Investigations in the application of homotopy theory

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This project deals with three separate problems in homotopy theory. The first is the study of algebraic K-theory of fields and motivic homotopy theory. We will continue the study of the representation theoretic model for the algebraic K-theory of fields using derived completion constructions, and attempt to use motivic homotopy theory and algebraic K-theory to study the Halperin-Carlsson conjectures on free finite group actions on finite complexes. The second is internal to homotopy theory, and will carry out an analysis of higher topological cyclic homology constructions in the hopes of obtaining good geometric versions of phenomena of higher chromatic levels. The third problem is the study of various aspects of "generalized persistence", which provides a method for studying finite metric spaces, and therefore many data sets coming out of science and engineering. Specifically, we wish to study barcode invariants of certain quivers which will help to assess the "strength" of a topological signal. This project deals with applications of certain geometric techniques, referred to as topological, to other parts of mathematics and eventually even outside of mathematics. Topological techniques capture qualitative features of geometric objects, such as the presence of loops, spheres, and other surfaces within them in a systematic way which allows it to be used as a tool for geometric pattern recognition in a broad sense. Interpreted this way, these techniques have applications in areas as diverse as number theory and the solution of algebraic equations on the one end to problems in data analysis on the other end. In addition, we will pursue an interesting direction in the study of homotopy theory, where we hope to make concrete and geometrical certain constructions which at the moment are less sharply defined.