Bong Lian (Brandeis) on Homological Mirror Symmetry

CMSA recently hosted a three-day Simons Collaboration workshop on Homological Mirror Symmetry. We interviewed Professor Bong Lian (Brandeis University), one of the workshop’s organizers, about the event and the field.

Could you please introduce yourself and tell us a bit about your academic/scientific background?

My name is Bong Lian. I received my PhD in Physics from Yale University in 1991, and am currently on the faculty of the Mathematics Department at Brandeis. My research has been in String Theory, and I’m particularly interested in algebraic and geometric aspects of physical models arising from this theory.

How did you come to be involved in the field of homological mirror symmetry? Why is this subject of interest to researchers?

Mirror Symmetry was a phenomenon discovered by physicists in the late 1980s, as a result of an intriguing ambiguity in interpreting a certain class of string theory models in two apparently different but physically equivalent ways. Both physicists and mathematicians have been working since to give a precise and general mathematical framework for this remarkable equivalence or “mirroring”, and in the process have inspired a wealth of unexpected, yet 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 I was lucky enough to have been part of those efforts. Because of its many surprising and deeply consequential predictions, proving the HMS Conjecture is expected to require revolutionary ideas that would likely transcend traditional boundaries between mathematical subjects, perhaps even boundaries between mathematics and physics. As such, this problem has attracted significant interest of researchers from traditionally very different backgrounds and expertise.

What prompted the organization of the Simons Collaboration workshop on homological mirror symmetry at CMSA? How did you come to be involved in organizing it?

It’s been an honor for me to be a member of a team of nine investigators from eight different institutions that proposed the HMS project — a collaboration whose singular goal was to solve the conjecture. The project is directed by T. Pantev of UPenn, and our scientific program is led by M. Kontsevich of IHES and S.-T. Yau of Harvard. We were very fortunate to have been selected by the Simons Foundation as one of two “inaugural” projects funded by the Simons Collaboration grant in September 2015. The initial phase of our project was to organize and gather expertise from relevant branches of mathematics and physics, and bring experts from the cutting edge of those areas to a few central locations including UPenn, the University of Miami, UC Berkeley, Harvard and the IHES. The goal was to try to first flesh out each others’ ideas and approaches to the problem, learn from one another, while “comparing notes” on the state of the art in the process. The Simons Collaboration workshop at the Harvard CMSA was in fact one of the gatherings (the fourth since last November) we’ve had with this goal in mind. Brandeis and Harvard have agreed to co-organize and sponsor this workshop at the CMSA, and it’s been a pleasure to be the co-organizer from Brandeis.

Is there anything in particular going on within the field that excites you? Any applications or new directions that you find personally interesting?

In recent months, the HMS conferences and workshops have given us opportunities to hear many new ideas being introduced in symplectic, traditional algebraic, as well as derived algebraic geometry, not to mention physics itself. We have witnessed a remarkable synthesis of these ideas in a range of mirror-symmetry-related contexts — from deriving results in enumerative geometry for Calabi-Yau manifolds, to describing the mirror equivalence for elliptic orbifolds and their generalizations, to establishing new insights into quantum geometry, wall-crossing phenomenon, and Lagrangian intersection theory. I personally find this coming together of a wealth of exciting new ideas for the one goal — proving HMS — quite fascinating. While my current research with my collaborators tends to focus on understanding some aspects of quantum geometry from a certain viewpoint, we recognize this is but one small piece of a much larger mirror symmetry puzzle, and we look forward to understanding its connections to other approaches.

Do you find that there are benefits to participating in a program like this that involves researchers from a variety of backgrounds?

Certainly. Most of us have been raised mathematically in somewhat conventional approaches that “respect” traditional boundaries of mathematics and/or physics subjects. But advances in mirror symmetry have all but eviscerated those boundaries. We feel that being immersed in this program has been very beneficial to us in understanding questions from many classical domains — questions that were long thought to be firmly embedded in one subject or another. Counting problems in enumerative and arithmetic geometry are one such example; Lagrangian intersection theory in symplectic geometry is yet another such example. We believe that mirror symmetry will continue to be a fertile ground for researchers like ourselves to discover new ways to solve old problems for generations to come.

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More information on the Simons Collaboration on Homological Mirror Symmetry can be found at https://schms.math.berkeley.edu/events/harvard2016/