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Algebraic Topology
- Algebraic Topology finds applications in sensor network design, coverage analysis for sensor networks, and in expanding data analysis techniques to give better visualizations for large data sets.
- It has also been applied to computer vision, pattern recognition algorithms and topological data analysis.Algebraic Topology can be used in robotics. Motion planning and behavioral algorithms for robotics have been studied with topological tools.
- Knot theory is used when dealing with protein folding and other analysis of DNA function. There are enzymes called 'topoisomerases' that change the knottedness of loops of DNA. In fact, when bacteria (which have circular 'chromosomes' called plasmids) reproduce, they make use of an enzyme whose specific role to to unlink Hopf links! There are antibiotics that target this enzyme.
- Model categories have been used in the study of concurrency. See this paper by Gaucher.
- Nash's proof (Ann. of Math, Vol. 54, No.2; 1951) that every finite non-cooperative game has an equilibrium point in mixed strategies is a direct application of Brouwer's fixed point theorem, and spurred a great deal of interest in applications of game theory to economics (cf. this survey article). Game theory itself has applications in computer science and mathematical finance.
Group Theory
- Group theory provides methods for understanding the Rubik's cube, and for generating algorithms for solving the cube remarkably quickly from any state the cube may be in.
- Groups find various applications in chemistry, eg. in the study of crystal structures and spectroscopy.
- Cryptography - various hard algorithmic problems about groups are used to design crypto-systems.
- Groups of symmetries are used to reduce the dimension of parameter spaces in engineering models to make model verification more tractable.
- Potentially fast matrix multiplication;
- Card tricks that don't work by sleight of hand, but via the arrangements of the cards. e.g. Sim Sala Bim. If you think about it, the symmetric group explains the trick and shows you how you to extend it past three piles of seven cards, but to N piles of M cards.
Differential geometry
- Lie groups are used in robotics (to find the most efficient way to maneuver a robotic arm, for instance).
- Spherical trigonometry is essential for navigation (a few centuries ago, this was THE application of mathematics to the real world - naval empires were built upon this!)
- Finsler geometry can be used in planning shipping routes when ocean currents and winds (as well as the earth's curvature) need to be taken into account to conserve fuel.
- Differential geometry (Riemann metrics + stress tensors) is used in mechanical engineering to study the properties of large membranes, for example, how one should go about building a large tent. Keyword for literature search "elastic membrane", "continuum mechanics".
- Without taking into account the effects of general relativity on the orbiting satellites that make up the GPS system, the locations reported by GPS receivers would accumulate errors of around 10km each day, rendering the system useless.
- Nonlinear control theory makes heavy use of differential geometry
- Quantum theory of atoms in molecules is an application of Morse theory in quantum chemistry.
Analysis of PDE
- Partial differential equations are used a lot for modelling systems in biology and medicine, and help describe e.g. animal coat pattern formation (zebras, leopards...), wound healing, tumor growth, spread of a virus in a population, predator-prey systems in ecology, predicting the variations of concentrations of chemicals (hormones, drugs...) within an organ over time...
- PDEs are used in climate modelling, from atmospheric dynamics to ocean currents.
- Radar imaging is based on solving an inverse problem. The recent buzz about metamaterials and invisibility is based on understanding variable-coefficient elliptic problems.
Representation Theory
Much of modern particle physics is related to representations of Lie algebras. For instance, Gell-Mann's "Eightfold Way" comes from the representation theory of SU(3) and its associated algebra.
Almost every application of theoretic physics in solid state physics extensively uses representation theory in description of periodic and quasi-periodic media as crystals, semiconductors etc. In fact solutions of Schroedinger Equation in such cases, numerical or analytical has to be carried in accordance with representation of crystal/quasi cristal symmetry group.
There are applications of representation theory to three-dimensional Cryo-Electron Microscopy - there is a recent paper of Hadani and Singer about this in the Annals.
The study of the orbits of the permanent and the determinant (thought of as points in the space of polynomials) is the central idea in Valiant's algebraic version of P vs NP, and the representation theory of the relevant coordinate rings of the orbit closures is a leading approach by Mulmuley and Sohoni.
Algebraic geometry
- Elliptic curve cryptoghy
- Motion planning: Configuration spaces of robot arms are semi-algebraic sets, and algebraic geometry (especially the Cylindrical Algebraic Decomposition) has been used to understand their geometry and design algorithms.
- Algebraic geometry over finite fields is used to construct error correcting codes.
- Statistical models are often semi-algebraic sets, and algebraic geometry can be used to devise tests for the correctness of the model or to fit parameters.
- Birational geometry can be used in the design of NURBS and CAD tools.
- Projective geometry and implicitization is used in 3D image reconstruction from multiple camera views.
- Geometry and Representation theory of tensors can be used in Physics, Computer Science, Statistics, Phylogenetics, Psycometrics.
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math.CV Complex Variables
- Conformal mapping simplifies various problems about heat conduction and fluid flow, such as calculating steady temperatures. Modelling the flow of fluids around solid bodies (for example aircraft wings!) can also be simplified by appropriate conformal mappings.
- Tools from complex analysis (phasors, argument principle, conformal mapping) are widely used in analyzing electrical circuits, and stress and strain analysis in mechanical engineering.
- The uniformization theorem for discrete complex analysis (circle packings) allows for efficient and pleasant visualization of geometric data that can be related to surface triangulations.
