Paper extends matrix completion to nonlinear algebraic varieties.
problem Matrix completion for nonlinear algebraic varieties.
method Tensorization and Kronecker product approach.
result New method outperforms existing state-of-the-art methods.
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48 results for “nonlinear algebraic varieties”
One has believed that low energy effective theories of the Higgs branch of gauged linear sigma models correspond to supersymmetric nonlinear sigma models, which have been already investigated by many works. In this paper we discuss a explicit derivation of supersymmetric nonlinear sigma models from gauged linear sigma …
Study algebraic invariants from lightning self-attention models.
problem Understanding polynomial coefficients of self-attention mechanisms.
method Identify algebraic invariants using polynomial coefficients and coordinate geometry.
result Found linear and nonlinear families of algebraic invariants.
Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.
problem Measuring complexity of finding nearest points in Grassmannian space.
method Uses Lipschitz critical point theory and o-minimal geometry.
result Establishes fundamental properties of GDC, including bounds and finiteness conditions.
We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e. each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional…
Surveying recent work on Kähler metrics and algebraic variety stability.
problem Understanding canonical Kähler metrics on algebraic varieties.
method Analyzing recent developments in algebraic geometry.
result Relation between canonical Kähler metrics and stability in algebraic geometry.
The article examines twisted cohomologies on algebraic and analytic varieties.
problem Understanding and comparing twisted cohomologies on algebraic and analytic varieties.
method Comparison and definition of twisting parameters in both categories, algebraic and analytic.
result Reviewed isomorphisms of twisted cohomologies for cohomologous twisting parameters.
New proof of divisibility property for certain algebraic varieties.
problem Divisibility property for LQEL varieties.
method Construction of Clifford algebra representations to Severi varieties.
result New proof of Russo's Divisibility Property for LQEL varieties.
The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.
problem Understanding gradings on nilpotent Lie algebras associated with algebraic varieties.
method Analyzing lattice structures in nilpotent Lie groups and their fundamental groups.
result Conditions for a lattice to be the fundamental group of a smooth complex algebraic variety.
Study character varieties of surfaces using cluster algebras and Poisson structures.
problem Character varieties of surfaces and their Poisson structures.
method Use Bonahon-Wong's trace map and cluster algebras associated with ideal triangulations.
result Recover Goldman Poisson algebra from cluster algebra structure and show automorphisms.
BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on toric varietiesmath.AG
Study BV operators on holomorphic polyvector fields on toric varieties.
problem Existence of BV operators in Gerstenhaber algebras.
method Analyzing BV operators on holomorphic polyvector fields on smooth compact toric varieties.
result Necessary and sufficient condition for BV operators existence.
Quantum trace maps abelian character varieties to SL2 character varieties.
problem Classifying irreducible representations of Chekhov-Fock algebras.
method Non-commutative deformation of algebraic morphisms.
result Induces birational morphism between torus and SL2 character variety.
The paper verifies predictions about algebraically hyperbolic varieties and their maps.
problem Verifying Demailly's conjecture about algebraically hyperbolic varieties.
method Defining and proving stability of algebraically hyperbolic varieties, and studying boundedness of maps.
result Finite automorphism groups and surjective endomorphisms of projective varieties.
Investigates fundamental groups and path lifting for algebraic varieties.
problem Understand fundamental groups and path lifting properties of algebraic varieties.
method Analyzes three fundamental questions about fundamental groups of algebraic varieties.
result Identifies conditions for surjectivity on fundamental groups and path lifting properties.
Study identifies subvarieties of projective varieties mapping to models.
problem Understanding mappings of subvarieties to models on projective varieties.
method Analyzes smooth projective varieties with holomorphic locally homogeneous structures.
result Determines all subvarieties mapping to the model.
Study complex slices on real algebraic varieties and their properties.
problem Understanding the properties and bounds of complex slices on real algebraic varieties.
method Analyzing intersections and cohomology classes of complex subvarieties with real subvarieties.
result Proved an upper bound for the linking number of certain slices.
Study tests if a probability measure is near a real algebraic variety.
problem Deciding if a probability measure is near a real algebraic variety.
method Proved upper bound on sample complexity, reduced to semialgebraic decision problem, studied Hausdorff geometry of real algebraic varieties.
result Upper bound on sample complexity for testing variety hypothesis.
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.
New method recovers matrices with nonlinear structures using optimization on Grassmann manifold.
problem Recovering high-rank matrices with nonlinear structures like subspaces or clusters.
method Formulated as rank minimization of a nonlinear feature map, approximated by constrained non-convex optimization on the Grassmann manifold, using Riemannian and alternating minimization schemes.
result Global convergence and worst-case complexity bounds for alternating minimization scheme, leading to unique limit point.
Advances in Koszul modules and syzygies of algebraic varieties.
problem Understanding Koszul modules and syzygies in algebraic geometry.
method General theory and applications to specific problems.
result Progress in understanding Koszul modules and syzygies, including applications to Chen ranks and line bundles.
The paper introduces pseudo-quotients for algebraic actions and applies them to character varieties.
problem Characterizing algebraic actions and their quotients.
method Introducing pseudo-quotients as a weak version of quotients for algebraic actions, focusing on purely topological properties.
result Pseudo-quotients are unique up to virtual class in characteristic zero and can be used to compute character varieties.
Proves algebraicity of maps to hyperbolic varieties using specialization techniques.
problem Analytic maps to hyperbolic varieties over complex numbers.
method Transcendental specialization technique.
result Proves equivalence of algebraicity and Borel hyperbolicity for certain maps.
We define a quandle variety as an irreducible algebraic variety endowed with an algebraically defined quandle operation . It can also be seen as an analogue of a generalized affine symmetric space or a regular -manifold in algebraic geometry. Assume that is normal as an algebraic variety and that the a…
New knot invariant from braided Hopf algebra.
problem Developing a new knot invariant.
method Non-commutative generalization of knot groups using braided Hopf algebra.
result New quantum character variety as an alternative to skein module.
This thesis explores algebraic cycles and moduli spaces over real numbers.
problem Understanding the cycle class map and its image in real algebraic geometry.
method Constructing integral Fourier transforms on Chow rings of abelian varieties over any field.
result Proof of integral Hodge conjecture for real abelian threefolds and moduli space properties.
Toric co-Higgs sheavesmath.AG
Study co-Higgs sheaves on toric varieties, finding explicit examples.
problem Characterizing and understanding co-Higgs sheaves on toric varieties.
method Characterization and explicit computation of examples.
result Explicit examples of co-Higgs sheaves on toric varieties computed.
Kähler-Ricci flows' tangent cones are algebraic varieties.
problem Understanding the structure of Kähler-Ricci flows' tangent cones.
method Analyzing tangent cones as normal affine algebraic varieties and using Hörmander's estimate.
result The regular set of tangent cones coincides with the algebraic regular set.
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.
Constructs a new graded variety from algebraic data.
problem Creating a -graded extension of differential varieties.
method Algorithm using homotopy retract data of Koszul-Tate resolution.
result Significantly reduced number of homological computations.
The main object of our study is a four dimensional Lie algebra which describes the symmetry properties of a nonlinear Black-Scholes model. This model implements a feedback effect which is typical for an illiquid market. The structure of the Lie algebra depends on one parameter, i.e. we have to do with a one-parametric …
We apply the Nash-Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space $H_{k-nil}^2(\math…
We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero.
Surveying -homotopy theory and contractible varieties.
problem Understanding the relationship between -homotopy theory and contractible varieties.
method Exploring the interplay between -homotopy theory and affine algebraic geometry.
result Highlighting the connection between -homotopy theory and contractible varieties.
New methods for constructing submersions between different types of spaces.
problem Creating submersions between Euclidean, Minkowski, and Finsler spaces.
method Homogeneous nonlinear splittings and nonlinear lifts.
result New examples of Finsler functions on reductive homogeneous spaces.
New invariant derived from skein algebra representations at roots of unity.
problem Understanding skein algebras of open surfaces and their invariants.
method Extended skein algebras, constructed isomorphisms, embedding and isomorphism constructions.
result Invariant associated to each skein algebra representation class.
New method studies discriminantal loci of algebraic varieties.
problem Understanding discriminantal loci of algebraic varieties.
method Efficient use of groupoids to describe monodromy.
result New insights into discriminantal loci of hypersurfaces.
The Bäcklund problem is solved for both the compact and noncompact versions of the Ishimori (2+1)-dimensional nonlinear spin model. In particular, a realization of the arising Bäcklund algebra in the form of an infinite-dimensional loop Lie algebra of the Kač--Moody type is provided.
Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced, motivated by the so called space-time foam structures in General Relativity with dense singularities, and by Quantum Gravity. A variety of applications of these algebras has been presented, among them, …
The Unique Tangent Cone Property for Weakly Holomorphic Maps into Projective Algebraic Varietiesmath.DG
Paper proves unique tangent maps for complex maps into algebraic varieties.
problem Proving uniqueness of tangent maps for complex maps into algebraic varieties.
method Developed techniques to prove uniqueness of tangent maps for weakly holomorphic and locally approximable maps.
result Established unique tangent cone property for weakly holomorphic maps into projective algebraic varieties.
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no referenc…
Criteria for extending degree-2 Azumaya algebras with C2-actions over curves.
problem Determining when degree-2 Azumaya algebras with C2-actions extend to entire curves.
method Criteria for extension of algebra and new condition for extension with action, testable by computer algebra systems.
result New conditions for extending degree-2 Azumaya algebras with C2-actions over curves.
We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov …
This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas …
The Bianchi varietymath.DG
The totality Lie(V) of all Lie algebra structures on a vector space V over a field F is an algebraic variety over F on which the group GL(V) acts naturally. We give an explicit description of Lie(V) for dim V=3 which is based on the notion of compatibility of Lie algebra structures.
Graph manifold study confirms quasi-projective links.
problem Characterizing algebraic links with quasi-projective fundamental groups.
method Analysis of characteristic varieties of graph manifolds.
result Simple proof of Papadima's question on quasi-projective links.
These lecture notes concern the algebraic geometry of the character variety of a finitely generated group in SL(2,C) from the point of view of skein modules. We focus on the case of surface and 3-manifolds groups and construct the Reidemeister torsion as a rational volume form on the character variety.
Study the geometry of torus link character varieties, finding unexpected relations.
problem Understanding the geometry of torus link character varieties.
method Developed an intrinsic stratification to relate the geometry with torus knots, computed E-polynomial for SL2(C) and SL3(C).
result Unexpected relation with the number of strands of the link.
Finite automorphisms of SL(2,C) character varieties identified.
problem Character varieties of surfaces with negative Euler characteristic.
method Algebraic automorphisms and mapping class groups studied.
result Finite extension of mapping class group identified.