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Much of his work is on the topic of sequences of random events, what we call "stochastic processes." That is, each event in the string of events has an associated probability. You would study the value of each event (the outcome) across time and/or space. This mathematical concept is frequently used in studying things like behavior of the stock market or growth of bacteria.
One type of stochastic process is called a random walk; random walks describe a path within a mathematical space. It can be used to describe literal movement, such as the path of an animal looking for food, or more figurative movement, such as the financial gains and losses of a gambler. Though the term random walk was created by Karl Pearson, Lévy did a great deal of research into this concept, identifying special cases of random walks (such as the so-called Lévy flight).
Lévy also identified an interesting probability puzzle known as a martingale: a random process where the expected value on the next observation is equal to the previously observed value. For example, the best guess of what an interest rate will be tomorrow is what it is today.
There are some interesting stories about where the name "martingale" comes from, with some arguing that it comes from the device used with horses; a martingale hooks around the horse's head and connects to a strap on the neck, to keep the horse from moving its head too far up or down. But wherever the name comes from, when Lévy described it, he drew upon a particular approach to gambling made popular in 18th century France. What it involves is doubling one's bet with each loss, so the goal is to recoup lost money while also making a profit.
Theoretically, this strategy is winning, because if the game is fair, I won't lose every time, and I'll get the money I lost back. The problem in practice is that the gambler could go broke before he or she gets far enough to win anything. Sure, one good hand would turn everything around. But each bad hand gets the gambler deeper into debt. Of course, I could also bankrupt the house in the meantime. This strategy isn't so much a strategy; it just depends on the game being fair and probability doing its thing (eventually). The martingale is one of the reasons casinos place limits on how much you can bet.
To tie these two concepts together, a random walk could be a martingale if it has no trend. That is, if each step one direction is counteracted by a step in the opposite direction, the trend line will be flat. So the expected value is always the same: 0.
A few more facts on Lévy:
Like many statisticians, he was called upon to assist with the war effort during World War I.
In addition to the information above, he contributed to topics of functional analysis, differential equations, and partial differential equations. And though today, he is considered the forefather of many modern concepts, he wasn't viewed as very important in his time. (There was quite a bit of snobbery from pure mathematicians about statistics. It was considered glorified arithmetic, inspired by such low-brow activities as gambling.)
During World War II, he was fired from his job as professor at École Polytechnique, because of laws discriminating against Jews. His job was reinstated, though, and he remained there until retiring in 1959.
Both his daughter (Marie-Hélène Schwartz) and son-in-law (Laurent Schwartz) were also mathematicians.
Some of his research, which was considered esoteric at the time, has turned out to have incredibly important applications. This is why I argue with people when they say we should fund applied rather than basic research - you never know when or how basic research will end up being useful.