Towards a theory of singular symplectic varieties

Singular algebraic (sub)varieties are fundamental to the theory of smooth projective manifolds. In parallel with his introduction of pseudo-holomorphic curve techniques into symplectic topology 30 years ago, Gromov asked about the feasibility of introducing notions of singular (sub)varieties suitable for this field. I will describe a new perspective on this question and motivate its appropriateness in the case of normal crossings singularities. It leads to multifold versions of symplectic sum and cut constructions expected by Gromov and notions of one-parameter families of degenerations of symplectic manifolds and logarithmic tangent bundles in the spirit of the Gross-Siebert program. In our category, the standard triple point condition of algebraic geometry is the only obstruction to the smoothability of NC singularities. This is joint work with M. Farajzadeh Tehrani and M. McLean, some completed and some in preparation.