In 2014, the Simons Foundation, one of the world’s largest and most prominent basic science philanthropies, launched an unprecedented program to fund multi-year, international research collaborations in mathematics and theoretical physics. These are $10M grants over four years, renewable, that aim to drive progress on fundamental scientific questions of major importance in mathematics, theoretical physics, and theoretical computer science. There were 82 proposals in this first round. In September 2015, two were funded. Both involve Brandeis.
Matthew Headrick (Physics) is deputy director of the Simons Collaboration It from Qubit, which involves 16 faculty members at 15 institutions in six countries. This project is trying from multiple angles to bring together physics and quantum information theory, and show how some fundamental physical phenomena (spacetime, black holes etc.) emerge from the very nature of quantum information. Fundamental physics and quantum information theory remain distinct disciplines and communities, separated by significant barriers to communication and collaboration. “It from Qubit” is a large-scale effort by some of the leading researchers in both communities to foster communication, education and collaboration between them, thereby advancing both fields and ultimately solving some of the deepest problems in physics. The overarching scientific questions motivating the Collaboration include:
- Does spacetime emerge from entanglement?
- Do black holes have interiors?
- Does the universe exist outside our horizon?
- What is the information-theoretic structure of quantum field theories?
- Can quantum computers simulate all physical phenomena?
- How does quantum information flow in time?
Bong Lian (Mathematics) is a member of the Simons Collaboration on Homological Mirror Symmetry, which involves nine investigators from eight different institutions in three countries. Mirror Symmetry, first discovered by theoretical physicists in late ‘80s, is a relationship between two very different-looking physical models of Nature, a remarkable equivalence or “duality” between different versions of a particular species of multidimensional space or shape (Calabi-Yau manifolds) that seemed to give rise to the same physics. People have been trying to give a precise and general mathematical description of this mirroring ever since, and in the process have generated a long list of very surprising and far-reaching mathematical predictions and conjectures. The so-called “Homological Mirror Symmetry Conjecture” (HMS) may be thought of as a culmination of these efforts, and Lian was a member of the group (including S.-T. Yau) that gave a proof of a precursor to HMS in a series of papers in the late ‘90s.
Lian and his Simons collaborators are determined to prove HMS in full generality and explore its applications. One consequence of HMS says that if one starts from a “complex manifold” (a type of even-dimensioned space that geometers have been studying since Riemann described the first examples in 1845), then all its internal geometric structures can in fact be described using a certain partner space, called a “symplectic manifold”. The latter type of space was a mathematical edifice invented to understand classical physics in the mid-1900s. This connection goes both ways: any internal geometric structure of the symplectic partner also has an equally compelling description using the original complex partner. No one had even remotely expected such a connection, especially given that the discoveries of the two types of spaces — complex and symplectic — were separated by more than 100 years and were invented for very different reasons. If proven true, HMS will give us ways to answer questions about the internal geometric structure of a complex manifold by studying its symplectic partner, and vice versa.
Proving HMS will also help resolve many very difficult problems in enumerative geometry that for more than a century were thought to be intractable. Enumerative geometry is an ancient (and until recently moribund) branch of geometry in which people count the number of geometric objects of a particular type that can be contained inside a space. Mirror symmetry and HMS have turned enumerative geometry into a new way to characterize and relate shapes and spaces.