Perspectives: Research and Creative Activities at SIUC, Fall 1997 



by Marilyn Davis

As an expert on probability theory, Salah Mohammed knew the odds were steep. 

But late last year the SIUC mathematics professor learned that he'd won a choice plum in his discipline: a visiting professorship at the renowned Mathematical Sciences Research Institute in Berkeley, Calif. Some 400 mathematicians from around the world applied; only 15 were offered positions. 

This academic year Mohammed is spending his sabbatical at the institute--giving lectures, mentoring junior colleagues, and doing intensive research with one of his recent collaborators, Denis Bell, a math professor at the University of North Florida (who was also awarded a visiting professorship). 

For a mathematician, it's about as close to nirvana as you can get. And it's been a long journey from the arid scrubland of Sudan, where Mohammed was born, to the hills of Berkeley--or Southern Illinois, for that matter. 

A soft-spoken man who's patient with interviewers, Mohammed undercuts the stereotype of the mathematician as an ivory-tower hermit oblivious to the real world. 

When I first meet him, he's a bit preoccupied with the fact that his water heater is acting up. Though his office is devoid of decoration, his office door bears newspaper clippings: a tribute to a famous number theorist, Paul Erdös, who died last year; a letter to the New York Times from a mathematician concerning the Unabomber case. He's developed his own web site, which is user-friendly and attractively designed. He follows politics. 

And his work, though highly theoretical, relates directly to the richness and unpredictability of the world around us. 

Mohammed's specialization is stochastic analysis, the mathematical study of the dynamics of random variables. It has applications in fields that affect our daily lives and the planet's future: engineering, finance, meteorology, and ecology, among others. 

Stochastic analysis is concerned with random change over time, or random evolution. It is an extension of basic calculus into the area of probability theory. Like calculus, it deals with dynamical (changing) systems. Unlike calculus, those systems are affected by random influences--and that brings probability into play. 

If you bet $1,000 that a tossed coin will come up heads, you're in the realm of elementary probability--in this case, a 50-50 chance of winning the bet. The act of the coin toss involves a random variable, but not random evolution. 

If instead you buy a five-year, fixed-rate certificate of deposit with that $1,000, you're in the realm of elementary calculus (although you don't need calculus to compute your interest). You've bought into a dynamical system, but one in which you can quantify where you're going to end up. The system is deterministic: it evolves, but not randomly. At any point the state of the system--the value of your CD--is known with certainty. 

Now suppose you invest that $1,000 in the stock market instead. This time you've taken the plunge into a stochastic system, whose fluctuations depend on an untold number of factors, including random effects over time. Guarantees? None, except the certainty of change. 

Life itself, of course, is a stochastic process. 

"Stochastic" is from the Greek, meaning roughly "to aim at; to guess at." Much of the natural and manmade world is characterized by randomness and uncertainty. Stochastic systems--from the weather to the stock market--are influenced by many factors, including chance events. 

Some of those random factors might be intrinsic to the system and others extrinsic. And the interaction of internal and external influences produces random effects as well. 

To use a real-world analogy: populations, whether of humans or other species, are highly stochastic. Genetic mutations and patterns of mating within a population are internal kinds of randomness. Immigration--the arrival of newcomers--is an example of an external random influence. The interaction of these and other factors also introduces randomness in the system. 

The variables in a stochastic system are not always quantifiable, or even known. But the highly theoretical work of Mohammed and his colleagues is improving our mathematical understanding of such systems. Scientists and engineers then can take the equations stochastic analysis generates, and use them to model real-world systems. 

In some fields, the goal is to design more-reliable systems by controlling for stochastic effects as much as possible. 

"Engineers call stochastic effects noise," Mohammed explains. In signal processing, for instance, a better mathematical understanding of noise--the chance occurrence of distortion, or interference--allows the engineer to filter out that noise more effectively. 

In other fields, the goal is not to design a better system, but to understand a system better--in order to make more-accurate predictions or to intervene in the system somehow. 

Understanding the effects of climate variables better could improve long-range weather forecasting. Finding better ways to predict fluctuations in a population could help biologists save endangered species and help economists estimate labor-force needs years into the future. And one of Mohammed's doctoral students, Mercedes Arriojas, has developed formulas for modeling the behavior of financial markets. 

The idea of memory is important to much of Mohammed's work. Some stochastic systems can be analyzed without knowing their past. Such systems, termed Markov processes after a Russian mathematician, are said to have no memory. 

One very important type of Markov system is called white noise. Mathematically speaking, it's pretty manageable. That's because at any point in time, the probability of the system being in a given state is described by a bell-shaped curve. 

A standard example of white noise is the random jiggling that tiny particles suspended in a liquid or a gas undergo as molecules collide. The phenomenon is called "Brownian motion," and it provides a good mathematical approximation of the behavior of many real-world systems. 

"It's close enough to reality to allow you to make some qualitative inferences," says Mohammed. And it's a classic case of a Markov process--a process without memory. 

Think of a plume of smoke from a cigarette, Mohammed suggests. Ruling out any unusual influences, such as a gust of wind, the meandering fluctuations of the smoke are a type of Brownian motion. 

"You can take that trail of smoke and you can cut it off at any point--you can remove its history--and still it's going to evolve the same way, probabilistically," he explains. "You can say, 'With a certain probability, this particle of smoke is going to take this region, this space.' You can calculate that probability." 

He gives another, more startling example: stock market fluctuations, generally speaking, fit the characteristics of Brownian motion. Although a trail of smoke and the stock market are totally different kinds of phenomena, "they represent more or less the same mathematical idea," he says. 

But most of Mohammed's work concerns stochastic systems with memory. The term "hereditary" is used to describe such systems, because the history of the system must be taken into account in understanding its likely future evolution. That's true, for instance, of population dynamics. 

The notion of memory, or history, means that the effects of some of the random influences on the system aren't seen immediately, but only after a delay. An example from population dynamics is the gestation period in producing offspring. Another example is the incubation period in infectious diseases. 

That time lag is what makes stochastic hereditary systems so complex to analyze. Unlike white noise, they are non-Markovian processes--and much more difficult to grapple with mathematically. 

To handle them, Mohammed took a mathematical technique used in non-stochastic analysis and adapted it to stochastic analysis. The method basically "recasts" a non-Markovian process as a Markovian process. It works by segmenting the history of a system into units that are treated as single values. 

"You treat each segment as an entity, a representation of the state of the system," says Mohammed. That allows a workable way to analyze the system mathematically and infer things about its probable evolution. 

Much of Mohammed's latest research on stochastic hereditary systems has been done with Michael Scheutzow, a mathematics professor at the Technical University of Berlin. One of their interests is determining whether certain non-Markovian systems, under certain conditions, can stabilize, or reach equilibrium. 

As one possibility, for example, a system's randomness might eventually "regularize" into a white-noise pattern. 

Stability, in this case, doesn't mean deterministic. The system is still dynamical and still subject to random influences. 

"It becomes fixed in the probabilistic sense," says Mohammed. At equilibrium, its variables have a fixed distribution, be it the classic bell-shaped distribution or some other curve. 

Some systems will stabilize under certain conditions and others won't. Being able to distinguish between them has practical applications in modeling real-world systems. It's especially important for engineers to know the rate of stabilization of systems they're designing. 

"If you have to wait for a very long time for the system to stabilize, that's not efficient," Mohammed points out. "You want to design the system in such a way that it reaches equilibrium very quickly." 

Mohammed grew up in a small town on the Blue Nile, two hours southeast of Khartoum. His father, a railroad worker, urged him to become a doctor. In a poor land--and Sudan, Mohammed says, has poverty more profound than almost anything seen in America--medicine promises relative prosperity. 

During high school, however, an enthusiastic teacher from Egypt had captivated Mohammed with the beauty of Euclidean geometry. It has all the ingredients of higher mathematics, he explains--definition, theorem, and proof--but "because the material itself is not so sophisticated, it's available to a high school kid." 

Often, after the teacher explained a proof, he would ask the students to find a better one. "Sometimes you could come up with maybe a shorter proof, maybe a more elegant proof," says Mohammed. The challenge appealed to him. 

Because in those days Sudan gave its brightest students scholarships to the University of Khartoum, Mohammed was not denied opportunity because of poverty. He tried engineering school, partly to placate his father, but threw it over after only a week for math.

He found drafting tedious, he says, but he had a larger dissatisfaction: "The mathematical component is subservient [in engineering]. It is not the center of attention." 

After earning his bachelor's degree in 1970, Mohammed went to the University of Dundee in Scotland for his master's. The climate, so unlike his hot, dry homeland, chilled him to the bone, and although he already spoke English, the Scottish accent threw him for a loop. 

"The first few days I was completely lost," he confides. But he adjusted quickly. "It was culture shock, definitely, but it was also exciting because it was different. I like to go to different places." 

Mohammed did well abroad, earning his master's in 1972 and then his doctorate from the University of Warwick (England) in 1976. He returned to Sudan and for several years taught at the University of Khartoum, where he produced two pioneering monographs. 

The first concerned the geometric structure of certain types of non-random dynamical systems. This fit with Mohammed's doctorate, which concerned higher aspects of geometry and had nothing to do with matters stochastic. 

But Mohammed soon found himself more intrigued by probability theory and stochastic analysis. His second monograph, published in 1984, was the first book on the theory of stochastic hereditary systems. 

"Very little had been done in this area," he says. He's now working on a third major monograph, this one on the geometric structure of stochastic hereditary systems. He expects it to break new ground as well. 

In the early 1980s, he met SIUC mathematician Ronald Grimmer during a visit to England and learned that the Math Department was looking for a visiting professor. Mohammed was interested. 

"I think I wanted to try the experience of the States, you know--to explore America," he says. He also realized that, to advance beyond a certain level in his career, it was necessary to leave his homeland. 

"Sudan is very isolated scientifically, and it is difficult to travel away from Khartoum," he explains. "I needed to communicate more with other mathematicians." America offered even more opportunities than Europe did for that, he adds. 

So in 1984 he and his family came to Carbondale for what he thought would be a one-year stay. During his visit, however, a tenure-track position came open in the department, and he was invited to apply. He was hired in 1985 as an associate professor, was promoted to full professor in 1989, and three years later became a U.S. citizen. He stresses proudly that he is an African American. 

The details of Mohammed's work--especially his discoveries about the intricacies of stochastic hereditary systems--are inaccessible to nonmathematicians. But the importance of the research has been recognized by many organizations. 

The National Science Foundation has funded his work since 1989. His current grant runs through the year 2000. He's also had funding from NATO (with Scheutzow), has held several short-term research fellowships from the Alexander von Humboldt Foundation in Germany, and early in his career had visiting fellowships with the British Science and Engineering Research Council. 

Tall and lanky, Mohammed hardly seems to fit in his small office, where even the floor is double-parked with books and papers. On his desk hums a UNIX workstation, which functions as a web server as well as a PC. 

Mohammed's web site shows the premium he places on communication. For his students, he posts syllabi, announcements, and grades. For his colleagues, he posts preprints of his journal articles so that they don't have to wait for the published versions to read his results. 

In this high-tech world, though, it's somehow reassuring to learn that the bulk of his mathematical work is done the old-fashioned way: calculations scrawled with pencil on paper, or chalk on blackboard. 

One of Mohammed's favorite pastimes is reading Arabic poetry (his first language is Arabic). Some of it dates from pre-Islamic times. 

"It's very rhythmic, formal, highly structured," he says. "It's very musical: everything has to balance very well. It's very difficult to write; I can't write it myself. But when I see a piece of poetry which is nice, I just immediately attune to it." 

That the patterns of poetry should appeal to a mathematician is not incongruous. Math presents an aesthetic challenge not unlike that of poetry: finding the elegant solution, the good fit. Conquering that challenge is a thrill for Mohammed. 

"You try something and you work on it for months, maybe, and you're almost despairing, saying no, I don't see it; any way you turn it, it seems to be a dead end," he muses. 

"And then somehow, you just click, and suddenly it opens up. It's very pleasing. 

"That's much more of an art, you see, than a science. It's like doing a picture and looking at it here, and here, and here, and saying, 'Oh, this can't be right. It has to be this way to fit, to be OK, to be nice, and pleasant.' And then it fits, and it's right. 

"It's absolute ecstasy when you get that." 

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