We prove that the foam and matrix factorization universal rational sl3 link homologies are naturally isomorphic as projective functors from the category of link and link cobordisms to the category of bigraded vector spaces.
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48 results for “SL3 local systems”
We define the universal sl3-link homology, which depends on 3 parameters, following Khovanov's approach with foams. We show that this 3-parameter link homology, when taken with complex coefficients, can be divided into 3 isomorphism classes. The first class is the one to which Khovanov's original sl3-link homology belo…
This thesis splits into two major parts. The connection between the two parts is the notion of "categorification" which we shortly explain/recall in the introduction. In the first part of this thesis we extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex…
Quantum cluster algebra constructed from web skein relations on surfaces.
problem Quantization of cluster structures on moduli spaces of SL3 local systems.
method Constructing a quantum cluster algebra inside the skew-field of a skein algebra of unpunctured surfaces.
result Laurent expressions of webs in clusters have positive coefficients.
Characterizes flag geometries for Hitchin representations in SL3(R).
problem Understanding flag geometries associated with Hitchin representations in SL3(R).
method Geometric characterization based on invariant foliations and refraction flows.
result Constructs refraction flows for positive roots in general sl_n(R), with highest root flows being C^1+α.
Intersections of Dual -Websmath.GT
Interprets SL3-web intersections on surfaces.
problem Interpreting SL3-web intersections on surfaces.
method Intersection pairing between reduced SL3-webs and tropical sets.
result Provides a new proof of flip equivariance.
Let G be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of G, which can be also seen as the space of maximal flats of the symmetric space of G. We define its Chabauty compactification…
New coordinates for SL3-web graphs on surfaces defined by Fock-Goncharov.
problem Characterizing functions on SL3-character varieties of surfaces.
method Introduced tropical Fock-Goncharov coordinates based on Knutson-Tao rhombus inequalities and congruence conditions.
result Tropical coordinates naturally index the commutative algebra of functions on SL3-character varieties.
Geometric model of unbounded sl3 laminations with tropical coordinates.
problem Modeling unbounded laminations in cluster varieties.
method Introducing tropical cluster coordinates and geometric gluing procedures.
result Established a geometric gluing procedure for unbounded sl3 laminations.
Quantum traces for : The case math.GT
Extends quantum trace map to SL3(C) for 3D surfaces.
problem Generalizing quantum trace map to higher dimensions.
method Definition of SL3(C) quantum trace invariant.
result Construction of SL3(C) quantum trace map.
On regular subgroups of math.GR
Regular subgroups of SL3(R) are identified and ruled out.
problem Identifying and characterizing regular subgroups of SL3(R).
method Using Kapovich–Leeb–Porti and Guichard–Wienhard divergent subgroups criteria, and Oh's results.
result Regular subgroups of SL3(R) are precisely lattices in minimal horospherical subgroups.
Natural coordinates for SL3-webs on surfaces are shown to be consistent under triangulation changes.
problem Natural coordinates for SL3-webs on surfaces.
method Showed natural coordinates are consistent under triangulation changes using cluster transformations.
result Natural coordinates for SL3-webs on surfaces are consistent under triangulation changes.
The stable Khovanov-Rozansky homology of torus knots has been conjecturally described as the Koszul homology of an explicit non-regular sequence of polynomials. We verify this conjecture against newly available computational data for sl(3)-homology. Special attention is paid to torsion. In addition, explicit conjectura…
We exhibit a certain infinite family of three-stranded quasi-alternating pretzel knots which are counterexamples to Lobb's conjecture that the sl_3-knot concordance invariant s_3 (suitably normalised) should be equal to the Rasmussen invariant s_2. For this family, |s_3| < |s_2|. However, we also find other knots for w…
Study real forms and GIT quotients in algebraic varieties.
problem Linking real points of complex GIT quotients to real GIT quotients.
method Explore actions of real forms on complex algebraic varieties and prove lifting properties.
result Some real points of complex GIT quotients can be lifted to real GIT quotients under certain conditions.
Paper proves a Cohen-Dimca-Orlik type theorem for Z-local systems of hyperplane arrangements.
problem Proving a Cohen-Dimca-Orlik type theorem for -local systems.
method Analyzing local system cohomology groups of hyperplane arrangements complements.
result Proves a Cohen-Dimca-Orlik type theorem for -local systems.
