Kuo-Tsai Chen (1923-1987) is best known to the mathematics community
for his work on iterated integrals and power series connections in
conjunction with his research on the cohomology of loop spaces. His
work is intimately related to the theory of minimal models as
developed by Dennis Sullivan, whose own work was in part inspired by
the research of Chen.
An outstanding and original mathematician, Chen's work falls naturally
into three periods: his early work on group theory and links in the
three sphere; his subsequent work on formal differential equations,
which gradually developed into his most powerful and important work;
and his work on iterated integrals and homotopy theory, which occupied
him for the last twenty years of his life. The goal of Chen's
iterated integrals program, which is a de Rham theory for path spaces,
was to study the interaction of topology and analysis through path
integration.
The present volume is a comprehensive collection of Chen's
mathematical publications preceded by an article, "The Life and Work
of Kuo-Tsai Chen," placing his work and research interests into their
proper context and demonstrating the power and scope of his
influence.
[1] Integration in free groups.- [2] Commutator calculus and link invariants.- [3] Isotopy invariants of links.- [4] A group ring method for finitely generated groups.- [5] Iterated integrals and exponential homomorphisms.- [6] On the composition functions of nilpotent Lie groups.- [7] Integration of paths, geometric invariants and a generalized Baker—Hausdorff formula.- [8] Integration of paths, a faithful representation of paths by noncommutative formal power series.- [9] Exponential isomorphism for vector spaces and its connection with Lie groups.- [10] Free differential calculus IV.- [11] Linear independence of exponentials of Lie elements.- [12] Formal differential equations.- [12A] Formal differential equations.- [13] Decomposition of differential equations.- [13A] Decomposição de Equações Diferencias.- [14] An expansion formula for differential equations.- [15] Decomposition and equivalence of local vector fields.- [16] Expansion of solutions of differential systems.- [17] On local diffeomorphisms about an elementary fixed point.- [18] Equivalence and decomposition of vector fields about an elementary critical point.- [19] Local diffeomorphism—C? realization of formal properties.- [20] On a generalization of Picard’s approximation.- [21] On nonelementary hyperbolic fixed points of diffeomorphisms.- [22] Iterated path integrals and generalized paths.- [23] Algebraization of iterated integration along paths.- [24] Normal forms of local diffeomorphisms on the real line.- [25] Algebraic paths.- [26] Homotopy of algebras.- [27] Covering-space-like algebras.- [28] An algebraic dualization of fundamental groups.- [29] Exact dynamic systems are tree-like and vice versa.- [30] A sufficient condition for non-abelianness of fundamental groups of differentiable manifolds.- [31] Algebras of iterated path integrals and fundamental groups.- [32] Differential forms and homotopy groups.- [32A] Iterated integration and loopspace cohomology.- [33] On Whitehead products.- [34] Free subalgebras of loop space homology and Massey products.- [35] Iterated integrals of differential forms and loop space homology.- [36] Fundamental groups, nilmanifolds and iterated integrals.- [37] Solvability on manifolds by quadratures permitting only integrals.- [38] Connections, holonomy and path space homology.- [39] Iterated integrals, fundamental groups and covering spaces.- [40] Reduced bar constructions on de Rham complexes.- [41] Extension of C? function algebra by integrals and Malcev completion of ?1.- [42] Iterated path integrals.- [43] Pullback de Rham cohomology of the free path fibration.- [44] Path space differential forms and transports of connections.- [45] Circular bar construction.- [46] Poles of maps onto Pn(C) and Whitehead integrals.- [47] Pullback path fibrations, homotopies and iterated integrals.- [48] The Euler operator.- [49] On the Hopf index theorem and the Hopf invariant.- [50] Pairs of maps into complex projective space.- [50A] On differentiable spaces.- [51] Degeneracy indices and Chern classes.- [52] On the Bézout theorem.- [53] Loop spaces and differential forms.- [54] Smooth maps, pullback path spaces, connections, and torsions.