Glossary
We add here some definitions (often truncated and simplified) of some mathematical names, words, and phrases alluded to in the interviews. (We beg the indulgence of professional mathematicians who may find them painfully inadequate.) For some of the more technical terms and theorems cited, please refer to google.com or, more specifically, to mathworld.wolfram.com or to planetmath.org, since to describe them here would involve far too many additional definitions to make them useful to most readers.
Algebraic geometry: The study of the geometry of curves, surfaces, and higher-dimensional objects defined by systems of polynomial equations.
Algebraic topology: The study of topological properties using methods from abstract algebra.
Analytic function: An analytic (holomorphic) function is one that can be expressed in terms of all of its higher-order derivatives.
Analytic number theory: The study of the whole numbers using methods and results from calculus and the theory of functions of a complex variable.
Archimedean solids: The thirteen semiregular solids, originally discovered by Archimedes, whose faces are all regular polygons and whose vertices are all surrounded by the same arrangement of faces.
Bourbaki: Collective pen name of a group of (mainly French) mathematicians who set out to write a formal and rigorous account of “elementary” mathematics. Bourbaki also runs a seminar that meets three times a year in Paris to discuss recent mathematical developments.
Brauer-Fowler theorem: A crucial step in the long project of classifying the finite simple groups, those that have no nontrivial normal subgroups.
Cauchy-Bunyakovskii-Schwarz inequality: If ai, bi ≥ 0, i = 1, 2, . . . , n, then
Cayley graphs: A pictorial representation of the structure of a group.
Chebyshev inequality: An inequality in probability theory that states that in any probability distribution “almost all” of the values are “close to the mean.”
Combinatorics: The branch of mathematics concerned with counting, often making use of methods from calculus and algebra.
Complex function theory: The exploration of properties of functions where the variables are no longer real numbers but complex numbers, those of the form a + bi, where a and b are real and i2 = −1.
Composite number: A positive whole number that is not a prime.
Conformal mappings: Transformations that preserve the size and the sign of angles and hence the overall shape of a geometric figure.
Dedekind cut: A technical device used to define irrational numbers in terms of rational numbers.
Dense set (of real numbers): A set of numbers sufficiently large to provide arbitrarily good approximations for all other numbers.
Differential geometry: The study of curves and surfaces using techniques from differential calculus.
Dirichlet series: A series of the form
a1/1s + a2/2s + a3/3s + . . . .
The Riemann zeta function and its generalizations are examples of functions defined by Dirichlet series.
Dynamical systems: Any system that evolves over time, particularly if described by differential equations.
Entire function: A function of a complex variable that is holomorphic at every point in the finite complex plane.
Erdős-Mordell theorem (sometimes referred to as the Erdős-Mordell inequality): For a point P inside a triangle ABC, the sum of the three distances from P to the vertices is greater than or equal to twice the sum of the distances from P to the three sides.
Erdős numbers: An assignment of numbers to mathematicians that goes as follows: if a person has coauthored a paper with Paul Erdős the number assigned to that person is 1; if another person (who has not coauthored a paper with Erdős) has coauthored a paper with a person having ErdŐs number 1, then the second person has ErdŐs number 2; and so on. It sets up a partial ordering of mathematicians that depends on their “distances” from ErdŐs.
Euler’s formula: eiθ = cos θ + i sin θ, where i2 = −1; a formula connecting the exponential function with trigonometric functions.
Fermat’s equation (“last theorem”): A famous conjecture for which Fermat claimed a proof in 1670, but with no details provided: there are no x, y, and z, positive whole numbers, that satisfy the equation xn + yn = zn, for n greater than 2. Finally, Andrew J. Wiles provided a proof in 1994.
Field: A set of numbers for which addition, subtraction, multiplication, and division work as they do for the real numbers.
Fields Medal: Widely viewed as the highest honor awarded to mathematicians, this prize was initiated with funds from the International Congress of Mathematicians in Toronto in 1924 and named for the president of that Congress, John Charles Fields. It was first awarded in 1936 at the Congress in Oslo, where one of the first two recipients was Lars Ahlfors.
Floer theory: A part of differential geometry with implications in symplectic geometry, topology, and mathematical physics.
Functional analysis: The study of linear spaces, especially linear spaces of functions.
Game theory: The mathematical theory involved in the choice of strategy in decision making.
Geometric function theory: The study of the geometric properties of analytic functions. A principal result in this area is the Riemann mapping theorem that says that under certain conditions there is a conformal function that maps a region in the complex plane into a disk.
Graph theory: The study of the properties of networks formed by choosing a set of vertices and connecting some of them by edges.
