Related papers: Signature for flat unitary bundles over surfaces w…
Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of $4$, and can be computed using an element of $H^2(\mathsf{Sp}(2g, \mathbb{Z}),\mathbb{Z})$. Denoting by $1 \to \mathbb{Z} \to…
We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic,…
The theory of signature invariants of links in rational homology spheres is applied to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, an explicit formula is derived to compute…
For a closed topological manifold M with dim (M) >= 5 the topological structure set S(M) admits an abelian group structure which may be identified with the algebraic structure group of M as defined by Ranicki. If dim (M) = 2d-1, M is…
Let $G$ be a complex semisimple Lie group and $\mathfrak g$ its Lie algebra. In this paper, we study a special class of cyclic Higgs bundles constructed from a $\mathbb Z$-grading $\mathfrak g = \bigoplus_{j=1-m}^{m-1}\mathfrak g_j$ by…
This paper gives a construction for all minimal immersions $f$ of the Poincar\'{e} disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $\rho$ of a hyperbolic surface…
We provide a one-parameter family of Lorentz-Riemann signature-change models of metric manifolds. This family generalizes the Kossowski's signature type-changihg stablished in [9]. Simple local expressions are sought around the hypersurface…
We study the index of the APS boundary value problem for a strongly Callias-type operator D on a complete Riemannian manifold $M$. We show that this index is equal to an index on a simpler manifold whose boundary is a disjoint union of two…
We investigate the properties of the Dirac operator on manifolds with boundaries in presence of the Atiyah-Patodi-Singer boundary condition. An exact counting of the number of edge states for boundaries with isometry of a sphere is given.…
Abelian Chern-Simons theory relates classical theta functions to the topological quantum field theory of the linking number of knots. In this paper we explain how to derive the constructs of abelian Chern-Simons theory directly from the…
We use the G-signature theorem to define an invariant of strongly invertible knots analogous to the knot signature.
We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index of our previous paper. Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary…
The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class…
Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this…
We define a noncommutative space we call the quantum solid torus. It is an example of a noncommutative manifold with a noncommutative boundary. We study quantum Dirac type operators subject to Atiyah-Patodi-Singer like boundary conditions…
New extrinsic lower bounds are given for the classical Dirac operator on the boundary of a compact domain of a spin manifold. The main tool is to solve some boundary problems for the Dirac operator of the domain under boundary conditions of…
This is the author's Ph.D. thesis. We introduce two related invariants for local (and standard graded) rings called differential and syzygy symmetric signature. These are defined by looking at the maximal free splitting of the module of…
We generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to…
The construction of invariants of three-dimensional manifolds with a triangulated boundary, proposed earlier by the author for the case when the boundary consists of not more than one connected component, is generalized to any number of…
In this work we propose to study the general Robin boundary value problem involving signed smooth measures on an arbitrary domain $\Omega$ of $\mathbb R^d$. A Kato class of measures is defined to insure the closability of the associated…