rectification; Abhyankar-Sathaye embedding problem; variable Virasoro conformal blocks; Painlevé transcendents; Stokes matrices; Stokes sectors; monodromy; Riemann-Hilbert problem (RHP); Fredholm determinant; Whittaker functions; Szegő constant; Toeplitz determinant; Young diagrams; Ponsot-Teschner Formula; quantum dilogarithms; connection formulas; connection constants; regular conformal blocks; irregular conformal blocks; Alday-Gaiotto-Tachikawa (AGT) relation; conformal field theory (CFT); 4D supersymmetric gauge theory; Nekrasov functions; Whittaker vectors; Whittaker module; Virasoro generators; Weyl symmetry group; Chiral vertex operator; tau function; Lax pair; Jimbo-Miwa-Ueno tau function; connection problem; Barnes \(G\)-function; Verma module; Virasoro algebra 