Cobordism for Hamiltonian loop group actions and flat connections on the punctured two-sphere

We develop a theory of symplectic cobordism and a Duistermaat-Heckman principle for Hamiltonian loop group actions. As an application, we construct a symplectic cobordism between moduli spaces of flat connections on the three holed sphere and disjoint unions of toric varieties. This cobordism yields formulas for the mixed Pontrjagin numbers of the moduli spaces, equivalent to Witten's formulas in the case of symplectic volumes. Received June 15, 1998     

Grope cobordism of classical knots

Motivated by the lower central series of a group, we define the notion of a grope cobordism between two knots in a 3-manifold. Just like an iterated group commutator, each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to our notion of grope cobordism. Thus our results can be viewed as a geometric interpretation of finite type invariants.The derived commutator series of a group also has a three-dimensional analogy, namely knots modulo symmetric grope cobordism. On one hand this theory maps onto the usual Vassiliev theory and on the other hand it maps onto the Cochran-Orr-Teichner filtration of the knot concordance group, via symmetric grope cobordism in 4-space. In particular, the graded theory contains information on finite type invariants (with degree h terms mapping to Vassiliev degree 2^h), Blanchfield forms or S-equivalence at h=2, Casson-Gordon invariants at h=3, and for h=4 one finds the new von Neumann signatures of a knot.     

On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups

In [Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In this work, by using Quinn's Transversality Theorem [Proc. Sympos. Pure. Math. 15 (1970) 213-222], it will be shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It will be shown that the Thom isomorphism in this theory will be satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps will be proved. After we discuss the relation between this theory and classical cobordism, we describe some applications to the complex cobordism of flag varieties of loop groups and we do some calculations.     

Bordism of semi-free <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-actions

We calculate geometric and homotopical bordism rings associated to semi-free S¹ actions on complex manifolds, giving explicit generators for the geometric theory. The classification of semi-free actions with isolated fixed points up to cobordism complements similar results from symplectic geometry.     

THE CALCULUS OF GENERATING FUNCTIONS AND THE FORMAL ENERGY FOR HAMILTONIAN ALGORITHMS

1.DarbouxTransformationConsidercotangentbundleT*R"acRZnwithnaturalsymplecticstructureandtheproductofcotangentbundles(T*R")x(T*R")=R4nwithnaturalproductsymplecticstructureCorrespondingly,weconsidertheproductspaceR"xR"rsRZn.ItscotangentbunT*(R"xR")=T*Rzn=R'nhasnaturalsymplecticstructure')PreparedbyQinMengzhaoChoosesymplecticcoordinatesz~(p,q)onthesymplecticmanifold,thenforsymplectictransformationg:T*R"~T*R",wehaveitisaLagrangiansubmanifoldofT*R"xT*RninR4n~(R'",J4.).NotethatonR4nth…     

The Lagrangian cobordism group of $$T^2$$

We compute the Lagrangian cobordism group of the standard symplectic 2-torus and show that it is isomorphic to the Grothendieck group of its derived Fukaya category. The proofs use homological mirror symmetry for the 2-torus.     

The origins and history of the fields medal

The Fields Medal is the most distinguished unternational award in mathematics. John Charles Fields felt strongly the lack of a Novel Prize in mathematics. He was also disturbed, as Chairman of the Committee of the 1924 International Mathematical Congress, by the scars left from the Versailles Treaty which marred the international character of the Congress. This paper traces the evolution of the award from incidents in Fields' background, through its formal inception at meetings of the University of Toronto Committee fir the 1924 Congress, to its eventual establishment and the first awards at the Oslo International Congress in 1936. Included are a list of all Fields Medal winners to date, the text of Fields' proposal for the establishment of the award, and relevant excerpts from minutes of the Toronto Committee, and a photograph of the medal.     

A wall-crossing formula for the signature of symplectic quotients

We use symplectic cobordism, and the localization result of Ginzburg, Guillemin, and Karshon to find a wall-crossing formula for the signature of regular symplectic quotients of Hamiltonian torus actions. The formula is recursive, depending ultimately on fixed point data. In the case of a circle action, we obtain a formula for the signature of singular quotients as well. We also show how formulas for the Poincaré polynomial and the Euler characteristic (equivalent to those of Kirwan can be expressed in the same recursive manner.

     

On the cobordism and commutative monoid with cancellation approaches to conformal field theory

In the late 1980s, Graeme Segal axiomatized conformal field theory in terms of a cobordism category. In that same preprint he outlined a more symmetric trace approach, which was recently rigorized in terms of pseudo algebras over a 2-theory. In this paper, we treat the cobordism approach in the pseudo algebra context. We introduce a new algebraic structure on a bicategory, called a pseudo 2-algebra over a theory, as a means of comparison for the two approaches. The main result states that the 2-category of pseudo algebras over a fixed 2-theory is biequivalent to the 2-category of pseudo 2-algebras over a fixed theory in certain situations.     

Stable étale realization and étale cobordism

We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A¹-homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero.On the other hand we get a natural setting for étale cohomology theories. In particular, we define and discuss an étale topological cobordism theory for schemes. It is equipped with an Atiyah-Hirzebruch spectral sequence starting from étale cohomology. Finally, we construct maps from algebraic to étale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element.     

