The Simons Collaboration program in Homological Mirror Symmetry at Harvard CMSA and Brandeis University is part of the bigger Simons collaboration program on Homological mirror symmetry (https://schms.math.berkeley.edu) which brings to CMSA experts on algebraic geometry, Symplectic geometry, Arithmetic geometry, Quantum topology and mathematical aspects of high energy physics, specially string theory with the goal of proving the homological mirror symmetry conjecture (HMS) in full generality and explore its applications. Mirror symmetry, which emerged in the late 1980s as an unexpected physical duality between quantum field theories, has been a major source of progress in mathematics. At the 1994 ICM, Kontsevich reinterpreted mirror symmetry as a deep categorical duality: the HMS conjecture states that the derived category of coherent sheaves of a smooth projective variety is equivalent to the Fukaya category of a mirror symplectic manifold (or Landau-Ginzburg model). We are happy to announce that the Simons Foundation has agreed to renew funding for the HMS collaboration program for three additional years.

A brief induction of the Brandeis-Harvard CMSA HMS/SYZ research agenda and team members are as follow:

Directors:

Shing-Tung Yau (Harvard University)

Born in Canton, China, in 1949, S.-T. Yau grew up in Hong Kong, and studied in the Chinese University of Hong Kong from 1966 to 1969. He did his PhD at UC Berkeley from 1969 to 1971, as a student of S.S. Chern. He spent a year as a postdoc at the Institute for Advanced Study in Princeton, and a year as assistant professor at SUNY at Stony Brook. He joined the faculty at Stanford in 1973. On a Sloan Fellowship, he spent a semester at the Courant Institute in 1975. He visited UCLA the following year, and was offered a professorship at UC Berkeley in 1977. He was there for a year, before returning to Stanford. He was a plenary speaker at the 1978 ICM in Helsinki. The following year, he became a faculty member at the IAS in Princeton. He moved to UCSD in 1984. Yau came to Harvard in 1987, and was appointed the Higgins Professor of Mathematics in 1997. He has been at Harvard ever since. Yau has received numerous prestigious awards and honors throughout his career. He was named a California Scientist of the Year in 1979. In 1981, he received a Oswald Veblen Prize in Geometry and a John J. Carty Award for the Advancement of Science, and was elected a member of the US National Academy of Sciences. In 1982, he received a Fields Medal for “his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex MongeAmpre equations”. He was named Science Digest, America’s 100 Brightest Scientists under 40, in 1984. In 1991, he received a Humboldt Research Award from the Alexander von Humboldt Foundation in Germany. He was awarded a Crafoord Prize in 1994, a US National Medal of Science in 1997, and a China International Scientific and Technological Cooperation Award, for “his outstanding contribution to PRC in aspects of making progress in sciences and technology, training researchers” in 2003. In 2010, he received a Wolf Prize in Mathematics, for “his work in geometric analysis and mathematical physics”. Yau has also received a number of research fellowships, which include a Sloan Fellowship in 1975-1976, a Guggenheim Fellowship in 1982, and a MacArthur Fellowship in 1984-1985. Yau’s research interests include differential and algebraic geometry, topology, and mathematical physics. As a graduate student, he started to work on geometry of manifolds with negative curvature. He later became interested in developing the subject of geometric analysis, and applying the theory of nonlinear partial differential equations to solve problems in geometry, topology, and physics. His work in this direction include constructions of minimal submanifolds, harmonic maps, and canonical metrics on manifolds. The most notable, and probably the most influential of this, was his solution of the Calabi conjecture on Ricci flat metrics, and the existence of Kahler-Einstein metrics. He has also succeeded in applying his theory to solve a number of outstanding conjectures in algebraic geometry, including Chern number inequalities, and the rigidity of complex structures of complex projective spaces. Yau’s solution to the Calabi conjecture has been remarkably influential in mathematical physics over the last 30 years, through the creation of the theory of Calabi-Yau manifolds, a theory central to mirror symmetry. He and a team of outstanding mathematicians trained by him, have developed many important tools and concepts in CY geometry and mirror symmetry, which have led to significant progress in deformation theory, and on outstanding problems in enumerative geometry. Lian, Yau and his postdocs have developed a systematic approach to study and compute period integrals of CY and general type manifolds. Lian, Liu and Yau (independently by Givental) gave a proof of the counting formula of Candelas et al for worldsheet instantons on the quintic threefold. In the course of understanding mirror symmetry, Strominger, Yau, and Zaslow proposed a new geometric construction of mirror symmetry, now known as the SYZ construction. This has inspired a rapid development in CY geometry over the last two decades. In addition to CY geometry and mirror symmetry, Yau has done influential work on nonlinear partial differential equations, generalized geometry, Kahler geometry, and general relativity. His proof of positive mass conjecture is a widely regarded as a cornerstone in the classical theory of general relativity. In addition to publishing well over 350 research papers, Yau has trained more than 60 PhD students in a broad range of fields, and mentored dozens of postdoctoral fellows over the last 40 years.

