DERIVED CATEGORIES AND ALGEBRAIC K-THEORY OF SINGULARITIES

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Singularities are ubiquitous in mathematics and physics. In the realm of the physical world a good example of a singularity is a black hole. In mathematical terms having a singularity usually means that a denominator is becoming zero in a coefficient or in a solution to a differential equation. This project is about singularities in more abstract area of mathematics: Algebraic Geometry. Here the non-singular (that is smooth) objects are much better understood than singular ones, and yet singularities play a crucial role in the modern Algebraic Geometry such as in the Minimal Model Program (Caucher Birkar Fields Medal 2018). One way to study geometric objects and shapes in Algebraic Geometry (called algebraic varieties) is to attach algebraic invariants to them, such as numbers, rings, or categories. One of the central such modern invariants is the so-called derived category of coherent sheaves. Derived categories of coherent sheaves are much better understood for non-singular varieties, than for singular ones, for the basic reason that singularities provide a new layer of complications to deal with. In this proposal I suggest a systematic study of derived categories of singular algebraic varieties, their decomposition into simpler pieces (semiorthogonal decompositions), their numerical properties (algebraic K-theory) and the relationship between derived categories of singular varieties and their nonsingular replacements (resolutions of singularities). The study is connected to several areas of modern pure mathematics: Algebra, Algebraic Geometry, Homological Algebra, Category Theory, Algebraic K-theory, and mixes these in new and meaningful ways in order to enhance our understanding of singularities.