Glossary

A

algebraic geometry

Traditionally, the geometry of solutions in the complex numbers to polynomial equations. Modern algebraic geometry is also concerned with algebraic varieties, which are a generalization of such solution sets, as well as solutions in fields other than complex numbers, for example finite fields.

algebraic topology

The branch of topology concerned with homology and other algebraic models of topological spaces.

algebraic variety

A space which is locally the solution locus to a set of polynomial equations. Algebraic varieties are for algebraic geometry{algebraic geometers} what topological spaces are for topology{topologists}. Indeed, many algebraic varieties are (complex) manifolds. However, algebraic varieties may also have complicated singular sets and may be parametrized with rings other than the complex numbers. (For the technical reason that the real numbers are not algebraically closed, one does not consider algebraic varieties over the real numbers in the straightforward sense.)

alternating-sign matrix

A matrix of 0's, 1's, and -1's such that, if the zeroes are deleted from any row or column, the remaining entries alternate in sign and begin and end with 1.

almost complex manifold

A manifold with the property that each tangent space has the structure of a complex vector space, but the complex structures are not necessarily compatible with true complex coordinates as they are for a complex manifold.

analysis

One of the three traditional branches of mathematics, along with algebra and geometry. It is the branch concerned with estimates, inequalities, differential and integral calculus, and properties of the real numbers. It includes such areas as real analysis, functional analysis, operator theory, measure theory, differential equations, and special functions.

analytic

In analysis, a function or a structure described by functions is analytic its Taylor series converges to it. Since the function must first have a Taylor series, it is in particular smooth if it is analytic.

applied mathematics

Mathematics for the sake of its use to science or society.

asymptotics

A general term in mathematics referring to the properties of an object as key parameters such as the dimension, the non-linearities, or the length scale, become very large or very small.

Attouch-Wets topology

A certain topology on functions or compact sets on an infinite-dimensional vector space which is useful in analysis and optimization problems. A sequence converges in this topology if it converges (uniformly, or in the Hausdorff metric) in each finite ball centered at the origin.

B

bacterial flagellar motor

The motor in a cell or bacterium that turns a flagellum, a rigid corkscrew tail that the organism uses to swim.

band-connected sum

A knot formed from two other knots by connecting them along two parallel segments called a band. Although the original two knots cannot be entangled with each other (they must be separated by a sphere), the band can meander among them in a complicated way. A band-connected sum in which the band is in the simplest possible position is called a connected sum of knots.

basis

In mathematics, usually means basis in the sense of linear algebra; a minimal set of vectors that spans a vector space.

Brownian motion

The most common type of continuous random motion of a particle, one in which the particle's vibrations have more and more energy at short length and time scales. It models the motion of a particle in a fluid, fluctuation of stock prices, and many other processes.

C

calculus of variations

Calculus problems, especially differentiation and maximization, involving functions on a set of functions of a real variable. For example, finding the shape of a cable suspended from both ends.

capillary wave

A small wave in a body of water whose behavior is governed by surface tension rather than gravity.

category theory

The study of abstracted collections of mathematical objects, such as the category of sets or the category of vector spaces, together with abstracted operations sending one object to another, such as the collection of functions from one set to another or linear transformations from one vector space to another.

causality

Given an event X in a physical system or a corresponding feature in a partial differential equation, a trichotomy which divides other events into those that may have caused or modified X, those which X can cause or alter, and those which are causally independent from X.

Cayley numbers/octonions

A non-associative generalization of the quaternions and the complex numbers involving numbers with one real coefficient and seven imaginary coefficients.

cellular automaton

A mathematical model consisting of a grid of cells, a notion of neighboring cells, and a list of states for the cells. The state of each cell at a given iteration of time is a function of its state and that of its neighbors at the previous iteration.

characteristic class

A kind of homological model for a decoration or property of a manifold or other topological space. The simplest characteristic class describes how a manifold fails to be orientable, that is, in which directions a being can travel in the manifold and reverse its handedness.

Chern-Simons form

A differential 3-form computed from a connection or gauge field on a manifold. Since it is a 3-form, it is in effect a function that can be naturally integrated over a 3-manifold, and it plays an important role in 3-dimensional topology and quantum field theory.

circle packing

An arrangement of round disks in the euclidean-space{Euclidean} or hyperbolic plane or on the round sphere such that no two disks overlap with non-zero area. Depending on the context, the circles may or may not be the same size. A theorem of Koebe, revived by Thurston, states that given any planar graph, there is a circle packing with a circle for each vertex of the graph and kissing circles for each edge.

classical

In physics and mathematical physics, the term classical sometimes has the narrow meaning of non-quantum; equations of motion interpreted by means of ordinary dynamical systems rather than statistical quantum rules.

classification

The goal in a branch of mathematics of providing an exhaustive list of some type of mathematical object with no repetitions. Example: The classification of 3-manifolds is one of the outstanding problems in topology. With the advent of computers, one weak but precise way to state a classification problem is to ask whether there is an algorithm to determine whether two given objects are equivalent.

codimension

In general, if a mathematical object sits inside or is associated to another object of dimension n, then it is said to have codimension k if it has dimension n-k.

combinatorial geometry

The visual study of discrete and finite structures, and the study of discrete and finite possibilities for the arrangement or features of geometric objects.

complement

A word with a relatively specific meaning in mathematics. The complement of a subset X in a set Y is Y-X, the set of all things in Y but not X.

complex analysis

The study of functions, especially analytic functions, of one or several complex variables, and related questions concerning Riemann surfaces or complex manifolds.

complex manifold

A manifold with complex coordinates; its ordinary or real dimension is then twice its complex dimension.

compressible fluid

An actual or mathematically abstracted fluid that can change its volume.

computational complexity

The absolute minimum amount of time (or sometimes space) that a computer must take to perform a particular computational task. For example, sorting n numbers has time complexity proportional to n(log n).

conformal

Angle-preserving or angle-defining. The Mercator map is a conformal map of the Earth because angles are true. A conformal structure on a manifold defines angles between curves segments on the manifold but not their lengths.

connected sum

1. A manifold formed from two others by removing balls and gluing along the resulting spherical boundary.

2. The analogous operation for knots; a band-connected sum in which the band connects the knots in the simplest possible way (by piercing the separating sphere only once).

conservation law

A Partial differential equation that expresses the fact that some physical quantity is locally conserved in a fluid or other continuous physical system, such as energy, momentum, or the quantity of fluid itself. Typically the behavior of such a system is completely determined by its conservation laws.

constant relative width

Two convex bodies have constant relative width if the width of one in any given direction is the length of a chord of the other in the same direction.

constant width

A convex body has constant width if any two parallel hyperplanes that touch the convex body on opposite sides are the same distance apart.

control

In mathematics, a control is a time-dependent function u(t) that influences a dynamical system dy/dt = F(u,y), with the understanding that u should be chosen to minimize some value of the solution y(t) or otherwise optimize some related quantity.

control theory

The mathematics of programming robots and other machines to respond to changing conditions so that they maintain self-control. For example, the problem of programming an automatic pilot of an airplane so that the plane doesn't crash. See also

convex

A region in the plane, in Euclidean space, or in some other geometry with lines such as hyperbolic space, is convex if it always contains the line segment connecting two points if it contains the two points themselves. A convex body is, technically, a closed and bounded convex set with non-zero volume.

convex geometry

The study of convex shapes, usually in Euclidean space.

correlation

1. The relationship between any two random variables which may or may not be independent. It may be expressed in terms of conditional probabilities or the mutual probability distribution of the random variables.

