An odd unbounded (respectively, $p$-summable) Fredholm module for a unital Banach $*$-algebra, $A$, is a pair $(H,D)$ where $A$ is represented on the Hilbert space, $H$, and $D$ is an unbounded self-adjoint operator on $H$ satisfying:

(1) ${{(1+{{D}^{2}})}^{-1}}$ is compact (respectively, Trace $\left( {{\left( 1+{{D}^{2}} \right)}^{-(p/2)}} \right)<\infty $, and

(2) $\{a\in A|\,\,[D,a]\,\,\text{is}\,\,\text{bounded}\}\text{ }$ is a dense $*$- subalgebra of $A$.

If $u$ is a unitary in the dense $*$-subalgebra mentioned in (2) then

$$uD{{u}^{*}}=D+u[D,{{u}^{*}}]=D+B$$

where $B$ is a bounded self-adjoint operator. The path

$$D_{t}^{u}:=(1-t)D+tuD{{u}^{*}}=D+tB$$

is a “continuous” path of unbounded self-adjoint “Fredholm” operators. More precisely, we show that

$$F_{t}^{u}:=D_{t}^{u}{{\left( 1+{{\left( D_{t}^{u} \right)}^{2}} \right)}^{-\frac{1}{2}}}$$

is a norm-continuous path of (bounded) self-adjoint Fredholm operators. The spectral flow of this path $\left\{ F_{t}^{u} \right\}$ (or $\{D_{t}^{u}\}$) is roughly speaking the net number of eigenvalues that pass through 0 in the positive direction as $t$ runs from 0 to 1. This integer,

$$\text{sf}\left( \left\{ D_{t}^{u} \right\} \right):=\text{sf}\left( \left\{ F_{t}^{u} \right\} \right),$$

recovers the pairing of the $K$-homology class $[D]$ with the $K$-theory class $[u]$.

We use I.M. Singer's idea (as did E. Getzler in the $\theta $-summable case) to consider the operator $B$ as a parameter in the Banach manifold, ${{B}_{\text{sa}}}\left( H \right)$, so that spectral flow can be exhibited as the integral of a closed 1-form on this manifold. Now, for $B$ in our manifold, any $X\,\in \,{{T}_{B}}({{B}_{\text{sa}}}(H))$ is given by an $X$ in ${{B}_{\text{sa}}}(H)$ as the derivative at $B$ along the curve $t\,\mapsto \,B\,+\,tX$ in the manifold. Then we show that for $m$ a sufficiently large half-integer:

$$\alpha (X)=\frac{1}{{{{\tilde{C}}}_{m}}}\text{Tr}\left( X{{\left( 1+{{\left( D+B \right)}^{2}} \right)}^{-m}} \right)$$

is a closed 1-form. For any piecewise smooth path $\left\{ {{D}_{t}}=D+{{B}_{t}} \right\}$ with ${{D}_{0}}$ and ${{D}_{1}}$unitarily equivalent we show that

$$\text{sf}\left( \left\{ {{D}_{t}} \right\} \right)=\frac{1}{{{{\tilde{C}}}_{m}}}\int_{\text{0}}^{\text{1}}{\text{T}}\text{r}\left( \frac{d}{dt}\left( {{D}_{t}} \right){{\left( 1+D_{1}^{2} \right)}^{-m}} \right)dt$$

the integral of the 1-form $\alpha $. If ${{D}_{0}}$ and ${{D}_{1}}$ are not unitarily equivalent, wemust add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form:

$$\text{sf}\left( \left\{ {{F}_{t}} \right\} \right)=\frac{1}{{{C}_{n}}}\int_{\text{0}}^{\text{1}}{\text{T}}\text{r}\left( \frac{d}{dt}\left( {{F}_{t}} \right){{\left( 1-{{F}_{t}}^{2} \right)}^{n}} \right)dt$$

for $n\ge \frac{p-1}{2}$ an integer. The unbounded case is proved by reducing to the bounded case via the map $D\mapsto F=D{{(1+{{D}^{2}})}^{-\frac{1}{2}}}$ We prove simultaneously a type II version of our results.

Let $R$ be a countable, principal ideal domain which is not a field and $A$ be a countable $R$-algebra which is free as an $R$-module. Then we will construct an ${{\aleph }_{1}}$-free $R$-module $G$ of rank ${{\aleph }_{1}}$ with endomorphism algebra $\text{En}{{\text{d}}_{R}}\,G=A$. Clearly the result does not hold for fields. Recall that an $R$-module is ${{\aleph }_{1}}$-free if all its countable submodules are free, a condition closely related to Pontryagin’s theorem. This result has many consequences, depending on the algebra $A$ in use. For instance, if we choose $A\,=\,R$, then clearly $G$ is an indecomposable ‘almost free’module. The existence of such modules was unknown for rings with only finitelymany primes like $R={{\mathbb{Z}}_{\left( p \right)}}$, the integers localized at some prime $p$. The result complements a classical realization theorem of Corner’s showing that any such algebra is an endomorphism algebra of some torsionfree, reduced $R$-module $G$ of countable rank. Its proof is based on new combinatorialalgebraic techniques related with what we call rigid tree-elements coming from a module generated over a forest of trees.

