# Penn Mathematicians Win $10 Million Grant to Prove Homological Mirror Symmetry

By Patrick Ammerman

A team of researchers led by University of Pennsylvania mathematical physicists Tony Pantev and Ron Donagi have received a $10 million Simons Collaboration Grant to prove the Homological Mirror Symmetry Conjecture, one of mathematics’ outstanding open problems. Solving this has potential applications in fields from particle physics to geometry.

“Homological mirror symmetry has generated a lot of deep research and interesting theorems,” said Pantev, a professor of Mathematics in the School of Arts & Sciences. “The ideas have gestated enough that we can really push and converge on a method that would solve it.”

The conjecture concerns what are called Calabi-Yau spaces, tiny, six-dimensional curved spaces whose properties were originally hypothesized in 1957 by Eugenio Calabi, a now-retired Penn mathematician, and proven 21 years later by Shing-Tung Yau. According to string theory, all matter is made up of vibrating strings wrapped around these Calabi-Yau spaces, strings that create musical notes we “hear” as electrons, protons, photons and gravitons.

It did not take long for physicists to realize the overwhelming importance of these spaces in string theory. One famous paper showed that the properties of these “musical” notes are similar to the properties of the particles physicists detect in particle accelerators. Physicists also noticed that very often Calabi-Yau spaces came in pairs, which they called “mirror spaces.” Though the geometry of a mirror space looks nothing like that of the original, these spaces have an identical effect on particle physics. It’s as if a violin and tuba played the exact same music and a listener could not tell which instrument was being used.

Mathematicians, however, dismissed these “mirror spaces” because no known geometric operations related pairs of spaces in such a way. Then in 1991 a group of physicists used mirror symmetry to propose a revolutionary approach to enumerative geometry, the branch of mathematics that counts solutions to algebraic equations, solving a century-old open problem. Mathematicians could no longer ignore them, leading to a period of collaboration between the two disciplines.

Homological mirror symmetry, proposed in 1994 by collaboration team member Maxim Kontsevich, goes a step further, mathematically formulating the existence of Calabi-Yau mirror pairs and looking at the relation between them. When this conjecture first emerged, Pantev was finishing his doctoral work at Penn in algebraic geometry and mathematical physics. Donagi, his adviser, was working on a related problem in the field. Before long, they combined forces to work the problem together. Now, 20 years later, they have forged a partnership with the leading researchers in the field from Harvard, Brandeis, Columbia and Stony Brook universities, the Institut des Hautes Etudes Scientifiques, the University of Miami and the University of California, Berkeley, to hopefully reach a solution.

“It’s just a wonderful synthesis of so many different streams within modern mathematics,” said Donagi, a professor in Penn’s Mathematics and Physics & Astronomy departments.

Pantev said he believes the time is right for this new partnership, which includes experts in three different areas of math, each of whom has previously contributed to solving a piece of the Homological Mirror Symmetry Conjecture. By combining these approaches, the team hopes to answer what’s so far remained a mystery.

Nearly 100 research teams sought funding from The Simons Collaboration, the first such call-for-proposals from The Simons Foundation. From that pool, Pantev’s team was one of two to receive a grant after making in-person presentations. The other award went to a Stanford University–based group of physicists that also includes Vijay Balasubramanian, the Cathy and Marc Lasry Professor of Physics at Penn.

The Simons Collaboration grant provides $10 million over five years for research projects in mathematics, physics and computer science, with the possibility of a three-year renewal and an additional $7.5 million.