2. The quadratic term in the relationship between two real-valued random variables; the expectation of the product minus the product of the expectations, suitably normalized.

cyclically symmetric, self-complementary plane partition/CSSCPP

A plane partition in a cube which is symmetric under sending the x axis to the y axis, y to z, and z to x, and at the same time is equal to its complement; the set of cubes in the box not in the plane partition considered again as a plane partition by inverting all three axes. Equivalently, a lozenge tiling of a regular hexagon invariant under 60 degree rotation.

curvature

In Riemannian geometry, usually means the intrinsic curvature of a manifold with a Riemannian metric. The curvature at a point is positive (negative) if the sum of the angles of a small approximate triangle at that point is greater than (less than) 180 degrees.

cusp

A trumpet-shaped salient of a hyperbolic structure or Riemannian metric which is typically infinitely long. The neighborhood of an ideal vertex is a kind of cusp.

D

decision problem

The problem of producing a computational algorithm that answers some yes/no question in a finite amount of time. For example, the question of whether two given knots are equivalent, or the question of whether an algebraic equation has a solution in the rational numbers.

degenerate diffusion

A diffusion process that only allows diffusion in prespecified directions instead of in every direction.

Dehn filling

Another view of Dehn surgery due to Thurston and motivated by hyperbolic geometry. If the complement of a knot in a 3-manifold has a family of hyperbolic structures, then typically many of them can be completed or filled in to realize different Dehn surgeries along the knot.

Dehn surgery

A modification of a 3-manifold in which a solid torus around a knot is removed and replaced by a solid torus in a different position.

diffeomorphism

A bijection between two manifolds that preserves all smooth structure.

differential geometry

The general study of smooth manifolds decorated by continuous structures such as foliations, Riemannian metrics, and symplectic structures. Riemannian geometry is a disproportionate part of differential geometry.

diffusion equation

A partial differential equation that models the statistics or distribution of many particles undergoing Brownian motion, or the diffusion of one fluid in another fluid, or the diffusion of heat.

distribution/probability distribution

The defining description of a random variable; the set of all possible values of a random variable together with the probability of attaining each value or the probability that the value lands in any given range.

Donaldson theory/Seiberg-Witten theory

A theory invented by Donaldson and revolutionized by Seiberg and Witten that derives many topological properties of smooth 4-manifolds using gauge theory.

Dugundji extension theorem

A generalization of the Tietze extension theorem to functions taking values in a Banach space. The gist of the result is that if X is closed in a metrizable Y, then the extension of a real valued function f on X to Y can be linear and continuous as a function of f.

dynamical system

A flow on a space or an iterative function from a space to itself. By Platonist convention and because of scientific applications, a dynamical system is said to move the points of a space around over time, and dynamical systems are studied to determine trajectories under this motion.

E

earthquake

A discontinuous, surjective map from the hyperbolic plane to itself which moves different pieces isometricaly, and the pieces are separated by hyperbolic lines that make a lamination. A homeomorphism of the circle at infinity of the hyperbolic plane to itself, if it is sufficiently nice in a natural sense, extends uniquely to an earthquake.

ensemble

In physics, a probability distribution on the states of a complicated system being studied. For example, the thermal probability distribution of a physical object is called the canonical ensemble.

enumeration

The goal in combinatorics of finding a simple formula for the number of some type of mathematical object. For example, there are n! permutations of n distinct letters.

enumerative combinatorics

The branch of mathematics concerned with finite enumeration or counting problems.

epi-convergence

A notion of convergence of a sequence of real-valued functions useful in minimization problems and the calculus of variations. More technically, a sequence of functions F_1, F_2, ... epi-converges to F at a point x if some subsequence of some F_1(x_1), F_2(x_2), ... converges to F(x), but no subseqence of any F_1(x_1), F_2(x_2), ... converges to a number smaller than F(x), where in both cases x_1, x_2,... converges to x.

epi/hypo-convergence

A more complicated version of epi-convergence suitable for sequences of saddle functions.

essential

In low-dimensional topology, a non-trivial kind of curve or surface in a manifold that only sometimes exists. An essential sphere in a 3-manifold, for example, is a sphere that does not bound a ball.

Euclidean space

A finite-dimensional vector space, such as ordinary 3-dimensional space, together with a metric that satisfies the Pythagorean theorem.

Euler characteristic

An integer associated to a manifold or other topological space which is particularly easy to compute. Example: The Euler characteristic of a surface is given by the number of faces minus edges plus vertices.

Euler relation

An identity that may be satisfied by a function f on convex polytopes in Euclidean space. The positive Euler relation says that which f(C) is the sum of its values for all faces of C of all dimensions; the negative relation says that it is the alternating sum.

exactly solvable

A statistical mechanical model is called exactly solvable if its main basic parameters, such as its entropy, have simple formulas or can be quickly calculated. It is does not mean that the model is completely understood.

excitation

In quantum physics, an object such as an atom or an elementary particle is exicted if it has more than its minimum possible energy.

exponential growth

A group, geometric space, or other mathematical object is said to have exponential growth if its points or elements have some kind of measure of complexity such as length and the number of elements with complexity n exponentiates as n grows. For example, the set of words pronouncable by an English speaker has exponential growth in the length of the word.

F

fiber bundle

A topological space that divides into locally parallel fibers that may twist globally. The set of fibers is itself a topological space called the base space. For example, a Möbius strip is a fiber bundle over a circle with line segments as fibers.

Fields medal

By convention, the most prestigious award for research in mathematics. It is awarded every four years to between two and four mathematicians under the age of 40.

fixed-point free

A flow or map is fixed-point free if it does not send any point to itself; any such point would be a fixed point.

flow

A vector field in Euclidean space or on a manifold which gives the velocity of a particle moving through the manifold as a function of its position. More technically, an autonomous ordinary differential equation.

foliation

A decoration of a manifold in which the manifold is partitioned into sheets of some lower dimension, and the sheets are locally parallel. (More technically, the foliated manifold is locally homeomorphic to a vector space decorated by cosets of a subspace).

Fredholm determinant

A complex analytic function which generalizes the characteristic polynomial of a matrix. It is defined for those operators which have continuous kernels, i.e., kernels in the sense of analysis.

functional analysis

The study of differentiation, integration, estimates, and asymptotics of functions of real numbers. The modern research version of calculus.

fundamental group

A group, often non-abelian, that rather fully describes the periodicity or the 1-dimensional holes in a topological space.

G

G_2

The group of symmetries of the Cayley numbers. It is a compact, 14-dimensional Lie group which is not naturally part of an infinite sequence of Lie groups, hence it is an exceptional Lie group.

gauge field/connection

A force field in nature, or an analogous vector field in mathematics with an enomormous amount of symmetry that expresses the redundancy or ambiguity of many parameters. The simplest example is the electromagnetic field; its inherent symmetry implies that in electrical circuits, relative voltages are measureable but absolute voltages are an arbitrary convention. The most interesting gauge fields have non-abelian symmetry groups. Also called a connection in differential geometry.

Gelfand-Fuchs cohomology

An ad hoc modification of certain homology{cohomology} groups in various infinite-dimensional settings that is more relevant and easier to compute than ordinary cohomology.

general topology

The study of abstract, formal, and foundational properties of topological spaces. General topology is related to logic, functional analysis, and dynamical systems.

general relativity

Einstein's model of gravity as the curvature of space and time. General relativity implies that spacetime is a 4-manifold.

genus

1. In geometric topology, the number of holes of a surface. Usually this means the maximum number of disjoint circles that can be drawn on the surface such that the complement is connected. If there are no such circles, the surface is planar, and the genus then sometimes means the maximum number of disjoint arcs with the same property.

2. A measure of the twistedness of a fiber bundle. Technically, if X is a topological space with a fiber bundle F, it is the minimum number of open sets that cover X such that F is trivial over each open set.

3. Some other number which generalizes or is analogous to the topological genus, such as the arithmetic genus of an algebraic curve.

geodesic

On a Riemannian manifold or some other metric space, a curve which is the shortest path between any two points on it that are sufficiently close together.

geometric function theory

The study of the geometry in the complex plane of complex analytic functions, in particular the relation between the image of the unit disk of an analytic function and its power series.

geometrical optics/WKB approximation

An approximation of a wave equation which assumes that the physical environment of a wave varies slowly compared to its wavelength. It predicts that the wave follows the trajectory of a refracting ray, such as a light ray.

Geometrization Conjecture

The conjecture of Thurston that, after cutting along essential spheres and tori, every compact 3-manifold admits a special Riemannian metric known as a geometry, usually hyperbolic geometry. The conjecture subsumes the Poincaré conjecture and many other standard conjectures about 3-manifolds, and constitutes a classification of 3-manifolds.

Geometrization Theorem

The Geometrization Conjecture for Haken manifolds, proved by Thurston. (The technical heart of the theorem, the lemma that the skinning map has a unique fixed point, was independently proved by McMullen.)

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Glimm's method

A method of proving existence of solutions to certain partial differential equations by discretely approximating space by a lattice and then taking the limit as the lattice spacing goes to zero.

global analysis

The study of partial differential equations (and other structures from analysis) on manifolds.

Grand Unified Theory

A theory in fundamental physics that unifies the three forces of elementary particles: Electromagnetic forces, weak interactions, and strong interactions. The first are already unified in the electroweak theory. The electroweak and strong forces in ununified form, together with a short list of particles that interact via these forces, is called the Standard Model and may be a precursor to a Grand Unified Theory.

graph

In research mathematics, usually means set of points with another set of edges connecting them.

graph theory

The study of graphs, either for their own sake, or as models of such diverse things as groups (in pure mathematics) or computer networks.

Greenberg-Hastings model

A cellular automaton in which cells may be diseased or healthy or in one of several states in between. A fully diseased cell infects neighboring cells, and a cell which is not completely healthy becomes one unit healthier at each turn. This automaton tends to form whorls and spirals given suitable initial conditions.

Gromov norm

An invariant associated with the homology of a topological space that measures how many simplices are needed to represent a given homology class.

H

Haefliger structure

A relatively complicated structure on an n-manifold which consists of a k-plane field plus auxiliary information derived from the diffeomorphism group of (n-k)-dimensional Euclidean space. A foliation has a parallel Haefliger structure just as it has a parallel plane field, but a Haefliger structure may have a singular locus where it is not parallel to a foliation.

Haken

A noted mathematician with the privilege of having an adjective named after him. A 3-manifold is Haken if it is closed, orientable, and has no essential sphere, but has an incompressible surface.

Hamiltonian

An expression that represents the equations of motion of a classical physical system via Hamilton's equations. The equations are related to the Hamiltonian via phase space or symplectic geometry. The solutions conserve the Hamiltonian if it is time-independent, and the value of the Hamiltonian can be interpreted as the energy of the system.

Heegaard splitting

A division of a 3-manifold into two handlebodies. An early result in topology states that every closed 3-manifold (closed meaning that the manifold is finite and connected but has no boundary) has a Heegaard splitting and a resulting description in terms of a Heegaard diagram, which describes how the two handlebodies are glued together. The surface lying between the two handlebodies of the splitting is a Heegaard surface.

homeomorphism

A bijection between two topological spaces that preserves all continuous structure; the basic notion of equivalence in topology.

homological algebra

The algebraic study of the homology and cohomology of manifolds and other mathematical objects. Homological algebra is a grand generalization of linear algebra.

homology/cohomology

Homology and cohomology are algebraic objects associated to a manifold or other mathematical object which give one measure of the number of holes of the object. The homology of a topological space has a relatively technical definition, but it is relatively easy to compute and study with tools from linear algebra.

Hopf algebra

An abstract algebraic object, generalizing a group or a Lie algebra, with enough structure to have a representation theory.

horocycle/horosphere

A generalized (round) circle or sphere in the hyperbolic plane or space. A horosphere has infinite radius and meets the sphere at infinity at one point, and its geometric center is the same point.

hyperbolic geometry

In modern terms, hyperbolic geometry is the study of manifolds with Riemannian metrics with constant negative curvature. The hyperbolic plane is a particular hyperbolic manifold which is in a sense universal among hyperbolic surfaces; similarly there is also hyperbolic n-space. Escher's circle limit prints are excellent illustrations of the hyperbolic plane.

hyperbolic PDE

A partial differential equation that resembles a wave equation with one time coordinate, one or several space coordinates, and a finite speed of propogation for features of solutions.

hyperplane/hypersurface

A high-dimensional plane or submanifold in a vector space or manifold with codimension 1.

