We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

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Non-commutative differential geometry

The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genus g. By this we mean setting up a Chow ring for the moduli space M g of curves of genus g and its compactification M g, defining what seem to be the most important classes in this ring and calculating the class of some geometrically important loci in M g in terns of these classes. We take as a model for this the enumerative geometry of the Grassmannians. Here the basic classes are the Chern classes of the tautological or universal bundle that lives over the Grassmannian, and the most basic cycles are the loci of linear spaces satisfying various Schubert conditions: the so-called Schubert cycles. However, since Harris and I have shown that for g large, M g is not unir.ational [H-M] it is not possible to expect that M g has a decomposition into elementary cells or that the Chow ring of M g is as simple as that of the Grassmannian. But in the other direction, J. Harer [Ha] and P. Miller [Mi] have strong results indicating that at least the low dimensional homology groups of M g behave nicely. Moreover, it appers that many geometrically natural cycles are all expressible in terms of a small number of basic classes.

/pdf/towards-an-enumerative-geometry-of-the-moduli-space-of-2m7yfw99qf.pdf

Towards an Enumerative Geometry of the Moduli Space of Curves

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

We announce a reformulation of the conjecture in [8,9,10]. The advantage of the new version is that it is simpler and applies more generally than the earlier statement. A key point is to use the universal example for proper actions introduced in [10]. There, the universal example seemed somewhat peripheral to the main issue. Here, however, it will play a central role.

http://www.personal.psu.edu/ndh2/math/Papers_files/Baum,%20Connes,%20Higson%20-%201994%20-%20Classifying%20space%20for%20proper%20actions%20and%20K-theory%20for%20group%20C*-algebras.pdf

Classifying Space for Proper Actions and K-Theory of Group C*-algebras

The basic tool for a general Riemann-Roch theorem is MacPherson’s graph construction, applied to a complex E. of vector bundles on a scheme Y, exact off a closed subset X. This produces a localized Chern character1 ch x y (E.) which lives in the bivariant group $ A{\left( {X \to Y} \right)_\mathbb{Q}} $ For each class α∈A * Y, this gives a class 

$$ ch_X^Y\left( {E.} \right) \cap \alpha \in {A_ * }{X_\mathbb{Q}} $$

whose image in $ {A_ * }{Y_\mathbb{Q}} $ is $ {\sum {\left( { - 1} \right)} ^i}ch\left( {{E_i}} \right) \cap \alpha $ The properties needed for Riemann-Roch, in particular the invariance under rational deformation, follow from the bivariant nature of ch x y E.

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Riemann-Roch for Singular Varieties

/pdf/singularities-of-holomorphic-foliations-40fc9euayw.pdf

Singularities of holomorphic foliations

Publisher Summary This chapter discusses the Chern character for discrete groups. H j (X;Q) is the j-th Cech cohomology group of X with coefficients the rational numbers Q. The key property of this classical Chern character is that it is a rational isomorphism. Cyclic cohomology can be used to define the delocalized equivariant cohomology of X. The traditional homotopy quotient Chern character gives a map, which is always surjective, for compact X. The chapter also discusses twisted homology and K homology. K homology is the homology theory associated to the Z × BU spectrum. A concrete realization of this theory is obtained by using the K-cycle definition.

Paul Baum

Papers

Classifying Space for Proper Actions and K-Theory of Group C*-algebras

Riemann-Roch for Singular Varieties

Singularities of holomorphic foliations

Chern character for discrete groups

The cohomology of quotients of classical groups