1. Introduction & Purposes

Matrix Factorizations act as a powerful bridge between commutative algebra, algebraic geometry, and mathematical physics, converting the abstract homological study of singularities into concrete linear algebra. Specifically, by a theorem of Orlov, they give an explicit, computable model for the singularity category of a hypersurface, which is the part of the derived category that "only sees" the singularities. Since their introduction, matrix factorizations have found far-reaching applications in the study of hypersurface singularities, the description of B-branes in Landau–Ginzburg models in homological mirror symmetry, and in providing the necessary algebraic infrastructure for Khovanov–Rozansky link homology.

This minicourse will be an introduction to these ideas, with an emphasis on concrete computations.

2. Outline & Descriptions

(1) Regularity and the Singularity Category

(2) Cohen-Macaulay modules on hypersurface singularities

(3) D-Branes in Landau-Ginzburg Model and Orlov’s equivalence

(4) Example Computations

3. Prerequisites

●   Fundamentals of algebraic geometry and homological algebra

●   Basic knowledge of the derived category (of coherent sheaves)

4. Registration

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