I

ideal vertex

A vertex at infinity of a hyperbolic polyhedron. Geometrically, the faces of the polyhedron taper together at an exponential rate as they extend towards the ideal vertex.

incompressible

1. In physics, refers to a fluid that cannot change volume or a flow that is possible for such a fluid.

2. In 3-dimensional topology, refers to a surface in a 3-manifold with the property that no essential circle in the surface bounds a disk in the manifold.

incompressible fluid

A fluid that does not change its volume except under extreme conditions, or a mathematically abstracted fluid that strictly conserves its volume.

infinite dimensions

In mathematics, the concept of an infinite-dimensional space considered literally. It is a vector space with an infinite basis or a space with infinitely many coordinates.

initial condition

The state of a system at its initial time of consideration. Equivalently, a prespecified value or set of values of a function satisfying a differential equation at an initial time.

instanton

A solution to a partial differential equation, especially a non-abelian gauge field equation, which is localized in all directions.

integrable system

An ordinary or partial differential equation or system of equations with a maximal number of conserved quantities such as energy and momentum. In a sense, an integrable system can be completely solved.

integrate/integrable

An infinitesimal structure of any of various kinds can often be integrated, meaning pieced together into a geometric whole. This process is usually abstracted from the prime example: integrating a function in calculus. Typically the infinitesimal structure could have pervasive inconsistencies that would prevent it from being integrable.

integro-differential equation

An equation relating a function to both its derivatives and antiderivatives.

interacting particle system

A dynamical system consisting of a large number of particles which collide or otherwise interact when they are close to each other. For example, a microscopic model of a molecular liquid or gas.

invariant

In topology, a number, polynomial, or other quantity associated to a topological object such as a knot or 3-manifold which depends only on the underlying object and not on its specific description or presentation.

Ising model

The first and simplest interesting model in statistical mechanics. It consists of a grid of (abstracted) atoms, each of which can be in either of two states, spin up or spin down. The energy of a given state of the grid is given by the number of atoms which are spin up or down and by the number of pairs of neighboring atoms whose spins agree or disagree.

isospectral deformation

A perturbation (continuous change) of the entries of a matrix, or the coefficients of a differential operator or other linear operator, that does not change its eigenvalues.

J

Jacobian variety

A space associated to a Riemann surface defined most succinctly as the complex cohomology (as a vector space) divided by the integral cohomology (as a lattice). It is simultaneously a complex manifold, an algebraic variety, and a Lie group.

Jones polynomial

A famous invariant of knots and links discovered by Vaughan Jones. It has several extremely elementary definitions and at the same time involves deep mathematics.

K

kernel

1. In algebra, the set of vectors annihilated (sent to zero) by a matrix, linear operator, homomorphism, or any similar function.

2. In analysis, a continuous analogue of a matrix. Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold.

Klein-Gordon equation

The partial differential equation Nabla u + u = 0 in Euclidean space or Minkowski space or a variant. The Klein-Gordon equation is the second-simplest PDE which is invariant under isometries of space; it arises in classical and quantum field theory in physics.

knot/link

A link is a collection of disjoint circles lying in a 3-manifold, often but not always Euclidean 3-space or the 3-sphere. A knot is link which happens to be a single circle. Manifold topologists usually study tame knots and links, which can be represented by smooth or polygonal curves, but there are also wild links which are infinitely knotted. Ordinary knots and links were topologically classified (in a certain sense) by Haken and Thurston.

Korteweg-de Vries equation/KdV equation

The partial differential equation u_t + uu_x + u_xxx = 0 which is important both in applied mathematics, because of the physical phenomena it models, and in pure mathematics, because of the structure of its solutions.

H

hierarchy

An infinite system of equations whose truncations (meaning the first n equations for some n) are meaningful and have similar properties. Usually hierarchies refer to systems of partial differential equations generalizing a single equation of interest. For example, the KP hierarchy generalizes the KP equation.

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K

KP equation

The partial differential equation u_yy - (u_t - u_xxx - uu_x)_x = 0, an important integrable system in mathematical physics.

L

L-shaped method

A method for two-stage linear stochastic programming problems in which one solves for all optimal recourses in bunches and uses the solutions to form a linear programming problem for the initial choice. It is based on the Benders decomposition method for mixed linear programming problems.

Lagrangian

An expression that represents the equations of motion of a classical physical system via the Euler-Lagrange equations. The solutions to the Euler-Lagrange equations and the dynamics of the system correspond to minima, maxima, and other critical points of the Lagrangian.

lamination

A decoration of a manifold in which some subset is partitioned into sheets of some lower dimension, and the sheets are locally parallel. It may or may not be possible to fill the gaps in a lamination to make a foliation.

large deviations/theory of large deviations

A collection of methods for estimating and proving results about astronomically unlikely events in probability theory, usually large deviations from likely outcomes dictated by the law of large numbers and the central limit theorem.

lattice

A periodic arrangement of points such as the vertices of a tiling of space by cubes or the positions of atoms in a crystal. More technically, a discrete abelian subgroup of an n-dimensional vector space which not contained in an n-1-dimensional vector space. Lattices play a central role in the theory of Lie groups, in number theory, in error-correcting codes, and many other areas of mathematics.

lattice path

A sequence of points in a lattice such that each point differs from its predecessor by a finite list of allowed steps. Random lattice paths are an interesting model for the random motion of a particle and lattice paths are also important in enumerative combinatorics.

law of large numbers

A basic result in probability theory which dictates that a large number of samples chosen from the same probability distribution have approximately the same proportions as the distribution. The law also estimates the strength of this approximation and is related to the chi-squared test.

least-area surface

A surface (or manifold) lying in space or in a manifold which minimizes area (surface volume) among a class of similar surfaces. Example: The round sphere has least area among surfaces in Euclidean space that enclose a fixed volume.

leopard spot

In the proof of the Geometrization Theorem, when the skinning map is applied to a conformal structure A to produce a new one B, B is made up of many (unrolled) copies of A that resemble leopard spots.

level spacing

In mathematical physics, the difference between consecutive elements in some set of real numbers, in particular the difference between consecutive energy levels or eigenvalues of a matrix or linear operator.

Lie algebra

An algebraic structure on a vector space which describes multiplication of elements of a Lie group which are very close to the identity (infinitesimal transformations). Lie algebras are almost as important as their comrades-at-arms, Lie groups.