Professor Bong Lian (Brandeis University)

Born in Malaysia in 1962, Bong Lian completed his PhD in physics at Yale University under the direction of G. Zuckerman in 1991. He joined the permanent faculty at Brandeis University in 1995, and has remained there since. Between 1995 and 2013, he had had visiting research positions at numerous places, including the National University of Taiwan, Harvard University, and Tsinghua University. Lian received a J.S. Guggenheim Fellowship in 2003. He was awarded a Chern Prize at the ICCM in Taipei in 2013, for his “influential and fundamental contributions in mathematical physics, in particular in the theory of vertex algebras and mirror symmetry.” He has also been co-Director, since 2014, of the Tsinghua Mathcamp, a summer outreach program launched by him and Yau for mathematically talented teenagers in China. Since 2008, Lian has been the President of the International Science Foundation of Cambridge, a non-profit whose stated mission is “to provide financial and logistical support to scholars and universities, to promote basic research and education in mathematical sciences, especially in the Far East.” Over the last 20 years, he has mentored a number of postdocs and PhD students. His research has been supported by an NSF Focused Research Grant since 2009. Published in well over 60 papers over 25 years, Lian’s mathematical work lies in the interface between representation theory, Calabi-Yau geometry, and string theory. Beginning in the late 80’s, Lian, jointly with Zuckerman, developed the theory of semi-infinite cohomology and applied it to problems in string theory. In 1994, he constructed a new invariant (now known as the Lian- Zuckerman algebra) of a topological vertex algebra, and conjectured the first example of a G algebra in vertex algebra theory. The invariant has later inspired a new construction of quantum groups by I. Frenkel and A. Zeitlin, as semi-infinite cohomology of braided vertex algebras, and led to a more recent discovery of new relationships between Courant algebroids, A-algebras, operads, and deformation theory of BV algebras. In 2010, he and his students Linshaw and Song developed important applications of vertex algebras in equivariant topology. Lian’s work in CY geometry and mirror symmetry began in early 90’s. Using a characteristic p version of higher order Schwarzian equations, Lian and Yau gave an elementary proof that the instanton formula of Candelas et al implies Clemens’s divisibility conjecture for the quintic threefold, for infinitely many degrees. In 1996, Lian (jointly with Hosono and Yau) answered the so-called Large Complex Structure Limit problem in the affirmative in many important cases. Around the same year, they announced their hyperplane conjecture, which gives a general formula for period integrals for a large class of CY manifolds, extending the formula of Candelas et al. Soon after, Lian, Liu and Yau (independently by Givental) gave a proof of the counting formula. In 2003, inspired by mirror symmetry, Lian (jointly with Hosono, Oguiso and Yau) discovered an explicit counting formula for Fourier-Mukai partners, and settled an old problem of Shioda on abelian and K3 surfaces. Between 2009 and 2014, Lian (jointly with Bloch, Chen, Huang, Song, Srinivas, Yau, and Zhu) developed an entirely new approach to study the so-called Riemann-Hilbert problem for period integrals of CY manifolds, and extended it to general type manifolds. The approach leads to an explicit description of differential systems for period integrals with many applications. In particular, he answered an old question in physics on the completeness of Picard-Fuchs systems, and constructed new differential zeros of hypergeometric functions.

Denis Auroux (Harvard University)

Denis Auroux’s research concerns symplectic geometry and its applications to mirror symmetry. While his early work primarily concerned the topology of symplectic 4-manifolds, over the past decade Auroux has obtained pioneering results on homological mirror symmetry outside of the Calabi-Yau setting (for Fano varieties, open Riemann surfaces, etc.), and developed an extension of the SYZ approach to non-Calabi-Yau spaces.After obtaining his PhD in 1999 from Ecole Polytechnique (France), Auroux was employed as Chargé de Recherche at CNRS and CLE Moore Instructor at MIT, before joining the faculty at MIT in 2002 (as Assistant Professor from 2002 to 2004, and as Associate Professor from 2004 to 2009, with tenure starting in 2006). He then moved to UC Berkeley as a Full Professor in 2009.
Auroux has published over 30 peer-reviewed articles, including several in top journals, and given 260 invited presentations about his work. He received an Alfred P. Sloan Research Fellowship in 2005, was an invited speaker at the 2010 International Congress of Mathematicians, and in 2014 he was one of the two inaugural recipients of the Poincaré Chair at IHP. He has supervised 10 PhD dissertations, won teaching awards at MIT and Berkeley, and participated in the organization of over 20 workshops and conferences in symplectic geometry and mirror symmetry.

Senior Personnel: