Simons Collaboration on Homological Mirror Symmetry

The Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring 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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:

  1. categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
  2. the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
  3. Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.

The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:

  • Jacob Bourjaily (Neils Bohr Institute)  4/1/2018 – 4/14/2018
  • Colin Diemer (IHES)  2/25/2018 – 3/10/2018
  • Baohua Fu (Chinese Academy of Sciences4/15/2018 – 4/28/2018
  • Andrew Harder (University of Miami4/15/2018 – 4/28/2018
  • Shinobu Hosono (Gakushuin University) 2/25/2018 – 3/10/2018
  • Adam Jacob (UC Davis) 3/5/2018/ – 3/16/2018
  • Ivan Loseu (Northeastern University) 1/21/2018 – 2/3/2018
  • Cheuk-Yu Mak (Cambridge University) 4/1/2018 – 4/15/2018
  • Daniel Pomerleano (Imperial College) 3/19/2018 – 3/23/2018
  • Mauricio Romo (Tsinghua University) 4/1/2018 – 4/18/ 2018
  • Emanuel Scheidegger (Albert Ludwigs University of Freiburg) 2/22/2018 – 3/22/2018
  • Dmytro Shklyarov (Technische Universität Chemnitz) 3/4/2018 – 3/17/2018
  • Matt Young (Chinese University of Hong Kong) 1/15/2018 – 2/9/2018
  • Jeng-Daw Yu (National Taiwan University) 4/2/2018 – 4/6/2018
  • Minxian Zhu (Yau Mathematical Sciences Center, Tsinghua University) 1/ 22/2018 – 2/25/2018

The Center will also be hosting a four-day Simons Collaboration Workshop on Homological Mirror Symmetry and Hodge Theory on January 10-13, 2018, and another April 5-7. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. Click here for more information.