Lie group

A group (in the sense of abstract algebra) which is at the same time a manifold. Example: The group of rotations in n dimensions is a Lie group of dimension n(n-1)/2. Lie groups are fundamental objects in mathematics and physics, especially quantum field theory.

life/Conway's game of Life

A popular 2-dimensional cellular automaton in which cells may be present or absent, a cell is born if in the previous iteration three cells (out of the nearest eight neighbors) were adjacent to an empty space, and a cell stays alive if two or three living cells were adjacent.

linear differential operator

An operator which takes functions to functions by taking a linear combination of different derivatives. For example, d/dt + t d^2/dt^2. It is an ordinary differential operator if there is only a single independent variable, i.e., if it corresponds to ordinary differential equations.

linear programming

The problem, and associated area of mathematics, of maximizing or minimizing a linear function on a convex set, especially a polytope. Equivalently, maximizing a linear expression in some number of variables subject to linear equalities and inequalities.

localized solution

A solution of a differential equation, or a similar mathematical object, which is confined to a small region even though it has the freedom to spread out, like a lump in a carpet.

locally convex

For a topological vector space, the property that there exist arbitrarily small open regions around the origin (and consequently any other point) which are convex sets.

logic

The branch of mathematics in which mathematical assertions and reasoning are studied as formal mathematical objects.

long-time solution

A solution to a differential equation that exists, or has been proven to exist, for a long interval of the equation's time parameter, although not necessarily for infinite time.

loop space

Given a topological space X, its loop space is the topological space of all continuous functions from a circle to X. Loop spaces are important examples of new topological spaces formed from old ones, as well as examples of infinite-dimensional spaces in mathematics.

loop and sphere theorems

Two foundational theorems in 3-manifold topology, proved by Papakyriakopoulos in the 1950's. The loop and sphere theorems generalize Dehn's lemma, which asserts that if a knot in three dimensions bounds a disk which intersects itself in its interior, the disk can be simplified to remove all self-intersection and the knot must be trivial.

lozenge

A rhombus with a 60 degree angle.

M

MacDonald identities

A set of simple formulas conjectured by MacDonald in the 1970's which give a signed enumeration of lattice paths in certain lattices related to Lie algebras. It was understood from computer evidence that they were almost certainly true, but a conceptual proof of the most general identities had to await sophisticated ideas in the theory of infinite-dimensional Lie algebras.

Mach reflection

In ordinary reflection of waves, the incoming and reflected waves fronts make a V shape. Mach reflection is a non-linear type of reflection where the two waves merge a distance away from the surface of reflection to make a Y.

magnetic monopole

A particle which has been predicted but not observed from which magnetic field lines radiate, just as electric field lines radiate from a charged particle.

magnon

A quasi-particle in a crystal which manifests a local distortion of the crystal's magnetic field.

Mandelbrot set

A certain subset of the complex plane which is ubiquitous in popular accounts of research in mathematics.

manifold

A fundamental mathematical object which locally resembles a line, a plane, or space (is locally homeomorphic to a vector space). For example, a sphere or a doughnut, the playing field of the video game Asteroids, and (in the theory of general relativity) physical space are all manifolds. The term n-manifold means a manifold which locally resembles n-dimensional space, not one which might lie in n-dimensional space.

mathematical biology

The application of mathematics to problems in biology. Since mathematicians are biological entities, some mathematical biologists are motivated by the desire to write equations about themselves.

mathematical physics

The study of mathematical concepts used in physics, especially statistical mechanics and quantum field theory. Some say that modern mathematical physics is all areas of mathematics other than classical mathematical physics.

matrix model

A kind of quantum field theory in two dimensions involving matrix-valued fields which is related to random triangulations.

Matukuma's equation

The non-linear partial differential equation Nabla u + u^p/(1+v^2) = 0, where u is a function of a vector v in n dimensions. The equation was proposed by Matukuma as a model for a spherical cluster of stars.

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metric

A distance function on some space or set; an assignment of distance to every unordered pair of points that satisfies the triangle inequality.

metrizable

For a topological space, the property that there exists a metric compatible with the topology. To say that a topological space is metrizable is to treat it as a metric space, but without distinguishing any specific or preferred distance function.

min cut, max flow principle

The principle that, given a flow of some substance through a network of pipes with a maximum allowed flow in each pipe, the maximum possible total flow equals the capacity of the most constricted set of pipes which, if cut, would separate the source from the drain.

minimal

A widely used word in mathematics that generally means atomic or unsimplifiable. For example, prime numbers are minimal with respect to factorization of whole numbers.

minimal surface

A surface (or manifold) which locally minimizes surface area (or surface volume), which means that one cannot replace small patches of the surface and decrease the area. See also least-area surface.

Minkowski space

A finite-dimensional vector space, especially a 4-dimensional one, together with an indefinite inner product with one positive or timelike direction and many negative or spacelike directions. In particular, ordinary spacetime in the theory of Special Relativity.

moduli space

Given a topological object, usually low-dimensional, with a continuous family of possible geometric structures, the moduli space is the set of these structures considered itself as a geometric object, usually high-dimensional. Two structures that differ by a topological automorphism of the low-dimensional object are represented by the same point. In particular, the space of conformal structures on a surface.

morphogenesis

The period of an embryo's existence where it determines the shape, organs, and body parts of the organism which it will become.

N

Navier-Stokes equation

The basic equation modelling the motion of a uniform fluid with no viscosity, but possibly varying density.

non-abelian

Non-commutative or order-dependent. For example, the group of manipulations of the Rubik's cube is non-commutative because the state of the cube depends greatly on the order that moves are performed on it.

non-equilibrium statistical mechanics

The study of thermal and statistical properties of systems that are out of equilibrium, e.g., that are changing temperature or moving.

non-linear

Referring to an equation, model, or solution, not amenable to linear algebra or having terms which are not strictly proportional to values of the variables.

nonanticipativity

A solution to a stochastic programming problem, in the form of a sequence of responses to random events, is nonanticipative if each given responses depends only on past events and not on future events.

nucleation

A process in a physical system, or a mathematical model such as a cellular automaton or a statistical model, whereby a bubble or other structure appears spontaneously at a random or unpredictable spot.

number theory

The study of properties of integers, generalizations of integers, and relations between them, especially Diophantine equations (equations in integers) and prime numbers.

O

optimal control

The study of problems that combine optimization with control theory; how to set the behavior machines so that they respond as efficiently as possible to variable or time-dependent conditions.

orbifold

A topological object defined by Thurston which is locally modelled by Euclidean spaces divided by finite groups of symmetries. Orbifolds are manifolds with singularities such as reflection surfaces, where they resemble manifolds with boundary, and cone lines, where they are modelled (in the direction perpendicular to the cone line) by a cone with an angle of 360/n degrees for some n.

ordinary differential equation/ODE

Any equation relating a function of one variable to its derivatives.

