| Group Theory |
Finite Groups |
Study of groups with a finite number of elements |
Sylow Theorems, Simple Groups |
Classification of Finite Simple Groups |
| Group Theory |
Algebraic Groups |
Geometric structures that are also groups |
Lie Groups, Linear Algebraic Groups |
Connections with algebraic geometry |
| Group Theory |
Transformation Groups |
Groups of transformations of a mathematical object |
Symmetries, Automorphisms |
Applications in Physics |
| Ring Theory |
Noncommutative Rings |
Rings in which multiplication is not commutative |
Division Rings, Skew Fields |
Applications to quantum mechanics |
| Ring Theory |
Commutative Rings |
Rings in which multiplication is commutative |
Ideals, Prime Ideals |
Research in algebraic geometry |
| Ring Theory |
Noetherian Rings |
Rings satisfying the ascending chain condition on ideals |
Cohen-Macaulay Rings, Artinian Rings |
Applications in commutative algebra |
| Field Theory |
Galois Theory |
Study of the symmetries of roots of polynomials |
Galois Groups, Field Extensions |
Applications to coding theory |
| Field Theory |
Transcendental Extensions |
Field extensions that are not algebraic |
Algebraically Closed Fields, Transcendental Numbers |
Studies in function fields |
| Linear Algebra |
Matrix Theory |
Study of matrices and their properties |
Eigenvalues, Determinants |
Numerical Linear Algebra |
| Linear Algebra |
Representation Theory |
Study of abstract algebraic structures by linear transformations |
Modules, Characters |
Applications in physics and chemistry |
| Linear Algebra |
Algebraic Geometry |
Study of solutions of systems of polynomial equations |
Varieties, Schemes |
Connections with topology |
| Commutative Algebra |
Ideals |
Study of structures in commutative rings |
Prime Ideals, Maximal Ideals |
Use in algebraic geometry |
| Commutative Algebra |
Local Rings |
Rings with a unique maximal ideal |
Discrete Valuation Rings, Completion |
Applications in number theory |
| Algebraic Geometry |
Affine Varieties |
Geometric objects defined as the solution set of polynomials |
Coordinate Rings, Nullstellensatz |
Connections to computational algebra |
| Algebraic Geometry |
Projective Varieties |
Geometric objects defined in projective space |
Homogeneous Coordinates |
Applications in computer graphics |
| Algebraic Geometry |
Algebraic Surfaces |
Two-dimensional algebraic varieties |
Singularities, Rational Points |
Research on moduli spaces |
| Homological Algebra |
Chain Complexes |
Sequence of abelian groups or modules connected by homomorphisms |
Exact Sequences, Homology |
Applications in topology |
| Homological Algebra |
Derived Categories |
Categories that arise from homological algebra concepts |
Functoriality, Triangulated Categories |
Research in geometry |
| Category Theory |
Higher Categories |
Categories where morphisms can also have morphisms |
2-Categories, Infinity Categories |
Connections to homotopy theory |
| Category Theory |
Functors |
Mappings between categories |
Covariant Functors, Contravariant Functors |
Applications in various fields |
| Category Theory |
Limits and Colimits |
Generalized notions of subobjects and quotients |
Pullbacks, Pushouts |
Applications in logic |
| Number Theory |
Algebraic Number Theory |
Study of algebraic structures related to integers |
Number Fields, Ideals |
Research on Diophantine equations |
| Number Theory |
Analytic Number Theory |
Application of analytic methods to number theory |
Dirichlet Series, L-functions |
Research on prime number distribution |
| Number Theory |
Transcendental Number Theory |
Study of numbers that are not roots of any non-zero polynomial |
Transcendental Numbers, Baker's Theorem |
Connections to algebraic geometry |
| Combinatorial Algebra |
Design Theory |
Study of combinatorial designs |
Block Designs, Turán's Theorem |
Applications in coding theory |
| Combinatorial Algebra |
Graph Theory |
Study of graphs as mathematical structures |
Planar Graphs, Trees |
Applications in computer science |
| Polynomial Algebra |
Polynomial Rings |
Rings formed from polynomials over a given coefficient ring |
Roots, Factorization |
Applications in computer algebra systems |
| Polynomial Algebra |
Algebraic Functions |
Functions defined by polynomial equations |
Rational Functions, Algebraic Curves |
Connections with algebraic geometry |
| Universal Algebra |
Algebraic Structures |
Studies the common features of all algebraic structures |
Groups, Rings, Fields |
Research on varieties |
| Universal Algebra |
Algebraic Equations |
Equations involving algebraic structures |
Algebraic Laws, Identities |
Applications in logic |
| Computational Algebra |
Algorithmic Algebra |
Algorithms for solving algebraic problems |
Polynomial Factorization, Groebner Bases |
Computational complexity aspects |
| Computational Algebra |
Symbolic Computation |
Using symbols to compute algebraic expressions |
Computer Algebra Systems, Symbolic Integration |
Research in automated reasoning |
| Representation Theory |
Lie Algebras |
Algebras associated with Lie groups |
Root Systems, Weight Systems |
Applications in physics |
| Representation Theory |
Module Theory |
Study of modules over rings |
Simple Modules, Projective Modules |
