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


Introduction

Algebraic geometry is a branch of mathematics that combines abstract algebra, particularly commutative algebra, with geometry. It is a discipline where algebra meets geometry and where spaces whose defining equations are given by polynomials are studied in depth.

This field is responsible for many developments in both algebra and geometry, and has influence in a variety of fields, including number theory and string theory. Algebraic geometry provides a powerful set of tools for understanding the intrinsic properties of solutions of polynomial equations.

Basic concepts

To understand algebraic geometry in depth we need to understand some basic concepts:

Affine varieties

The affine variety is the set of solutions to a system of polynomial equations in a finite number of variables. These solutions can be viewed as geometric shapes. For example, consider the set of equations:

x^2 + y^2 = 1

The above equation describes a circle in the affine plane. More generally, if we have variables f_1, f_2, ..., f_m x_1, x_2, ..., x_n f_m, then the set:

V(i) = { (x_1, x_2, ..., x_n) | f_1(x_1, x_2, ..., x_n) = 0, ..., f_m(x_1, x_2, ..., x_n) = 0 }

is an affine variety.

Algebraic sets

An algebraic set is similar to an affine variety but is defined more generally. It is any subset of an affine space that is the solution set of a system of polynomial equations. The set of all algebraic sets forms a topology called the Zariski topology.

In the example shown, a circle is depicted, which represents the algebraic variety given by the equation x^2 + y^2 = 1.

Zariski topology

The basis of the Zariski topology on algebraic sets is formed by the complements of algebraic sets. In some ways, the Zariski topology can be seen as "coarse" or "finer" than the standard topology, depending on the context.

Projectile varieties

These are types of variety that refine our understanding beyond the affine variety. They deal with the issue of points 'at infinity'. Similar to the affine variety, the projective variety is defined as the zero set of homogeneous polynomials.

Consider the projective plane P^2. A point in P^2 is given by coordinates (x : y : z) that are not all zero, where two sets of coordinates are considered equal if they differ by a non-zero scalar multiplier.

Example of a projective variety

x^2 + y^2 - z^2 = 0

Here is an equation that represents a projective variety, which is often represented in geometry as a conic section, such as an ellipse or hyperbola.

Algebraic curves

Algebraic curves are of the one-dimensional variety. They form an important area of study in algebraic geometry. Perhaps the simplest example of an algebraic curve is a line, given by a linear polynomial in the affine plane.

A more complex example is the elliptic curve:

y^2 = x^3 + ax + b

This equation describes an algebraic curve in two variables, where special properties of the coefficients a and b determine the shape and nature of the curve.

The above path represents an elliptic curve, which can be characterized by closed loops.

Intersection theory

This aspect of algebraic geometry deals with the intersection of two or more varieties in a meaningful way, often providing information about dimensions and shared properties.

Sheaf principles and schemes

Sheaf theory advances the idea of actions on space and plays an important role in modern algebraic geometry. Schemes are an advancement of variations, allowing a seamless union of algebraic and topological methods.

Basic sheaf

Consider the collection of open sets of a topological space; a sheaf provides consistent data on all open sets, such as functions satisfying certain properties.

Singular points and solutions

When dealing with variations, singular points are where the normal rules of geometry do not apply, such as cusps or crossing points on curves. Resolving these singularities is a fundamental task in algebraic geometry.

Example of a singular point

y^2 = x^2(x + 1)

This curve presents a cusp at the origin (0,0), which represents a typical singular point scenario.

Conclusion

In short, algebraic geometry combines algebra and geometry to investigate the geometric properties of solutions to polynomial equations. It takes advantage of many sophisticated tools and concepts, such as varieties, sheaves, and schemes, to solve problems within mathematics.

It is a field rich in theory and application, making significant contributions to subjects ranging from theoretical physics to cryptography. Given its history and development, algebraic geometry remains a vibrant and essential area of study in mathematics.


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Algebraic Geometry

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