Related papers: Topological boundary maps in physics: General theo…
We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski…
I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume…
In this paper, we classify completely hyperbolic 3-manifolds corresponding to geometric limits of Kleinian surface groups isomorphic to $\pi_1(S)$ for a finite-type hyperbolic surface $S$. In the first of the three main theorems, we…
Competition among repetitive measurements of noncommuting observables and unitary dynamics can give rise to a wide variety of entanglement phases. Here, we propose a general framework based on Lyapunov analysis to characterize topological…
This paper is based on my talk at ICM on recent progress in a number of classical problems of linear algebra and representation theory, based on new approach, originated from geometry of stable bundles and geometric invariant theory.
We give a group-theoretic interpretation of relativistic holography as equivalence between representations of the anti de Sitter algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the…
This paper attempts to relate some ideas of Grothendieck in his Esquisse d'un programme and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss the Teichm\"uller theory, the mapping class groups,…
We study the low-energy properties of a truncated Haldane pseudopotential with maximal half filling, which describes a strongly correlated system of spinless bosons in a cylinder geometry. For this Hamiltonian with either open or periodic…
We study topological boundedness of order-to-topology bounded and order-to-topology continuous operators from ordered vector spaces to topological vector spaces. The uniform boundedness principle for such operators is investigated.
J.H.C. Whitehead introduced the concept of crossed modules in the early 20th century. These crossed modules are crucial for algebraic models of 2-type homotopy, which involve connected spaces with no higher than second-degree homotopy…
These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum…
Topological materials are renowned for their ability to harbor states localized at their peripheries, such as surfaces, edges, and corners. Accompanying these states, fractional charges appear on peripheral unit cells. Recently,…
Variational principles in mechanics, field theory and geometric analysis are usually formulated on closed admissible classes, where boundary variations are either fixed or independently cancelled through natural boundary conditions.…
In this paper, applications of the connection between the soliton theory and the commuting nonselfadjoint operator theory, established by M.S. Liv\v{s}ic and Y. Avishai, are considered. An approach to the inverse scattering problem and to…
We investigate topological properties of density matrices motivated by the question to what extent phenomena like topological insulators and superconductors can be generalized to mixed states in the framework of open quantum systems. The…
We define a new topological invariant of line arrangements in the complex projective plane. This invariant is a root of unity defined under some combinatorial restrictions for arrangements endowed with some special torsion character on the…
We study effectively one-dimensional systems that emerge at the edge of a two-dimensional topologically ordered state, or at the boundary between two topologically ordered states. We argue that anyons of the bulk are associated with…
Starting from a dynamical system $(\Omega,G)$, with $G$ a generic topological group, we devise algorithms that generate families of patterns in the Euclidean space, which densely embed $G$ and on which $G$ acts continuously by rigid shifts.…
By studying scattering Lie groups and their associated Lie algebras, we introduce a new method for the characterisation of collision invariants for physical scattering families associated to smooth, convex hard particles in the particular…
We extend the twisted gauge theory model of topological orders in three spatial dimensions to the case where the three spaces have two dimensional boundaries. We achieve this by systematically constructing the boundary Hamiltonians that are…