"The fundamental questions in mathematics often end up applying somewhere in ways that we never really imagined."
Assistant professor of mathematics
A solid grounding in grammar, accented with a firm understanding of logic, all topped off with a healthy dose of intuition. Those three basic skills will benefit Bucknell students studying any discipline. They might be surprised to find themselves learning them in Assistant Professor of Mathematics Stephen Wang's mathematics courses.
Grammar is essential for drafting a coherent proof. "You must write in complete sentences," Wang says. "You are presenting an argument to your reader and it needs to be readable, and one reads in sentences." The verb might be an equal sign, and the nouns might be single letters to designate variables, but the thought must be complete. X = y is fine; x + 3 is not. "Precise language is extremely important once you get into the argument-based mathematics," he says. Likewise, logical argument is crucial for creating a meaningful proof. Wang challenges his students to think about the construction of foolproof arguments.
While the rules of grammar and logic are essential to writing a correct proof, gut feeling can help a mathematician figure out what approaches to try. "Intuition is incredibly useful," he says. "It tells you what you might want to prove." That is where the artistry comes in. As in other fields, mathematical intuition can develop with experience, but it is extremely difficult to teach.
In his own research, Wang studies geometric group theory. Advanced geometry is not necessarily about the shapes and angles learned in grade school. "Geometry isn't just about circles and polygons and polyhedra," he says. Instead, the objects Wang studies are typically more abstract.
"One generally doesn't apply geometric group theory techniques to anything that one can actually touch. The sphere is boring from a geometric group theory perspective," he says.
Mathematical research might focus on intangible objects, but that abstraction does not make its truth any less real. Just as pioneers in number theory asked questions centuries ago that today are used in computer science and cryptography, Wang's work could find tangible applications.
"The fundamental questions in mathematics often end up applying somewhere in ways that we never really imagined," Wang says. However, focusing on future usages is not really the point. "I like being able to say that things are true," he says. "There is beauty to it."
Posted Sept. 22, 2009