Intersection theory for graphs

An intersection theory developed by the author for matroids embedded in uniform geometries is applied to the case when the ambient geometry is the lattice of partitions of a finite set so that the matroid is a graph. General embedding theorems when applied to graphs give new interpretations to such invariants as the dichromate of Tutte. A polynomial in n + 1 variables, the polychromate, is defined for graphs with n vertices. This invariant is shown to be strictly stronger than the dichromate, it is edge-reconstructible and can be calculated for proper graphs from the polychromate of the complementary graph. By using Tutte's construction for codichromatic graphs (J. Combinatorial Theory 16 (1974), 168–174), copolychromatic (and therefore codichromatic) graphs of arbitrarily high connectivity are constructed thereby solving a problem posed in Tutte's paper.     

Intersection theory on non-commutative surfaces

Consider a non-commutative algebraic surface, , and an effective divisor on , as defined by Van den Bergh. We show that the Riemann-Roch theorem, the genus formula, and the self intersection formula from classical algebraic geometry generalize to this setting.

We also apply our theory to some special cases, including the blow up of in a point, and show that the self intersection of the exceptional divisor is . This is used to give an example of a non-commutative surface with a commutative which cannot be blown down, because its self intersection is rather than . We also get some results on Hilbert polynomials of modules on .

     

Intersection theory of algebraic obstructions

Let A be a noetherian commutative ring of dimension d and L be a rank one projectiveA-module. For 1≤r≤d, we define obstruction groups E^r(A,L). This extends the original definition due to Nori, in the case r=d. These groups would be called Euler class groups. In analogy to intersection theory in algebraic geometry, we define a product (intersection) E^r(A,A)×E^s(A,A)→E^r+s(A,A). For a projective A-module Q of rank n≤d, with an orientation , we define a Chern class like homomorphism

w(Q,χ):E^d−n(A,L^′)→E^d(A,LL^′),     

Intersection theory of projective linear spaces

We study the intersection theory of a class of projective linear spaces (generalizations of projective space bundles in which the fibres are linear but of varying dimensions). In particular we give exact sequences for the Chow and Chow cohomology groups reminiscent of those for regular blowups. During this research the author was supported by a Sloan foundation doctoral disertation fellowship     

Intersection theory over quantum ruled surfaces

In this paper, we will extend several results on intersection theory over commutative ruled surfaces to quantum ruled surfaces. Typically, we define the fiber of a closed point, the quasi-section, and the quasi-canonical divisor on a quantum rules surface, and study how these “curves” on a quantum ruled surface intersect with each other.     

Intersection theory on an open variety

Summary The concepts of algebraic equivalence and virtual intersection are extended to an ambient variety from which a subvariety has been removed.     

Intersection homology theory via rectifiable currents

Here is given a rectifiable currents version of intersection homology theory on stratified subanalytic pseudomanifolds. This new version enables one to study some variational problems on stratified subanalytic pseudomanifolds. We first achieve an isomorphism between this rectifiable currents version and the version using subanalytic chains. Then we define a suitably modified mass on the complex of rectifiable currents to ensure that each sequence of subanalytic chains with bounded modified mass has a convegent subsequence and the limit rectifiable current still satisfies the perversity condition of the approximating chains. The associated mass minimizers turn out to be almost minimal currents and this fact leads to some regularity results.Received: 21 April 2001, Accepted: 15 April 2003, Published online: 1 July 2003     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

Multiparameter spectral theory in Hilbert space

The object of this paper is to present a unified approach to multiparameter spectral theory of linear operators in Hilbert space. The theory is applicable to both bounded and unbounded operators and has application in the study of multiparameter spectral problems of ordinary differential operators. The main results include a Parseval equality and an eigenfunction expansion theorem.     

Intersection theory of incidence varieties and polygon spaces

In this paper smooth parameterizing spaces for polygons in projective space are introduced and their intersection theory is studied. In particular we give an expression for the Chow ring as a quotient of a polynomial ring. In addition the Chow cohomology rings of various incidence varieties are computed.     

Ljusternik-Schnirelman theory in partially ordered Hilbert spaces

We present several variants of Ljusternik-Schnirelman type theorems in partially ordered Hilbert spaces, which assert the locations of the critical points constructed by the minimax method in terms of the order structures. These results are applied to nonlinear Dirichlet boundary value problems to obtain the multiplicity of sign-changing solutions.

     

Jones index theory by Hilbert C-bimodules and K-theory

In this paper we introduce the notion of Hilbert -bimodules, replacing the associativity condition of two-sided inner products in Rieffel's imprimitivity bimodules by a Pimsner-Popa type inequality. We prove Schur's Lemma and Frobenius reciprocity in this setting. We define minimality of Hilbert -bimodules and show that tensor products of minimal bimodules are also minimal. For an - bimodule which is compatible with a trace on a unital -algebra , its dimension (square root of Jones index) depends only on its -class. Finally, we show that the dimension map transforms the Kasparov products in to the product of positive real numbers, and determine the subring of generated by the Hilbert -bimodules for a -algebra generated by Jones projections.

     

Hilbert inequality on grand function spaces

In this paper, we study the Hilbert inequality with conjugate exponents in the framework of fully measurable grand Lebesgue spaces and grand Bochner Lebesgue spaces.