FAQ for non-mathematicians

Mathematics textbooks have not appeared out of nowere, and they are not the only textbooks that will ever be written. Indeed, the people writing them, or those writing other works used by those textbooks' authors, have been doing research in mathematics, i.e. they discovered* "new mathematics" that was not known before. For instance, what you may know as "X's theorem" (where X is a name) wasn't necessarily known before X; the mathematician** X who stated and "proved" such theorem therefore did "research" in mathematics. Accordingly, mathematicians have continuously discovered* new results during history, and they (we) will keep doing it in the future.

In general, mathematics is built starting from some axioms (e.g. those about natural numbers), using which one can give definitions (e.g. "the sum of natural numbers") and formally prove very simple theorems (i.e. "the sum of 2 and 3 is 5"). In turn, using those definitions and those theorems, one can define more complicated concepts and prove more complicated theorems, and so on. During the centuries, mathematicians proved a huge amount of results in arithmetics, algebra, geometry, analysis, etc. 

What people usually learn in school is barely a glimpse of what is known in mathematics, and most curricula don't usually go any further than the birth of mathematical analysis in the XVII century. At university people study more modern mathematics, up to the point they are put in contact with open problems which are not (yet) solved and specialised areas which are not (yet) explored. Research in mathematics is about solving those problems and exploring those areas, which in turn will lead to formulating new problems and discovering new areas of future research.

* the debate if mathematics is discovered or created will be probably never settled, but I personally side with the Platonists in this regard

** note that, especially going back in the millennia, the history of mathematics can be much more complex: for instance, Pythagoras' theorem was very well known already by the Egyptians, the Indians, the Chinese, etc.

As all mathematicians, I spend most of my research time either sitting at a desk with pen and paper or standing in front of a blackboard, both alone or in small groups (in presence or online). A few mathematicians, working on more "computational" fields, might need some specialised (computer algebra, computer graphics and/or numerical) software in their research, and a few of them might also need to program some algorithms. Nevertheless, most of us (including me) use computers only to write our articles (using Latex), and most of the "numbers" you might read there will be 0 and 1.

The reason of our unconventional (when compared to most scientists) working style is that we do not work with data, experiments, patients, etc., but rather with definitions, conjectures, theorems, and proofs. See here for a well written description of what mathematics research looks like.

I work in differential geometry. Modern geometry investigate abstract, higher-dimensional, "curved" spaces called manifolds. The adjective "differential" is related to the fact that one can compute derivatives of functions on manifolds similarly to what one does in ordinary mathematical analysis with functions of one or several variables.

My main interests revolves around Lie groupoids and Lie algebroids. Lie groupoids arise by looking at local "symmetries" of manifolds, while Lie algebroids are their infinitesimal counterpart, i.e. what one obtains after taking the "linear approximation" of Lie groupoids.

tl;dr: I work on symmetries of abstract curved spaces

Short and provocative answer: this is a bit like saying "You’re a doctor? I don’t really know anything about medicine, but can you explain exactly how the endocrine system works in two minutes or less?" or "You’re an engineer? I used to play with building blocks when I was a kid. Show me exactly what you’re designing and, if there’s anything I don’t understand, it’s probably your fault" (taken from If I Reacted to Other People’s Careers the Way They React to Me Becoming a Mathematician by Jordy Greenblatt).

Longer but more satisfying answer: unlike many other sciences, it is much more complicated to convey what pure mathematics is about, hence being specific about one's research area. An average educated person will have at least a rough intuition and/or a visual idea of what an atom, the DNA, or an electromagnetic field is, but will usually have no idea what a vector space, a topology, or an abelian group is (these are some basic concepts which mathematics students learn in their first or second year, on which most of pure mathematics is built). See the blog article "Science communication as a pure mathematician" for a similar discussion by a colleague, and Soapbox Science as a pure mathematician for her testimony of an outreach activity.

At the end of my PhD thesis (see pages 229-235 for the English version and pages 237-244 for the Dutch one) I have written a "summary for muggles", where I tried to explained in a few pages the ideas behind the results of my PhD thesis. Precisely for the issues pointed out in the paragraph above, I actually discuss very little about my own results, and mostly give an intuitive explanation about what differential geometry is and which objects I use in my research. Below is an extract.

Students in high school learn how to describe 2-dimensional geometric pictures by means of algebraic equations. For instance, x^2+ y^2 = 1 defines a circle of radius 1; similar formulae can be used for lines, parabolae, hyperbolae, etc. The unknowns x and y define the so-called Cartesian coordinate system for the standard Euclidean plane (i.e. the infinite piece of paper where we imagine to draw our axes and our pictures).

Cartesian geometry can be studied also in dimension 3, where we deal with pictures in a coordinate grid given by three axes, x, y and z, which define the standard Euclidean space. This point of view is easily generalisable to any dimension, leading to the notion of Euclidean space of dimension n: any point in it can be described by the coordinates x_1, ... , x_n .

A manifold is a "curved space" that locally looks like the Euclidean space of a certain dimension n. In other words, for any point on the manifold, we can provide n coordinates to describe a small region around it. One should think of these as "curved coordinates grids".

