Representation Theory of Finite Dimensional Algebras

PI将通过各种几何、同调和组合方法研究有限维代数的表示，后者包括对Groebner技术的改编。她的第一个项目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），并在格拉斯曼品种中进行投影设置。结合不同的观点，她将探索具有固定顶部或固定激进层的表示的模问题，并对给定表示的退化进行分类，这些退化使顶部或更强烈地说，激进层保持不变。第二个项目是同系学的，专注于一类单调代数（弦代数），它首先出现在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（在这一点上很容易理解）扩展到无限维表示。特别是，她将讨论有限射影维模块的结构，以及在整个模块类别中，这些模块的完整子类别的位置（以内外映射的方式）。第三个项目针对派生类别。在这个方向上，PI将继续对链配合物的几何结构进行先前的研究。她希望应用于被称为“无环猜想”的同调问题。从更广泛的角度来看，该项目可以定位如下：我们的大多数宇宙物理模型都存在于有限维向量空间中。这些空间中最有趣的带有附加结构，例如李或关联代数乘法。联想有限维代数是研究者研究的对象。理解它们的一种方法是探索它们的“表示”，这可以被认为是代数的线性化快照，在矩阵的代数中实现——粗粒度和细粒度快照都对突出代数的不同特征感兴趣。

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.