RUI: Interactions between Homotopy Theory and Algebra

对于每一类经典代数（结合代数，交换代数，李代数），都有一个与之相关联的一类简单代数。这些范畴的对象由一系列代数以及满足一定关系的相邻层之间的面和简并映射组成。简单代数的相关同伦理论为与原始代数类型相关的同伦代数的建模提供了一种设置（从而为派生代数的研究提供了基础）。因此，这个类别提供了一个自然的地方来研究这些代数利用标准同调装置，如高扭转，高扩展，和切复。PI继续他对简单代数的研究，涉及到将代数实现为空间的同伦或上同调的问题，以及与切复的猜想刚性性质有关的问题。为此，PI将致力于进一步阐述代数的Andre-Quillen上同调的一般内部结构。在与D. Blanc和M. Johnson的持续合作中，PI将使用这种结构来确定和研究李代数的Andre-Quillen上同调中所表示的高同伦运算的代数性质。这反过来将用于阐明通过同伦群拓扑实现给定李代数的阻碍理论，并寻求使这些障碍可计算。PI还将通过Andre-Quillen上同调可能具有的刚性性质继续研究局部完全交代数和局部Gorenstein代数的划分。这将涉及进一步阐述由简单交换环建模的衍生交换代数的结构。再一次，Andre-Quillen上同调的内部结构将有助于阐明那些可能属于这种划分的衍生代数的性质，并使用这些性质来表征和分类它们的类型。最后，PI将开始在派生代数几何的背景下进行进一步的特殊调查。衍生代数出现在代数拓扑、代数几何和交换代数的各种背景中。从理论物理流入代数拓扑和代数几何的最新概念（如拓扑模形式，椭圆对象和手性代数）可以在代数几何的背景下基于某些类型的派生代数进行研究。这个项目将增强我们对衍生代数和衍生代数几何理论如何通过复杂的同调装置为研究这种复杂的几何结构提供一个背景的理解。PI与本科生合作，并在夏季为他们提供从事研究的机会。PI邀请来自全国各地的数学家在校园举行的学术研讨会上发言。研讨会上的演讲让学生有机会认识来自不同大学的研究人员，并接触到研究生阶段所追求的数学类型。

To every category of classical algebras (associative, commutative, Lie) there is an associated category of simplicial algebras. The objects of such categories consist of a sequence of algebras together with face and degeneracy maps between adjacent levels, satisfying certain relations. The associated homotopy theory of simplicial algebras provides a setting for modeling the homological algebra associated to the original algebra type (and, thus, providing a basis for the study of derived algebras). Thus this category provides a natural place to study such algebras utilizing standard homological devices such as higher torsion, higher extensions, and the tangent complex. The PI continues his research on simplicial algebras in relation to the problem of realizing algebras as the homotopy or cohomology of spaces and in relation to the conjectural rigidity properties of the tangent complex. To that end, the PI will aim to further elaborate the general internal structure of Andre-Quillen cohomology of algebras. In a continuing collaboration with D. Blanc and M. Johnson, the PI will use this structure to determine and study the algebraic properties of higher homotopy operations as they are represented in the Andre-Quillen cohomology of (a variation of) Lie algebras. This in turn will be used to elucidate the obstruction theory for topologically realizing a given Lie algebra via homotopy groups and seek to make these obstructions computable. The PI will also continue studying the divide between locally complete intersection algebras and locally Gorenstein algebras through the possible rigidity properties possessed by Andre-Quillen cohomology. This will involve elaborating further the structure of derived commutative algebras as modeled by simplicial commutative rings. Again the internal structure of Andre-Quillen cohomology will serve to help elucidate the properties of those derived algebras that possibly fall within this divide and use these properties to characterize and classify their types. Finally, the PI will begin carrying further this particular investigation within the context of derived algebraic geometry.Derived algebras appear in various contexts in algebraic topology, algebraic geometry, and commutative algebra. Recent concepts flowing into algebraic topology and algebraic geometry from theoretical physics (such as topological modular forms, elliptic objects and chiral algebras) can be studied from within the context of algebraic geometry based on certain types of derived algebras. This project will enhance our understanding of how the theory of derived algebra and derived algebraic geometry can provide a setting for studying such complex geometric structures through sophisticated homological devices. The PI works with undergraduates and opportunities are provided for them to engage in research during the summers. The PI invites mathematicians from around the country to speak at a colloquium held on campus. Talks at the colloquium give students a chance to be introduced to researchers from various universities and be exposed to the type of mathematics being pursued at the graduate level.