Representation Theory of Finite Dimensional Algebras

PI将调查表示有限维algebrasby各种几何，同调，和组合的方法，后者包括适应Groebner技术。 她的第一个项目的目标是固定维度和固定顶部的表示。为此，PI正在补充该领域的经典几何方法-主要是仿射-与格拉斯曼品种内的投影设置。 结合不同的观点，她将探索与固定的顶部或固定的激进分层和工作的那些退化的一个给定的表示，离开顶部或更强烈，激进分层不变的分类表示的模量问题。 第二个项目是同构的，重点关注一类驯服的代数（弦代数），它首次出现在Gelfand-Ponomarev关于洛伦兹群表示的工作中。 她的目标是将有限维表示的同调行为（在这一点上很好理解）扩展到无限维表示。 特别是，她将解决结构模块的有限投射尺寸和位置-在和传出的地图方面-的完整的子类别的这种模块在整个模块类别。 第三个项目是针对派生类别。 在这个方向上，PI将继续先前对链状配合物几何结构的研究。她期望应用于被称为“无圈猜想”的长期同调问题。在更广泛的角度下，该项目可以定位如下：我们对宇宙的大多数物理模型都生活在有限维向量空间中。这些空间中最有趣的是带有额外的结构，例如Lie或结合代数乘法。 结合有限维代数是研究者的研究对象。理解它们的一种方法是探索它们的“表示”，可以认为是在矩阵代数中实现的代数的线性化快照-粗粒度和细粒度快照都是为了突出代数的不同特征。

The PI will investigate representations of finite dimensional algebrasby a variety of geometric, homological, and combinatorial methods,the latter including adaptations of Groebner techniques. Her firstproject targets representations with fixed dimension and fixed top.For that purpose, the PI is supplementing the classical geometricmethods in the field -- predominantly affine -- with a projectivesetting inside Grassmannian varieties. Combining the differentviewpoints, she will explore moduli problems for representations withfixed top or fixed radical layering and work on a classification ofthose degenerations of a given representation which leave the top or,more strongly, the radical layering unchanged. The second project ishomological and focuses on a class of tame algebras (stringalgebras), which first surfaced in work of Gelfand-Ponomarev onrepresentations of the Lorentz group. Her goal is to extend the (atthis point well-understood) homological behavior of finitedimensional representations to infinite dimensional ones. Inparticular, she will address the structure of modules of finiteprojective dimension and the placement -- in terms of in- andoutgoing maps -- of the full subcategory of such modules within theentire module category. The third project is aimed at derivedcategories. In this direction, the PI will continue prior researchon the geometry of chain complexes. She expects applications to alongstanding homological problem known as the `No Loop Conjecture'.Under a wider angle, the project can be located as follows: Themajority of our physical models for the universe live in finitedimensional vector spaces. The most interesting of these spaces carryadditional structure, such as a Lie or associative algebramultiplication, for example. Associative finite dimensional algebrasare the objects of the investigator's research. One approach towardsunderstanding them is to explore their 'representations', which canbe thought of as linearized snapshots of the algebra which arerealized within algebras of matrices -- both coarse-grained andfine-grained snapshots are of interest to highlight differentfeatures of the algebra.