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math.CT Category Theory
- Category Theory helps design modern and novel programming languages, that end up being able to do optimizations based on mathematical theorems, and even allow provably correct code with less effort than using other techniques. For example, the language Haskell makes use of many ideas from category theory.
- There are deep connections between category theory and logic, in the sense of computer science.
- Feynman diagrams form a (monoidal) category.
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Statistics
- Accurate polling
- Fraud detection: whether financial (e.g. via Benford's law) or voter fraud.
- Used in the world of finance, economics and gambling on a daily basis.
- Predicting consumer preferences (e.g. Netflix prize)
- Experimental design and hypothesis testing (e.g. testing of medical hypotheses)
- Machine learning
- Quality control
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Rings and Algebras
- Google's Pagerank algorithm is based, in part, on the singular value decomposition.
- Fourier analysis / transforms and linear algebra is at work in the world millions of times per second (video, audio). In particular, creating or displaying a JPEG image requires the discrete fourier transform.
- Quaternions are used in 3D modeling and animation software to represent rotations in a more robust form than Euler angles (helping to avoid transition issues like gimbal lock).
- Every simulation using Finite Element Method uses algebra in very extensive way.
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Classical Analysis
- Fourier analysis allows one to precisely divide up the electromagnetic spectrum, leading of course to radio, television, wireless, and so forth.
- The fast Fourier transform (and relatives, such as the fast Wavelet transform) is an essential component of many signal processing algorithms.
- MRI is based on inverting the Radon transform.
- Wavelets are used in signal processing (e.g. image compression)
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Number Theory
- From Number Theory comes the ideas and theoretical basis for modern cryptography, used to secure communications everywhere from banking to cellphones.
- A more quirky one is SETI (the primes in binary would be a very clear indication of a signal non-natural origin, and would be a starting point for communication).
- There is a gamma ray telescope design using mod p quadratic residues to construct a mask. Gamma rays cannot be focused, so this design uses a redundant array of detectors separated from the mask to reconstruct directional information.
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Probability
- Stochastic processes such as branching processes and HMMs are used to model speciation and extinction (the Tree of Life), evolution of molecular sequences, cell proliferation, and other things in biology. For example, see `Branching Processes in Biology' by Kimmel and Axelrod.
- Here's a somewhat frivolous one (but one that casinos greatly care about): the number of times one needs to shuffle a deck before it truly randomizes.
- used in the world of finance, economics and gambling on a daily basis.
- Random number generation is a key component of many efficient algorithms, and also plays an important role in cryptography.
- Stochastic calculus is used to price options (Black-Scholes formula) and to hedge against risk. (Of course, it is not always applied wisely...)
- Markov chains are used to find uniformly random objects. This, among other things, makes designing an experiment fairest and crypto-systems based on designs securest.
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IT Information theory
- Compression; efficient use of bandwidth
- Error-correcting codes protect against digital data corruption from noise, packet loss, physical damage, etc.
- Used in machine learning
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Numerical analysis
Linear programming algorithms are used in compressed sensing, which is now being used in MRI and imaging to increase resolution and/or decrease the number of measurements required.
Numerical analysis is what makes calculators work. (And so much more!)
We use numerical linear algebra to approximate solutions to discretized versions of complicated PDEs.
At the heart of Google's Pagerank algorithm is a relatively simple numerical eigenvector computation called the Power Method. The study of large complicated networks (e.g. Facebook) is done using tools from graph theory which again comes back to using the tools of numerical linear algebra.
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Functional Analysis
- Used in signal processing for modeling and design.
- Used in machine learning in the design of classifiers (e.g. spam filters)
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History and overview:
understanding mathematics as a social and human endeavor
considering alternative approaches that historically have been used to explore quantifiable relationships
to teach us that no mathematical ideas appear in empty social space: every idea is a child of its time. It is application of HO to mathematics itself
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Dynamical systems
- Modeling flow of liquids, like the animated flow of lava in the movie "Volcano"
- Solving classical few-body problems, though it doesn't help with the modern notorious "two-body problem"
- Heteroclinic trajectories are used in space mission design
- Heteroclinic tangles are used by chemical engineers to get well-mixed reactants
- Pseudo-Anosov braids are used to design efficient methods for stirring viscous liquids
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Spectral theory
- Spectroscopy (of course)
- To avoid bridge collapse, knowing where the resonant frequencies are is extremely important. :-)
- Shape and model recognition
- Network analysis and security (spectral graph theory)
- The design of quantum wave guides
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Symplectic Geometry
Symplectic integrators are used for numerical simulation of Hamiltonian mechanics. Prominent applications include molecular dynamics, solar system dynamics, computer animation, and a wide variety of problems in mechanical engineering.
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Operator Algebras
- (In "Nature") Operator algebras, and, more broadly, operator theory, appear in mathematical models for quantum phenomena.
- (In Engineering) Completely positive maps are used in quantum information theory. There are also many connections between operator algebras and wavelets, which is useful in electrical engineering.
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Logic
- Lambda calculus, the theoretical basis for functional programming (Lisp in particular), was developed by Alonzo Church in the 1930's as part of his research on recursion and the foundations of mathematics.
- Formal verification is used to verify software and hardware in which failure rates need to be as close to zero as possible (e.g. avionics)
- Finite Model Theory is used to design and improve database query systems.
- logical reasoning is widely and sometime non-trivial used in law application in courts, and in should be in parliaments during setting a law systems
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