By Tom Johnson
At a panel discussion at the Marian Miner Cook Athenaeum last Wednesday on new ideas in mathematics, the professors were out to prove something that had nothing to do with complex theorems and the kinds of postulates only numbers geeks can love.
Instead, in “Math for the New Millennium: Ideas that Change the World,” CMC mathematics professors Asuman Aksoy P ‘05, Blake Hunter, Chiu-Ken Kao and Lenny Fukshansky endeavored to show how – in mathematics – what goes around, comes around.
Further, in introducing the panel, Michelle Goodwin 16, a mathematics major at CMC whose thesis advisor is Professor Fukshansky, summed it up when she said: “Math is the cornerstone for every single thing you do day to day. Everything that you’ll hear about today relates to your life in some special way and everything you do should lead you back to mathematics concepts.”
Indeed, mathematics occupies a unique place among the vast variety of academic disciplines: it can be classified as both an art and a science. Nowadays, the process of mathematical discovery is at the heart of both fundamental intellectual pursuit and technological innovation leading to breakthroughs in engineering and computer science, digital communications and even artificial intelligence.
Asuman Aksoy, Crown Professor of Mathematics and Roberts Fellow (who's been a CMC faculty member since 1987), spoke about the Invariant Subspace Problem -- a major problem in an area of mathematics called Functional Analysis, her specialty.
"For all the people in the room that have trouble trying to find the “value of X,” this is really a tough one even for brilliant mathematicians,” she said. “Mathematicians enjoy the advantages but suffer the penalties in being able to talk in a special language of numbers.”
Quoting Paul Erdos, a famous Hungarian mathematician, Prof. Aksoy said the definition of a mathematician is “someone who can turn coffee into theorems.” In another rather enigmatic quote, she cited German mathematician David Hilbert’s definition of the discipline as: “Mathematics is what confident people understand the word to mean.”
“When I first read that quote, I didn’t think it was very inclusive,” Prof. Aksoy said. “but there is a kind of reckoning of it if we look at the invariant subspace problem.”
Prof. Aksoy said that in the field of functional analysis, the invariant subspace problem is a partially unresolved problem. It asks whether every bounded operator on a Banach space sends some non-trivial closed subspace to itself.
The original form of the problem posed by Paul Halmos was in the special case of polynomials with compact square. It was successfully resolved for a more general class of polynomially compact operators, by Allen R. Bernstein and Abraham Robinson in 1966.
“Mathematics research is not a monologue, but rather a dialogue,” Prof. Aksoy said, acknowledging the tradition that advances in mathematics are developed – often over intervening decades – by theorists who build on what came before. “Bit by bit, they bring more light onto the problem.”
… so that competent people can understand!
Blake Hunter, Assistant Professor of Mathematics at CMC, spoke about the field of Data Mining, which has many applications in the analysis of social networks, Twitter, image databases and even movie recommendation filtering. With the advent of computers, people have a wealth of information literally at their fingertips; the challenge is how to cut through it all to get the precise information they need.
“Topic modeling digs through all that information and tries to add structure back into what is largely unstructured raw data so that a searcher can find the exact book, page, whatever that they need to reference,” he said.
Prof. Hunter said that Topic Modeling starts out with data – such as books – or perhaps some digital database (a bunch of files from Twitter); maybe S&P 500 stock analytics or company analysis; possibly data from biological or chemical experiments – almost anything.
“You want to find a way to process all this data, but it can get messy,” he said. “We try to find hidden themes or thematic structures in these massive data sets by using linear algebra techniques so we can search, summarize and make predictions from the data.”
Echoing what Prof. Aksoy said, Prof. Hunter mentioned that ideas like Topic Modeling encompass exactly what mathematicians do. “We take the idea and try to extend it into further and further ideas and try to abstract them out. And Topic Modeling is just one abstract.”
Chiu-Yen Kao, Associate Professor of Mathematics (Applied Mathematics, a specialty) talked about level set methods which capture the contours of dynamical implicit surfaces which, in turn, occur everywhere. “Think of an ocean wave,” she said. “If you think about the interface between the air and water, it’s really a dynamical surface because they move and change with respect to time.”
Prof. Kao also included snowflakes which form due to relative temperature and humidity as dynamical surfaces and even brains which form more complicated surface structures over time.
Lenny Fukshansky, Professor of Mathematics spoke about the ABC Conjecture, which is a major open problem in Diophantine Analysis, a central branch of Number Theory. In particular, he explained how this conjecture impacts Fermat's Last Theorem.
The abc conjecture in number theory is stated in terms of three positive integers, a, b and c, which have no common factors greater than 1 and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d is usually not much smaller than c.
In other words: if a and b are composed from large powers of primes, then c is usually not divisible by large powers of primes.
In number theory, Fermat's Last Theorem states that "no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two. Here a^n, b^n, c^n means "a to the power n", "b to the power n", "c to the power n", respectively.
This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica by the ancient mathematician Diophantus of Alexandria. He claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians.
"The fact that great numbers of mathematicians have thought about the abc conjecture and Fermat's Last Theorm is vital regardless if solutions have been reached or not," said Prof. Fukshansky. "The discussion motivated the development of several central areas of modern mathematics, including abstract algebra and number theory, as we know them today."