The adjective smooth refers to the absence of singularities - there are no edges, corners or any kind of sharp points. More formally, let us look at any two different points on a manifold: the coordinates describing the regions around them will be different, but they will be related by some functions, the changes of coordinates. A manifold is smooth when all these functions admit derivatives of any order (singularities would arise in points where we cannot derivate).

Let us look at the most common examples of smooth manifolds in low dimensions.

• Curves are smooth manifolds of dimension 1. Indeed, any arc of a curve can be approximated by small line segments, which can be described by one coordinate (the length).

• Surfaces are smooth manifolds of dimension 2. Indeed, any area of a surface can be approximated by small rectangles, which can be described by two coordinates (length and width). For instance, the surface of the Earth is a 2-dimensional manifold because it can be fully encoded by two well-known numbers: the latitude and the longitude. The same applies to the surface of a doughnut without sprinkles (mathematically known as a torus), of a maschera di carnevale cookie (a genus 2 torus), of a pretzel (a genus 3 torus), or of other smooth pastries with any number of holes.

• Our universe (from a Newtonian point of view) is a smooth manifold of dimension 3: any point in the space can be described by three global coordinates, after having fixed an origin (for instance, the Sun).

Manifolds of higher dimensions are more difficult to picture graphically for humans with a three-dimensional mindset and can be described only mathematically. However, this is one of the many instances where a general definition eventually pays off the price of its abstractness. Indeed, higher dimensional manifolds appear naturally, and are very relevant, in modern physics: for instance, the space-time of general relativity is a manifold of dimension 4, and in (some version of) string theory, physicists see the universe as a manifold of dimension 11.

One can perform different kinds of transformations on manifolds: for instance, they can be moved, rotated, stretched, compressed, or even cut, pinched, etc. Of course, a random transformation may change our manifold into a very different kind of space. In geometry one is often interested in transformations, called diffeomorphisms, which preserve the smoothness of our manifold, as well as its "shape". This means, among other things, that the dimension is preserved; moreover, the transformation admits an "inverse transformation" satisfying the same properties.

For instance, a function transforming a circle in a square is not a diffeomorphism (it does not preserve smoothness), whereas transforming a circle into an oval is a diffeomorphism (roughly speaking, this means that there are no differences between them from the point of view of smooth manifold). More generally, translations and rotations are diffeomorphisms of any manifold, while cutting or pinching are not.

A diffeomorphism does not necessarily have to act on the whole manifold, i.e. being global. For instance, instead of rotating an entire sphere, one can rotate only the northern hemisphere. This is an example of a locally defined diffeomorphism, or (more imprecise but shorter) local diffeomorphism.

[...]

A geometric structure on a smooth manifold is an extra piece of information that allows one to perform specific tasks on that manifold. Let us describe a few of them.

• The most intuitive class of geometric structures are (Riemannian) metrics; they are tools which enable us to measure distances between points on a manifold. For the Euclidean space, the standard way of measuring lengths of straight lines is an example. On a curved space, a metric simply associates a positive number to any two points. For instance, the distance between any two points on the surface of a sphere can be computed as the length of the shortest arc of curve between them (this leads to the concept of geodesics, which is used to determine the best trajectory for planes around the Earth).

• A slight variation of this notion is a pseudo-Riemannian metric: this arises in general relativity, where it has the purpose of measuring distances between two events in the 4-dimensional space-time universe. There can be counterintuitive consequences: unlike a Riemannian metric, here the distance between two points is not necessarily a positive number, but can also be negative (if the temporal separation between the two events is greater than the spatial separation).

• Another important geometric structure coming from physics, more precisely Hamiltonian mechanics (a reformulation of the classical Newtonian mechanics, due to R. Hamilton (1833)), is a symplectic structure. It can be defined only on manifolds of dimension 2n, which are interpreted as the phase space of a particle moving in an n-dimensional space: the first n coordinates of the manifold defines its position, the other n its velocity. The role of a symplectic structure is to describe the dynamics of the particle in terms of the energy of the physical system. More generally, a symplectic structure can take any function in input, and gives as output a trajectory where the function does not change; in the previous physical example, the symplectic structure tells us where the energy of the system is conserved.

• The last class of geometric structures we describe in this introduction are (regular) foliations. These are used to partition a manifold into "smaller" manifolds, called "leaves". The adjective regular refers to the fact that all leaves must have the same dimension. An easy example (which explains the name of this structure) is given by the three-dimensional Euclidean space, which is foliated by an infinite amount of parallel two-dimensional Euclidean planes (imagine an infinitely high stack of infinitely large sheets of papers). Another example is given by the foliation of the surface of a torus in parallel circles of the same radius (think of a slinky toy - or the Slinky character from Toy Story, and remove head, tail and paws - where you join the two ends).
  As a counterexample, note that the parallels of the Earth (circles of constant latitude, e.g. the Equator or the Tropics) do describe a partition of its surface, but not a regular foliation. The reason is that, while most leaves are 1-dimensional (since they are standard circles), the north and the south poles are covered by two circles of radius zero, which consist only of one point, and are therefore 0-dimensional manifolds. The meridians of the Earth do not describe a regular foliation either: they do not even form a partition, since through the north and the south poles there is more than one leaf.

