The main goal of this paper is to prove the following two conjectures for genus up to two:
1. Witten's conjecture on the relations between higher spin curves and Gelfand--Dickey hierarchy.
2. Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds.
The main technique used in the proof is the invariance of tautological equations under loop group action.
Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.Feng QuInternational Center for Mathematical Research (BICMR), Beijing (Peking) University, Beijing 100871
The purpose of this short article is to prove a product formula relating the log Gromov-Witten invariants of V×W with those of V and W in the case the log structure on V is trivial.
D. ArcaraDepartment of Mathematics, Saint Vincent College, Latrobe PA 15650Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112
We prove that all monomials of κ-classes and ψ-classes are independent in $R^k(\ocM_{g,n})/R^k(\partial\ocM_{g,n})$ for all k≤[g/3]. We also give a simple argument for κ_l≠0 in $R^l(\ocM_g)$ for l≤g−2.
Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112R. VakilDepartment of Mathematics, Stanford University, Stanford (Palo Alto), CA 94305
We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems.
Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112R. PandharipandeDepartment of Mathematics, Princeton University, Princeton, New Jersey, 08540
The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over the rationals) of the corresponding cobordism groups over Spec(C) for all dimensions of varieties and ranks of bundles. The basis consists of split bundles over products of projective spaces. Moreover, we prove the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism.