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 ↓


Differential Geometry


Differential geometry is a field of mathematics that uses the tools of calculus and linear algebra to study the geometric properties of curves, surfaces, and other higher-dimensional analogs. This fascinating branch of mathematics has its roots in the study of geometry, which deals with the properties and relationships of points, lines, surfaces, and solids. In differential geometry, the focus is on understanding these concepts at a more abstract level, allowing for the exploration of complex structures that arise in both pure and applied mathematical contexts.

Basic concepts

The primary objects of study in differential geometry are differential manifolds. A manifold is a topological space that locally resembles Euclidean space. For example, when viewed up close a circle looks like a line, and a sphere looks like a plane. Differential manifolds are manifolds that are equipped with a structure that allows us to perform calculus on them.

Curves and tangents

Consider a smooth curve on a two-dimensional plane, such as a circle or an ellipse. A fundamental concept in differential geometry is the tangent vector. Given a point on a curve, the tangent vector is a vector that "touches" the curve at that point and points in the direction following the curve.

Tangent vector = derivative of the curve at the point

For a parameterized curve c(t) = (x(t), y(t)) in R 2, the tangent vector at a point is given by (dx/dt, dy/dt).

ABtangent line

Surface

A surface in differential geometry is a two-dimensional differentiable manifold. Examples include spheres, tori (doughnut shapes), and the surfaces of various solids. An essential aspect of studying surfaces is understanding how they curve in space. Two notable concepts are Gaussian curvature and mean curvature.

Gaussian curvature

Gaussian curvature gives an exact quantitative measure of the curvature of a surface. For any point on a two-dimensional surface, the Gaussian curvature is the product of the maximum and minimum curvature at that point. If you have seen a saddle-shaped surface, you have seen a surface with negative Gaussian curvature.

Gaussian curvature, K = k1 * k2

Here, k1 and k2 are the principal curvatures. For a sphere of radius r, the Gaussian curvature is simply 1/r^2 which is positive.

Positive curvatureNegative curvature

Higher dimensions

While curves and surfaces are relatively easy to visualize, differential geometry also extends to higher dimensions. A three-dimensional manifold is something that locally resembles three-dimensional space; you can think of it like a "3D surface." These higher-dimensional objects, often called "hypersurfaces" when they generalize the concept of surfaces, are important in general relativity and advanced physics.

Applications of differential geometry

Differential geometry has wide applications in many fields, including physics, engineering, and computer science. Here, we highlight some important applications.

General relativity

One of the most profound applications of differential geometry is found in the theory of general relativity. Introduced by Albert Einstein, general relativity is a theory of gravity that describes gravity as a geometric property of space and time, or spacetime. The theory uses Riemannian geometry, an extension of differential geometry, to model the curvature of spacetime and its effect on the motion of objects.

Einstein's Field Equations: G_{μν} = 8πGT_{μν}

Here, G_{μν} represents the Einstein tensor, which describes spacetime curvature, and T_{μν} is the energy–momentum tensor.

Computer graphics

Differential geometry also plays an important role in computer graphics. It helps render curves and surfaces smoothly. Surfaces can be modeled using mathematical functions, and differential geometry provides the tools needed to calculate how light interacts with different surfaces, allowing for realistic rendering and shading.

Engineering

In engineering, the principles of differential geometry are applied in the design of objects with complex shapes, especially in the automotive and aerospace industries. Curved surfaces must be carefully designed and analyzed for aerodynamic properties, emphasizing the importance of curvature and shape.

Conclusion

Differential geometry is a remarkable field of mathematics that bridges the gap between algebraic theories and real-world applications. From the beautiful simplicity of curves to the profound complexities of spacetime, this subject provides essential insights into the geometric structure of the universe. Its ideas span a variety of fields, demonstrating both the beauty and utility of mathematical abstraction.


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Differential Geometry

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