Schubert calculus via cluster categories

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Deep mathematics can often arise from seemingly simple questions. The field of enumerative geometry examines techniques to study moduli problems, that is, questions involving the number of possible intersections of lines or curves with some fixed points or curve, and higher-dimensional versions of these. In one direction, this led to the introduction of Schubert calculus and, from there, to the study of cohomology rings, whose algebraic structure encodes the intersection theory. This has been very successful but powerful as they are, generalisations can sometimes leave behind unresolved fundamental questions. This proposal concerns one such stubborn question: a conjecture that each of the algebras in a family that has been introduced as part of these theories is finite-dimensional. Investigating this conjecture in the smallest cases has brought to light some mysterious numerology. Each algebra is graded and for those cases where the algebra is known to be finite-dimensional, the highest occurring degree of an element is precisely equal to an important number appearing in the representation theory of a very different class of algebras, the preprojective algebras. Existing technology is unable to explain this but we will use cutting-edge tools to make a link between the setting of the conjecture and very recent progress in another area, that of cluster theory, in which the preprojective algebras play an important role. Through the link, we will be able to transfer information between the two areas, remove the mystery and so resolve the conjecture.