Application in both algebra and geometry |
| Algebraic Topology |
Homotopy Theory |
Study of topological spaces up to continuous deformation |
Homotopy Groups, CW-complexes |
Research in algebraic invariants |
| Algebraic Topology |
Homology Theory |
Investigation of topological spaces through algebraic constructs |
Singular Homology, Čech Homology |
Connections to manifold theory |
| Algebraic Topology |
Cohomology Theory |
Dual to homology, providing algebraic invariants |
Cohomology Rings, Poincaré Duality |
Research in topology and geometry |
| Applied Algebra |
Cryptography |
Use of algebra in secure communication |
Elliptic Curves, Finite Fields |
Development of new cryptographic protocols |
| Applied Algebra |
Coding Theory |
Study of error-correcting codes using algebraic techniques |
Linear Codes, Algebraic Geometry Codes |
Applications in data transmission |
| Applied Algebra |
Game Theory |
Mathematical study of strategies in games using algebraic structures |
Nash Equilibria, Cooperative Games |
Connections with economics |
| Algebraic Logic |
Algebraic Structures in Logic |
Study of logics using algebraic techniques |
Lattices, Boolean Algebras |
Connections with computer science |
| Algebraic Logic |
Modal Logic |
Study of necessity and possibility through algebra |
Kripke Semantics, Frame Construction |
Applications in philosophy |
| Noncommutative Geometry |
Noncommutative Spaces |
Studying spaces where coordinates do not commute |
C*-Algebras, Quantum Geometry |
Connections to physics |
| Noncommutative Geometry |
Quantum Groups |
Algebraic structures that generalize groups in quantum mechanics |
Hopf Algebras, Quantum Symmetries |
Applications in theoretical physics |
| Noncommutative Algebra |
Braided Categories |
Categories where the morphisms are braided" |
Braided Monoidal Categories |
Applications in topological quantum field theory |
| Algebraic Dynamics |
Dynamical Systems |
Study of algebraic actions over fields |
Automorphisms, Iterate Maps |
Applications in number theory |
| Algebraic Dynamics |
Iteration Theory |
Study of sequences defined by iterating functions |
Fixed Points, Bifurcations |
Research in complex dynamics |
| Arithmetic Geometry |
Arithmetic Schemes |
Geometric techniques applied to arithmetic problems |
Rational Points, Diophantine Geometry |
Applications in number theory |
| Arithmetic Geometry |
Motives |
Abstract objects relating algebraic cycles to cohomology theories |
Category of Motives, Chow Groups |
Research in algebraic cycles |
| Gröbner Bases |
Algorithm for Ideals |
Computational method for solving systems of polynomial equations |
Reduced Gröbner Basis, Ideal Membership |
Applications in computer algebra |
| Gröbner Bases |
Applications in Algebraic Geometry |
Using Gröbner bases in geometric contexts |
Projective Varieties, Affine Varieties |
Simplifying polynomial system study |
| Categorical Logic |
Categorical Foundations |
Logic as a branch of category theory |
Topoi, Sheaf Theory |
Connections to set theory |
| Higher Dimensional Algebra |
Enriched Categories |
Categories enriched over a specific category |
Monoidal Enriched Categories |
Research in higher algebra |
| Universal Algebra |
Varieties of Algebras |
Classes of algebraic structures defined by identities |
Equational Class, Algebraic Theories |
Applications in model theory |
| Algebraic Geometry |
Tropical Geometry |
Study of piecewise-linear structures arising from algebraic geometry |
Tropical Varieties, Toric Varieties |
Emerging connections to combinatorics |
| Quantum Algebra |
Quantum Groups and Rings |
Algebraic structures that generalize classical notions in quantum physics |
Kac-Moody Algebras, Quantum Symmetries |
Applications in theoretical physics |
| Linear Algebra |
Numerical Linear Algebra |
Study of algorithms for both direct and indirect linear algebra methods |
Matrix Factorization, Eigenvalue Problems |
Connections to machine learning |
| Algebraic Statistics |
Algebraic Methods in Statistics |
Statistical techniques derived from algebraic structures |
Algebraic Models, Bayesian Networks |
Emerging field with connections to computation |
| Coherent Sheaves |
Sheaf Theory |
Categorical language for local-global principles around functions |
Coherent Sheaves, Support |
Applications in algebraic geometry |
| Birational Geometry |
Birational Maps |
Study of rational maps between algebraic varieties |
Birational Equivalence, Minimal Models |
Contemporary research on surfaces |
| Homotopical Algebra |
Homotopical Methods in Algebra |
Study of algebraic structures using homotopy theory |
Model Categories, Homotopy Limits |
Research in topology |
| Excitation Theory |
Mathematical Finance |
Application of algebraic structures to financial models |
Risk Models, Stochastic Processes |
Emerging research area |
| Combinatorial Algebra |
Polytopes |
Study of combinatorial properties of polytopes |
Face Lattice, Polyhedral Combinatorics |
Connections to optimization |
| Arithmetic Algebraic Geometry |
L-functions |
Study of complex analytic functions tied to number theory |
Elliptic Curves, Modular Forms |
Emerging connections with algebraic topology |
| Geometric Representation Theory |