Group: An algebraic structure consisting of a set with a single operation, often used to study problems involving symmetry.
Hamiltonian circuit: In a (connected) graph, a Hamiltonian circuit is a path (a sequence of edges) that passes in turn through each of the points (vertices) once and returns to the starting point. The problem of finding the shortest possible Hamiltonian circuit is often called the “Travelling Salesman Problem” (still unsolved) that would allow a salesman to visit each of his customers in a way involving the least travel distance or, in other words, the least time to make his calls.
Hypercube: The analogue of a cube in spaces of dimension higher than three.
Kleinian groups: A class of groups that arise in the study of conformal (angle-preserving) transformations of the unit ball in 3-space.
Magic square: A square array of integers (whole numbers) in which the sums of the integers in each row, each column and each diagonal are all the same.
Modern algebra: Often called “abstract algebra,” this is the study of algebraic structures such as groups, rings, fields, vector spaces, and modules, among others.
Nachlass: A legacy, in particular, in the case of a scholar or scientist, the collection of notes, manuscripts and correspondence related to the person’s work.
Number theory: The branch of mathematics involving properties of integers (whole numbers).
Ordinary differential equation: An equation involving a function of one variable and its derivatives, the solution of which is a function or a set of functions. In many examples, the variable is time, yielding a dynamical system.
Peano axioms: A small set of defining properties that characterize the natural numbers: 0, 1, 2, 3, . . . .
Pell’s equation: The problem of finding integers x and y such that x2 − ny2 = 1 where n is an integer (a whole number) that is not a square.
Point set topology: Sometimes called general topology, it is the study of those properties of sets of points and functions between them, especially continuity, dimension, and connectedness.
Pólya (enumeration) theorem: Sometimes named for Federov—who discovered it first—or Redfield—who published it first but after which it remained unknown—it was promulgated by George Pólya. This theorem allows one, for example, to count colorings of constituent parts of polyhedra that appear to be different but are in fact the same, just seen from different viewpoints. Commonly applied in chemistry in the study of molecular structures.
Polyominoes: A generalization of the shape of a domino but with larger numbers of squares placed together to form more elaborate tiles; a concept from recreational mathematics.
Prime: A prime, p, is a positive whole number ≥2 that has only 1 and p as factors.
Prime Number Theorem: If π(x) is the number of primes up to and including x, then π(x) is asymptotically (that is, for very large x) x/ln x.
Quasiconformal mappings: In complex variables these are mappings that are “almost” conformal, that is, angle distortion is limited within certain technical constraints.
Riemann hypothesis: A conjecture by Bernhard Riemann in 1857 that the (nontrivial) zeros (points in the complex plane where the Riemann zeta function takes on the value 0) lie on a vertical line in the plane where the real part is equal to ½. The conjecture remains unproved and is widely viewed as the most important and tantalizing unsolved problem in pure mathematics.
Riemann zeta function: The Dirichlet series where ai = 1 for i = 1, 2, 3, . . . The zeta-function “encodes” many properties of the prime numbers. (See Dirichlet series.)
Ring: An algebraic structure with two operations motivated by the study of the integers or sets of polynomials.
Selberg sieve: A generalization of Eratosthenes’s sieve, a method in analytic number theory for locating primes among the positive integers.
String theory: An active area of research in particle physics that is concerned with reconciling quantum mechanics and general relativity. The field is often cited in referring to the work of Edward Witten, the only physicist to have received the Fields Medal in mathematics.
Summability: A field of mathematics concerned with techniques assigning values to divergent series.
Symplectic geometry: A branch of differential geometry or differential topology, with origins in classical mechanics.
Tauberian theorems: A family of theorems that serve as crucial tools in analytic number theory and complex analysis.
Theory of numbers: (See Number theory.)
Topology: The study of properties of geometric figures that are unchanged under certain transformations; sometimes called “rubber sheet geometry.”
Uniform convergence: A property of a set of functions on a given set that converge at the same rate for all members of the set.
Van der Pol’s equation: The ordinary differential equation
u″ + α(u2 − 1)u′ + βu = 0
that has one and only one periodic solution.
Vector geometry: In n-dimensional space, the study of geometry using vectors, objects that, unlike numbers, have not only magnitude but direction as well.
Voting theory: The study of various systems for setting up elections, where the choice of the system can affect the outcome. One of the principal and most surprising results is the Arrow impossibility theorem, sometimes called Arrow’s paradox, which says, roughly, that no “fair” system can be devised that satisfies three conditions, each of which seems entirely reasonable to expect of any voting system. Kenneth Arrow won the Nobel Prize in Economics in 1972 for related work.