String theory predicts its own gravity rather than GR. In General Relativity the metric is the only geometric and gravitational field, whereas in string theory the closed-string massless sector comprises a skew-symmetric B-field and the string dilaton in addition to the metric. Furthermore, these three fields transform into each other under T-duality. This hints at a natural augmentation of GR: upon treating the whole closed string massless sector as stringy graviton fields, Double Field Theory may evolve into `Stringy Gravity'. Equipped with an O(D,D) covariant differential geometry beyond Riemann, we spell out the definitions of the stringy Einstein curvature tensor and the stringy Energy-Momentum tensor. Equating them, all the equations of motion of the closed string massless sector are unified into a single expression which we dub the Einstein Double Field Equations.

How many H2O molecules are needed to form water? While the precise answer is not known, it is clear that the answer should be a finite number rather than infinity. We revisit with care the ideal Bose gas confined in a cubic box which is discussed in most statistical physics textbooks. We show that the isobar of the ideal gas zigzags on the temperature-volume plane featuring a `boiling-like' discrete phase transition, provided the number of particles is equal to or greater than a particular value: 7616. This demonstrates for the first time how a finite system can feature a mathematical singularity and realize the notion of `Emergence', without resorting to the thermodynamic limit. ref: arXiv:1310.5580

To the full order in fermions, we construct D = 10 type II supersymmetric double field theory. We spell the precise N = 2 supersymmetry transformation rules as for 32 supercharges. In terms of a stringy differential geometry beyond Riemann, the constructed action unifies type IIA and IIB supergravities in a manifestly covariant manner with respect to O(10, 10) T-duality and a ‘pair’ of local Lorentz groups, or Spin(1, 9) × Spin(9, 1), besides the usual general covariance of supergravities or the generalized diffeomorphism. The distinction of IIA and IIB may arise after a diagonal gauge fixing of the Lorentz groups. They are identified as two different types of ‘solutions’ rather than two different theories. References: arXiv:1210.5078 (N=2) arXiv:1206.3478 (bosonic N=2) arXiv:1112.0069 (N=1)

Four dimensional N=4 super Yang-Mills theory contains a bigger superalgebra than AdS or superconformal algebra, su(2,2/4). It corresponds to a noncentral extension of the latter. The talk is for both physicsts and mathematicans interested in a novel way of obtaining noncentral extensions of Lie algebras.