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.
We consider SU(2)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form M×S3/Γ, where M is a smooth manifold and S3/Γ is a three-dimensional Sasaki-Einstein orbifold. We obtain new quiver gauge theories on M whose quiver bundles are based on the affine ADE Dynkin diagram associ…
Triangulates permutahedra for Coxeter groups, revealing braid group connections.
problem Triangulating permutahedra for Coxeter groups.
method Constructs triangulations using total linear stability and height functions.
result Explicitly relates two braid group presentations.
We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Pruefer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs…
We construct a finite dimensional quiver algebra from the non-simply laced type B Dynkin diagram, which we call the type B zigzag algebra. This leads to a faithful categorical action of the type B braid group A(B), acting on the homotopy category of its projective modules. This categorical action is a…
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.
This paper introduces cluster exchange groupoids for Coxeter-Dynkin diagrams and finds their fundamental groups are braid groups.
problem Understanding the fundamental groups of cluster exchange groupoids for Coxeter-Dynkin diagrams.
method Introduced cluster exchange groupoids for Coxeter-Dynkin diagrams and showed the fundamental group isomorphic to braid groups.
result The fundamental group of the exchange groupoid for a Coxeter-Dynkin diagram is the braid group associated with the diagram.
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.
New Dynkin condition for manifolds with boundary yields bi-Lipschitz equivalence and spectral properties.
problem Characterizing smooth Riemannian manifolds with boundary.
method Introducing a Dynkin-type condition and proving its equivalence to a weighted manifold.
result Bi-Lipschitz equivalence and various spectral properties of manifolds with boundary.
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.
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.
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.
Paper defines saddle points in asymmetric Dynkin games using martingale theory.
problem Tackles saddle point conditions in asymmetric Dynkin games with partial information.
method Uses martingale theory to identify super and submartingales related to equilibrium payoffs.
result Characterizes saddle point strategies in terms of equilibrium payoffs' dynamics and Doob-Meyer decompositions.
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.
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.
Enhances psyquandle invariants for singular and pseudoknots.
problem Counting invariants for singular knots and pseudoknots.
method Uses quivers to extend in-degree polynomial invariants.
result Obtains biquandle coloring quivers and in-degree polynomial invariants.
We show that every flag variety contains a naturally defined homogeneous cominuscule subvariety. From the Dynkin diagram of the flag variety, we compute the Dynkin diagram of that subvariety.
When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on R3×S1 are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these…
We study a doubly reflected backward stochastic differential equation (BSDE) with integrable parameters and the related Dynkin game. When the lower obstacle L and the upper obstacle U of the equation are completely separated, we construct a unique solution of the doubly reflected BSDE by pasting local solutions and…
Symmetrizes 4d and 3d BPS quivers for Argyres-Douglas theories.
problem Understanding the relationship between 4d and 3d BPS quivers.
method Analyzes geometric backgrounds and uses skein modules to derive quiver partition functions.
result Proves isomorphism between 4d wall-crossing and unlinking of symmetric quivers.
Let G be a Lie group and Q a quiver with relations. In this paper, we define G-valued representations of Q which directly generalize G-valued representations of finitely generated groups. Although as G-spaces, the G-valued quiver representations are more general than G-valued representations of finitely generated group…
Knot invariants and quiver stability linked through full twists.
problem Relating knot invariants to quiver stability under twists.
method Using HOMFLY-PT skein relations and linking/unlinking operations on symmetric quivers.
result Full twists on knots correspond to unlinking or linking of augmented symmetric quivers, confirming stable growth in both.
We start briefly surveying research on optimal stopping games since their introduction by E.B.Dynkin more than 40 years ago. Recent renewed interest to dynkin's games is due, in particular, to the study of Israeli (game) options introduced in 2000. We discuss the work on these options and related derivative securities …
Introduces tensor product for quiver representations and applies to stable bundles and character varieties.
problem Stability and classification of quiver bundles and their subvarieties.
method Definition of tensor product for quiver representations and application to stability and character varieties.
result Tensor products of polystable quiver bundles are polystable and provide insights into character varieties.
Study of conformal limits in Nakajima quiver varieties.
problem Understanding the conformal limits of Nakajima quiver varieties.
method Defined and studied a conformal limit construction for Nakajima quiver varieties, proving it is a limit of a one-parameter family and gives a biholomorphic map.
result Proved the conformal limit is a biholomorphic map between Lagrangian submanifolds of different quiver varieties.
The relation between open topological strings and representation theory of symmetric quivers is explored beyond the original setting of the knot-quiver correspondence. Multiple cover generalizations of the skein relation for boundaries of holomorphic disks on a Lagrangian brane are observed to generate dual quiver desc…
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.
We construct two infinite sequences of immersions of the 3-sphere into 4-space, parameterized by the Dynkin diagrams of types A and D. The construction is based on immersions of 4-manifolds obtained as the plumbed immersions along the weighted Dynkin diagrams. We compute their Smale invariants and bordism classes of im…
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.
Classifies totally geodesic submanifolds in exceptional symmetric spaces.
problem Classifying totally geodesic submanifolds in exceptional symmetric spaces.
method Classification and introduction of an invariant (Dynkin index) for totally geodesic embeddings.
result Existence of a totally geodesic submanifold of minimal codimension with specific properties.
Forbidden moves categorify fused links into quivers.
problem Categorify fused links to invariant of links.
method Use forbidden moves to categorify fused links.
result Obtain three polynomial invariants of links.
Study proves value of non-Markovian games with partial, asymmetric info.
problem Value of non-Markovian Dynkin games with partial and asymmetric information.
method Probabilistic and functional analytic approach based on Sion's min-max theorem.
result Existence of optimal strategies for both players in randomised stopping times.
Define quiver representation-valued invariants for classical and virtual knots
problem Define quiver representation-valued invariants for classical and virtual knots
method Define an infinite family of quiver representation-valued invariants of classical and virtual knots associated to a choice of data vector consisting of a biquandle, abelian group, set of biquandle arrows weights with values in the abelian group, coefficient ring and set of biquandle endomorphisms.
result Extract four new polynomial invariants as decategorifications
We argue how to identify supersymmetric quiver quantum mechanics description of BPS states, which arise in string theory in brane systems representing knots. This leads to a surprising relation between knots and quivers: to a given knot we associate a quiver, so that various types of knot invariants are expressed in te…
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.
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.
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.
New polynomial invariants for knots and links.
problem Defining new invariants for knot theory.
method Infinite family of quiver representations.
result Infinite family of two-variable polynomial invariants.
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 …
The paper connects hyperpolygon spaces to Higgs bundle moduli spaces via degenerations.
problem Modeling hyperkähler 4-manifolds and their degenerations.
method Using parabolic SL(2,C)-Higgs bundles and Nakajima quiver varieties.
result ALG-D4 spaces degenerate to ALE-D4 spaces under a limit. We define an equivalence relation on graphs with signed edges, such that the associated adjacency matrices of two equivalent graphs are congruent over Z. We show that signed graphs whose eigenvalues are larger than −2 are equivalent to one of the simply laced Dynkin diagrams: An, Dn, E6, $E_…
New invariants for virtual knots and links defined via quiver representations.
problem Defining new invariants for virtual knots and links.
method Quiver representations associated to virtual biquandles and rings.
result New polynomial invariants for virtual knots and links.
This paper studies a 2-players zero-sum Dynkin game arising from pricing an option on an asset whose rate of return is unknown to both players. Using filtering techniques we first reduce the problem to a zero-sum Dynkin game on a bi-dimensional diffusion (X,Y). Then we characterize the existence of a Nash equilibrium…