Study local structure of knot group representations into SL(n,C).
problem Understanding the local structure of knot group representations.
method Analysis of tangent cone and use of Luna's slice theorem.
result Local structure of representation variety at diagonal representations.
Stratifies representation varieties of twisted Hopf links.
problem Stratifying representation varieties of twisted Hopf links.
method Using stratification of AGLr(C)-representation varieties of the fundamental group of the complement of a twisted Hopf link. result Explicit description and computation of motives for ranks 1 and 2.
Study representation varieties of twisted Hopf links using combinatorial and Hodge theory.
problem Representation theory of Hopf link complements with n twists.
method Combinatorial problem and equivariant Hodge theory.
result Close formulas for E-polynomials of representation and character varieties for ranks 2 and 3.
Study on representation varieties of RAAGs for different Lie groups.
problem Investigating connectedness of representation varieties for RAAGs under various Lie groups.
method Examined G-representation varieties of RAAGs for SU(n),Sp(n),U(n) and compared to other Lie groups. result Connectedness of representation varieties for SU(n),Sp(n),U(n), but not for SO(n),Spin(n) for n≥3. Study genus-three Torelli maps and their fixed point sets in representation varieties.
problem Understanding fixed point sets and representation varieties of genus-three Torelli maps.
method Analyzing fixed point sets and representation varieties of powers of bounding pair maps.
result Determined the number of connected components of fixed point sets and representation varieties.
The Ptolemy variety for SL(2,C) is an invariant of a topological ideal triangulation of a compact 3-manifold M. It is closely related to Thurston's gluing equation variety. The Ptolemy variety maps naturally to the set of conjugacy classes of boundary-unipotent SL(2,C)-representations, but (like the gluing equation var…
Computes dimensions of representation and character varieties for 2 and 3-dimensional orbifolds.
problem Computing dimensions of representation and character varieties for orbifolds.
method Analyzes varieties of representations and characters of hyperbolic and non-hyperbolic orbifolds, provides tools for computation.
result Shows that the dimension of a specific component of characters equals half the dimension of the variety of characters of the boundary.
Anosov representations of surface groups are complex manifolds.
problem Characterizing the geometry of Anosov representations of surface groups.
method Analyzing the character varieties of Anosov representations into $\SL(n , \C)$.
result Character varieties of Anosov representations are complex manifolds of specific dimension.
Researchers describe character varieties for Hopf links, proving geometric properties.
problem Character variety geometry of Hopf links with n twists. method Geometric descriptions of irreducible and totally reducible representations.
result Complete geometric description of SU(2)-character variety for r=2. The study examines dynamics on SU(2)-representation varieties for surfaces and non-orientable surfaces.
problem Dynamics of group actions on SU(2)-representation varieties of surfaces and non-orientable surfaces.
method Description and analysis of group actions generated by Dehn twists on SU(2)-representation varieties.
result Explicit invariant rational functions on SU(2)-representation varieties for specific cases of surfaces and non-orientable surfaces.
We give a description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL(3,C) or SL(3,C). We moreover obtain an explicit parametrization of matrices generating the representation and a description of the projection of the representation variety into the charac…
Character varieties of knot groups into SU(3) are stratified and analyzed.
problem Characterizing representations of knot groups into SU(3).
method Stratification of character variety into reducible, 2D+1D, and irreducible representations.
result Homotopy equivalence between SU(3) and SL(3,C) character varieties.
New compactification for character varieties with good topological properties.
problem Compactification of character varieties with good topological properties.
method Announced a new compactification with interpretations of ideal points.
result Relates to Weyl chamber length compactification and applies to maximal and Hitchin representations.
Minimal action of mapping class group on character variety.
problem Character variety of Deroin-Tholozan representations.
method Geometric perspective using symplectic structure.
result Infinite mapping class group orbits are dense.
The paper studies how the singularity of character varieties changes when representations are extended.
problem Understanding the singularity of character varieties for extended representations.
method Analyzes the variety of characters of fundamental groups in different matrix groups.
result Character varieties become singular when irreducible representations are extended, and the singularity is described.
Springer varieties appear in both geometric representation theory and knot theory. Motivated by knot theory and categorification Khovanov provides a topological construction of (n/2,n/2) Springer varieties. We extend Khovanov's construction to all two-row Springer varieties. Using the combinatorial and diagrammatic …
The paper connects hyperbolic Dehn surgery and Higgs bundles to construct model objects in representation varieties.
problem Constructing model objects in representation varieties for Higgs bundles.
method Reviewing hyperbolic Dehn surgery and bending procedures, and applying them to Higgs bundles.
result Explicit examples of model objects in representation varieties are produced.
Positive representations of surface groups in PO(p,q) form connected components of character varieties.
problem Characterizing representations of surface groups in special orthogonal groups PO(p,q).
method Using Anosov representations and root versus weight collar lemmas.
result Connected components of character varieties are formed by Θ-positive Anosov representations. Study character varieties for 3-punctured sphere group representations in PU(2,1).
problem Characterize representations with special elliptic elements.
method Analyze relative character varieties and momentum maps.
result Representations form fibers of the momentum map.
Classifies invariant measures on specific character varieties.
problem Classifying invariant probability measures on character varieties.
method Measure disintegration along transverse Lagrangian tori fibrations.
result Ergodic measures are either counting measures on finite orbits or Liouville measures.
The paper derives formulas for symplectic volume forms on surface representation varieties.
problem Calculating symplectic volume forms on surface representation varieties.
method Multiplicative gluing formulas and Heusener-Porti results.
result Symplectic volume form on Σg,0 is a product of forms on Σ2,1 and Σ2,2. The paper reinterprets knot group invariants using affine transformations.
problem Alexander invariants of knots and their geometric interpretation.
method Representation varieties of knot groups into extrmAGL1(C). result Alexander polynomial as the singular locus of a coherent sheaf.
Study Riemannian geometry of maximal surface group representations in pseudo-hyperbolic space.
problem Characterize the geometry of maximal surface group representations in pseudo-hyperbolic space.
method Introduced a scalar product on the first cohomology group, leading to a Riemannian metric on the smooth locus.
result Found totally geodesic sub-varieties and orbifold structures in the space of representations.
Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as E3 (Euclidean 3-space), H3 (hyperbolic 3-space) and E2,1 (Minkowski 3-space), using quaternion algebra theory, are studied. We study the different representations of a 2-generator group in which th…
For a lattice Γ of a simply connected solvable Lie group G, we describe the analytic germ in the variety of representations of Γ at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by G. By this description, under…
We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called {\em weakly maximal} representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore we prove that t…
Generalizes Toledo invariant to singular klt varieties, proving Milnor-Wood inequality.
problem Proving Milnor-Wood inequality for singular klt varieties.
method Generalized Toledo invariant to singular klt varieties and proved Milnor-Wood inequality.
result Milnor-Wood inequality for singular klt varieties.
Given an ideal triangulation of a connected 3-manifold with non-empty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject to Thurston's gluing equations. From this, one can recover a representation of th…
This is a survey on Reidemeister torsion for hyperbolic three-manifolds of finite volume. Torsions are viewed as topological invariants and also as functions on the variety of representations in SL2(C). In both cases, the torsions may also be computed after composing with finite dimensional r…
Given a hyperbolic knot K and any n≥2 the abelian representations and the holonomy representation each give rise to an (n−1)-dimensional component in the SL(n,C)-character variety. A component of the SL(n,C)-character variety of dimension ≥n is called high-d…
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.
Characterizes Anosov reducible representations in terms of eigenvalues.
problem Understanding Anosov representations in reducible settings.
method Characterizes Anosov representations using eigenvalue magnitudes of irreducible block factors.
result Connected components of character varieties do not contain reducible representations for many non-elementary hyperbolic groups.
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.
The paper calculates the motive of a specific knot's character variety.
problem Computing the motive of the character variety of torus knots representations.
method Introduced a stratification of the variety based on a canonical filtration, reducing the motive computation to a combinatorial problem.
result Computed the motive of the character variety for torus knots.
The paper proves a quadratic formality for Sasakian manifolds' representation varieties.
problem Analyzing the variety of representations of fundamental groups of Sasakian manifolds.
method Proving almost-formality of de Rham complex and vanishing cup product theorem.
result Quadratic formality of analytic germs of representation varieties.
Study character varieties of arborescent knots and hyperbolic knots.
problem Computing and understanding mSL(2,C)-character varieties of knots. method Described a procedure for computing mSL(2,C)-character varieties of arborescent knots, clarified facts about arborescent tangles, and studied character varieties of hyperbolic knots. result Found that each hyperbolic knot with essential surfaces has a 1-dimensional character variety.
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.
Given a finite volume hyperbolic 3-manifold, we compose a lift of the holonomy in SL(2,C) with the n-dimensional irreducible representation of SL(2,C) in SL(n,C). In this paper we give local coordinates of the SL(n,C)-character variety around this representation. As a corollary, this representation is isolated among al…
Springer varieties are studied because their cohomology carries a natural action of the symmetric group Sn and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties Xn as subvarieties of the product of spheres (S2)n. We show that…
For each oriented surface Σ of genus g we study a limit of quantum representations of the mapping class group arising in TQFT derived from the Kauffman bracket. We determine that these representations converge in the Fell topology to the representation of the mapping class group on $\boH(Σ)$, the space of regular f…
We give explicit equations that describe the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C). This has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irred…
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.
The paper studies mapping class group actions on character varieties of surfaces.
problem Understanding the dynamics of mapping class group actions on relative extPSL(2,R)-character varieties. method Definition and proof of simple-stability and primitive-stability of representations.
result Holonomies of hyperbolic cone surfaces are simple-stable and primitive-stable.
We study properties of irreducible and completely reducible representations of finitely generated groups Gamma into reductive algebraic groups G in in the context of the geometric invariant theory of the G-action on Hom(Gamma,G) by conjugation. In particular, we study properties of character varieties, X_G(Gamma)=Hom(G…
There is a well-known correspondence between the symplectic variety of representations of the fundamental group of a punctured Riemann surface into a compact Lie group G, with fixed conjugacy classes at the punctures, and a complex variety of holomorphic bundles on the unpunctured surface with a parabolic structure at …
New method finds Fuchsian representations dominating others in surface group representations.
problem Finding Fuchsian representations that dominate others in surface group representations.
method Straightening the pleated plane and applying strip deformations.
result There exists a Fuchsian representation that strictly dominates a given non-Fuchsian representation.
We propose a new non-commutative generalization of the representation variety and the character variety of a knot group. Our strategy is to reformulate the construction of the algebra of functions on the space of representations in terms of Hopf algebra objects in a braided category (braided Hopf algebra). The construc…
Study shows mapping class group action is ergodic on specific representations.
problem Ergodicity of mapping class group action on specific representations.
method Applied symplectic methods developed by Goldman and Xia.
result The action is ergodic.