Geometry and Representation Theory |
Study of representations of groups using geometric methods |
Character Theory, Geometry of Representations |
Applications in modern physics |
| Boolean Algebra |
Applications of Algebra in Logic |
Study of algebraic structures to model logical deductions |
Lattice Theory, Boolean Algebras |
Connections to computer science and Boolean functions |
| Modular Representation Theory |
Group Representations Modulo Prime |
Study of representations of groups over fields with characteristic p |
Modular Characters, Block Theory |
Applications in finite group theory |
| Functional Analysis |
Algebras of Operators |
Study of algebraic aspects of functional spaces |
Banach Algebras, C*-Algebras |
Applications in quantum mechanics |
| Algebraic Combinatorics |
Combinatorial Structures |
Study of algebraic structures arising from combinatorial problems |
Posets, Polynomials |
Emerging applications |
| Computational Combinatorics |
Algorithmic Combinatorial Techniques |
Algorithms applied to combinatorial optimizations |
Combinatorial Enumeration, Graph Algorithms |
Developments in optimization |
| formal Group Theory |
Study of Groups Defined by Power Series |
Groups where operations are defined by formal series |
Formal Groups, Cauchy Products |
Emerging research area |
| Topological Groups |
Groups with Topological Structure |
Group theory where the group is equipped with a topology |
Locally Compact Groups, Lie Groups |
Application in analysis |
| Stability Theory |
Algebraic Stability |
Understand algebraic equations' stability under perturbations |
Lyapunov Functions, Stability Conditions |
Relations to differential equations |
| Free Algebras |
Algebras with no relations |
Study of algebraic structures that lack constraints |
Free Groups, Free Modules |
Connection to combinatorial algebra |
| Classical Algebra |
Traditional Prominence in Modern Concepts |
Historical algebraic constructs and their modern applications |
Algebraic Identities, Historical Analysis |
Continuing impact on Algebra |
| Matrix Representations |
Matrices as Linear Maps |
Study of matrices portraying linear transformations between spaces |
Matrix Decomposition, Spectral Analysis |
Applications in computer graphics |
| Group Actions |
Algebras of Group Actions on Sets |
Understanding sets acted upon by group structures |
Orbits, Fixed Points |
Connections to symmetry |
| Robust Data Analysis |
Algebraic Structures in Computation |
Using algebraic structures for robust data management and analysis |
Data Polymorphisms, Algebraic Topology |
Emerging computational techniques |
| Geostatistics |
Algebraic Methods in Geostatistics |
Study of spatial phenomena through algebraic modeling |
Kriging, Variograms |
Connections to environmental sciences |
| Polynomial Ideals |
Study of Ideals Generated by Polynomials |
Applications of ideals within polynomial rings |
Radical Ideals, Primary Decomposition |
Methodologies in algebraic geometry |
| Perturbation Theory |
Understanding Algebraic Systems under Variations |
Study of solutions under perturbations of parameters |
Stability, Sensitivity |
Applications in physics |
| Symplectic Algebra |
Algebra of Symplectic Groups |
Understanding structures in mechanics through algebra |
Symplectic Groups, Canonical Forms |
Connections to classical mechanics |
| Geometric Group Theory |
Study of Groups via Geometric Principles |
Study how groups can be understood through geometrical constructs |
Cayley Graphs, Hyperbolic Groups |
Research on low-dimensional topology |
| Order Theory |
Study of Algebraic Structures Under Orderings |
Connections with lattice structures and their orderings |
Ordered Sets, Lattices |
Applications in theoretical computer science |
| Infinity-Categorical Algebra |
Study of Higher-Dimensional Algebraic Structures |
Understanding complex algebraic frameworks under higher structures |
(∞,1)-Categories, Simplicial Categories |
Emerging research area |
| Action Logic |
Algebraic Study of Logic and Actions |
Logic framework based on algebraic principles of actions |
Dynamic Systems, Temporal Logic |
Connections with computer sciences |
| Abstract Algebra |
Group and Ring Structures |
Defining arithmetic properties through algebraic structures |
Modules, Algebras |
Emerging research methodologies |
| Function Algebra |
Study of Algebras Associated with Functions |
Connections between various function systems and algebraic properties |
Algebraic Function Fields, Rational Functions |
Emerging applications in computational systems |
| Commutative Geometry |
Exploring Geometry through Commutative Algebra |
Algebra's principles applied in geometric frameworks |
Affine Varieties, Projective Varieties |
Expansion in modern algebraic applications |
| Representation of Algebras |
Understanding Complex Structures via Representation |
Connecting algebraic entities via representation theories |
Artin-Wedderburn Theorem, Representation Moduli |
Applications in modern theoretical physics |
| Abstract Homology |
Homology Studies Outside Classical Contexts |
Abstracting homological studies into more general settings |
Mayer-Vietoris Sequence, Spectral Sequences |
Increasing importance in modern algebra |