The paper generalizes Hodge theory to semisimple local systems and proves a geometric Decomposition theorem.
problem Generalizing Hodge theory to semisimple local systems.
method Establishing a canonical isomorphism and proving a global invariant cycle theorem.
result A new geometric proof of the Decomposition theorem for semisimple local systems.
We establish a structure theorem for the integral points on moduli of special linear rank two local systems over surfaces, using mapping class group descent and boundedness results for systoles of local systems.
Graphs and local systems count multiwebs.
problem Counting multiwebs in graphs with local systems.
method Using Kasteleyn matrices and web-traces.
result Determinant of Kasteleyn matrix counts multiwebs.
We study bi-Hamiltonian systems of hydrodynamic type with non-singular (semisimple) non-local bi-Hamiltonian structures and prove that such systems of hydrodynamic type are diagonalizable. Moreover, we prove that for an arbitrary non-singular (semisimple) non-locally bi-Hamiltonian system of hydrodynamic type, there ex…
Analog forecasting uses local dynamics to predict chaotic systems.
problem Theoretical connections between analog forecasting and dynamical systems are overlooked.
method Local approximations of the system's dynamics, linear regression, and estimation of analog forecasting errors.
result Analog forecasting performances are highly linked to the local Jacobian matrix of the flow map.
We investigate local configuration controllability for mechanical control systems within the affine connection formalism. Extending the work by Lewis for the single-input case, we are able to characterize local configuration controllability for systems with degrees of freedom and input forces.
The paper studies cohomological Donaldson-Thomas theory for local systems on a 3-torus.
problem Cohomological Donaldson-Thomas theory for local systems on the 3-torus.
method Using exponential maps and the tripled Jordan quiver, the paper proves cohomological integrality for GL_n and SL_n local systems.
result The paper proves Langlands duality statements for SL_n and PGL_n cohomological Donaldson-Thomas invariants for prime n.
Generalizes abelianization for framed local systems over surfaces.
problem Understanding framed local systems over punctured surfaces for various groups.
method Analysis of spectral networks, triangulations, and matrix reinterpretation of path lifting rules.
result Parametrizations of moduli spaces of decorated and framed local systems.
Local systems in diffeologymath.AT
Introduces a framework for rational homotopy theory in diffeological spaces.
problem Challenges in rational homotopy theory for smooth spaces with arbitrary fundamental groups.
method Utilizes local systems over simplicial sets and a model structure for diffeological spaces.
result Establishes an equivalence between fibrewise rational diffeological spaces and algebraic local systems.
Minimal energy local systems on curves are compact components of character varieties.
problem Characterizing local systems on surfaces with minimal energy.
method Study of minimal energy local systems on surfaces of genus g with d punctures.
result Minimal energy local systems form compact components of real relative character varieties.
Holonomies match for higher local systems and principal 2-bundles.
problem Matching holonomies for higher local systems and principal 2-bundles.
method Higher Riemann-Hilbert correspondence and principal 2-bundles.
result Holonomies coincide for both formalisms.
Data-Informed Decomposition for Localized Uncertainty Quantification of Dynamical Systemsphysics.comp-ph
Bayesian framework for identifying localized regions of interest in dynamical systems.
problem Identifying regions of high-resolution uncertainty quantification in complex dynamical systems.
method Bayesian inference with Gaussian process surrogate and polynomial chaos expansion.
result Unified computational scheme reduces overall cost for uncertainty quantification.
The paper studies limit sets on using stationary measures.
problem Investigating the Hausdorff dimension of limit sets on for .
method Using stationary measures to generalize the Patterson-Sullivan formula and establish dimension formulas.
result Sharp lower bounds and Hausdorff dimensions for Anosov representations and the Rauzy gasket.
Model financial default cascades on sparse graphs via hitting times.
problem Capturing systemic risk in large, sparsely-connected financial networks.
method Dynamic particle systems with hitting times and convergence theory.
result Characterization of default time distribution in tree-like networks.
The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top degree, and bounded chambers form a basis of the remaining cohomology group. We det…
Proposes local coordinate frames for improving model performance in complex dynamical systems.
problem Improving model performance in complex, non-linear, and time-dependent dynamical systems.
method Introduces roto-translation invariant local coordinate frames for geometric graphs.
result The approach outperforms state-of-the-art models in various complex scenarios.
