What sort of tasks does a mathematician commonly do in their job?
While I know the image of "people in suits crunching numbers" is wholly false, I'm not actually all that sure about what mathematicians (whether they're studying topology, physics, or any branch beyond) commonly do as a part of their occupation. Is it more akin to individual work or research in a team? Is it something concrete like solving real world problems or more pure number theory?
As you might expect, the work you might do as a mathematician can be quite varied.
Most mathematicians engage in research. Broadly construed, "research" refers to the process of attempting to make progress in solving problems that currently have no solution. There are countless unsolved problems in mathematics (look up "millennium problems" for a handful of famous examples), and it's often the job of the mathematician to learn more about these problems. This involves extended study into mathematical topics, usually beginning at the college level (such as calculus, linear algebra, real and complex analysis, abstract analysis, probability theory, and many others). This is often followed by graduate studies in mathematics, which usually involve a combination of additional study into broad areas (like the ones above), plus focused research into a more narrow set of topics (based on the interests you've developed until that point).
Generally speaking, a research endeavor begins with a question. For example, "what is the shortest path between two points, assuming the path is confined to a plane?" One often starts by trying to gain some intuition for the question being asked. This might involve drawing a picture or writing down and quickly evaluating some first guesses at a solution. If you're lucky, you hit upon the correct solution, and you're able to prove it's correct. Otherwise, you might start cycling through a process that involves thinking about the problem, reading what others have written about the problem, and updating your guess at a solution and/or its proof. If you're lucky, the problem has already been solved. You might use this work going forward (always giving credit to the original authors), or use it to inform your own, new method of solving the problem. Once you've made a concrete step of progress, your job is to document how you did this, and share that work with others; your peers will review your work and ensure it uses sound methodology; your work is then generally published for the benefit of the broader scientific community.
Broadly speaking, there are two "types" of math -- pure and applied. Pure mathematicians tend to work on mathematical problems for the sake of mathematical progress; they find mathematical puzzles interesting enough to spend their lives learning about and trying to solve them. Their work may or may not have obvious or indirect relationships to real-world phenomena (though often such connections are made much later). Applied mathematicians tend to work on mathematical problems as a means of understanding or solving a problem in another domain (such as physics, data science, finance, medicine, etc.). Their techniques often require expertise not just in mathematics but also in the relevant external domain; working knowledge of programming languages are often useful for simulations and for achieving results usable in the external domain.
Your other questions are very good ones to ask. Research may be carried out individually or as part of a team; sometimes it is carried out as part of a geographically distributed team (ie, you might work with colleagues across the country or across the world). This really depends on the details of your particular situation. Most often, you'll do research as part of your graduate studies, and follow that group for at least a few years. You'll likely build connections from there and make decisions about whether to branch out.
One advantage of this field is that it is often possible to make progress with pen, paper, and computer (ie, no need for expensive laboratory equipment and lots of space or power to run it). This makes your expertise flexible and portable.
A mathematician might also choose to teach (often as part of the same position that pays him or her to do research). But I presume you've already seen what math teachers do, and so I've tried to focus here on the details of what you likely haven't seen a mathematician doing.
To sum up: mathematics can be thought of as the formalized study of patterns, with the aim of generating insight into a wide variety of phenomena (mathematical and otherwise). It comes complete with its own language, which happens to be the most precise and useful way to describe a staggering array of puzzles. Studying mathematics requires copious amounts of both time and work, but it is, in its own right, a fascinating and rewarding endeavor.
If you have any additional questions about being a mathematician, please continue asking -- curiosity is a critical characteristic of any good mathematician!
Source: I am an applied mathematician.
Nonetheless, there are “pure” mathematicians who would devote their life in the strengthening the foundation of this language, extension and further development of existing systems or structures, or the exploration of new possibilities, i.e., new mathematical systems and structures.
Others are “applied” mathematicians who will tackle practical problems in physical sciences, engineering, statistics, and many other applied fields. For example, number theory can be applied to cryptography, and topology can be used in improving network operations.
In most universities, the mathematics department is separate from the probability and statistics department. But for our purposes, a concentration on number is enough of a similarity to consider the two as one. Statistics as a profession means not only designing experiments and interpreting the data collected from them, but reaching conclusions that have a profound effect on the direction of industry. In other words, all the fun stuff that the actuaries don't get to do. At a median salary of $72,830, it's good to know that there's a corresponding reward for all that cogitation.
This is an appropriate career for someone who majors in mathematics. The Wall Street Journalfamously cited "mathematician" as the best job in America in 2009, an honor that probably deserves an asterisk. (Criteria included safety on the job, which is why lumberjack ended up dead last in the same study.) Being a mathematician is a profession that requires its practitioners to solve problems. What's the optimal way to get these pallets from the distribution center to the retail outlets? How much money could we save by having the trucks drive a different route? Or switching to diesel? Or using trains instead? That's just one extremely narrow example, but mathematicians have their place throughout dozens of industries. As a decisive knockout punch to the apologists for people who major in the humanities, a typical professional mathematician can expect to earn $94,160 annually.
For thousands, this is the culmination: the ideal job for the student who wants to not only study math, but tie it to the tangible rewards of the marketplace. "Financial analysis" doesn't take a formal definition, but among other things it means knowing how to compile, prepare, read and evaluate the all-important financial statements that all public companies must create and disseminate. At the higher levels, it means determining what capital investments to make, how to pay for them and where to finance them. When a company decides to issue stock, or buy it back, or split it, it doesn't assign the task to just anyone. A numerate college graduate with the relevant three-letter acronym "CFA" (Chartered Financial Analyst) is the person they'll put on the job. In fact, there are a number of job opportunities for CFA holders.
The role of financial analyst is one of the few occupations on the list that comes with its own formal designation and licensure. The requisite college degree is nice to have, but it's only a starting point to becoming a CFA. This means more study, more examinations and if you're willing to put in the time, four years of work experience. However, once you become a CFA, you'll be recognized (and employable) all over the world. One-hundred-and-thirty-ei
ght countries recognize the designation, and most of the rest are too small to support or warrant a vibrant CFA community. The median salary for a financial analyst in the United States, at least according to the latest round of statistics (compiled, no doubt, by someone with a mathematics degree), is $74,350.
The morbidity of estimating how many people are going to die, and when, is countered by the comfortable pay most actuaries receive. Fittingly, it takes an actuarial recruiting firm to quantify the data for us. According to Chicago-based D.W. Simpson, actuarial salaries have something of a linear relationship to tenure. Measured by total compensation (including insurance and pension), the average actuary makes $66,839 a year, plus $7,459 for each year in the profession.
Being proficient with numbers and their operations (or more accurately, the study of quantity in all its forms) doesn't imply a talent for things mechanical. For those math students adept at applying their knowledge to the physical world, designing planes and spacecrafts is about as worthwhile and prestigious a career as there is. Amediansalary of $102,420 is nothing to downplay, either.