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I decided to approach this question via a sort of Fermi estimate. My goal was to get some sort of estimate on how much mathematics do I know, and then to use that goalpost to try and estimate how much mathematics does the average person know. Bear in mind that these are going to be very rough estimates—I’m mostly trying to get an answer on roughly the right order of magnitude.The way I attacked this was by making use of the AMS Mathematics Classification Chart. This is the standard system used for identifying what subject areas a math paper actually belongs to. It is very complete—it is 44 pages long, as I recall. I threw out every part of this chart that did not directly belong to mathematics—so, for instance, anything about mathematical education was right out. After that, I went through it and selected anything with which I had even the barest familiarity. This got me to about 463 topics out of 3992. So, I estimate that I have some very broad knowledge of about 10% of all known mathematics. I estimate that I have reasonably good familiarity with about 10% of that, and I have good enough familiarity to actually be able to do research in about 10% of that. So I would estimate that I know no more than about 0.1% of all mathematics that exists.I would further estimate that I know about a hundred times more mathematics than the average person. This puts the average mathematical knowledge at about 0.001% of all that is known. This is a small enough percentage that any differences between Europe and America can probably be ignored.

I was initially going to write that this question is tricky to answer, because who are “we”? Are “we” professional mathematicians, or the average layperson? And then I realized that it actually completely doesn’t matter, because the answer is “yes, absolutely”, regardless. (EDIT: It seems Alan Bustany already had this realization in his answer.)Let’s get the obvious out of the way: the average layperson knows roughly zero mathematics. I swear I’m not trying to be insulting—this is the honest truth. For the most part, the average layperson knows some subset of the mathematics that was discovered prior to 1637. There are a few exceptions to this general rule: the average layperson might know (or at least, have seen) more about probability and statistics than was known at the time, and they might be familiar with matrix multiplication, which was introduced by Cayley in the 19th century.However, there is plenty of mathematics that was formalized (in at least rudimentary versions) prior to 1637 that many laypeople aren’t particularly familiar with, such as continued fractions, the Euclidean algorithm, the Pell equation, inversive geometry, and projective geometry. And, more importantly, mathematical knowledge has been progressing at a roughly exponential rate, so in the 300+ years since we have discovered far more than the preceding 4000+ years.As near as I can tell, most laypeople have not heard of any of the following:Algebraic structures (e.g. groups, rings, fields, modules)Galois theoryGraph theoryMetric spacesComplex analysisFunctional analysisTopology (although apparently there was at some point kids’ programming that introduced some very basic topological ideas)Algebraic topology (though they might have heard of homology or cohomology)Category theoryHomological algebraMeasure theoryModel theoryFirst order logicAlgebraic geometryArithmetic geometry (though they might have heard of elliptic curves)Additive number theoryAlgebraic number theoryAnalytic number theoryAlgebraic extensions of the rationalsLocal and global fields (e.g. [math]p[/math]-adic numbers)I could easily go on—these were just some of the first things that popped into my head. And that is just material that I am familiar with, at a high level (meaning that most everything listed above can be subdivided several more times). This is obviously not going into anything that I have not heard of before. But I am just one mathematician, with one narrow segment of material that I am familiar with, with little knowledge of what happens outside of it.And that’s the thing: no one is going to be familiar with all of mathematics. There is just too much. Just for fun, I decided to look through the AMS classification chart to see how long it would take before I found anything that I had not heard of. As I suspected, it was basically immediate:Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)Algebraic logicFractional graph theory, fuzzy graph theoryThue-Mahler equationsDessins d’enfantsLinkage, complete intersections and determinantal idealsNone of these words mean anything to me (and, believe me, this is not even close to an exhaustive list of what I don’t know and don’t understand, but merely a small sample produced by randomly looking through the aforementioned chart before I got bored). I wouldn’t be able to tell you the first thing about any of this.So, the conclusion is: most laypeople know roughly zero mathematics. Most mathematicians know slightly more than zero mathematics. The vast majority of mathematics is mathematics that we have never heard of. It is the rule, rather than the exception.

Joseph Anthony's response is spot-on!!I've been to Australia (or, "Oz", as the locals call it) three times. It's one of the safest countries on the planet!This does not mean that you should leave your brains at home, when you board the plane. Of course, as with everywhere else, common sense should guide you, no matter where you go.But, when compared to other countries, Australia is as safe as a church. In fact, Oz is soooooooooo far from other countries (like USA), that the two don't even belong on the same classification chart !!