New method constructs nilpotent Lie algebras from quivers.
problem Constructing nilpotent Lie algebras from quivers.
method Using paths within quivers to construct nilpotent Lie algebras.
result Constructs a broad family of Ricci soliton nilmanifolds.
New Lie algebras from quivers lead to rigid Ricci solitons.
problem Constructing Lie algebras from quivers to study geometric structures.
method Using finite quivers without cycles to construct solvable Lie algebras and proving their geometric properties.
result Simply-connected Lie groups corresponding to these Lie algebras admit left-invariant Ricci solitons, and when quivers are oriented multi-trees, these groups are rigid.
Study connects bordered theory to Khovanov-Seidel quiver algebras.
problem Connecting bordered theory to quiver algebras.
method Investigates relationship between bordered theory and Khovanov-Seidel algebras.
result Shows isomorphism and homotopy equivalence between algebras and bimodules.
Floer homology connects to quiver Hecke algebras in Coulomb branches.
problem Understanding the representation theory of quiver Hecke algebras.
method Using Lagrangian Floer homology in multiplicative Coulomb branches.
result Cylindrical KLRW algebras are realized by Floer homology.
New identities lift q-dilogarithm to a more complex algebra.
problem Extending q-dilogarithm identities to a more complex algebra.
method Showed lift to HOMFLYPT-skein algebra of a genus n handlebody.
result New identities associated to unidirectional A_n-quiver.
Study quantized Coulomb branches of Jordan quiver gauge theories and their connections to Cherednik algebras.
problem Understanding the structure of quantized Coulomb branches in Jordan quiver gauge theories.
method Proved isomorphisms between quantized Coulomb branches and spherical graded Cherednik and cyclotomic rational Cherednik algebras.
result Quantized Coulomb branches are deformations of subquotients of Yangians of affine gl(1). Study links curve singularities to quiver mutations.
problem Understanding the relationship between curve singularities and quiver mutations.
method Investigates the connection between the topology of curve singularities and the mutation equivalence of quivers associated with their morsifications.
result Established a connection between the topology of isolated curve singularities and the mutation equivalence of quivers.
Introduces Nakajima bundles on algebraic curves, generalizing quiver representations and bundles.
problem Generalizing quiver representations and bundles on algebraic curves.
method Assigns complex vector bundles and sections/connections to nodes and edges of a quiver, using gauge-theoretic characterizations.
result Proves Hitchin-Kobayashi correspondence between stable quiver bundles and Nakajima bundle representations.
Neural networks are mathematically represented via quiver representations.
problem Understanding how neural networks process data and create representations.
method Representing neural networks as quiver representations with activation functions.
result Neural networks' computations can be studied algebraically and geometrically.
Nilpotent Lie algebras obtained from ordered sets and quivers are algebraic Ricci solitons.
problem Constructing nilpotent Lie algebras as algebraic Ricci solitons
method Using transitively and antisymmetrically ordered sets (TAOSs) and incidence algebras
result Nilpotent Lie algebras with arbitrarily high degrees of nilpotency are algebraic Ricci solitons
Theory of smooth relative connections on quiver bundles developed.
problem Existence of smooth relative connections over quiver bundles.
method Developed a theory over RQ on smooth twisted quiver bundles, provided obstructions and necessary/sufficient conditions. result Established a necessary and sufficient condition for the existence of smooth relative connections on tree-type quiver bundles.
Categorical action of braid group on projective modules.
problem Action of extended braid group on projective modules.
method Constructed a trigraded quiver algebra to represent the group action on the homotopy category of projective modules.
result Intersection numbers from topological origin match trigraded dimensions of morphism spaces.
New algebraic structures help categorify link invariants.
problem Classifying and distinguishing links and virtual links.
method Introducing mc-biquandles and categorifying homsets.
result New link invariants defined via mc-biquandle coloring quivers.
Paper develops Morse-theoretic approach to quiver varieties convolution.
problem Convolution in quiver varieties via Morse theory.
method Morse complex and cup product on smooth space of representations.
result Topological information encoded in cup product of Morse complex.
Study links and quivers, proving polynomial equality conjecture.
problem Link and quiver invariants and their relations.
method Cluster algebra invariants, point count polynomials, skein relations.
result Equality conjecture between plabic graph link polynomial and quiver point count polynomial proved for specific cases.
