The topological index of a surface was previously introduced by the first author as the topological analogue of the index of an unstable minimal surface. Here we show that surfaces of arbitrarily high topological index exist.
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48 results for “topological index”
Study index theory on Lie group homogeneous spaces using topological and analytic methods.
problem Index theory on homogeneous spaces of Lie groups.
method Topological and analytic approaches: Riemann-Roch formula and heat kernel methods.
result Local index formula representing higher indices of equivariant elliptic operators.
The disk complex of a surface in a 3-manifold is used to define its {\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical surfaces. The main result is that one may always isotope a surface with top…
Constructs hyperbolic manifolds with surfaces of high topological index.
problem Creating hyperbolic manifolds with surfaces of high topological index.
method Using graph theory and topological properties, constructs surfaces in hyperbolic manifolds.
result Proves existence of hyperbolic manifolds containing surfaces of arbitrarily high topological index.
Study topological quantum mechanics on orbifolds with geometric interpretation.
problem Quantum mechanical models on symplectic orbifolds.
method Explicit orbifold version of quantum HKR map and exact semi-classical approximation.
result Geometric and quantum field theoretic interpretation of orbifold algebraic index.
Atiyah-Singer theorem links math fields, predicts topological insights.
problem Understanding the interplay between analysis, geometry, and topology.
method Analyzes and generalizes topological invariants in differential geometry.
result Predicts the index of elliptic operators based on topology.
Bridge surfaces remain topologically minimal after slight changes.
problem Preserving topological minimality of bridge surfaces after small perturbations.
method Analyzing how bridge surfaces behave under small changes (perturbations).
result After perturbation, bridge surfaces maintain topological minimality with index at most one more.
Study finds minimal hypersurfaces grow linearly in index, contrary to 3D.
problem Understanding index growth of minimal hypersurfaces.
method Partitioning methods for compact Lie groups, applied to hypersurfaces.
result Linear index growth for hypersurfaces of fixed topological type.
Bounds on minimal surfaces' ends and stability index.
problem Understanding the topology and stability of minimal surfaces.
method Proving bounds on the number of ends and stability index of minimal surfaces.
result Minimal surfaces of genus have a finite number of ends and a bounded stability index.
A new method tracks index using topological data analysis for sparse portfolios.
problem Sparse index tracking with robust risk management.
method Topological learning via Vietoris-Rips filtration for sparse regularization.
result The method outperforms state-of-the-art techniques in various market conditions.
New Bailey pairs derived for tetrahedron index, linking knot invariants.
problem Deriving knot invariants using Bailey pairs for the tetrahedron index.
method Developed new Bailey pairs in terms of q-series.
result New Bailey pairs express the pentagon identity of the tetrahedron index.
The abstract discusses connecting quantum mechanics and algebraic index theories.
problem Exploring the connection between quantum mechanics and algebraic index theories.
method Explains how the classical algebraic index theorem can be proved in terms of BV quantization of topological quantum mechanics and 2d chiral CFT.
result Shows how the generating function of all genus Gromov-Witten invariants on elliptic curves is mirror equivalent to an elliptic chiral index.
Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…
3D-index connects to normal surfaces theory.
problem Constructing invariants of 3-manifolds.
method 3D-index as a generating series of normal surfaces.
result 3D-index is a connection to normal surface theory.
This is an expository article. It discusses an approach to hypoelliptic Fredholm index theory based on noncommutative methods (groupoids, C*-algebras, K-theory). The paper starts with an explicit index theorem for scalar second order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-bro…
CIFs replace single bijections with continuous families to avoid topological limitations.
problem Normalising flows struggle with targets with complex topologies.
method Propose Continuously Indexed Flows (CIFs) replacing single bijections with a continuous family.
result CIFs avoid topological limitations and perform better empirically.
New proof of index theorem for topological manifold bundles.
problem Index theorem for fiber bundles of compact topological manifolds.
method Use of a convenient framework for bivariant theories and recent results on the homotopy type of the topological cobordism category.
result Refinement of the assembly map for an extended A-theory characteristic.
Paper introduces Lie algebroid index theory and a generalized Riemann-Roch theorem.
problem Generalizing Riemann-Roch theorem for manifolds with regular foliations.
method Developed Lie algebroid index theory and applied it to obtain a generalized Riemann-Roch theorem.
result Obtained a generalized Riemann-Roch theorem for manifolds with regular foliations.
The fixed point index of topological fixed point theory is a well studied integer-valued algebraic invariant of a mapping which can be characterized by a small set of axioms. The coincidence index is an extension of the concept to topological (Nielsen) coincidence theory. We demonstrate that three natural axioms are su…
Bounds on the index of free boundary minimal surfaces.
problem Understanding the index of free boundary minimal surfaces.
method Comparison of energy and area indices, combining with previous work.
result Area index is bounded by a linear function of genus and boundary components.
Proves formula for 3D index change with Dehn filling.
problem Transforming 3D index under Dehn filling.
method Relative 3D index, gluing principle, inductive framework, q-hypergeometric functions.
result Rigorous proof of Gang-Yonekura formula.
On the spectrum and index of expanding and translating solitons of the mean curvature flow in math.DG
The study classifies translating and self-expanding solitons in 3D space.
problem Characterizing the topology and index of solitons in mean curvature flow.
method Analyzing the spectrum and index of expanding and translating solitons in .
result Translating and self-expanding solitons have finite topology under certain conditions.
Estimates translator stability via topological features.
problem Quantifying stability of translators in geometric flows.
method Adapted Li-Tam theory to weighted settings, estimating nullity of stability operator.
result Quantitative index bounds for translators via topology.
