Any sufficiently-complicated (more than basic arithmetic) formal system of mathematics with finite axioms is necessarily incomplete. That is, there are statements which are true (making this a constructive theorem) that cannot be proved within the system. An example of this is the Continuum Hypothesis in traditional ZFC set theory.
There are various variations on infinity, known as transfinite numbers. The continuum, also known as the reals, is of cardinality (size) aleph 1 whereas the natural numbers are of cardinality aleph null (named by Georg Cantor, one of the greatest geniuses of all time). The Continuum Hypothesis involves discovering whether or not there is a transfinite number between aleph null and aleph 1. These transfinite numbers were discovered using a method known as a diagonal slash.
You cannot comb a hairy ball smooth. This is part of the reason there is always a small/minor/major hurricane on Earth (air currents).
1729 is the first number that can be produced as the cube of two different positive integers (1, 12) (9, 10).
In any gravitational field, the work done is independent of the path taken and depends only on the starting point and ending point. This is an example of a conservative force field, as referred to in vector calculus.
In any graph (mapped onto a plane), the number of “faces” (completely surrounded sections + the outside) minus the number of edges plus the number of vertices is equal to 2. The number 2 is known as the Euler characteristic of the surface, and is invariant for equivalent topological surfaces.
A coffee cup is equal to a donut, according to topology (commonly known as “rubber sheet geometry”). Proving theorems about objects in three dimensions (or 4 or 2 or 1) is often the hardest for topologists, which is why there is a special part of topology known as “low dimensional topology”.
Any map can be colored with 4 or fewer colors. This fact was proven by computer, and later rechecked independently, and is considered the first “proof by computer”.
Fermat’s last theorem was proven by uniting the study of elliptic curves to that of modular forms, two very disparate branches of mathematics. The initial flaws in the proof (in 92 or 93) were corrected and released in 1995, after several hundred years of effort. The initial claims of Fermat that he “had a proof” are now mostly discredited, as his notes have revealed he searched for a proof for cases 3 and 4 (which would not be necessary if he had a general proof). The proof was very much a 20th century mathematical proof.
Fourier analysis is the breakdown of complex periodic graphs into elementary terms of sines and cosines. It is often used in digital signal processing (for ever increasing approximation) and in loads of other EE applications (circuits, etc).
There are certain types of computational problems that are not polynomial in terms of the complexity involved to solve them. It is not known if these algorithms are not optimal, but what is true about a certain class of algorithms, known as NP-complete algorithms, is that if one can become P (polynomial) in run-time, then all the others will immediately follow. Also, you win a million bux from the Clay Math Institute, but who cares about that?
Any computing device is equivalent to any other in terms of its ability to solve problems, discounting memory and speed. Given enough time, and using the limitations of its input, any computing machine with certain well-defined mathematical operations can do anything any other computing machine can. This result is known as the Church-Turing thesis, but it isn’t known to help Adobe Acrobat open any faster or give any extra frames per second in Doom 3.
There is a branch of analysis (calculus) in which an infinitesimal is a well-defined number, as opposed to standard analysis where it is considered an illusion, easy to use for the purpose of teaching.