The overall theme of the investigator's research is the application of homotopy theory to other areas of mathematics and to physics, in particular to the understanding of strings and similar structures. Among these structures, most directly related to of string theory is conformal field theory, which also appears to be closely connected to elliptic cohomology. This in turn has implications in the area of automorphic forms, and relations with Borcherds' Moonshine module. There is also a remarkable connection with the absolute Galois group of Q, through Grothendieck's program of dessins d'enfants, and the action of Grothendieck-Teichmueller structures on the formalism that make up conformal field theories. By way of analogy, studying the space of loops in a topological space leads to the string topology, which is closely related to operads and deformation theory in more abstract contexts. To approach these questions, the investigator uses categorical Koszul duality. Related to Koszul duality is the notion of Grothendieck/Verdier duality in various contexts, which led back to A^1- and equivariant stable homotopy theory, including Real-oriented homotopy theory. This is directly analogous to the A^1-homotopy theory constructed by Morel and Voevodsky.

The overall theme of the investigator's research is the interaction between the area of algebraic topology in mathematics, and string theory in physics. String theory emerged in physics in the 70's and 80's as a theory providing new hope toward gaining a unified theory of all the forces of nature; its basic idea was that the fundamental units of the universe are not point-like particles, but tiny one-dimensional strings, which may form a closed loop, or may be open (with two endpoints). The theory since underwent a tumultuous development in which many additional structures emerged. Today it is still believed in physics that the unification program may be facilitated by some theory closely related to structures discovered by string theory. Despite the long history, many fundamental aspects of string theory are not well mathematically understood, which, it seems, has somewhat hindered the physical theory. The investigator studies some of these issues both directly, and by way of analogy 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 for studying topological spaces, by associating to them certain algebraic/numerical invariants. In physics, spacetime itself can be considered as a topological space. However, to understand strings in spacetime, 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 in topological spaces. This includes conformal field theory and string topology, as well as certain structures from the area of algebra, such as operads and Hochschild cohomology. The investigator also considers the structures arising from loops in certain purely abstract, general context, for instance that of categorical Koszul duality.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
0503814
Program Officer
Joanna Kania-Bartoszynska
Project Start
Project End
Budget Start
2005-07-01
Budget End
2010-06-30
Support Year
Fiscal Year
2005
Total Cost
$88,712
Indirect Cost
Name
Wayne State University
Department
Type
DUNS #
City
Detroit
State
MI
Country
United States
Zip Code
48202