Positive scalar curvature is incompatible with certain topological invariants of spin manifolds. The theorem shows that if the $\hat{A}$-genus of a spin manifold is not zero, no metric on it can have positive scalar curvature everywhere. The proof is short: positive curvature forces the Dirac operator to have no kernel, and the index theorem turns this into a statement about topology.

The lemma splits space into a part where $f$ is small, at most $\alpha$, and a collection of cubes where $f$ is large on average, but not too large. The large part occupies little volume, at most $\norm{f}_{L^1}/\alpha$. It is the basic tool for studying functions that are only integrable, without any better bound.

A closed surface in space without self-intersections, however crumpled, has an inside and an outside. To get from the inside to the outside you must cross the surface, and every point of the surface touches both sides.

A strictly convex function is a function whose graph lies strictly below each chord, except at the endpoints of the chord. Its graph contains no line segments. The node strictly convex function gives the precise definition.

A driver in a car tries to run over a pedestrian in a large empty parking lot. The car is faster, but it cannot turn sharply, while the pedestrian can change direction at will. The question is when the driver can be sure to succeed, and how both should move. It is a classical test problem for the theory of pursuit and evasion.

The conjugacy classes of a $p$-group that have more than one element all have sizes divisible by $p$. The order of the group is also divisible by $p$. So the classes with one element, which form the centre, cannot be just the identity.

In a world where all randomness comes from one Brownian motion, every random payoff is a constant plus a stochastic integral against that motion. In betting terms, any such payoff can be produced from its mean by a suitable trading strategy in the Brownian motion.

A Cayley table of a finite set with a binary operation is a square table. It shows the result of the operation for each pair of elements. For a finite group, it shows the full multiplication of the group.

Feed a unit vector $x$ into $T$ and measure how much of $Tx$ points back along $x$; that complex number is $\ip{Tx}{x}$. The set of all these numbers is the numerical range. The theorem says that this set is convex: the segment between any two of its points lies inside it.

The kernel of a linear map is the set of vectors that the map sends to zero. It is a subspace of the domain.

On the real line, the sets in one piece are exactly the intervals. The proof needs the completeness of $\mathbb{R}$, because $\mathbb{Q}$ has gaps where it can be split.

Landweber iteration solves $A x = y$ by repeatedly correcting the guess in the direction that reduces the misfit $\norm{A x - y}$. With exact data it converges to the best solution. With noisy data it first improves and later starts to fit the noise, so one stops it at the right moment.

Real closed fields are the fields that behave like the real numbers for all algebraic purposes. Every positive number has a square root and odd-degree polynomials always have a root. Take any statement written with $+$, $\cdot$, $<$ and quantifiers over the elements of the field. Model theory shows that it is true in a real closed field exactly when it is true in $\R$.

If the bottom part and the top part of a group can both be built from abelian layers, then so can the whole group. You stack the layers of the top part on the layers of the bottom part. For example, $S_4$ is built from $V$, $A_4/V$ and $S_4/A_4$.

Let $X$ be a compact space. Let $C(X)$ be the space of bounded continuous functions on $X$, which is a Banach space under the supremum norm.

To arrange $n$ different objects in a row, you have $n$ choices for the first place, then $n - 1$ for the second, and so on. The product of these numbers is $n!$.

The Hilbert symbol answers a yes-or-no question: does a certain quadratic equation have a nonzero solution in the field? Over the real numbers, $z^2 = a x^2 + b y^2$ has a nonzero solution unless both $a$ and $b$ are negative. So $(a, b)_\R = -1$ exactly when $a < 0$ and $b < 0$. Over $\Q_p$ the answer depends on Legendre symbols modulo $p$, and it can be computed by a short formula.

A Lie algebra is a vector space with a bracket, the infinitesimal shadow of a Lie group. The theorem says that every such shadow, in finite dimension, really comes from a group. Among all groups with the given shadow there is a best one, the simply connected one, and it is unique.

To find the chance that several things all happen, take them one at a time. Multiply the chance of the first by the chance of the second given the first. Then multiply by the chance of the third given the first two, and so on.

Applying $\Hom(-, N)$ to a short exact sequence loses exactness on the right. The Ext groups record exactly what is lost, in a long sequence of correction terms. To compute them, replace $M$ by a projective resolution, apply $\Hom(-, N)$, and take cohomology. For example, $\operatorname{Ext}^1_\Z(\Z/n\Z, \Z) \cong \Z/n\Z$, which reflects the extension $0 \to \Z \xrightarrow{n} \Z \to \Z/n\Z \to 0$ that does not split.