# Oberwolfach Reports

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

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

Matrix Factorizations in Algebra, Geometry, and Physics

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

^{[2]}, Henning Krause

^{[3]}and Christoph Schweigert

^{[4]}(1) Department of Computer and Mathematical Sciences, University of Toronto at Scarborough, 1265 Military Trail, ON M1C 1A4, Toronto, Canada

(2) Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwanoha, 277-8568, Kashiwa, Japan

(3) Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

(4) Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, 20146, 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