We formulate and analyze a multi-agent model for the evolution of individual and systemic risk in which the local agents interact with each other through a central agent who, in turn, is influenced by the mean field of the local agents. The central agent is stabilized by a bistable potential, the only stabilizing force…
We give a new proof for the local existence of a smooth isometric embedding of a smooth -dimensional Riemannian manifold with nonzero Riemannian curvature tensor into -dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce …
In the previous paper we constructed the local system of Khovanov complexes on the Vassiliev space of knots and extended it to the singular locus. In this paper we introduce the definition of the homology theory (local system) of finite type and prove the first finiteness result: the Khovanov local system restricted to…
Identifying the location of a disturbance and its magnitude is an important component for stable operation of power systems. We study the problem of localizing and estimating a disturbance in the interconnected power system. We take a model-free approach to this problem by using frequency data from generators. Specific…
Categorifies Chern-Weil theory for infinite local systems.
problem Categorifying Chern-Weil theory for infinite local systems.
method Using DG functors and -natural isomorphisms.
result Provides a categorification of the Chern-Weil homomorphism.
Locally adaptive nearest neighbors improve automated systems' performance and are easier to interpret.
problem Improving automated systems' performance and interpretability.
method Developed a method for k nearest neighbors algorithms to learn locally adaptive metrics.
result Locally adaptive metrics improve performance and are interpretable.
Study limits of quasi-local angular momentum at infinity of gravitating systems.
problem Understanding limits of quasi-local angular momentum at infinity of gravitating systems.
method Based on optimal isometric embedding and quasilocal mass theory, the study defines and analyzes the limits of quasi-local angular momentum at spatial and null infinity.
result Limits of quasi-local angular momentum are discussed at spatial and null infinity of an isolated gravitating system.
Twisted local systems on surfaces of finite type appear often in geometry and physics. Most of them arise geometrically as local systems of charts for pleated hyperbolic structures. Bonahon and Thurston's "shear-bend coordinates" parameterize these local systems of charts. On a surface …
Electrostatic systems with specific tensors are locally conformally flat.
problem Understanding the geometry of electrostatic systems with special tensors.
method Proving local conformal flatness for electrostatic manifolds with divergence-free Bach tensor.
result Three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat.
Model interpretability is an increasingly important component of practical machine learning. Some of the most common forms of interpretability systems are example-based, local, and global explanations. One of the main challenges in interpretability is designing explanation systems that can capture aspects of each of th…
Paper studies rigidity phenomena for elliptic systems on Riemannian manifolds.
problem Local radial rigidity of elliptic systems on Riemannian manifolds.
method Reduction to singular ordinary differential equations of Euler type.
result Local uniqueness and existence results for solutions with prescribed initial jets.
We propose in this paper a constructive procedure that transforms locally, even at singular configurations, the kinematics of a car towing trailers into Kumpera-Ruiz normal form. This construction converts the nonholonomic motion planning problem into an algebraic problem (the resolution of a system of polynomial equat…
Proves local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction.
problem Local bi-integrability of bi-Hamiltonian systems.
method Bi-Poisson reduction to prove local bi-integrability.
result Constructs a complete set of functions in bi-involution for bi-Hamiltonian systems.
New method constructs axial vector fields and defines quasi-local spin-angular momentum.
problem Constructing axial vector fields on Riemannian two-spheres.
method Using centre-of-mass unit sphere reference systems and Lie-propagated unit sphere reference systems.
result Constructive definition of quasi-local spin-angular momentum and balance relations.
Developed a new symmetric hyperbolic formulation for Einstein-Yang-Mills system.
problem Future stability of solutions of the Einstein-Yang-Mills system with arbitrary dimension.
method Tensorial symmetric hyperbolic formulation and local well-posedness for Cauchy problem.
result Established local well-posedness for the Cauchy problem of EYM equations in the temporal gauge.
Proves properties of complex algebraic varieties and local systems.
problem Properties of complex algebraic varieties and local systems.
method Analytic Zariski open subsets and algebraic maps.
result Trivializing covering spaces of complex algebraic varieties.