Schubert calculus via cluster categories

深奥的数学往往可以从看似简单的问题中产生。枚举几何领域考察了研究模问题的技术，即涉及与某些固定点或曲线的直线或曲线可能相交的数量的问题，以及这些问题的高维版本。在一个方向上，这导致了舒伯特微积分的引入，并从那里开始了对上同环的研究，其代数结构编码了交集理论。这是非常成功的，但尽管它们很强大，但归纳有时会留下未解决的基本问题。这个提议涉及到一个如此顽固的问题：一个猜想，作为这些理论的一部分引入的一个代数族中的每一个代数都是有限维的。在最小的案例中调查这一猜想，揭示了一些神秘的数字命理学。每个代数都是分级的，对于那些已知代数是有限维的情况，元素的最高出现度正好等于一个重要的数字，出现在一个非常不同的代数类别的表示理论中，预投影代数。现有的技术无法解释这一点，但我们将使用尖端的工具将猜想的设置与另一个领域的最新进展联系起来，即聚类理论，其中预投影代数发挥了重要作用。通过链接，我们将能够在两个区域之间传递信息，消除神秘感，从而解决猜想。

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.