Searcharxiv⌕ Search

arXiv subjects

Bing Kwan So

Publications and source records attributed to Bing Kwan So.

9 recordsLinked to original sources
arXiv2024

Deformation and $K$-theoretic Index Formulae on Boundary Groupoids

Boundary groupoids were introduced by the second author, which can be used to model many analysis problems on singular spaces. In order to investigate index theory on boundary groupoids, we introduce the notion of {\em a deformation from the pair groupoid}.Under the assumption that a deformation from the pair groupoid $M \times M$ exists for Lie groupoid $\mathcal{G}\rightrightarrows M$, we construct explicitly a deformation index map relating the analytic index on $\mathcal{G}$ and the index on the pair groupoid. We apply this map to boundary groupoids of the form $\mathcal{G} = M_0 \times M_0 \sqcup G \times M_1 \times M_1 \rightrightarrows M=M_0\sqcup M_1$, where $G$ is an exponential Lie group, to obtain index formulae for (fully) elliptic (pseudo)-differential operators on $\mathcal{G}$, with the aid of the index formula by M. J. Pflaum, H. Posthuma, and X. Tang. These results recover and generalize our previous results for renormalizable boundary groupoids via the method of renormalized trace.

math.KT↗
arXiv2024

Renormalized Index Formulas for Elliptic Differential Operators on Boundary Groupoids

We consider the index problem of certain boundary groupoids of the form $\mathcal{G} = M _0 \times M _0 \cup \mathbb{R}^q \times M _1 \times M _1$. Since it has been shown that for the case that $q \geq 3$ is odd, $K _0 (C^* (\mathcal{G})) \cong \bbZ $, and moreover the $K$-theoretic index coincides with the Fredholm index, we attempt in this paper to derive a numerical formula for elliptic differential operators on $\mathcal{G}$. Our approach is similar to that of renormalized trace of Moroianu and Nistor \cite{Nistor;Hom2}. However, we find that when $q \geq 3$, the eta term vanishes, and hence the $K$-theoretic and Fredholm indices of elliptic (respectively fully elliptic) pseudo-differential operators on these groupoids are given only by the Atiyah-Singer term. As for the $q=1$ case we find that the result depends on how the singularity set $M_1$ lies in $M$.

math.OA↗
arXiv2022

Analytic surgery and gluing of the Bismut-Lott torsion form and eta form

Given a fiber bundle with closed connected fibers, and a family of separating hypersurfaces, we study the behavior of the Bismut-Lott analytic torsion form, and the eta form for a duality bundle, under analytic surgery in the sense of Hassell, Mazzeo and Melrose. We find that under the surgery limit, the rescaled heat kernel is non-singular, while both the Bismut-Lott analytic torsion form and eta form can be written as the sum of a logarithmic term, which satisfies the Igusa additivity property, the b- Bismut-Lott analytic torsion form (respectively the b- eta form), and an error term coming from the reduced normal operator. Hence we obtain a gluing formula for these invariants.

math.DG↗
arXiv2018

K-theory and index formulas for boundary groupoid C*-algebras

We compute explicitly the K-groups of some boundary groupoid C*-algebras with exponential isotropy subgroups. Then we derive index formulas that computes the K-theoretic and Fredholm indexes of elliptic (respectively totally elliptic) pseudo-differential operators on these groupoids.

math.KT↗
arXiv2017

Non-commutative analytic torsion form on the transformation groupoid convolution algebra

Given a fiber bundle $Z \to M \to B$ and a flat vector bundle $E \to M$ with a compatible action of a discrete group $G$, and regarding $B / G$ as the non-commutative space corresponding to the crossed product algebra, we construct an analytic torsion form as a non-commutative deRham differential form. We show that our construction is well defined under the weaker assumption of positive Novikov-Shubin invariant. We prove that this torsion form appears in a transgression formula, from which a non-commutative Riamannian-Roch-Grothendieck index formula follows.

math.DG↗
arXiv2016

Regularity of analytic torsion form on families of normal coverings

We prove the smoothness of the L^2-analytic torsion form on some fiber bundles with non-compact fibers of positive Novikov-Shubin invariant. We do so by generalizing the arguments of Azzali-Goette-Schick to an appropriate Sobolev space, and proving that the Novikov-Shubin invariant remains positive in the Sobolev settings, using an argument of Alvarez Lopez-Kordyukov.

math.DG↗
arXiv2010

Pseudo-differential operators, heat calculus and index theory of groupoids satisfying the Lauter-Nistor condition

In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the calculus of uniformly supported pseudo-differential operators that is analogous to the calculus with bounds defined on manifolds with boundary. We derive a Fredholmness criterion for operators on the Bruhat sphere, and prove that their parametrices up to compact operators lie inside the extended calculus; we construct the heat kernel of perturbed Laplacian operators; and prove an Atiyah-Singer type renormalized index formula for perturbed Dirac operators on the Bruhat sphere using the heat kernel method.

math.AP↗