P

Painlevé equation/Painlevé function

The Painlevé equations are six families (denoted by Roman numerals) of second-order ODEs whose solutions have relatively simple behavior in the complex plane. (Namely, their branch points as complex functions depend only on the equation and not on the particular solution) The solutions are called Painlevé functions. The simplest such equation (Painlevé I) is y'' = 6y^2 + x.

partial differential equation/PDE

Any equation relating a function of several variables to its partial derivatives in different directions.

periodic trajectory

A trajectory of a flow that repeats, or makes a topological circle.

permanent

A function of a matrix which has the same terms as the determinant, but all terms are added and none are subtracted.

permanent-determinant method

A scheme, found by P. W. Kasteleyn, to change the signs of the entries of a matrix with planar sparseness in such a way that the permanent becomes the determinant.

phase transition

A self-induced reorganization of a uniform physical system or a statistical model of such a system. Crystallization, melting, boiling, and magnetization are examples of phase transitions.

phonon

A quantum of sound, usually observed in a crystal. It is to sound exactly what a photon is to light.

PL flow

A motion on a space or a manifold, akin to a flow given by a vector field, in which every particle in a given simplex of some triangulation moves with constant velocity and in the same direction, so that the particle trajectories are polygons.

planar sparseness

A matrix M has planar sparseness if the graph formed by connecting row i to column j whenever M_ij is non-zero is planar.

plane field

A decoration of a n-manifold that assigns a tangent k-dimensional plane to every point in a continuous fashion. The general notion includes line fields and hyperplane fields; line fields are very similar to vector fields such as the magnetic field.

plane partition

A stack of unit cubes in a rectangular box or in the positive octant in space such that to the left, behind, and below every cube lies either another cube or a wall. A plane partition in a box is equivalent to a lozenge tiling of a hexagon in the plane.

Platonism

The belief that mathematical objects exist independent of physical models. It is a useful pretense in mathematics, especially in geometry.

pleated surface

A surface in Euclidean or hyperbolic space which resembles a polyhedron in the sense that it has flat faces that meet along edges. Unlike a polyhedron, a pleated surface has no corners, but it may have infinitely many edges that form a lamination.

Poincaré conjecture

The conjecture that a closed, simply-connected 3-manifold must be homeomorphic to the 3-sphere. Many mathematicians, including Poincaré himself, have presented incomplete and incorrect proofs of the conjecture.

polyhedron

1. A region in Euclidean space which consists of flat facets with flat edges. More technically, a polyhedron must locally be a cone over a lower-dimensional polyhedron. It is sometimes but not always implicitly assumed that a polyhedron is a manifold, a topological sphere or ball, or a convex set.

2. An abstract space with properties analogous to that of a polyhedron, such as a simplicial complex.

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polytope

The n-dimensional generalization of a polygon or a polyhedron.

probability theory

The mathematical study of random chance and tools from analysis used to model it.

progressive hedging

An algorithm in stochastic programming to find the optimal sequence of responses to random events. For each response, the algorithm progressively stiffens penalties for dependence on future events to enforce nonanticipativity constraints.

projective plane

A 2-manifold obtained by gluing a Möbius strip and a disk along their circle boundaries.

Prym variety

An algebraic variety associated to a conformal map between two Riemann surfaces. Like the Jacobian variety, it is a quotient of a certain complex vector space by a lattice; in particular, if the target Riemann surface is the Riemann sphere, it is the Jacobian variety.

pure mathematics

Mathematics for the sake of its internal beauty or logical strength.

Q

quantum field theory

The study of force fields such as the electromagnetic field in the context of quantum mechanics, and often special relativity also. The mainstay of modern high-energy physics.

quantum G_2 link invariant

A polynomial invariant of knots and links, similar to the Jones polynomial, and associated with the Lie group or Lie algebra G_2, in particular its quantum group.

quantum gravity

The study of general relativity as a quantum field theory; the theory of integration of quantities over the space of Riemannian or Minkowskian metrics on a manifold.

quantum group

A generalization of a group which has an associative multiplication law but does not have group elements in the usual sense. As a space, a quantum group is defined by non-commuting operators which are analogous to coordinates on a Lie group in the same way that non-commuting operators represent measurable quantities in quantum mechanics.

quantum mechanics

A theory of physics that says that all phenomena in the universe follow non-standard probabilistic laws called the quantum rules. The quantum rules radically affect the behavior of any physical system with few available states, such as atoms and elementary particles.

quasi-particle

In quantum physics, any excitation that acts as a particle. For example, a phonon. All physical particles, including photons, electrons, and protons, can be understood as quasi-particles.

R

random matrix

A matrix whose entries are random variables, often statistically independent. One then considers such questions as the likely eigenvalues or expected determinant of such a matrix.

random matrix theory

The study of matrices chosen at random from some specific probability distribution, especially real, complex, or quaternionic (symplectic) matrices whose entries are independent Gaussian random variables.

random triangulation

A triangulation of a surface or other manifold chosen by a random process. In a particularly interesting case, one considers all triangulations with a fixed, large number of triangles to be equally likely.

rational map

A map from some field such as the real or complex numbers (plus infinity) to itself given by a rational polynomial function.

reciprocal process

A stochastic process whose state between any two times A and B depends only on the states at times A and B and not on any events before or after.

recourse problem

A two-stage stochastic optimization problem in which one provides an initial choice, then a random event is revealed, and then one chooses a recourse. For example, an investor may select a stock to purchase, then observe the stock price for a period of time, and then sell part or all of the stock.

representation/linear representation

A realization of a group, Lie group, or Lie algebra by matrices or linear transformations. More technically, a homomorphism from a group to a group of matrices.

representation theory

The study of linear representations of groups, Lie groups and Lie algebras.

Reuleaux polytope

A convex body in the plane or in higher dimensions which, like a Reuleaux triangle, consists of pieces of round spheres, each centered at one of the corners of the convex body.

Reuleaux triangle

A three-cornered curve consisting of three equal arcs of circles, each centered at the opposite corner.

Riemann matrix

In the \ddef{homology}cohomology of a Riemann surface, the integral cohomology has a 2g x 2g generating matrix. After a standard basis manipulation using a natural inner product on the cohomology and its structure as a complex vector space, one obtains a g x g matrix called the Riemann matrix. It determines the Riemann surface and is unique up to transformations coming from homeomorphisms of the surface.

Riemann-Schottky problem

The problem of determining whether a given complex matrix is the Riemann matrix of some Riemann surface.

