Kravets, O. H. Exceptional collections in categories of singularities of 3-dimensional Landau-Ginzburg models. (English. Russian original) Zbl 1375.13020 Russ. Math. Surv. 68, No. 6, 1136-1138 (2013); translation from Usp. Mat. Nauk 68, No. 6, 173-174 (2013). This is an announcement of a proof for a conjecture of Orlov in dimension less than or equal to 3. Given an invertible polynomial, it constructs a quiver with relations whose derived category of modules is equivalent to the derived category of singularities of the given polynomial.The key step is constructing a full strongly exceptional collection \((E_1,\ldots,E_m)\) of the derived category of singularities. Then by Bondal, the morphisms among \(E_i\) produce the quiver with the required property. There are two cases in dimension three: chain type and loop type. An exceptional collection is explicitly given in each case.The result is important to understand singularity theory and mirror symmetry for invertible polynomials. Reviewer: Siu Cheong Lau (Boston) Cited in 2 Documents MSC: 13D09 Derived categories and commutative rings 16E35 Derived categories and associative algebras 14J33 Mirror symmetry (algebro-geometric aspects) Keywords:Landau-Ginzburg model; exceptional collections; singularities; derived category; invertible polynomial; FJRW, ADE PDFBibTeX XMLCite \textit{O. H. Kravets}, Russ. Math. Surv. 68, No. 6, 1136--1138 (2013; Zbl 1375.13020); translation from Usp. Mat. Nauk 68, No. 6, 173--174 (2013) Full Text: DOI MNR