RUI: Interactions between Homotopy Theory and Algebra

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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.