Riemann sphere

A topological sphere consisting of the complex plane and the point at infinity; an example of a Riemann surface.

Riemann surface/complex curve

A surface with a conformal structure; a complex manifold with one complex dimension.

Riemannian geometry

The study of curvature and other properties of Riemannian metrics on manifolds. Sometimes also called differential geometry.

Riemannian metric

A metric on a manifold which is locally like ordinary distance in Euclidean space. Here ``locally'' is not meant in the usual sense that every point has a region around it that is identical to a region in Euclidean space; rather, a Riemannian metric agrees with Euclidean distance to first order in the sense of calculus.

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rotary engine/Wankel engine

An engine found in some motor vehicles, for example the Mazda RX-7, in which gasoline burns in crescent-shaped chambers, turning a rotating piston that drives an axle through its center. The piston is often a Reuleaux triangle.

round

In topology, the terms circle and sphere refer to topological objects and not geometric ones, so that the surface of an egg shape is a sphere. A round sphere, then, is a sphere with constant curvature; a sphere in the sense of geometry.

S

saddle function

A function F(x,y) of two vectors x and y (which typically lie in different vector spaces) is a saddle function if it is concave up in x and concave down in y.

scalar curvature

Part of the curvature of a Riemannian manifold at a given point p, or the corresponding real-valued function of p. It can be defined as the limiting discrepancy between the volume of a small metric ball at p and the volume in Euclidean space of a ball the same size, times a suitable proportionality constant.

scalar curvature equation

A partial differential equation arising in differential geometry. Given a function f on a manifold with a conformal structure, a solution to the equation produces a Riemannian metric compatible with the conformal sturcture and with scalar curvature f.

scenario analysis

A method in stochastic optimization in which one finds the optimum solution in a sample of cases (scenarios) and then iteratively reconciles the separate solutions to obtain a common solution to the stochastic programming problem.

Schrödinger equation

The partial differential equation i hbar psi_t = H psi, where H is some spatial linear differential operator called a Hamiltonian. It gives the evolution over time of a probability cloud in quantum mechanics.

semi-infinite form

A mathematical object suggested by the Dirac sea of electrons in quantum mechanics. More technically, an algebraic structure, often used as a vector space or a representation, of infinitely many anti-commuting variables divided into two camps, positive and negative. The typical vector is a product of all but finitely many of the negative variables and finitely many of the positive ones.

shallow water

Any of several mathematical models of waves in water that add terms caused by the finite depth of the water.

shock wave

Mathematically, a wave front in a non-linear PDE with a discontinuous change in pressure or some other parameter. Physically, a similar wave with a very abrupt change in pressure. Typically a shock wave can exist because the speed of communication behind it is greater than the speed of communication in front of it.

Siegel upper half-space

The set of n x n matrices for some n with positive-definite imaginary part. It is considered together with a group action taking Z to (AZ+B)/(CZ+D), where A,B,C, and D are certain integral matrices.

simple exclusion process

A statistical model and cellular automaton consisting of a collection of particles on a grid, usually filling a finite fraction of the grid. Each particle has a Poisson timer which triggers it to attempt to move to an adjacent location, and it will do so unless another particle is already there. In the second case, it does not move.

simple recourse

A recourse problem in which the optimal recourse is simply and explicitly determined by the information revealed after the initial choice.

simplex

The n-dimensional generalization of the triangle and the tetrahedron; a polytope in n dimensions with n+1 vertices.

simplicial

Made up of simplices. For example, a simplicial polytope has simplices as faces and a simplicial complex is a collection of simplices pasted together in any reasonable vertex-to-vertex and edge-to-edge arrangement.

singular perturbation

The addition of an extra small term in an equation, especially a differential equation, that qualitatively changes the nature of its solutions.

singular vector

A vector in a Verma module which is annihilated by all raising operators; a vector which generates a Verma submodule.

sine-Gordon equation/sinh-Gordon equation

The partial differential equation Nabla u + sin(u) = 0, or any of several variants, such as the one obtained by replacing sine by hyperbolic sine. All such equations are non-linear variants of the Klein-Gordon equation.

skein theory

An inductive definition of an invariant of knots and links which postulates a linear relation between the invariant of a given link and the invariant of the same link with a crossing switched or otherwise simplified. Sometimes skein theories also involve tangled graphs.

skinning map

An iterative map on conformal structures on a surface that appears in Thurston's proof of the Geometrization Theorem. In the proof, the hyperbolic structure of a cut-open manifold must be varied until it fits with itself when the manifold is glued back together. Such a hyperbolic structure is determined by a conformal structure at the wounds of the cut (ends). Given one such structure A, the skinning map produces a new one B chosen to match A; structure B also comes closer to matching itself.

smooth

1. Infinitely differentiable; possessing infinitely many derivatives. For example, sin(x) is a smooth function, while |x|^3 is not. More complicated mathematical objects such as manifolds are called smooth if they are defined or described by smooth functions.

2. Continuously differentiable; possessing a continuous tangent or derivative.

soliton

A solution to a partial differential equation which is localized in some directions but not localized in time, and which does not change its shape. An example of a traveling wave.

space

1. A general term referring to many kinds of mathematical objects. When an object is called a space, there is a Platonist connotation that some kind of being could exist in it and examine its properties as if it were a jail or the entire universe.

2. More specifically, 3-dimensional Euclidean space.

spectrum

1. In quantum physics, the set of allowed energy levels of a particle or system. It is directly related to bright or dark lines in a spectrum of light produced by a prism.

2. In mathematics, the set of eigenvalues of a linear transformation. By historical coincidence, it is equivalent to the notion of a spectrum in quantum mechanics.

sphere

In topology, any manifold equivalent (homeomorphic) to the usual round hollow shell in some dimension. A sphere in n+1-dimensional is called an n-sphere, because that is its dimension as a manifold.

statistical estimation

The problem of estimating a probability distribution given a set of random samples from the distribution. For example, estimating the distribution of income of people in a city given the results of a selective poll.

statistical mechanics

The study of statistical and thermal properties of physical materials and their idealized mathematical models.

stochastic differential equation

A differential equation whose coefficients change randomly in the time parameter of the equation. Typically it is solved by producing a range of solutions with a probability distribution.

stochastic optimization

The study of maximizing or minimizing (optimizing) desirable quantities (such as time or money) when the constraints on the allowed choices are chosen randomly. The function one wishes to maximize may also depend on random parameters. For example, the problem of choosing raw materials in manufacturing when their prices might change unpredictably.

stochastic process

A dynamical system with random fluctations at each iteration or influenced by random noise. A random variable which, at each stage in time, depends on its previous values and on further random choices. For example, the price of a stock is often modelled as a stochastic process.

stochastic programming

The theory of linear programming in the context of stochastic optimization; the problem of maximizing a randomly chosen linear function on a randomly chosen polytope.

stratifiable

A topological space is stratifiable if it can be decomposed into shells which are similar to the shells that are a fixed distance from a point in a metric space.

string theory

A physical theory in which one-dimensional loops travel through space and also merge and lyse as time elapses. This is in contrast to ordinary quantum field theory, which predicts point particles that emit and absorb each other. String theory is a candidate for a Theory of Everything.