Let $X$ be a graph with vertex set $V$ and let $A$ be its adjacency matrix. If $E$ is the matrix representing orthogonal projection onto an eigenspace of $A$ with dimension $m$, then $E$ is positive semi-definite. Hence it is the Gram matrix of a set of $\left| V \right|$ vectors in ${{R}^{m}}$. We call the convex hull of a such a set of vectors an eigenpolytope of $X$. The connection between the properties of this polytope and the graph is strongest when $X$ is distance regular and, in this case, it is most natural to consider the eigenpolytope associated to the second largest eigenvalue of $A$. The main result of this paper is the characterisation of those distance regular graphs $X$ for which the 1-skeleton of this eigenpolytope is isomorphic to $X$.

We consider a specific realization of the renormalization group $(\text{RG})$ transformation acting on functional measures for scalar quantum fields which are expressible as a polymer expansion times an ultra-violet cutoff Gaussian measure. The new and improved definitions and estimates we present are sufficiently general and powerful to allow iteration of the transformation, hence the analysis of complete renormalization group flows, and hence the construction of a variety of scalar quantum field theories.

We give upper bounds on the modulus of the values at $s\,=\,1$ of Artin $L$-functions of abelian extensions unramified at all the infinite places. We also explain how we can compute better upper bounds and explain how useful such computed bounds are when dealing with class number problems for $\text{CM}$-fields. For example, we will reduce the determination of all the non-abelian normal $\text{CM}$-fields of degree 24 with Galois group $\text{S}{{\text{L}}_{\text{2}}}\left( {{F}_{3}} \right)$ (the special linear group over the finite field with three elements) which have class number one to the computation of the class numbers of 23 such $\text{CM}$-fields.

We construct the tableaux realization of generalized Verma modules over the Lie algebra $\text{sl(3,}\,\mathbb{C})$. By the same procedure we construct and investigate the structure of a new family of generalized Verma modules over $\text{sl(}n,\,\mathbb{C}\text{)}$.

We continue in this paper our study of conjugacy classes of a reductive monoid $M$. The main theorems establish a strong connection with the Bruhat-Renner decomposition of $M$. We use our results to decompose the variety ${{M}_{\text{nil}}}$ of nilpotent elements of $M$ into irreducible components. We also identify a class of nilpotent elements that we call standard and prove that the number of conjugacy classes of standard nilpotent elements is always finite.

In this paper, we define the notion of ${{R}_{*}}\text{-LS}$ category associated to an increasing system of subrings of $\mathbb{Q}$ and we relate it to the usual $\text{LS}$-category. We also relate it to the invariant introduced by Félix and Lemaire in tame homotopy theory, in which case we give a description in terms of Lie algebras and of cocommutative coalgebras, extending results of Lemaire-Sigrist and Félix-Halperin.

Let $\pi :X\,\to \,S$ be a finite type morphism of noetherian schemes. A smooth formal embedding of $X$ (over $S$) is a bijective closed immersion $X\,\subset \,\mathfrak{X}$ , where $\mathfrak{X}$ is a noetherian formal scheme, formally smooth over $S$. An example of such an embedding is the formal completion $\mathfrak{X}={{Y}_{/X}}$ where $X\,\subset \,Y$ is an algebraic embedding. Smooth formal embeddings can be used to calculate algebraic De Rham(co)homology.

Our main application is an explicit construction of the Grothendieck residue complex when $S$ is a regular scheme. By definition the residue complex is the Cousin complex of ${{\pi }^{!}}{{O}_{S}}$, as in $[\text{RD}]$. We start with $\text{I-C}$. Huang's theory of pseudofunctors on modules with 0-dimensional support, which provides a graded sheaf ${{\oplus }_{q}}K_{X/S}^{q}.$ We then use smooth formal embeddings to obtain the coboundary operator $\delta :K_{X/S}^{q}\to K_{X/S}^{q+1}.$ We exhibit a canonical isomorphism between the complex $K_{X/S}^{\cdot },\delta $ and the residue complex of $[\text{RD}]$. When $\pi $ is equidimensional of dimension $n$ and generically smooth we show that ${{\text{H}}^{-n}}K_{X/S}^{\cdot }$ is canonically isomorphic to to the sheaf of regular differentials of Kunz-Waldi $[\text{KW}]$.

Another issue we discuss is Grothendieck Duality on a noetherian formal scheme $\mathfrak{X}$ . Our results on duality are used in the construction of $K_{X/S}^{\cdot }$.