Analysis of Hopf solitons as generalized fold mapscond-mat.soft
New framework models high-Hopf-index hopfions using generalized fold maps.
problem Modeling and understanding high-Hopf-index hopfions.
method Proposes a new mathematical framework based on generalized Hopf maps and fold maps.
result Establishes a robust bridge between singular maps and experimentally observed hopfion topologies.
Study index theory for foliated manifolds with boundary using blup groupoids.
problem Index theory for foliated manifolds with boundary.
method Use blup groupoids and pseudodifferential calculus to study index problems.
result Construct and prove new index theorems for fully elliptic operators.
In this paper, we survey recent results on index defects of elliptic operators on manifolds with boundary. Index defects are similar to the Hirzebruch signature defects in topology, where the defects appear as the correction terms to the signature formula on manifolds with boundary. For some natural classes of elliptic…
The paper simplifies string topology computations using Hochschild chain models.
problem Computing string topology operations efficiently.
method Hochschild chain models to simplify proofs and computations.
result New observations and detection of closed geodesics.
The paper bounds the energy index of harmonic Gauss maps on surfaces.
problem Bounding the energy index of harmonic Gauss maps on surfaces.
method Analyzes closed and complete non-compact surfaces in Lie groups with bi-invariant metrics.
result The energy index of the Gauss map is bounded by the topological genus.
Let be a (generalized) Dirac operator on a non-compact complete Riemannian manifold acted on by a compact Lie group . Let be an equivariant map, such that the corresponding vector field on does not vanish outside of a compact subset. These data define an element of -theory of the tran…
Paper proves index theorem for self-adjoint elliptic boundary problems.
problem Proving index theorem for self-adjoint elliptic boundary problems.
method Topological and pseudo-differential methods, generalized Atiyah-Singer approach.
result Removed technical assumption to prove index theorem.
The paper classifies stable hypersurfaces and gives bounds for Morse index.
problem Stability and instability of type-II partitioning problem.
method Complete classification of stable stationary hypersurfaces, topological restrictions, lower bound for Morse index.
result Complete classification of stable type-II stationary hypersurfaces in a ball.
The paper uses topological concepts to analyze neural networks, revealing complex structure and dynamics.
problem Understanding the structure and dynamics of deep learning models.
method Topological dynamical systems, index theory, and computational homology.
result Neurons correspond to simplexes in a simplicial complex, and topological invariants can be computed.
The notion of topological degree is studied for mappings from the boundary of a relatively compact strictly pseudo-convex domain in a Stein manifold into a manifold in terms of index theory of Toeplitz operators on the Hardy space. The index formalism of non-commutative geometry is used to derive analytic integral form…
Teaches Dirac operators for geometry and topology.
problem Interactions between geometry and topology.
method Families of Dirac operators and index theorems.
result Applications to metrics of positive scalar curvature and the three-dimensional Weinstein conjecture.
The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …
This paper introduces TDA and TSI for better business analytics.
problem Nonlinear, multi-scale business datasets under-represented by traditional tools.
method Topological Data Analysis (TDA) and Topological Stability Index (TSI).
result TSI reveals structural variability in business data.
Local index formula for Lorentzian Dirac operators on spacetimes.
problem Index theory for Lorentzian Dirac operators with nontrivial dynamics.
method Local index formula based on microlocal analysis.
result Established a local index formula for Lorentzian Dirac-type operators.
We establish a mod 2 index theorem for real vector bundles over 8k+2 dimensional compact pin manifolds. The analytic index is the reduced invariant of (twisted) Dirac operators and the topological index is defined through -theory. Our main result extends the mod 2 index theorem of Atiyan and Singer to non-o…
Explain Arnold's proof of the Morse index theorem using Maslov index.
problem Proving the Morse index theorem in Riemannian geometry.
method Using symplectic arguments and the Maslov index.
result Self-contained exposition of Arnold's proof.
Study on scalar curvature bounds and manifold topological complexity.
problem Understanding the topological complexity of manifolds with scalar curvature constraints.
method Introduced a small scale index theorem to establish bounds for Gromov's simplicial norm.
result Upper bound for Gromov's simplicial norm established in terms of scalar curvature, volume, and injectivity radius.
In this note we prove some results in flat and differential -theory. The first one is a proof of the compatibility of the differential topological index and the flat topological index by a direct computation. The second one is the explicit isomorphisms between Bunke-Schick differential -theory and Freed-Lott diff…
We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of Kneser and Haken, and more recent theorems of Rubinstein and Stocking, and is t…
Estimates total curvature of minimal hypersurfaces with bounded index and area.
problem Control total curvature of minimal hypersurfaces with bounded index and area.
method Proves qualitative estimates on total curvature in terms of index and area for hypersurfaces of dimension less than seven.
result Total curvature is quantised in terms of a limit surface and finite sums of Euclidean minimal hypersurfaces.
Physicists explain a mathematical theorem about topological insulators.
problem Mathematical formulation of APS index theorem not directly related to physical fermion system.
method Reformulated APS index theorem using η invariant of domain-wall Dirac operator.
result Equivalence between APS index and η invariant is generally true.
Defines a map connecting 3d-index and skein module.
problem Connecting mathematical physics predictions with topological quantum field theory.
method Defines a map from skein module to Laurent series ring.
result The map fulfills a supersymmetry prediction and is part of a conjectural topological quantum field theory.
The Atiyah-Singer index theorem is a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their…
Paper constructs first critical bridge spheres for nontrivial links.
problem Finding critical bridge spheres for nontrivial links.
method Equivalent combinatorial definition of critical surfaces.
result First known critical bridge spheres for nontrivial links.
We show that an -bridge sphere for the unknot is a topologically minimal surface of index at most .