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We use techniques from both real and complex algebraic geometry to study$K$-theoretic and related invariants of the algebra$C$($X$) of continuous complex-valued functions on a compact Hausdorff topological space$X$. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if$M$is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and$X$is contractible, then every finitely generated projective module over$C$($X$)[$M$] is free. The particular case $$ M = \mathbb{N}_0^n $$ gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic$K$-theory of$C$($X$) follows from the particular case $$ M = {\mathbb{Z}^n} $$ . We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on$C$^{*}-algebras, and for a homology theory of commutative algebras to vanish on$C$^{*}-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil$K$-theory implies that commutative$C$^{*}-algebras are$K$-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of$C$($X$). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.

Let $D={\mathbb H} \setminus \cup_{k=1}^N C_k$ be a standard slit domain where $\mathbb H$ is the upper half plane and $C_k$, $1\leq k\leq N$, are mutually disjoint horizontal line segments in $\HH$. Given a Jordan arc $\gamma\subset D$ starting at $\partial {\mathbb H},$ let $g_t$ be the unique conformal map from $D\setminus \gamma[0,t]$ onto a standard slit domain $D_t$ satisfying the hydrodynamic normalization. We prove that $g_t$ satisfies an ODE with the kernel on its righthand side being the complex Poisson kernel of the Brownian motion with darning (BMD) for $D_t$, generalizing the chordal Loewner equation for the simply connected domain $D={\mathbb H}.$ Such a generalization has been obtained by Y. Komatu in the case of circularly slit annuli and by R. O. Bauer and R. M. Friedrich in the present chordal case, but only in the sense of the left derivative in $t$. We establish the differentiability of $g_t$ in $t$ to make the equation a genuine ODE. To this end, we first derive the continuity of $g_t(z)$ in $t$ with a certain uniformity in $z$ from a probabilistic expression of $\Im g_t(z)$ in terms of the BMD for $D$, which is then combined with a Lipschitz continuity of the complex Poisson kernel under the perturbation of standard slit domains to get the desired differentiability.

No abstract uploaded!

No abstract uploaded!