arXiv research

A theory for cellular BF models on decomposed spaces.

Quantum cellular automata form a homology theory.

The paper studies Morse theory on manifolds with boundaries, constructing cellular structures and estimating critical points.

Study instantons on stratified spaces and their moduli spaces, linking to gauge theory and string theory.

Introduces a new relation between BF theory and gravity.

We extend our family rigidity and vanishing theorems in [{\bf LiuMaZ}] to the Spin^c case. In particular, we prove a K-theory version of the main results of [{\bf H}], [{\bf Liu1}, Theorem B] for a family of almost complex manifolds.

This paper analyzes in details the Batalin-Vilkovisky quantization procedure for BF theories on n-dimensional manifolds and describes a suitable superformalism to deal with the master equation and the search of observables. In particular, generalized Wilson loops for BF theories with additional polynomial B-interaction…

The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, González, and Rudyak [1009.1851] with the aim of constructing a cellular model of the configuration space of a sphere. In particular, it was shown that the classifying space (order complex) of the face poset of a totally …

The paper studies weak singular Hermite-Einstein structures on homogeneous vector bundles.

Abstract study shows Milnor metric from $BF$ theory partition function.

We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (${\bf H}_0$ is then the Hamiltonian) and risk control …

The paper explores handlebody versions of various diagram algebras.

Machine learning attacks mimic cellular decision-making, revealing new defense mechanisms.

After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point parti…

3D BF theory on certain 3-manifolds evaluated via residues and large k limits.

New surface observables yield 2-knot invariants in nonabelian theories.

We study the Chern-Simons topological quantum field theory with an inhomogeneous gauge group, a non-semi-simple group obtained from a semi-simple one by taking its semi-direct product with its Lie algebra. We find that the standard knot observables (i.e. traces of holonomies along knots) essentially vanish, but yet, th…

The paper introduces a new method to characterize cosmological models using observer-based invariants.

The paper extends cellular algebras with relative orderings and explores their properties.

New theorem shows gaps in magnetic Schrödinger operator spectra for large coupling.

New definition of Kuranishi spaces forms a 2-category.

The paper defines general-affine invariants for plane and space curves.

Extends CW topology to describe cellular homology on cell complexes.

Derives a general derivative identity for conditional mean in Gaussian noise.

We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (${\bf H}_0$ is then the Hamiltonian) and financial ris…

Finite Morse index implies finitely many ends and linear energy growth for solutions to the Allen-Cahn equation.

The study classifies cellular pseudomanifolds and their properties.

For any principal bundle $P$, one can consider the subspace of the space of connections on its tangent bundle $TP$ given by the tangent bundle $T{\cal A}$ of the space of connections ${\cal A}$ on $P$. The tangent gauge group acts freely on $T{\cal A}$. Appropriate BRST operators are introduced for quantum field theori…

This paper introduces a general perturbative quantization scheme for gauge theories on manifolds with boundary, compatible with cutting and gluing, in the cohomological symplectic (BV-BFV) formalism. Explicit examples, like abelian BF theory and its perturbations, including nontopological ones, are presented.

Algorithm recovers spike direction from modulo-reduced measurements in high dimensions.

Study homology of configuration spaces of finite cell complexes using subdivisional spaces and Morse theory.

The paper develops a theory of discrete Riemann surfaces using quadrilateral cells.

In the first part of this paper, we work out a perturbative Lagrangian formulation of semistrict higher gauge theory, that avoids the subtleties of the relationship between Lie 2-groups and algebras by relying exclusively on the structure semistrict Lie 2-algebra v and its automorphism 2-group Aut(v). Gauge transformat…

'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1106.4882), as the geometric structure on moduli spaces of $J$-holomorphic curves. An alternative to Kuranishi spaces is the 'polyfolds' of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185). Finding a s…

Geometric proof shows topological invariance of handle homology.

Tree almost automorphism groups have a cellular action on a contractible complex.

Optimizes natural frequencies of cellular composites with various microstructures.

The paper classifies algebraic curves in 4-balls and their boundaries.

For leveled spatial graphs, we find a surface embedding that allows cellular embedding.

The paper solves pseudo-convexity for special Lagrangian equations, with applications in mirror symmetry.

We give examples of harmonic cellular maps between negatively curved manifolds which are not diffeomorphisms but are homotopic to diffeomorphisms.

Study immersions of punctured 4-manifolds for quantum automata applications.

A new discretisation of a doubled, i.e. BF, version of the pure abelian Chern-Simons theory is presented. It reproduces the continuum expressions for the topological quantities of interest in the theory, namely the partition function and correlation function of Wilson loops. Similarities with free spinor field theory a…

The paper extends Gaussian processes to model complex interactions in cellular complexes.

In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincare formula and an equivariant Lefschetz theorem.

A saliency detection method for ECT images segments cellular components without supervision.

An observable for nonabelian, higher-dimensional forms is introduced, its properties are discussed and its expectation value in BF theory is described. This is shown to produce potential and genuine invariants of higher-dimensional knots.

We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincaré dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow through to the discrete case, we define the discrete analogs of period matrices, Ri…