Higher Spin Curves and Cohomological Field Theories

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The Investigator studies intersection theory on and thecohomological field theory arising from the moduli of algebraicspin curves and stable spin maps. Especially important in thisstudy is the exploration of the plethora of similarities betweenGromov-Witten theory and cohomological field arising from higherspin curves and spin maps. Also important in this research is thestudy of the many relations between these cohomological fieldtheories and integrable hierarchies, including significant recentprogress toward proving some remarkable conjectures like theW-algebra conjecture and Generalized Witten Conjecture.The research in this project is concerned with connections betweenalgebraic geometry and physics, and it has implications for bothsubjects. Algebraic geometry, which is the study of solutions topolynomial equations, and especially the sub-discipline ofalgebraic curves, which is one of the main foci of this research,have many applications. The most notable of these applicationsare in secure electronic communications (cryptography and errorcorrecting codes). Recent research has shown that algebraicgeometry also has significant ties to high-energy physics andplays an important role in helping us understand the fundamentalnature of the universe. Conversely, physics has helped us betterunderstand some important aspects of algebraic geometry and itsapplications. This research is focused on studying and furtherdeveloping some of these important links.