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ALGEBRA
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Description: This area includes content on all aspects of algebra and its application to related fields. Areas of research include linear and multilinear algebra, associative and non-associative rings and algebras, commutative and non-commutative rings and algebras, universal algebra and lattice theory, semigroups, homological algebra, representation theory, quantum algebra, quantum groups, skein theories, operator algebras, operadic and diagrammatic algebra, quantum field theory, and algebraic varieties.
ANALYSIS
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Description: This area includes content on all aspects of classical analysis, harmonic analysis, special functions, functional analysis, Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory, global analysis, orthogonal polynomials, calculus of variations, approximations, expansions, asymptotic, analysis of PDEs, existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics, complex variables, holomorphic functions, automorphic group actions and forms, pseudoconvexity, analytic spaces, and analytic sheaves.
COMPUTATIONAL & APPLIED MATHEMATICS
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Description: This area includes content on mathematical and computational techniques to solve scientific and engineering problems using methods such as numerical linear algebra, numerical optimization, numerical approximation, computational geometry, inverse problems, iterative methods, eigenvalue problems, nonlinear equations, nonlinear systems, inverse problems, numerical approximation, ordinary differential equations (ODEs), boundary value problems, partial differential equations (PDEs), difference equations, integral equations, optimization, operations research, linear programming, control theory, systems theory, optimal control, game theory, variational techniques, and modeling curves and surfaces.
DIFFERENCE & DIFFERENTIAL EQUATIONS
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Description: This area includes content on qualitative and quantitative theories of ordinary and partial differential equations and the existence and stability of difference equations. Areas of research include existence and uniqueness, boundary value problems on finite or infinite intervals, asymptotic behavior of the solutions, linearization methods, spectral theory, stochastic differential equations, and linear and nonlinear operators.
DISCRETE MATHEMATICS & COMBINATORICS
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Description: This area includes content on algebraic geometry and combinatorics, combinatorial optimization, discrete mathematics, enumeration, graph theory, lattices and lattice theory, and ordered algebraic structures.
DYNAMICAL SYSTEMS
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Description: This area includes content on the behavior of complex dynamical systems and the geometrical properties of trajectories using mathematical principles and tools. This includes chaos theory, logistic map dynamics, bifurcation theory, and self-organization processes.
ENGINEERING MATHEMATICS
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Description: This area includes content on applications of mathematical and computational tools in engineering and industrial research such as aerospace, atmospheric, bioengineering, biomedical and biomathematics, complex systems, chemical, electrical, fluid dynamics and turbulence, geophysical, informatics, mechanical, ocean, production, systems and transportation, fracture mechanics, nano-engineering, and robotics.
FINANCIAL MATHEMATICS
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Description: This area includes content on all areas of computational and theoretical finance. Research areas include mathematical economics, financial economics, securities, markets, stock, and international economics.
GEOMETRY
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Description: This area includes content on all aspects of geometry and its application to related fields. Areas of research include metric geometry, Euclidean, hyperbolic, discrete, convex, coarse geometry, symmetric spaces, complex geometry, differential geometry, Riemannian, pseudo-Riemannian and Finsler geometry, comparisons in Riemannian geometry, relativity, gauge theory, non-commutative geometry, K-theory and homology, manifolds and cell complexes, and analysis of manifolds.
HISTORY & PHILOSOPHY OF MATHEMATICS
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Description: This area includes content on the concept, history, and philosophy of mathematics, communication of mathematics, and ethics in mathematics.
LOGIC
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Description: This area includes content on logic, mathematical logic and foundations, set theory, point-set topology, and formal mathematics.
MATHEMATICAL BIOLOGY
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Description: This area includes content on the intersection of biology and mathematics, utilizing mathematical models and approaches to leverage understanding of complex biological systems. Research areas include mathematical modeling and analysis of brain and neuroscience, cell biology, epidemiology, evolution, immunology, multiscale systems, networks, physiology, population dynamics, systems biology, and computational biology.
MATHEMATICAL PHYSICS
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Description: This area includes content on applications of mathematical methods to investigate and analyze physical and chemical systems. Areas of research include mathematical approaches to problems of quantum and classical mechanics and field theories, as well as relativity and gravitation, statistical physics, and mathematical foundations of physical theories.
MATHEMATICS EDUCATOR: COURSES, CASES & TEACHING
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Description: This area includes content that supports high quality student learning in mathematics and related disciplines, and is designed for the sharing of teaching materials. This includes casebooks, reviews of casebooks, lecture notes, class summaries, outlines, supplementary materials (for your own or someone else's book), syllabi, problem sets, PowerPoint presentations, videos, and other teaching materials.
MATHEMATICS NEGATIVE RESULTS
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Description: This area includes content in which negative results were found. Negative results are instances in which the expected outcome of the research was not borne out by the results. The negative results area includes papers that have controversial or provocative results that challenge established theories as well as papers that replicate or support findings published elsewhere.
NUMBER THEORY
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Description: This area includes content on number theory, analytic number theory, algebraic number theory, prime numbers, diophantine equations, Galois theory, and arithmetic geometry.
NUMERICAL ANALYSIS
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Description: This area includes content on all aspects of numerical analysis and its application to related fields. Areas of research include numerical algorithms for problems in analysis and algebra and scientific computation.
OPTIMIZATION & CONTROL
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Description: This area includes content on optimization, operations research, linear programming, control theory, systems theory, optimal control, and game theory.
PROBABILITY & STATISTICS
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Description: This area includes content on the theory and application of probability and statistics. Areas of research include the foundations of probability and statistics theory, probability theory on topological structures, stochastic geometry, distribution theory, limit theorems, stochastic processes, stochastic analysis, Markov processes, Bayesian problems, parametric and nonparametric inference, multivariate analysis, regression, sequential methods, inference from stochastic processes, survival analysis, and data science.
THEORETICAL COMPUTER SCIENCE
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Description: This area includes content on all aspects of mathematics and computer science. Areas of research include analysis of algorithms, data structures and complexity, coding theory, computational complexity, computational geometry, machine learning theory, mathematical optimization, mathematical software, models of computations, network protocols, operations research, programming logic, parallel and distributed computation, probabilistic computation, quantum computation, automata theory, information theory, cryptography, program semantics and verification, and algorithmic game theory.
TOPOLOGY
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Description: This area includes content on all aspects of topology and its application to related fields. Areas of research include general topology, geometric topology, algebraic topology, topological groups, lie theory, continuum theory, point-set topology, dimension theory, local and global properties, manifolds, orbifolds, polyhedra, cell complexes, foliations, geometric structures, modules, ideals, computational aspects, invariant theory, and homotopy theory.
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