Applied homotopy theory

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The applications of homotopy theory form the most compelling aspects of the subject. Combinatorial models have always been particularly important, and now pervade the subject. More generally, simplicial structures exist throughout algebraic geometry, and homotopy types arise from geometric objects in many ways - the formulation of the higher algebraic K-theory of varieties was an early, provocative example. Calculational difficulties associated with K-theory led to the introduction of cohomological descent, which was formalized in the local homotopy theory of simplicial presheaves in the late 1980s. Local homotopy theory is the basis for motivic homotopy theory, and gives a conceptual context for the theory of stacks. Presheaves of spectra on the stack of formal group laws have emerged in defining elliptic cohomology theories and topological modular forms. Many of these ideas have applications in mathematical physics and computer science.