Unified 3D R-matrices from quantum cluster algebra.
problem Constructing new solutions to the tetrahedron equation.
method Symmetric butterfly quiver, quantum cluster algebra, quantum dilogarithms, q-Weyl algebra.
result Unified 3D R-matrices from various sources.
New relations link knot theory to quiver representations in 3d physics.
problem Exploring new connections between knot theory and quiver representations.
method Observing multi-cover skein relations and embedding them into M-theory.
result Obtained dualities of 3d N=2 theories associated to quivers. Constructs moduli stacks of quiver bundles and applies to Higgs bundles.
problem Classifying morphisms of vector bundles over a fixed base.
method General method for constructing moduli stacks of diagrams of vector bundles indexed by a simplicial set.
result Recovery of Nakajima quiver varieties and alternate construction of moduli stacks of Higgs bundles.
We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every …
We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix repres…
The paper constructs tilting modules for knots using algebraic structures.
problem Understanding the algebraic structure of knot invariants.
method Constructing modules over Jacobian algebras associated with knots.
result The constructed modules M are rigid and τ-rigid, and their endomorphism algebra is isomorphic to the Jacobian algebra. New C*-algebra connects cluster theory to Teichmüller space.
problem Connecting cluster theory with Teichmüller space.
method Introduced a C*-algebra A(x,Q) and proved its spectrum homeomorphic to a subset of Teichmüller space.
result Primitive spectrum of A(x,Q(T)) is homeomorphic to a generic subset of Teichmüller space.
In previous work a relation between a large class of Kac-Moody algebras and meromorphic connections on global curves was established---notably the Weyl group gives isomorphisms between different moduli spaces of connections, and the root system is also seen to play a role. This involved a modular interpretation of many…
For each integer n≥2 we describe the space of stability conditions on the derived category of the n-dimensional Ginzburg algebra associated to the A2 quiver. The form of our results points to a close relationship between these spaces and the Frobenius-Saito structure on the unfolding space of the A2 singul…
New link invariants derived from quandle filtrations.
problem Constructing link invariants from quandle structures.
method Decategorification of subquandle filtrations of quandle coloring quivers.
result Definition of disrespectful and graded disrespectful polynomials.
This paper studies posets associated with link diagrams and their algebraic properties.
problem Understanding the algebraic structure of posets derived from link diagrams.
method Associaed posets with link diagrams, proved distributivity, and described join irreducibles.
result Posets of Kauffman states are distributive lattices and isomorphic to coefficient quiver posets.
In this paper we investigate the convergence properties of the upwards gradient flow of the norm-square of a moment map on the space of representations of a quiver. The first main result gives a necessary and sufficient algebraic criterion for a complex group orbit to intersect the unstable set of a given critical poin…
New solutions to 3D integrability equations using quantum cluster algebras.
problem Constructing solutions to the tetrahedron and 3D reflection equations.
method Extending quantum cluster algebra approach to Fock-Goncharov quivers and investigating cluster transformations.
result Explicit formulas for matrix elements of solutions derived for typical representations.
Quantizes canonical bases for cluster varieties of type A.
problem Deforming algebra of functions on cluster varieties.
method Natural q-deformation of Fock and Goncharov's canonical basis. result Extension to quantum symplectic double.
Study of machine learning in quiver gauge theories and Seiberg duality.
problem Determining dualities in quiver gauge theories using machine learning.
method Defined and explored various questions related to binary and multi-class duality determination, evaluated performance of different classifiers, and analyzed effects of additional data.
result High accuracy and confidence achieved in determining dualities using machine learning.
Constructs quivers related to Weyl groups and higher Teichmüller spaces.
problem Understanding the structure of higher Teichmüller spaces.
method Constructs weighted quivers and computes cluster transformations.
result Establishes a correspondence between quivers and higher Teichmüller spaces.
Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.
problem Extending non-Abelian Hodge theory to moduli stacks of quiver connections.
method Formalizes and constructs moduli stacks of bundles with λ-connections over prestacks.
result Shows moduli stacks are algebraic and locally of finite presentation when base is smooth and projective.
