Zhaohu Nie

ON CHARACTERISTIC INTEGRALS OF TODA FIELD THEORIES

Characteristic integrals of Toda field theories associated to general simple Lie algebras are constructed using systematic techniques, and complete mathematical proofs are provided. Plenty of examples illustrating the results are presented in explicit forms.

A functor converting equivariant homology to homotopy

In this paper, an equivariant version of the classical Dold-Thom theorem is proved. Let G be a finite group, X a G-space, and k a covariant coefficient system onG. Then a topological abelian group GX GF k is constructed by the coend construction. For a G-CW complex X, it is proved

KAROUBI'S CONSTRUCTION FOR MOTIVIC COHOMOLOGY OPERATIONS

We use an analogue of Karoubis construction in the motivic situation to give some cohomology operations in motivic cohomology. We prove many properties of these operations, and we show that they coincide, up to some nonzero constants, with the reduced power operations in motivic cohomology originally constructed by Voevodsky. The relation of our construction to Voevodskys is, roughly speaking, that of a fixed point set to its associated homotopy fixed point set.

Classification of solutions to Toda systems of types C and B with singular sources

In this paper, the classification in Lin et al. (Invent. Math. 190(1):169–207, 2012) of solutions to Toda systems of type A with singular sources is generalized to Toda systems of types C and B. Like in the A case, the solution space is shown to be parametrized by the abelian subgroup and a subgroup of the nilpotent subgroup in the Iwasawa decomposition of the corresponding complex simple Lie group. The method is by studying the Toda systems of types C and B as reductions of Toda systems of type A with symmetries. The theories of Toda systems as integrable systems as developed in Leznov (Teoret. Mat. Fiz. 42(3):343–349, 1980), Leznov and Saveliev (Group-theoretical methods for integration of nonlinear dynamical systems, Progress in Physics, vol. 15. Birkhäuser, Verlag, 1992), Nie (J. Geom. Phys. 62(12):2424–2442, 2012), Nie (J. Nonlinear Math. Phys. 21(1):120–131, 2014), in particular the W-symmetries and the iterated integral solutions, play essential roles in this work, together with certain characterizing properties of minors of symplectic and orthogonal matrices.

On Sha's Secondary Chern-Euler Class

For a manifold with boundary, the restriction of Chern’s transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern– Euler class and was used by Sha to formulate a relative Poincare–Hopf theorem under the condition that the metric on the manifold is locally product near the boundary. We show that the secondary Chern–Euler form is exact away from the outward and inward unit normal vectors of the boundary by explicitly constructing a transgression form. Using Stokes’ theorem, this evaluates the boundary term in Sha’s relative Poincare–Hopf theorem in terms of more classical indices of the tangential projection of a vector field. This evaluation in particular shows that Sha’s relative Poincare–Hopf theorem is equivalent to the more classical law of vector fields. Department of Mathematics, Penn State Altoona, Altoona, PA 16601, USA Current Address: Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA e-mail: znie@psu.edu Received by the editors August 4, 2009. Published electronically May 13, 2011. AMS subject classification: 57R20, 57R25.

Liouville integrability of a class of integrable spin Calogero-Moser systems and exponents of simple Lie algebras

In previous work, we introduced a class of integrable spin Calogero-Moser systems associated with the classical dynamical r-matrices with spectral parameter, as classified by Etingof and Varchenko for simple Lie algebras. Here the main purpose is to establish the Liouville integrability of these systems by a uniform method based on evaluating the primitive invariants of Chevalley on the Lax operators with spectral parameter. As part of our analysis, we will develop several results concerning the algebra of invariant polynomials on simple Lie algebras and their expansions.

The Secondary Chern-Euler Class for a General Submanifold

We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study index for a vector field with non-isolated singularities on a submanifold. As an application, our studies give conceptual proofs of a classical result of Chern.

Blow-up formulas and smooth birational invariants

We prove that the blow-up formula for the singular homology of a complex smooth projective variety with a smooth center respects two natural filtrations, namely the topological and the geometric filtrations. This then enables us to establish some smooth birational invariants.