geometric-and-algebraic-topology

nLab join of simplicial sets Context Homotopy theory homotopy theory, (∞,1)-category theory, homotopy type theory flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed… models: topological, simplicial, localic, … see also algebraic topology Introductions Definitions Paths and cylinders Homotopy groups Basic facts Theorems Contents Idea The join …

Alain Connes (born on April 1, 1947) is a French mathematician, Fields medalist (1982), Crafoord prize winner (2001), Professor at IHÉS, Professor at Collège de France and part-time working as a Professor at Vanderbilt University. His interests include geometry, topology, especially K-theory and index theory, operator algebras, the connections between noncommutative geometry and number theory, an…

We study the poset of hyperbolic structures on Thompson’s group F and its generalizations F n for n ≥ 2 . The global structure of this poset is as simple as one would expect, with the maximal non-elementary elements being two quasi-parabolic actions corresponding to well-known ascending HNN-extension expressions of F n . However, the local structure turns out to be incredibly rich, in stark contr…

ABSTRACT Quantum annealing is emerging as a practical tool for large‐scale combinatorial optimization. We map continuum displacements and element densities to binary variables, turning both deformation analysis and topology optimization into quadratic unconstrained binary optimization (QUBO) problems that run on today's annealers. Two formulations are compared: one derived from the stiffness equa…

Abstract Additive manufacturing enables the production of structures conceived with optimal design freedom, realising the potential of conceptual gains. The design of composites with variable orientations, or variable stiffness design, holds unexplored potential for benefiting from optimisation techniques and additive manufacturing. This work presents an innovative two-level methodology for optim…

nLab cyclic loop space Contents Context Topology topology (point-set topology, point-free topology) see also differential topology, algebraic topology, functional analysis and topological homotopy theory Introduction Basic concepts - open subset, closed subset, neighbourhood - topological space, locale - base for the topology, neighbourhood base - finer/coarser topology - closu…

Nature Communications, Published online: 01 September 2026; doi:10.1038/s41467-026-77201-z Topological valley photonics enables robust on-chip light transport, but valley edge states lie outside the light cone, blocking free-space excitation. Here, authors fold these states into topological valley quasi-bound states in the continuum, enabling free-space coupling while preserving low-loss terahert…

his paper, we relate the space of line segments induced by a Jordancurve in the plane to the space of line segments induced by the plane byembedding the M¨obius band associated with the curve into the familyof toruses associated with the plane; we further demonstrate how thegeometric ”square condition” of the curve translates into a topologicalproperty through this embedding

Infinite group theory is the geometry of the endless: the groups whose word metrics shape their spaces—free groups' trees, hyperbolic groups' negative curvature, amenable groups' invariant means—and whose century's arc ran from Max Dehn's 1911 problems through Nielsen's transformations, von Neumann's and Day's amenability, Gromov's polynomial growth and 1987 hyperbolicity, Grigorchuk's intermedia…

The 6-sphere, as a smooth manifold is diffeomorphic to the coset space of G₂ (automorphism group of the octonions) by SU(3) (Fukami-Ishihara 55). For more see at G₂/SU(3) is the 6-sphere. The induced action of G₂ on induces an almost Hermitian structure which makes it a nearly Kaehler manifold?. Review in is in Agrikola-Borowka-Friedrich 17 coset space-structures on n-spheres: see also Spin(8)-su…

Nature Reviews Physics, Published online: 27 August 2026; doi:10.1038/s42254-026-00974-1 Topological pair density waves offer an insight into the interplay among finite-momentum pairing, topological matter and chiral superconductivity. This Perspective explores signatures of these states in kagome superconductors.

The Adams conjecture is a statement about triviality of spherical fibrations associated to certain formal differences of vector bundles (K-theory classes) via the J-homomorphism. The conjecture was stated in (Adams 63, conjecture 1.2), for vector bundles of rank up to two over a finite CW-complex, which was proven in (Adams 63, theorem 1.4). A general proof was then given in (Quillen 71). The Ada…