Related papers: On the Quillen determinant

In the paper one considers the local structure of the Fredholm joint spectrum of commuting $n$-tuples of operators. A connection between the spatial characteristics of operators and the algebraic invariant of the corresponding coherent…

Tangent categories provide an axiomatic approach to key structural aspects of differential geometry that exist not only in the classical category of smooth manifolds but also in algebraic geometry, homological algebra, computer science, and…

We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most…

I relate contextuality to line bundles. Line bundles are important in algebraic geometry, they determine through their global sections rational maps to projective spaces. I explain how such maps, if they exist, relate rationally the input…

On conformal manifolds of even dimension $n\geq 4$ we construct a family of new conformally invariant differential complexes. Each bundle in each of these complexes appears either in the de Rham complex or in its dual. Each of the new…

Let $(M,F)$ be a Finsler manifold. We construct a 1-cocycle on $\Diff(M)$ with values in the space of differential operators acting on sections of some bundles, by means of the Finsler function $F.$ As an operator, it has several…

The notion of a quasideterminant and a quasiminor of a matrix A=(a_{ij}) with not necessarily commuting entries was introduced recently by I.Gelfand and the second author. The ordinary determinant of a matrix with commuting entries can be…

Following the approach of Budzy\'nski and Kondracki, we define covariant differential algebras and connections on locally trivial quantum principal fibre bundles. We also consider covariant derivatives, connection forms and curvatures and…

We find the transition kernels for four Markovian interacting particle systems on the line, by proving that each of these kernels is intertwined with a Karlin-McGregor type kernel. The resulting kernels all inherit the determinantal…

We extend the classical trace (and determinant) known for the integral operators $$ ({\mathcal I}+)\int_{[0,1)^N}{\bf A}({\bf k},{\bf x}){\bf u}({\bf x})d{\bf x} $$ with matrix-valued kernels ${\bf A}$ to the operators of the form $$…

Recently L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (whose link $M$ is a rational homology sphere) with the Seiberg-Witten invariant of $M$…

The geometric quantization of a symplectic manifold endowed with a prequantum bundle and a metaplectic structure is defined by means of an integrable complex structure. We prove that its semi-classical limit does not depend on the choice of…

We generalise the construction of fuzzy CP^N in a manner that allows us to access all noncommutative equivariant complex vector bundles over this space. We give a simplified construction of polarization tensors on S^2 that generalizes to…

We characterize the groupoids for which an operator is Fredholm if, and only if, its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called {\em Fredholm}. Using results on the Effros-Hahn…

In \cite{baker-ozel}, by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation…

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Green's functions associated with closed ordinary differential…

The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein…

Let $(X,L,V)$ be a triplet where $X$ is an irreducible smooth complex projective variety, $L$ is an ample and spanned line bundle on $X$ and $V\subseteq H^0(X,L)$ spans $L$. The discriminant locus $\Cal D(X,V) \subset |V|$ is the algebraic…

We study a family of Fredholm determinants associated to deformations of the sine kernel, parametrized by a weight function w. For a specific choice of w, this kernel describes bulk statistics of finite temperature free fermions. We…

We associate determinant lines to objects of the extended abelian category built out of a von Neumann category with a trace. Using this we suggest constructions of the combinatorial and the analytic L^2 torsions which, unlike the work of…