Study on hyperconvex representations of hyperbolic groups in complex flag manifolds.
problem Characterizing hyperconvex representations of hyperbolic groups.
method Geometric and topological analysis of representations in mPSL(d,C). result Virtual isomorphism to Kleinian groups and flag manifold Hausdorff dimension restriction.
Study on hyperconvex representations of surface groups and their geometric properties.
problem Understanding the geometry of hyperconvex representations of surface groups.
method Holomorphic extension of Ahlfors--Bers map and analysis of limit sets.
result Limit set has Hausdorff dimension 1 if and only if representation is in PSL(d,R).
Collar lemma proven for certain surface group representations.
problem Proving a collar lemma for specific surface group representations.
method Using partial hyperconvexity properties and Anosov representations.
result 'Positivity properties' hold for partially hyperconvex representations.
In this paper, we establish Basmajian's identity for (1,1,2)-hyperconvex Anosov representations from a free group into PGL(n,R). We then study our series identities on holomorphic families of Cantor non-conformal repellers associated to complex (1,1,2)-hyperconvex Anosov representations. We show that the series …
Study extends Hausdorff dimension Hessian results to new hyperconvex representations.
problem Extending classical results on Hausdorff dimension Hessian.
method Analyzes (1,1,2)-hyperconvex representations and small complex deformations.
result Positive definiteness of Hessian of Hausdorff dimension for co-compact Γ in PO(n,1).
The paper proves rigidity for complex Kleinian groups.
problem Characterizing hyperconvex subgroups of complex Kleinian groups.
method Analyzing critical exponents and using representations of PSL(2, C).
result Uniform lattices in PSL(2, C) are the only (d-k)-hyperconvex subgroups with a specific critical exponent.
In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that t…
Inspired by the concept of hyperconvexity and its relation to curvature, we translate geometric properties of a metric space encoded by the curvature inequalities into the persistent homology induced by the Čech filtration of that space.
We show a very general existence theorem to the complex Monge-Ampère type equation on hyperconvex domains.
Maximal and Borel Anosov representations in Sp(4,R) are proven to be Hitchin.
problem Characterizing representations of surface groups into Sp(4,R) that are Borel Anosov and maximal. method Proving representations are Hitchin if they have maximal Toledo invariant and are Borel Anosov.
result Maximal and Borel Anosov representations in Sp(4,R) are Hitchin. We show that the notion of 3-hyperconvexity on oriented flag manifolds defines a partial cyclic order. Using the notion of interval given by this partial cyclic order, we construct Schottky groups and show that they correspond to images of positive representations in the sense of Fock and Goncharov. We construct poly…
Solves complex Monge-Ampère equation for measures with pluripolar parts.
problem Characterizing measures with complex Monge-Ampère equation solutions.
method Solves for measures with a pluripolar part in compact Kähler manifolds.
result Generalizes classical results in bounded hyperconvex domains.
We prove uniqueness of solutions to complex Monge-Ampère equations for small temperature.
problem Proving uniqueness of solutions to complex Monge-Ampère equations.
method Local and global analysis of bounded hyperconvex domains and compact complex manifolds.
result Uniqueness of solutions confirmed for small temperature parameters.
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Establishes a lower bound for Kähler hyperbolicity modulus in hyperconvex domains and bounded strongly pseudoconvex domains.
problem Kähler hyperbolicity modulus for simply-connected Kähler hyperbolic manifolds
method Computes the Kähler hyperbolicity modulus for bounded symmetric domains
result Establishes a lower bound for the Kähler hyperbolicity modulus in terms of the boundary behavior of the gradient length of a plurisubharmonic function
The paper connects Bergman-Calabi diastasis to Kähler metrics with constant holomorphic sectional curvature.
problem Characterizing domains with Bergman metrics of constant holomorphic sectional curvature.
method Using the Bergman-Calabi diastasis and its connection to the Bergman representative coordinate, the paper derives explicit formulas and proves properties of these metrics.
result Domains with Bergman metrics of constant holomorphic sectional curvature are hyperconvex or exhaustively biholomorphic to a ball.
Study confirms geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.
problem Confirming geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.
method Examined geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains.
result Affirmative answer to a longstanding open question about geodesic connectivity and rooftop envelopes.
Establishes a lower bound for Kähler-Einstein distance on certain domains.
problem Finding a lower bound for Kähler-Einstein distance on specific types of domains.
method Proves an analog of the Hopf lemma for Riemannian manifolds with Ricci curvature bounded from below.
result Establishes a lower bound for the Kähler-Einstein distance on pseudoconvex domains with positive hyperconvexity index.
In this paper, we show that the extremal length functions on Teichmüller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions. We also obtain alternative proofs of S.Krushkal's results that a function defined b…
For a bounded domain D and a real number p>0, we denote by Ap(D) the space of Lp integrable holomorphic functions on D, equipped with the Lp- pseudonorm. We prove that two bounded hyperconvex domains $D_1\subset \mc^n$ and $D_2\subset \mc^m$ are biholomorphic (in particular n=m) if there is a linear is…
The paper studies Kähler metrics from finite Monge-Ampère mass exhaustion functions.
problem Investigating the spectrum of complete Kähler metrics from finite Monge-Ampère mass exhaustion functions.
method Analyzing logarithmic potentials and the associated complete Kähler metrics, proving bounds on the spectrum using the finite Monge-Ampère mass condition.
result The lower bound of the spectrum of the Laplace-Beltrami operator is n2 under the finite Monge-Ampère mass condition. New representation theory for closed geodesic subflows.
problem Classifying representations with good geometric properties.
method Restricting to invariant closed geodesic subflows.
result Equivalent characterizations and properties of new representations.