We prove the existence of Lagrangian fillings for Dn-type Legendrian links.
problem Exact Lagrangian fillings of Legendrian links of Dn-type. method Legendrian weave calculus and construction of 1-cycles.
result Existence of a Lagrangian filling represented by a weave.
Study of quandle coloring quivers with dihedral quandles.
problem Link invariants and their enhancements using quandles.
method Introduced shadow quandle coloring quivers and cocycle quivers, studied equivalence with quandle coloring numbers and shadow quandle cocycle invariants.
result Equivalence of quandle coloring quivers with quandle coloring numbers and shadow quandle cocycle quivers with shadow quandle cocycle invariants for specific dihedral quandles.
Introduces generalized hyperpolygons and their geometric and algebraic properties.
problem Understanding moduli spaces of generalized hyperpolygons.
method Representation of a comet-shaped quiver, associated meromorphic Higgs bundles, Hitchin systems, and integrable Hamiltonian systems.
result Generalized hyperpolygons admit the structure of a completely integrable Hamiltonian system.
Classifies singularities in quiver varieties for specific Dynkin quivers.
problem Classifying singularities in quiver varieties.
method Classifies singularities using minimal imaginary roots and extended Dynkin quivers.
result Constructs hyper-Kähler cobordisms between binary polyhedral spaces.
The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebra…
The paper studies the geometry of Nakajima quiver varieties and their decompositions.
problem Understanding the geometry of Nakajima quiver varieties and their decompositions.
method Investigates the Białynicki--Birula decomposition of Nakajima quiver varieties, describing fixed points in terms of representations with relations of auxiliary quivers.
result Computes the motivic decomposition of Nakajima quiver varieties in terms of quiver-chain moduli spaces.
This review connects knot invariants to quiver representations.
problem Relating knot invariants to quiver representations.
method Relates symmetric quivers and their partition functions to quantum invariants of knots.
result Establishes a correspondence between knot invariants and quiver representations.
We define and calculate the HOMFLY polynomial for a specific type of quiver.
problem Calculating the HOMFLY polynomial for forest quivers.
method Recursive definition and closed-form expression for forest quivers.
result Closed-form expression for the HOMFLY polynomial of a forest quiver.
New polynomial invariants from quandle action quivers.
problem Classical and virtual knot and link invariants.
method Categorification of quandle counting invariant using quandle action quivers.
result Quandle action polynomials as decategorifications.
The study characterizes torus links' coloring quivers using dihedral quandles.
problem Characterizing the structure of coloring quivers for torus links.
method Exhaustively determining all possible numbers of colorings and their interconnections.
result The quiver structure varies based on the number of colorings.
Enhanced invariant for linkoids using quivers.
problem Counting invariants for linkoids.
method Use of quivers to generalize in-degree polynomial invariant.
result Introduced in-degree quiver polynomial matrix as a new invariant.
In-degree quiver polynomials for surface-links computed.
problem Computing in-degree quiver polynomials for surface-links.
method Defined using a quandle and set of endomorphisms, computed for surface-links with ch-index up to 10.
result Example computations for surface-links with ch-index up to 10.
We link Ginzburg algebras to Weinstein manifolds and Legendrian knots.
problem Understanding the relationship between Ginzburg algebras and Weinstein manifolds.
method Associated a stopped Weinstein manifold to a quiver and subquiver, proving quasi-isomorphism of relative Ginzburg algebra and Chekanov-Eliashberg dg-algebra.
result Relative Ginzburg algebra is quasi-isomorphic to Chekanov-Eliashberg dg-algebra of a singular Legendrian unknot link.
Explains biquandle brackets and quivers for a topology talk.
problem None explicitly stated; focuses on background information.
method Review of biquandle concepts and related structures.
result Clarifies understanding of biquandle bracket quivers.
New geometric proof for rational tangles links-quivers correspondence.
problem Recovering symmetric/antisymmetric colored HOMFLY-PT polynomials from a quiver.
method Geometric approach using winding numbers in punctured plane and its second configuration space.
result Explicit description of quivers for rational tangles.
Study on quandle coloring quivers for (p, 2)-torus knots and links.
problem Understanding quandle colorings of (p, 2)-torus knots and links.
method Introduced quandle coloring quivers and studied them for dihedral quandles.
result Characterized quandle coloring quivers for (p, 2)-torus knots and links.