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Algebraic Topology
The branch of mathematics that uses tools from abstract algebra to study topological spaces. Covers homotopy groups, homology, cohomology, and fiber bundles.
Pure Mathematics
Explore the foundational structures of modern mathematics. From abstract algebra and topology to mathematical logic and number theory, dive into rigorously verified research, historical developments, and contemporary breakthroughs.
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Functional Analysis
Studies vector spaces endowed with limit-related structure and linear operators acting upon them. Foundations for quantum mechanics and PDE theory.
Analytic Number Theory
Uses mathematical analysis as a tool to solve questions about the integers. Features the Prime Number Theorem, Riemann Zeta function, and sieve methods.
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Model Theory
The study of the relationship between formal theories and their models. Explores satisfiability, completeness, categoricity, and definability.
Differential Geometry
Applies techniques of calculus and linear algebra to problems involving curves, surfaces, and manifolds. Central to general relativity and gauge theory.
Category Theory
An abstract framework for mathematics that formalizes structure and relationships. Studies objects, morphisms, functors, and natural transformations.
Ergodic Theory
Studies dynamical systems with an invariant measure and related problems. Bridges probability, analysis, and statistical mechanics.
Set Theory & Foundations
Investigates the nature of sets, ordinals, cardinals, and axiomatic systems like ZFC. Addresses independence results, large cardinals, and forcing.
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Homological Algebra
Studies homology in a general algebraic setting. Covers chain complexes, exact sequences, derived functors, and cohomology theories.