Bubbles give insight into universe

By: Jerome Simons

Posted: 5/1/08

Imagine blowing soap bubbles when you were younger. Picture the thin, sparkling film of soap spreading out across a loop of plastic or metal wire, slowly building into a fragile sphere that comes loose and floats away.

It turns out there is an entire branch of mathematics devoted to the study of this sort of phenomenon. Patrick Zulkowski, a Hopkins grad student, along with his advisor Chikako Mese, are conducting research into the minimal surfaces between closed curves, a somewhat abstract problem with real-world consequences.

This question goes back to Plateau's Problem, a famous dilemma that dates to the 19th century. Joseph Plateau, a Belgian physicist, was interested in soap films and their surfaces in a closed loop of wire. His study of their properties led to the formulation of Plateau's laws, which describe the structure and behavior of their surfaces.

The essential law governing their behavior is that the soap film, due to its natural surface tension, wants to form a film with the smallest area possible. In a free-floating soap bubble, surface tension causes bubbles to form a perfect sphere. Things are more complicated while the growing bubble is still attached to the wire.

Zulkowski, who received a bachelor's and master's from the University of Delaware, studies these surfaces. But what would higher math at Hopkins be without some extra complexity? His studies take place in a multidimensional space, far beyond the normal realm of our three- (or four-) dimensional lives.

In order to explain his work, Zulkowski makes an analogy to our world. A geodesic is the shortest connection between two points. Here, geodesic is just a fancy way of saying straight line. But those lines vary as you add more spatial dimensions. The shortest connection between Baltimore and Los Angeles would definitely be a little bent, since our planet is a ball and not a flat plane.

As you add dimensions, finding the least distance, the geodesic keeps on becoming more complicated as you move on from the plane to a sphere all the way up to unimaginable spaces - spaces that do exist according to mathematicians and physicists.

Zulkowski and his advisor cannot draw pictures of these spaces, but they can set up curves that connect points in them using equations. "We don't know if a curve exists, we just know about its length," Zulkowski said.

In fact, the researchers are able to set up not just one, but an entire set of curves, which then becomes a sequence. If a shortest distance exists, this sequence of curves should converge on a solution.

"We need to prove that the sequence becomes a terminal curve which becomes the shortest connection, the geodesic," he said. This sounds fairly simple up to now, but modern mathematicians only have two "toys" to play with: a distance function between two points and simple knowledge about the length of the least distance.

This is not a whole lot considering that they deal with something so far beyond human perception. "Our basic objective is to say as much as we can with the little knowledge we have," Zulkowski said.

"In our moments with high self-confidence we say that success in our work could lead to the connection between quantum physics and general relativity," Zulkowski said.

Currently general relativity, the physics of large objects, does not comply with quantum physics, the physics of small objects. Why should they? Because it would make sense to get the same answers no matter how large or small the numbers being plugged in are.

General relativity assumes that mass and the subsequent mutual attraction between massive bodies are created through the curvature of space. This idea was one of Albert Einstein's greatest contributions to physics.

This is where Zulkowski's research comes in: Mathematicians like him try to gain an understanding of what the curvature looks like. This has been possible for about a hundred years, ever since Einstein formulated his famous equations.

But now, mathematicians take another feature into account when dealing with the curvature being important for general relativity: the quantum hypothesis. The latter assumes that space and time come in fixed portions - that they are discrete.

In the quantum world, you can walk one mile or two miles, but not one and a half miles. Everything in the quantum world, from matter to energy, is said to be quantized into discrete units.

All that mathematicians have to do now is figure out what mathematical systems look like if the universe really does work like that. Zulkowski's work would be successful if he could define curvature in discrete spaces.

He is working with coordinate systems with a few thousand axes, also known as manifolds. A typical axis you might work with in a math or physics class has just two or three axes, x, y and maybe z, each representing a different dimension.

If successful at defining curvature in discrete space-time, the basic criterion for general relativity would be fulfilled and all gravity would be a consequence of curvature. This time though, the quantum hypothesis is also on board.

"The fact that there are so many elegant and beautiful aspects of math makes me believe that math already exists and we discover it." Zulkowski said. "The logic that forms the basics of math is built into the fabric of our reality."