# $A_{∞}$-Structures and Moduli Spaces

### Alexander Polishchuk

University of Oregon, Eugene, USA

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This book discusses certain moduli problems related to $A_{∞}$-structures. These structures can be viewed as a way of recording extra information on cohomology algebras. They are useful in describing derived categories appearing in geometry, and as such, they play an important role in homological mirror symmetry.

The author presents some general results on the classification of $A_{∞}$-structures. For example, he gives a sufficient criterion for the existence of a finite-type moduli scheme of $A_{∞}$-structures extending a given associative algebra. He also considers two concrete moduli problems for $A_{∞}$-structures. The first is related to the moduli spaces of curves, while the second is related to the classification of solutions of an associative version of the Yang–Baxter equation.

The book will be of interest to graduate students and researchers working in homological algebra, algebraic geometry, and related areas.