A variety is irreducible if it cannot be expressed as the union of two proper sub-varieties. Each irreducible component represents a different component of the solution set.
  • Dimension: The dimension of a variety is an abstract concept, equal to the largest number of parameters needed to describe a point on the variety.
  • Degree: The degree of a variety represents the number of intersections of a normal line with that variety.

To gain a deeper understanding of affine and projective varieties, it is important to understand the nature of polynomials. These geometric units help algebraic geometers tackle fundamental problems in the field, such as determining possible shapes and the ways these shapes interact with each other.

Applications and further exploration

The study of affine and projective varieties has applications not only in theoretical mathematics but also in various fields such as robotics (for solving inverse kinematics), computer vision (for understanding image recognition) and physics (for concepts related to string theory).

Mathematicians explore these spaces using computational software to look at complex variations beyond the hand-drawn plane, and to push the boundaries of computation with symbolic systems that handle polynomial solutions.

This understanding of affine and projective varieties as foundational concepts inspired further explorations into the realm of algebraic geometry, ultimately leading to a better understanding of the geometric structures that govern mathematical theories and practical applications.


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