Convergence theory for the structured BFGS secant method with an application to nonlinear least squares

In 1981, Dennis and Walker developed a convergence theory for structured secant methods which included the PSB and the DFP secant methods but not the straightforward structured version of the BFGS secant method. Here, we fill this gap in the theory by establishing a convergence theory for the structured BFGS secant method. A direct application of our new theory gives the first proof of local andq-superlinear convergence of the important structured BFGS secant method for the nonlinear least-squares problem, which is used by Dennis, Gay, and Welsh in the current version of the popular and successful NL2SOL code.This research was sponsored by SDIO/IST/ARO, AFOSR-85-0243, and DOE-DEFG05-86 ER-25017.A portion of this work is contained in the second author's doctoral thesis under the supervision of the other two authors in the Department of Mathematical Sciences, Rice University. The second author would like to thank Universidad del Valle, Cali, Columbia, for support during his graduate studies.An early draft of this work was presented at the SIAM 35th Anniversary Meeting, October 12–15, 1987, Denver, Colorado.     

Base independent algebraic cobordism

The purpose of this article is to show that the bivariant algebraic A-cobordism groups considered previously by the author are independent of the chosen base ring A. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define the corresponding homology theory called algebraic bordism. We also generalize projective bundle formula, the theory of Chern classes, the Conner–Floyd theorem and the Grothendieck–Riemann–Roch theorem to this setting. The general definitions of bivariant cobordism are based on the careful study of ample line bundles and quasi-projective morphisms of Noetherian derived schemes, also undertaken in this work.     

Fundamental Classes in Algebraic Cobordism

We recall our construction of fundamental classes in the algebraic cobordism of smooth quasi-projective varieties, and show by example that it is not possible to extent this to fundamental classes, functorial for local complete intersection morphisms, for Cohen–Macaulay varieties.     

Cyclic cocycles on deformation quantizations and higher index theorems

We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the K-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes-Moscovici and its extension to orbifolds.     

Neo-Kantian foundations of geometry in the German Romantic period

While mathematics received relatively little attention in the idealistic systems of most of the German Romantics, it served as the foundation in the thought of the Neo-Kantian philosopher/mathematician Jakob Friedrich Fries (1773–1843). It fell to Fries to work out in detail the implications of Kant's declaration that all mathematical knowledge was synthetic a priori. In the process Fries called for a new science of the philosophy of mathematics, which he worked out in greatest detail in his Mathematische Naturphilosophie of 1822. In this work he analyzed the foundations of geometry with an eye to clearing up the historical controversy over Euclid's theory of parallels. Contrary to what might be expected, Fries' Kantian perspective provoked rather than inhibited a reexamination of Euclid's axioms. Fries' attempt to make explicit through axioms what was being implicitly assumed by Euclid while at the same time wishing to eliminate unnecessary axioms belies the claim that there was no concern to improve Euclid prior to the discovery of non-Euclidean geometry. Fries' work therefore serves as an important historical example of the difficulties facing those who wanted to provide geometry with a logically secure foundation in the era prior to the published work of Gauss, Bolyai, and others.     

Categorification of invariants in gauge theory and symplectic geometry

This is a mixture of survey article and research announcement. We discuss instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian correspondence as a morphism in the category of symplectic manifolds. During the year 1998–2012, those problems have been studied emphasizing the ideas from analysis such as degeneration and adiabatic limit (instanton Floer homology) and strip shrinking (Lagrangian correspondence). Recently we found that replacing those analytic approach by a combination of cobordism type argument and homological algebra, we can resolve various difficulties in the analytic approach. It thus solves various problems and also simplify many of the proofs.     

Archimedes and the spirals: The heuristic background

In his work, The Method, Archimedes displays the heuristic technique by which he discovered many of his geometric theorems, but he offers there no examples of results from Spiral Lines. The present study argues that a number of theorems on spirals in Pappus' Collectio are based on early Archimedean treatments. It thus emerges that Archimedes' discoveries on the areas bound by spirals and on the properties of the tangents drawn to the spirals were based on ingenious constructions involving solid figures and curves. A comparison of Pappus' treatments with the Archimedean proofs reveals how a formal stricture against the use of solids in problems relating exclusively to plane figures induced radical modifications in the character of the early treatments.     

Langages sur des alphabets infinis

This paper proposes three possible definitions of context-free languages over infinite alphabets. These are proved to be non-equivalent and, in fact, of increasing power. For each of them, the classical results about CFL's as well as the usual closure properties are looked at. This work will be followed by two forthcoming papers using these notions: one for defining Languages' Form (similar to Grammars Form), the other for beginning the study of limits of languages which plays an important role in the theory of formal languages.     

Smoothness of isotopy for symplectic pairs

Let M be a 2n-dimensional smooth manifold with a symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Similar to Moser's stability theorem for symplectic forms, one desires to establish a stability theorem for symplectic pairs. Some sufficient and necessary conditions are obtained by Bande, Ghiggini and Kotschick. In this article, we consider a technical problem relating to the stability theorem. To complete the proof of the stability theorem for symplectic pairs, we verify the smoothness of the isotopy which is ignored in the literature. The Hodge theory for Riemannian foliation is crucial to our discussion.     

Algebraic cobordism revisited

We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative Donaldson–Thomas theory. We use double point cobordism to prove all the degree 0 conjectures in Donaldson–Thomas theory: absolute, relative, and equivariant.