This paper explores the interactions between knot theory and quantum computing. On one side, knot theory has been used to create models of quantum computing, and on the other, it is a source of computational problems. Knot theory is often used to introduce topological idea to people without a formal mathematical backgr…
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48 results for “computability theory”
Survey on learning Boolean functions in computational theory.
problem Learning Boolean function classes in computational theory.
method Overview of known results in PAC and related models.
result Discussion of various learning results for Boolean functions.
TKFT models computation via smooth vector fields, simulating functions in a single dynamical step.
problem Modeling computation in a single step.
method Established Topological Kleene Field Theory (TKFT) as a new model of computation.
result Any computable function can be simulated in a single go of a dynamical system.
New theory challenges traditional machine learning assumptions.
problem Traditional machine learning theories are critiqued.
method A new theory is proposed and discussed.
result Learning true probabilities is not equivalent to other learning goals.
Computes immersions of -projective spaces using K-theory.
problem Computing immersions of equivariant projective spaces.
method Geometric filtration and localized slice spectral sequence.
result Obtained equivariant analogue of James periodicity.
Researchers compute differential K-theory for moduli stacks.
problem Computing differential K-theory for moduli stacks of principal G-bundles.
method Using homotopy theory of presheaves of spaces and spectra, they formulate results in terms of invariant polynomials and representation rings.
result They successfully compute the connective differential K-theory and differential cohomology of moduli stacks.
We prove that there is no parity anomaly in M-theory in the low-energy field theory approximation. Our approach is computational. We determine generators for the 12-dimensional bordism group of pin manifolds with a w_1-twisted integer lift of w_4; these are the manifolds on which Wick-rotated M-theory exists. The anoma…
Quantum map counts BPS states in special theories.
problem Computing protected spin characters in class S theories.
method Geometric approach from 5D supersymmetric Yang-Mills theory.
result Explicit computation of protected spin characters in various examples.
We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation.
Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise to a piecewise linear (PL) approximant. This is extended to isotopic convergenc…
By exploiting standard facts about and supersymmetric Yang-Mills theory, the Donaldson invariants of four-manifolds that admit a Kahler metric can be computed. The results are in agreement with available mathematical computations, and provide a powerful check on the standard claims about supersymmetric Yang…
Developed a real sutured Heegaard Floer theory.
problem Real sutured manifolds and their Floer homology.
method Real nice diagrams, combinatorial computation.
result New structural properties distinguishing from sutured Floer homology.
We develop the intersection theory at relative chain-cochain level, and apply it along with the use of Seifert disks for an oriented link to give a combinatorial algorithm to compute Massey's higher order linking numbers. It is subtle to compute higher-order linking numbers, and it has been a folklore to use the inters…
Quasi-conformal (QC) theory is an important topic in complex analysis, which studies geometric patterns of deformations between shapes. Recently, computational QC geometry has been developed and has made significant contributions to medical imaging, computer graphics and computer vision. Existing computational QC theor…
In the previous paper [GLM2018], we showed that the theory of harmonic maps between Riemannian manifolds may be discretized by introducing triangulations with vertex and edge weights on the domain manifold. In the present paper, we study convergence of the discrete theory to the smooth theory when taking finer and fine…
Novel theory combines combinatorial and topological elements.
problem Understanding combinatorial phenomena at the intersection of topology.
method Synthesizes combinatorial and topological approaches with a new framing concept.
result Framed combinatorial spaces exhibit better behavior than classical spaces.
We compute the equivariant -theory for a simply connected Lie group (acting on itself by conjugation). We prove that is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a non-simply connected Lie group , namely PSU(3),…
We compute the index of the real Cauchy-Riemann operator defined in FJRW theory in case of the smooth metric. For the cylindrical metric, we study the relation between the index of the linearized operator of Witten map and weights in weighted Sobolev space.
Develops obstruction theory for a specific 4-manifold index.
problem Computing the -index of 4-manifolds with free involution.
method Uses spectral sequences and cohomology with twisted coefficients.
result Computes the -index for various examples.
KnotMosaics package simplifies knot theory computations in SageMath.
problem Efficiently computing knot mosaic diagrams and their properties.
method Developed a SageMath package for knot mosaic diagrams, implementing validation, strand tracing, and computation algorithms.
result Enabled easy computation of knot mosaic diagrams and their properties.
We propose a way to define and compute invariants of general smooth 4-manifolds based on topological twists of non-Lagrangian 4d N=2 and N=3 theories in which the problem is reduced to a fairly standard computation in topological A-model, albeit with rather unusual targets, such as compact and non-compact Gepner models…
New invariant computed for Brieskorn homology spheres.
problem Computing an invariant for specific manifolds.
method Using Dijkgraaf--Witten invariants in topological K-theory.
result Computed invariant for Brieskorn homology spheres.
Geometrically computes superpotentials for certain 4D N=2 theories.
problem Computing effective twisted superpotentials for 4D N=2 theories.
method Spectral networks and abelianization to compute generating functions of brane opers.
result Geometric recipe for computing effective twisted superpotentials.
Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection, turn out simply to be…
A quantum field theory generalization, Baaquie, of the Heath, Jarrow, and Morton (HJM) term structure model parsimoniously describes the evolution of imperfectly correlated forward rates. Field theory also offers powerful computational tools to compute path integrals which naturally arise from all forward rate models. …
Fast algorithm for braid group Hecke representation, applied to knot invariants.
problem Computing topological invariants of knots efficiently.
method Representation-theoretic approach to braid group, leveraging quantum topology.
result Fast algorithm for Hecke representation of braid group, finding non-trivial braids.
