It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp extension, for all real p. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the Lp-integral curvature of a convex body. This problem is solved for positive p and is answered for negative p provided the given measure is even.

We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings, it is proved that these are exactly the torsion T such that the regular module has a special T-preenvelope. In particular, every torsion-enveloping class in Mod-R are of the form Gen(T) for a minimal silting module T. For the dual case, we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form Cogen(T), where T is a cosilting module.

We construct SYZ mirrors of the local Calabi--Yau manifolds of type $\widetilde{A}$ by developing an equivariant SYZ theory for the toric Calabi--Yau manifolds of infinite-type. The equations for the SYZ mirrors involve the Riemann theta functions and generating functions of the open Gromov--Witten invariants. We obtain explicit formulae for the generating functions which are open analogs of the Yau--Zaslow formula in dimension $2$, and show that they have nice modular properties. We also relate the SYZ mirror pairs with mirror symmetry for the abelian varieties and hypersurfaces therein.

Two functions Δ and Δ_{$b$}, of interest in combinatorial geometry and the theory of linear programming, are defined and studied. Δ($d, n$) is the maximum diameter of convex polyhedra of dimension$d$with$n$faces of dimension$d$−1; similarly, Δ_{$b$}($d,n$) is the maximum diameter of bounded polyhedra of dimension$d$with$n$faces of dimension$d$−1. The diameter of a polyhedron$P$is the smallest integer$l$such that any two vertices of$P$can be joined by a path of$l$or fewer edges of$P$. It is shown that the bounded$d$-step conjecture, i.e. Δ_{$b$}$(d,2d)=d$, is true for$d$≤5. It is also shown that the general$d$-step conjecture, i.e. Δ($d, 2d$)≤$d$, of significance in linear programming, is false for$d$≥4. A number of other specific values and bounds for Δ and Δ_{$b$}are presented.

We study the effective Batalin-Vilkovisky quantization theory for chiral deformation of two dimensional conformal field theories. We establish an exact correspondence between renormalized quantum master equations for effective functionals and Maurer-Cartan equations for chiral vertex operators. The generating functions are proven to have modular property with mild holo- morphic anomaly. As an application, we construct an exact solution of quan- tum B-model (BCOV theory) in complex one dimension that solves the higher genus mirror symmetry conjecture on elliptic curves.