String-related structures in homotopy theory

点击翻译按钮获取中文摘要

The overall theme of the investigator's research is the application ofhomotopy theory to other areas of mathematics and to physics, inparticular to the understanding of strings and similar structures. Amongthese structures, most directly related to of string theory is conformalfield theory, which also appears to be closely connected to ellipticcohomology. This in turn has implications in the area of automorphicforms, and relations with Borcherds' Moonshine module. There is also aremarkable connection with the absolute Galois group of Q, throughGrothendieck's program of dessins d'enfants, and the action ofGrothendieck-Teichmueller structures on the formalism that make upconformal field theories. By way of analogy, studying the space of loopsin a topological space leads to the string topology, which is closelyrelated to operads and deformation theory in more abstract contexts. Toapproach these questions, the investigator uses categorical Koszulduality. Related to Koszul duality is the notion of Grothendieck/Verdierduality in various contexts, which led back to A^1- and equivariant stablehomotopy theory, including Real-oriented homotopy theory. This is directlyanalogous to the A^1-homotopy theory constructed by Morel and Voevodsky.-------------------------------------The overall theme of the investigator's research is the interactionbetween the area of algebraic topology in mathematics, and string theoryin physics. String theory emerged in physics in the 70's and 80's as atheory providing new hope toward gaining a unified theory of all theforces of nature; its basic idea was that the fundamental units of theuniverse are not point-like particles, but tiny one-dimensional strings,which may form a closed loop, or may be open (with two endpoints). Thetheory since underwent a tumultuous development in which many additionalstructures emerged. Today it is still believed in physics that theunification program may be facilitated by some theory closely related tostructures discovered by string theory. Despite the long history, manyfundamental aspects of string theory are not well mathematicallyunderstood, which, it seems, has somewhat hindered the physical theory.The investigator studies some of these issues both directly, and by way ofanalogy and connection with similar structures elsewhere in mathematics.Topology can be thought of as geometry in its purely qualitative aspects.In particular, algebraic topology has built up many powerful methods forstudying topological spaces, by associating to them certainalgebraic/numerical invariants. In physics, spacetime itself can beconsidered as a topological space. However, to understand strings inspacetime, one needs to study not only the points in topological spaces,but loops in such spaces, which can be thought of as models for strings.The investigator studies the structures arising from the space of loops intopological spaces. This includes conformal field theory and stringtopology, as well as certain structures from the area of algebra, such asoperads and Hochschild cohomology. The investigator also considers thestructures arising from loops in certain purely abstract, general context,for instance that of categorical Koszul duality.