Associate professor Paul Hacking, mathematics and statistics, with his collaborator Sean Keel from the University of Texas at Austin, has been invited to deliver a lecture at the 2018 International Congress of Mathematicians (ICM) to be held on Aug. 1-9 in Rio de Janeiro. With Keel unable to attend, Hacking will deliver the lecture to members of the Algebraic and Complex Geometry section on Aug. 7.
The ICM, held roughly every four years since 1897, is one of the premier forums for presenting and discussing significant mathematical discoveries, say Hacking and colleagues. Some call it the Olympics of mathematics and the gold Fields Medal that is awarded there, the Nobel Prize of mathematics.
Hacking says, “It’s an unusual honor to be invited to give a speech at this international meeting. I very much appreciate the recognition of my peers. I’m looking forward to it, but it is a bit daunting. Many of my colleagues will be there and I look forward to catching up with them.” He believes the last time UMass Amherst was represented as a speaker there was 32 years ago in 1986, when Bill Meeks, now a distinguished professor, had the honor.
Hacking’s research area is algebraic geometry, one of two primary methods scientists use to study and define shapes. He explains, “Differential geometry uses the tools of calculus to solve geometric problems, whereas in algebraic geometry we use abstract algebra instead.”
“Geometry is often intuitive, and it’s easy for us to visualize the difference between the surface of a donut and a sphere,” he adds. “But to absolutely pin it down, you need to develop a language to rigorously describe these objects and how a ball is different than a donut. Mathematical language will nail it down precisely.”
Hacking has worked with two main colleagues, Mark Gross at Cambridge, U.K., and Keel at UT Austin, plus Maxim Kontsevich of the Institut des Hautes Etudes Scientifiques, Paris, to produce several papers and a survey of this research area over the past five years.
One field where algebraic geometry is useful is theoretical physics, Hacking says. String theory, which seeks to describe the fundamental forces of nature and how the universe operates, asserts that rather than the three dimensions plus time we are familiar with, there are instead 10 dimensions, and six of them are very, very small, in the quantum arena and not visible to the naked eye.
“Imagine a garden hose seen from a long distance away that appears to be one-dimensional, but as you get closer you see another dimension,” he explains. “String theory says if you were able to look at smaller scales you’d see extra dimensions. At very small scales, quantum mechanics plays a role in physics and weird phenomena will be explained by studying these six extra dimensions. There is a geometry of the very small object that would explain quantum behavior. This six-dimensional object is called a Calabi-Yau (C-Y) manifold, and it’s one of the objects we study in algebraic geometry.”
Understanding quantum physics requires precisely describing the shapes of these manifolds, Hacking says. “The other key idea is that the elementary particles of quantum physics are not points but little loops of string. This smooths out the interactions and makes the mathematics possible. Particle interaction, then, rather than an instantaneous collision, is gradual.” This leads on to mirror symmetry, which refers to paired C-Y manifolds, related in physics but mathematically very different, he adds.
Hacking, who came to campus in 2009 from the University of Washington, earned his undergraduate and advanced degrees at the University of Cambridge and completed a postdoctoral fellowship at the University of Michigan.