Imagine a map of the world that may stretch some regions more than others, but always keeps the angles between roads correct. A conformal geometry remembers exactly this: angles, not distances. The Mercator map is an example: it enlarges Greenland a lot, yet every angle between two streets is drawn true. Two shapes count as the same in conformal geometry when one can be bent onto the other by such an angle-keeping map.

Doing the Hilbert transform twice gives back the original function with a minus sign. So the transform loses no information, and its inverse is the same operation with a minus sign. On waves, it is a quarter-period shift, and two quarter-period shifts turn a wave upside down.

The Taft algebra is a small, explicit Hopf algebra that is "quantum" in both directions. It is built from a cyclic group element $g$ of order $n$ and a nilpotent element $x$ that do not commute but $q$-commute. Moving $x$ past $g$ costs a root of unity. Because everything is finite and explicit, it is the standard first test case for statements about finite-dimensional Hopf algebras.

The criterion decides whether all roots of a real polynomial lie inside the unit circle without finding the roots. It checks a few signs and inequalities that come from simple products of the coefficients. Engineers use it to see whether a digital filter or a sampled control loop is stable.

For an invertible matrix one has a number, the determinant. For a family of operators that are sometimes not invertible, one cannot have a single number, but one can have a line at each parameter where a determinant would live. When the operator is invertible, the line has a canonical nonzero element, the determinant itself. The twisting of this family of lines over the parameter space is a topological invariant of the family.

Let $a$ be algebraic over a field $K$. The minimal polynomial of $a$ over $K$ is the monic polynomial of smallest degree in $K[x]$ that has $a$ as a root. The definition of the minimal polynomial states it.

Add the numbers of Pascal's triangle along the shallow diagonals, and you get the Fibonacci numbers. Each term counts the ways to tile a strip with a fixed number of dominoes.

Cardinal numbers measure the size of sets, finite or infinite. They generalize the natural numbers as answers to the question "how many elements?".

Think of a machine that takes one solution of an equation and outputs another solution. The machine is itself a small system of equations of first order, which is easier to solve than the original equation. The simplest example is the pair of Cauchy–Riemann equations $u_x = v_y$, $u_y = -v_x$. If $u$ is harmonic, these equations give a harmonic conjugate $v$, so they carry one solution of Laplace's equation to another. For the sine-Gordon equation, applying the machine to the zero solution gives a kink, and applying it again gives solutions with two kinks.

If you know the totals $g(x)$ obtained by adding $f$ over everything below $x$, you can recover $f$. The recipe uses signed weights $\mu(y, x)$ that depend only on the shape of the poset. For ordinary numbers in a line it just says: take the difference of two consecutive running totals.

A tensor $u$ in $X \otimes Y$ can be written in many ways as a sum of simple products $x_i \otimes y_i$. Each way has a cost, the sum of the products of the lengths $\norm{x_i} \, \norm{y_i}$. The projective norm of $u$ is the cheapest cost over all ways to write it. For example, in $\ell^1 \otimes \ell^1$ this norm turns a tensor into a double sequence $(a_{mn})$ with norm $\sum_{m,n} \abs{a_{mn}}$.

The Gram matrix of a finite family of vectors in an inner product space is the square matrix of all their pairwise inner products.

The natural numbers cannot be reached from finite numbers: sums, products and powers of finite numbers stay finite. An inaccessible cardinal is an uncountable size with the same property. Taking power sets of smaller sets, or unions of fewer than $\kappa$ sets each smaller than $\kappa$, never gives a set of size $\kappa$. So everything built from below stays below, and the sets below $\kappa$ form a whole universe of their own.

A principal ideal domain is an integral domain in which every ideal is generated by one element. The integers and the polynomials over a field are examples. The definition of a principal ideal domain gives the conditions.

Integrating the holomorphic differentials of a surface along loops gives a lattice of periods in $\C^g$. Dividing $\C^g$ by this lattice gives a doughnut-shaped space of dimension $g$, which is the Jacobian. It is a group, so points of the curve, and formal sums of them, can be added inside it. For a torus $\C / \Lambda$ of genus $1$, the Jacobian is the torus itself.

The Euclidean $n$-space is the space of ordinary geometry in $n$ dimensions. A point is a list of $n$ coordinates. The dot product lets you measure lengths and angles, so the familiar facts of school geometry hold in each dimension.

A repair radius measures how far a colouring of a graph $G$ with a given profile of class sizes can be from a proper one.

The polynomial $x^n - a$ factors in two obvious ways: when $a$ is a perfect $p$-th power for a prime $p$ dividing $n$, and in one special case built from $x^4 + 4$. Capelli's theorem says that there are no other ways.

A long enough list of different numbers always contains a long subsequence that goes up or one that goes down. The terms need not be adjacent. For example, $3, 1, 4, 5, 2$ has the increasing subsequence $3, 4, 5$.

Take a collection of sets in which the union of any two sets of the collection is again in the collection. The conjecture says that some single element appears in at least half of the sets. It is very easy to state, yet it resisted all attempts for decades, and only recently was a constant fraction proved.