Accelerated Evaluation of Ollivier-Ricci Curvature Lower Bounds: Bridging Theory and Computationstat.ML
We simplify evaluation of Ollivier-Ricci curvature bounds in hypergraphs.
problem Computational challenges in evaluating Ollivier-Ricci curvature bounds in hypergraphs.
method Simplified approach with linear computational complexity.
result Significant improvements in evaluating Ollivier-Ricci curvature bounds.
Computations for prime knots up to 11 crossings.
problem Computing HOMFLY homology for prime knots.
method Direct computations for all prime knots up to 11 crossings.
result HOMFLY homology determined for all prime knots up to 11 crossings.
Quantum Character Theorymath.RT
Develops quantum character theory for complex reductive groups.
problem Quantum analogue of conjugation equivariant -modules.
method Schur-Weyl functor and double affine Hecke algebra.
result Computes endomorphism algebras of quantum Hotta-Kashiwara modules.
Extends Bayesian theory to handle complex interdependencies in multidimensional event spaces.
problem Complex interdependencies between events and hypotheses sets in real-world systems.
method Developed a mathematical formalism for modeling complex relationships through rigorous derivation and validated using analytical proofs, simulations, and case studies.
result MDSE theory improves prediction accuracy by 15-20% compared to standard Bayesian methods in high interdimensionality datasets.
We study refined topological string theory in the presence of orientifolds by counting second-quantized BPS states in M-theory. This leads us to propose a new integrality condition for both refined and unrefined topological strings when orientifolds are present. We define the SO(2N) refined Chern-Simons theory which co…
We compute Khovanov homology for tangles using TQFT.
problem Khovanov homology for tangles is not well studied or computed.
method Topological Quantum Field Theory (TQFT) construction.
result A comprehensive method for computing Khovanov homology of tangles.
Paper tackles decidability of subgroup discreteness problem.
problem Decidability of finitely generated subgroup discreteness in and .
method Examines different computational models to determine if the discreteness problem is decidable.
result The answer depends on the model of computation chosen.
Compute Bredon homology for a specific type of Artin groups.
problem Calculate Bredon homology for Artin groups of dihedral type.
method Compute Bredon homology groups of the classifying space for virtually cyclic subgroups with K-theory coefficients.
result Computed Bredon homology groups for Artin groups of dihedral type.
Develops trace class operators and inverse Laplacian theory for infinite dimensions.
problem Understanding trace class operators and inverse Laplacian on infinite dimensional spaces.
method Presentation of trace class operators and construction of inverse Laplacian on closed manifolds.
result Original trace computations involving the inverse Laplacian on the torus.
We show that the topological modular functor from Witten-Chern-Simons theory is universal for quantum computation in the sense a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern-Simons theory at a fifth ro…
This paper extends financial theory to measure learnable market structure under computational constraints.
problem Understanding learnable market structure under bounded computational capacity.
method Introduces financial epiplexity as a measure of learnable market structure, extending classical information theory.
result Proves that equal entropy does not imply equal epiplexity and derives thresholds for useful regimes.
We construct a spectral sequence converging to the Morava -theory of unordered configuration spaces and identify its E-page as the homology of a Chevalley-Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the -theory of the weight summands of iterated loop spaces of spheres (paramet…
Cobordisms of singular mapsmath.GT
Szűcs introduced cobordism of singular maps to compute groups of immersions and embeddings.
problem Computing cobordism groups of immersions and embeddings in dimensions where classical theory fails.
method Investigation of classifying spaces constructed by Szűcs and Rimányi.
result Collection and organization of results towards computation of cobordism groups of singular maps.
We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, wh…
Researchers compute invariants for knots and links in lens spaces using large N and k limits.
problem Computing invariants for knots and links in lens spaces.
method Large N and k limits, saddle point description, collective field theory.
result Invariants for the Hopf link and unknot are computed using on-shell collective field theory action.
New methods compute Alexander polynomials for complex knots.
problem Efficiently computing higher order Alexander polynomials for complex knots.
method Developed new algorithms to compute the Smith normal form of Alexander matrices.
result Computed Alexander polynomials for knots up to 100 crossings.
Topological Quantum Computationquant-ph
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liqu…
Homology and cohomology theory for topological quandles computed.
problem Computing invariants for knot diagrams using quandle cocycles.
method Introducing homology and cohomology theory for topological quandles, studying their relation to quandle groups, and using topological quandle cocycles to compute state sum invariants.
result State sum invariants computed using topological quandle cocycles.
Survey on computational models in dynamical systems, including new universality concepts.
problem Understanding the relationship between computational models and dynamical systems.
method Review of recent works on Turing universality, Topological Kleene Field Theories, and dynamical bordisms.
result Introduction of new perspectives on computability through dynamical systems.
We define and study the Burnside quotient Green ring of a Mackey functor. Some refinements of Dress induction theory are presented, together with applications to computation results for -theory and -theory of finite and infinite groups.
Computing Diffusion Geometrymath.DG
New framework for diffusion geometry simplifies complex calculations.
problem Challenges in applying calculus and geometry to real data.
method Reformulates calculus and geometry via diffusion processes.
result Improves precision, robustness, and computational efficiency.
We have developed a mathematical theory of the topological vertex--a theory that was original proposed by M. Aganagic, A. Klemm, M. Marino, and C. Vafa in hep-th/0305132 on effectively computing Gromov-Witten invariants of smooth toric Calabi-Yau threefolds derived from duality between open string theory of smooth Cala…