SU(2)

The 3-dimensional Lie group of 2 x 2 unitary matrices; the most common Lie group in mathematics and physics after the circle.

super-

A prefix that means that a mathematical structure is analogous to supersymmetry. Typically these structures have commuting and anticommmuting variables that are placed on an equal footing. For example, a supermanifold has anticommuting supercoordinates along with the usual commuting coordinates of its coordinate charts.

supersymmetry

A theory in physics that postulates a counterintuitive symmetric relationship between fermions, which are particles such as electrons that obey the Pauli exclusion principle, and bosons, which are particles such as photons that enjoy being in the same state as each other.

T

tangled graph

A graph in 3-dimensional space; equivalently, a graph drawn in the plane so that when edges cross, one edge goes over the other.

tau function

A form for the 2-point correlations of an integrable system that typically satisfies auxiliary differential equations and is also given by determinant formulas.

tensor

An object in linear algebra, generalizing a vector, an inner product, and a linear transformation, with a multi-dimensional matrix.

Theory of Everything

A unified theory of all fundamental forces and interactions in nature; a Grand Unified Theory that also includes gravity or general relativity.

Tietze extension theorem

The theorem that if X is a closed subset of a metrizable space Y (or more generally a normal space), a continuous, real-valued function on X can be extended to Y.

thin position

A representation of a knot or graph in Euclidean space, or some analogous structure, with the simplest possible horizontal slices. Between U-turns where the knot or link has a horizontal tangent (and between vertices of the graph), one counts the number of intersections with a horizontal plane. Thin position is achieved when the total of these counts is minimized.

threshold growth

A simple cellular automaton in which cells may be alive or dead, a cell is born if sufficiently neighbors are alive, and a cell, once alive, is immortal.

threshold vote

A cellular automaton in which each cell holds one of several opinions, and at each iteration it is moved to change its mind if sufficiently many neighbors hold a different opinion. If more than one opinion in the neighborhood meets the threshold, the cell retains its old opinion.

topological complexity

A lower bound on computational complexity introduced by Smale that involves topology. A task with high computational complexity requires a computer to make many decisions, sometimes arbitrary decisions, to untangle the topology of the space of possible inputs or outputs or the space of possibilities for an intermediate quantity.

topological quantum field theory

A quantum field theory, such as the Chern-Simons field theory in three dimensions, whose integrals for a manifold produce topological invariants.

topological space

The basic object of topology; a formal model of the qualitative way in which something is connected to itself. The technical definition is that a topological space is a set X together with distinguished subsets called open sets or open neighborhoods, such that the the empty set and X are open, the intersection of two open sets is open, and the union of an arbitrary collection of open sets is open. See also the definition of space.

topology

The study of how geometric objects are intrinsically connected to themselves. Since topologists are not concerned with the geometric measurements of objects, people often say that they study objects up to continuous deformation. But usually topologists consider spaces which have a topology (a qualitative shape or connectivity) but no predefined (quantitative) geometry. Knots and manifolds are typical examples of topological spaces.

train track

A graph drawn on a surface such that every vertex has degree three, and such that all three edges meeting at a vertex have a common tangent, two edges on one side and one on the other. Every lamination of a surface is approximately parallel to, or carried by, a train track.

transport

Any mechanism in physics by which particles or regions of fluid move around, or a mathematical model of such a mechanism such as a PDE.

transport coefficient

Any of various parameters in a mathematical model of transport of particles. More technically, a coefficient of a transport term in a PDE.

traveling wave

Any wave, such as a soliton, which does not change shape as it moves.

triangulation

A tiling of some object such as a manifold by simplices.

triple point

Any kind of point where three things meet: Three surfaces in space, three phases of a substance in a phase diagram, three shock waves, etc.

trivalent graph

A graph such that each vertex has three edges.

Turing machine

A mathematical model of a computer consisting of an automaton travelling along a tape. The automaton at any given time is in some state depending on its previous state and the data at its current position along the tape, and its state also determines whether it moves down the tape and what it writes to the tape at its current position.

U

unimodal

A function is unimodal if it goes up and then it goes down, once. For example, a bell curve.

V

valuation

In convex geometry, a function f on convex sets such that f(A) + f(B) = f(A cap B) + f(A cup B).

variation

In analysis, the amount that a function increases plus the amount that it decreases.

Verma module

A linear representation of a Lie group generated by applying ladder operators to a certain starting vector, called a highest-weight vector, in the same way that one constructs spin states of a particle in quantum mechanics starting from a state of maximal spin in the z direction. Unlike the set of spin states, a Verma module has no lowest-weight vector and is necessarily an infinite-dimensional vector space.

vertex operator algebra

An algebra with an associative and commutative addition operation and a non-associative, parameter-dependent multiplication operation. The operations satisfy axioms related to conformal structures on surfaces. Vertex operator algebras are a mathematical abstraction of 2-dimensional quantum field theories.

Virasoro algebra

A modified version (technically a central extension) of the Lie algebra of tangent vector fields (or infinitesimal motions) of a circle. It was first described abstractly in the physics literature by commutation relations in much the same way as rotation operators are discussed in quantum mechanics. It appears in the study of vertex operator algebras.

W

wave packet/wavelet

A wave with finite spatial extent or one that decays to zero outside of a finite region. Its frequency is necessarily indefinite but it may have relatively small uncertainty.

weak/weakly

In mathematics and especially in analysis, an object is called weak if it is of a generalized kind with fewer properties, and a property holds weakly if it holds in a lesser sense. For instance, a weak solution to an equation might be a discontinuous solution if a straightforward interpretation implies continuity.

weak shock wave

A shock wave with a relatively small discontinuous jump in pressure or another parameter.

Weierstrass approximation theorem

A foundational theorem that, given a smooth function, one can find a polynomial whose values and derivatives are arbitrarily close to those of the function.

Y

Yang-Baxter equation

An algebraic relation arising in statistical mechanics, topological quantum field theory, and quantum groups in which two tensors, one naturally represented by a right-side-up triangle and the other by an upside-down triangle, are equal.

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