Many interesting phenomena appear when studying transformations that preserve not only "the shape" of a manifold, but also an additional geometric structure. A (local) symmetry of a geometric structure on a manifold is precisely a (local) diffeomorphism which preserves the structure.

For instance, let us consider a Riemannian metric on a sphere. As we discussed earlier, translations and rotations are diffeomorphisms; moreover, they preserve the distances between any two points, hence they are symmetries of the metric (these are also known as isometries). On the other hand, if we rotate only the northern hemispere, we have a local diffeomorphism which is not an isometry (the distance between two points in two different hemisphere can change).

Similarly, stretching the sphere or the torus (entirely or partially) is a diffeomorphism, but is not an isometry. However, stretching the torus to a bigger one is a symmetry of the slinky toy foliation by parallel circles described earlier. In this case, each stretched circle deforms into another circle of bigger radius, which is still a manifold of dimension 1.

The reason we discussed the concept of symmetry is that a geometric structure on a manifold can be equivalently described via the symmetries of its local model.

[...]

This is probably the most common question I receive from non-mathematicians (which include also academics in other fields) - and I'm sure that the same holds for all my colleagues.

The usual answer I give is along the following lines. More than a century ago the theory of relativity was formulated. This constituted one of the biggest revolution in the history of physics, and yet it took many decades until people could see a "real-life" application of it, namely the GPS. In turn, general relativity relies heavily on differential geometry, a field of maths which was born more than a century earlier out of the pure curiosity of understanding the right notion of "curvature". This means that many mathematicians in the XIX century (and earlier) worked on apparently "useless" topics, which eventually found concrete applications only two centuries later. Therefore, in a couple of centuries, maybe what I (and my colleagues) do might help for a new revolutionary and "practical" invention as the GPS.

I quote here another possible answer from a reddit discussion:
The theoretical mathematicians develop theories which the applied mathematicians use in order to make tools for other researchers (engineering, economics, biology/chemistry/medicine,...), who then make products (software, machinery,...) that the technicians (engineers, systems administrators, medical technicians,...) use to give information to the practitioners (construction workers, doctors, brokers,...)

So for example:

The theoretical mathematician develops abstract invariant theory

The applied mathematician applies the theory for SO3 to computer imaging

The medical researcher/mechanical engineer/software engineer uses this to develop an improved MRT

The radiologist uses this machine to find out what’s wrong with the patient

The doctor interprets this information and prescribes medicine

Of course, history of mathematics (and of science) is much more complex: there are some examples of mathematics results which found concrete applications very quickly, but also many more which have not found any application so far (and maybe will never find it). The common theme is however that nature is written in the language of mathematics (Galileo Galilei), therefore experimental scientists are forced to rely directly or indirectly (e.g. via computer science) on mathematics in anything they do (see also The Unreasonable Effectiveness of Mathematics in the Natural Sciences). The mathematics they use, be it very basic or very advanced, may seem nowadays easily explained in a textbook, but it has required however decades or centuries of research before reaching that final, ready-to-use, stage. During that time, many mathematicians worked without any concrete goal in mind, but rather out of pure intellectual curiosity to explore wonderful worlds populated by abstract notions of spaces, patterns, symmetries, etc.

In conclusion, the impact of mathematics on society is enormous, but it is almost always indirect, and therefore, unfortunately, invisible. If we want science and technology to continue advancing and benefitting general society, us mathematicians need to be provided with time to pursue our theoretical research, without any pressure to obtain concrete and direct "impacts", and even at the "cost" of "wasting time" obtaining results which will never be "applied". The scientists of the next generations will however be thankful for that, exactly as the scientists of today profit of all the mathematics done in the past.

tl;dr: it takes decades or centuries of "abstract mathematical work" before some topics are ripe enough to be used by other scientists and engineers for practical applications. 

First of all, in my job I am also active in teaching and outreach - activities which have an immediate impact on society, and hopefully do not need any intrinsic justification for deserving public funding.

Still, research is an important component of my job. Regarding the public utility of curiosity-driven basic research, I refer to the letter Why Explore Space? by Ernst Stuhlinger. It talks about outer space, and not space in the mathematics' sense, but everything could be translated also to mathematics, via the arguments discussed in the previous question.

A very nice article precisely on this topic is Why Do We Pay Pure Mathematicians? by Ben Orlin. Another one is Why is math research important? by Cathy O’Neil.

I conclude with the following (provocative) quote:

It is clear, then, that we do not seek this science for the sake of any other end. For just as the man is free, we say, who exists for his own sake and not for another’s, so too this is the only free science, for it alone exists for its own sake.

(Aristotle, Metaphysics, Book I, Chapter 2 (982b 24–28)

It originally referred to philosophy, but it can be very well applied to mathematics as well (which, not surprisingly, was intrinsically tied with philosophy since antiquity).