Related papers: Reduction theorem for lattice cohomology

Lattice reduction is a NP-hard problem well known in computer science and cryptography. The Lenstra-Lenstra-Lovasz (LLL) algorithm based on the calculation of orthogonal Gram-Schmidt (GS) bases is efficient and gives a good solution in…

We introduce a pointfree theory of convergence on lattices and coframes. A convergence lattice is a lattice $L$ with a monotonic map $\lim_L$ from the lattice of filters on $L$ to $L$, meant to be an abstract version of the map sending…

The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it graph homology} is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of…

In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these…

We define a reduction covariant for the representations a la Vinberg associated to stably graded Lie algebras. We then give an analogue of the LLL algorithm for the odd split special orthogonal group and show how this can be combined with…

It has been recently conjectured by Boyer-Gordon-Watson that a closed, orientable, irreducible $3$-manifold $M$ is a Heegaard Floer $L$-space if and only if $\pi_1(M)$ is not left-orderable. In this article, we study this conjecture from…

We develop a framework that systematically casts the solvability and uniqueness conditions of linearized geometric boundary-value problems into cohomological terms. The theory is designed to be applicable without assumptions on the…

We construct the equivariant analytic lattice cohomology associated with the analytic type of a complex normal surface singularity whenever the link is a rational homology sphere. It is the categorification of the equivariant geometric…

This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of…

Topological defects play a fundamental role in the investigation of symmetries in quantum field theories. For conformal field theories in two space-time dimensions, it is possible to construct these defects using lattice models allowing…

In this paper we present a proof of the BMZ Reduction Lemma with a motivational perspective, and state this lemma for maps to manifolds using the classical definition of cohomological dimension. The lemma, proved and utilized in [4], gives…

We give a constructive proof that every weakly negative definite plumbing tree can be transformed into a negative definite one by a finite sequence of Neumann moves. The argument combines Neumann's plumbing calculus with the diagonalization…

We introduce an invariant of negative definite plumbed knot complements unifying knot lattice homology, due to Ozsv\'ath, Stipsicz, and Szab\'o, and the BPS $q$-series of Gukov and Manolescu. This invariant is a natural extension of…

We give cohomological criteria for logarithmic good reduction of elliptic surfaces up to modification. Along the way, we prove several more general results about such surfaces in positive characteristic, as well as about log smooth…

We construct a lattice cohomology ${\mathbb H}^*(C,o)=\oplus_{q\geq 0}{\mathbb H}^q(C,o)$ and a graded root ${\mathfrak R}(C,o)$ to any complex isolated curve singularity $(C,o)$. Each ${\mathbb H}^q(C,o)$ is a ${\mathbb Z}$-graded…

This paper explores a particular statistical model on 6-valent graphs with special properties which turns out to be invariant with respect to certain Roseman moves if the graph is the singular point graph of a diagram of a 2-knot. The…

We describe a canonical decomposition of the cohomology of the Dani-Mainkar metabelian Lie algebras associated with graphs. As applications, we obtain explicit formulas for the third cohomology of any Dani-Mainkar Lie algebra and for the…

We associate (under a minor assumption) to any analytic isolated singularity of dimension $n\geq 2$ the `analytic lattice cohomology' ${\mathbb H}^*_{an}=\oplus_{q\geq 0}{\mathbb H}^q_{an}$. Each ${\mathbb H}^q_{an}$ is a graded ${\mathbb…

In this paper, we develop a diamond graph theory and apply the theory to the (co)homology of the Lie algebra generated by positive systems of the classical semi-simple Lie algebras over the field of complex numbers. As an application, we…

Let $(C,o)$ be a complex analytic isolated curve singularity of arbitrary large embedded dimension. Its lattice cohomology ${\mathbb H}^*=\oplus_{q\geq 0}{\mathbb H}^q$ was introduced by \'Agoston and the author, each ${\mathbb H}^q$ is a…