Suppose bakeries send bread to cafés, and each pair (bakery, café) on the delivery list has a cost. The list is $c$-cyclically monotone if no group of bakeries can save money by passing their deliveries around in a circle. On the real line with cost $(x - y)^2$, sending $0 \to 0$ and $1 \to 1$ passes the test. Sending $0 \to 1$ and $1 \to 0$ fails, since swapping the two destinations lowers the cost from $2$ to $0$. An optimal transport plan can never contain such a wasteful cycle.

The equation $y'' + q y = 0$ describes a spring whose stiffness $q(t)$ changes with time. The solutions either keep crossing zero forever or eventually stay on one side. The theorem looks at a double average of the stiffness. If the solutions stop crossing zero and this average is bounded below, then the average must settle to a finite value. So an average that grows without bound, or that keeps swinging, forces endless oscillation.

Some important operators come in chains, where applying two consecutive operators always gives zero, as $d \circ d = 0$ for the exterior derivative. Then one studies the cohomology: what the next operator kills, modulo what the previous one produces. An elliptic complex is a chain of this kind that is elliptic in the sense that, at top order and in every nonzero frequency direction, the chain is exact. This makes the cohomology finite-dimensional on a closed manifold. For the de Rham complex, this cohomology is the de Rham cohomology, and its alternating sum of dimensions is the Euler characteristic.

Mehler's formula adds up an infinite series of products of Hermite polynomials and gets a single Gaussian expression. It is the Hermite analogue of summing a geometric series. The resulting kernel describes how a quantum harmonic oscillator evolves in time. It also describes a particle pulled back to the origin by a spring and shaken by noise.

The theorem says two descriptions of a random field agree. One description lists independence facts: each variable cares only about its neighbours. The other gives a formula: the probability is a product of terms, each involving only a group of mutually adjacent variables. This holds as long as no configuration has probability zero.

Curvature is not an arbitrary collection of numbers: it satisfies some built-in rules. The Bianchi identities are these rules. One is algebraic and says that a certain cyclic sum of curvatures is zero. The other says that a cyclic sum of derivatives of curvature is zero, much like the rule that the divergence of a curl is zero.

With coins of values $a$ and $b$, you can pay all large amounts when $a$ and $b$ are coprime. The theorem gives the largest amount that you cannot pay: $ab - a - b$.

Many angles describe the same direction, because full turns do not change it. The principal value picks one of them, the one between $-180$ and $180$ degrees, with $180$ allowed and $-180$ not.

To check that a group of matrices is a Lie group, one would normally need to build charts and check smoothness. This theorem removes the work: any subgroup of a Lie group that is closed as a set is automatically a smooth submanifold and a Lie group. So all the usual groups cut out by equations, such as orthogonal, unitary and symplectic groups, are Lie groups at once.

Picture a bundle as a family of fibres standing over the points of a base. At each point of the total space some directions go along the fibre; these are vertical and are given by the bundle. An Ehresmann connection chooses which of the remaining directions count as horizontal. For the product of a line and a circle, the obvious choice is to call the line directions horizontal. Then moving in the base lifts uniquely to moving horizontally upstairs.

A flow is a family of maps $\varphi_t$ of a space to itself, indexed by real time $t$. The map $\varphi_0$ is the identity, and $\varphi_s \circ \varphi_t = \varphi_{s+t}$. The flow of a vector field sends each point to the position, after time $t$, of the solution of $x' = f(x)$ through it. See the definition.

A semiprimary ring is semisimple up to a radical that dies after finitely many multiplications. This is the structure of a left Artinian ring, but without any chain condition. For example, let $R$ be the ring $k[x_1, x_2, \ldots]$ with all products $x_i x_j$ set equal to $0$. It is semiprimary: its radical is spanned by the $x_i$, it squares to zero, and the quotient is the field $k$. This ring is not Artinian, because the ideals $(x_n, x_{n+1}, \ldots)$ form an infinite descending chain.

Picture the diagram as a figure drawn on the floor and the apex $N$ as a point above it, with a string from $N$ down to each object. The strings must be consistent with the arrows $D(j) \to D(k)$ of the figure. Going down to $D(j)$ and then along the arrow gives the same result as going down directly to $D(k)$. For two sets $X$ and $Y$ with no arrows between them, a cone is just a set $N$ with two functions $N \to X$ and $N \to Y$.

A distributive lattice is a lattice in which meet and join distribute over each other. Lattices of sets and Boolean algebras are the main examples. The node distributive lattice gives the precise definition.

In the complex plane, multiplying by $i$ turns each arrow by a quarter turn. An almost complex structure gives such a quarter turn $J$ on every tangent plane of a manifold, with $J$ applied twice giving minus the identity. On $\R^2$ the rule $J(a, b) = (-b, a)$ is the quarter turn. Only manifolds of even dimension can carry such a structure.

Choose a few letters as generators and write down many random words in them as relations. The group you get is a random group. With few relations compared to the number of possible words, the group is almost surely infinite and looks like a tree at large scales. With too many relations, they collapse everything, and the group is almost surely trivial or has two elements.

An acyclic object is one on which a functor behaves as if it were exact. To compute derived functors one needs a resolution by objects of this kind, and injective objects are the universal choice. But they are large and hard to handle, so one uses objects adapted to the given functor. For sheaf cohomology, flasque sheaves, whose sections over open sets always extend, are acyclic for the global sections functor. For the tensor product, flat modules are acyclic, and this is why Tor can be computed with flat resolutions.

A principal bundle has, above each point of the base, a copy of a symmetry group whose origin has not been fixed. A connection tells how to move from one fibre to a nearby one, and it must respect the symmetry. The connection form does this by reading off, from each tangent vector, how fast it moves along the group direction. For the trivial bundle $M \times G$ there is a flat choice: moving in $M$ while staying at the same group element.

Over each point of a base space, a vector bundle provides a vector space. Given two bundles over the same base, place the two vector spaces side by side at each point to form their direct sum. The result is a new bundle whose rank is the sum of the two ranks. For the sphere, the tangent planes together with the normal lines give, at each point, the whole of $\R^3$.

Given a bunch of overlapping balls, one can keep a few of them that do not overlap at all. After blowing the kept balls up by a fixed factor, they cover everything the original balls covered. So the total size of a union of balls is controlled by the size of a disjoint subfamily.