# Oberwolfach Reports

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**Volume 10, Issue 3, 2013, pp. 2501–2552**

**DOI: 10.4171/OWR/2013/44**

Published online: 2014-06-01

Matrix Factorizations in Algebra, Geometry, and Physics

Ragnar-Olaf Buchweitz^{[1]}, Kentaro Hori

^{[2]}, Henning Krause

^{[3]}and Christoph Schweigert

^{[4]}(1) University of Toronto at Scarborough, Canada

(2) The University of Tokyo, Kashiwa, Japan

(3) Universität Bielefeld, Germany

(4) Universität Hamburg, Germany

Let $W$ be a polynomial or power series in several variables, or, more generally, a nonzero element in some regular commutative ring. A matrix factorization of $W$ consists of a pair of square matrices $X$ and $Y$ of the same size, with entries in the given ring, such that the matrix product $XY$ is $W$ multiplied by the identity matrix. For example, if $X$ is a matrix whose determinant is $W$ and $Y$ is its adjoint matrix, then $(X, Y)$ is a matrix factorization of $W$.

Such matrix factorizations are nowadays ubiquitous in several different fields in physics and mathematics, including String Theory, Commutative Algebra, Algebraic Geometry, both in its classical and its noncommutative version, Singularity Theory, Representation Theory, Topology, there in particular in Knot Theory.

The workshop has brought together leading researchers and young colleagues from the various input fields; it was the first workshop on this topic in Oberwolfach. For some leading researchers from neighboring fields, this was their first visit to Oberwolfach.

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Buchweitz Ragnar-Olaf, Hori Kentaro, Krause Henning, Schweigert Christoph: Matrix Factorizations in Algebra, Geometry, and Physics. *Oberwolfach Rep.* 10 (2013), 2501-2552. doi: 10.4171/OWR/2013/44