Proves EGF representations in specific geometric contexts.
problem Understanding representations of groups with hyperbolic properties.
method Analyzes projectively convex cocompact manifolds and convex projective manifolds with generalized cusps.
result Holonomy representations of specific geometric manifolds are EGF representations.
The paper establishes isomorphisms and constructs colored versions of Lawrence representations.
problem Understanding isomorphisms and colored versions of Lawrence representations.
method Explicit isomorphisms and construction of colored versions.
result Matrices for colored versions of BKL and Lawrence representations provided.
Study k-positive surface group representations and their degenerations.
problem Understanding the behavior of surface group representations under degenerations.
method Introduced k-positive representations and studied their degenerations using a limit theorem for positively ratioed representations.
result Degenerations of k-positive representations can lead to limits that are at least (k-3)-positive and irreducible limits are (k-1)-positive.
Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.
problem Understanding geometrically finite Fuchsian groups and their representations.
method Theory of Anosov representations, type-preserving deformations, limit maps, relative Anosov and dominated representations.
result Cusped Hitchin representations are Borel Anosov, stable under deformations, and limit maps vary analytically.
New representations defined for groups and graphs, with applications to stable representations.
problem Defining and constructing new types of representations for groups and graphs.
method Introducing (R,Λ)-directed Anosov representations and using Fock-Goncharov positivity to construct them. result Constructs large families of primitive stable representations from F2 to PGL(V), including non-discrete and non-faithful examples. Study subgroup actions on mapping class groups using Heisenberg representations.
problem Untwisting representations of mapping class groups on Heisenberg subgroups.
method Restrict and analyze twisted representations of mapping class groups to Heisenberg subgroups.
result Untwisting representations on Torelli group for any Heisenberg representation.
Researchers describe unitary representations of mixed braid groups.
problem Understanding unitary representations of mixed braid groups.
method Explicitly describe unitary representations on cohomology of Abelian branched covers.
result Image of the representation is generated by complex reflections and related to the multivariate Burau representation.
This paper addresses law invariant coherent risk measures and their Kusuoka representations. By elaborating the existence of a minimal representation we show that every Kusuoka representation can be reduced to its minimal representation. Uniqueness -- in a sense specified in the paper -- of the risk measure's Kusuoka r…
A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group B_n. In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this pape…
Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain …
In this paper, we introduce a study of prolongations of representations of Lie groups. We obtain a faithful (one-to-one) representation of TG where G is a finite-dimensional Lie group and TG is the tangent bundle of G, by using (not necessarily faithful) representations of G. We show that tangent functions of Lie group…
Polynomial representations found in surface braid and mapping class groups.
problem Homological representations of surface braid and mapping class groups.
method Study of homological representation functors and short exact sequences.
result Many homological representation functors are polynomial.
Paper addresses the disparity between sampled and mean representations in disentangled learning.
problem Disparity between sampled and mean representations in disentangled learning.
method Proposes a method to eliminate the disparity by proving and utilizing the relationship between total correlation of sampled and mean representations for multivariate normal distributions.
result Demonstrates that a factorized mean representation can have lower total correlation than the sampled representation.
New Θ-positive representations of surface groups discovered.
problem Generalizing Lusztig's total positivity to surface groups.
method Introducing Θ-positivity and proving properties of Θ-positive representations. result Discrete and faithful Θ-positive representations exist and form open sets in representation varieties. New representations for surface groups expand known Anosov classes.
problem Understanding new types of representations for surface groups.
method Introducing and studying simple Anosov representations of closed hyperbolic surface groups.
result Simple Anosov representations strictly contain Anosov representations.
New findings on cusped Borel Anosov representations and their properties.
problem Characterizing and understanding cusped Borel Anosov representations.
method Analyzing representations of lattices in PGL2(R) to PGLd(R). result Cusped Borel Anosov representations with specific properties are Hitchin representations.
The paper formalizes criteria for non-spurious and disentangled representations using causal methods.
problem Formalizing criteria for non-spurious and disentangled representations in representation learning.
method Causal perspective, counterfactual quantities, observable consequences of causal assertions.
result Computable metrics for assessing representation learning based on observed data.
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.
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermit…
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.
We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.
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…
We characterize groups admitting Anosov representations into SL(3,R), projective Anosov representations into SL(4,R), and Borel Anosov representations into SL(4,R). More generally, we obtain bounds on the cohomological dimension of groups admitting Pk-Anosov r…
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.
The paper studies conjugating complex representations into real ones.
problem Understanding representations of surface groups into complex Lie groups.
method Analyzes representations of finitely generated groups into PGL(k,C) and determines conjugacy conditions. result Identifies representations in the larger variety that are conjugate in PGL(k,C) to a representation in PGL(k,R).