We show that the detection of geometric intersection in an arbitrary representation of the mapping class group of surface implies the injectivity of that representation up to center, and vice versa. As an application, we discuss the geometric intersection in the Johnson filtration. Also, we further consider the problem…
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Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We introduc…
Surveying Hitchin representations of Fuchsian groups.
problem Understanding representations of Fuchsian groups.
method Survey and conjectural geometric description.
result Conjectural geometric picture of an augmented Hitchin component.
Study quandle modules over geometric quandles and their relation to Lie-Yamaguti representations.
problem Understanding quandle modules and their connection to Lie-Yamaguti representations.
method Examine quandle modules over quandle spaces, focusing on geometric structures.
result Modules over quandle spaces are linked to representations of Lie-Yamaguti algebras.
Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groupsmath.DG
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.
Special orthogonal representations from octonions have geometric properties linked to binary cubics.
problem Understanding geometric properties of special orthogonal representations from octonions.
method Using octonions and their derivations, spinors, and covariants to show geometric properties.
result Covariants and Mathews identities of these representations are related to the Fano plane and .
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.
New insights into Anosov representations of hyperbolic groups.
problem Understanding Anosov representations of relatively hyperbolic groups.
method Proving representations can be interpreted as restricted Anosov representations over flow spaces and showing stability under deformations.
result Representations of certain types are divergent, extended geometrically finite and stable under small deformations.
Neural nets learn robust geometric data representations.
problem Ensuring neural networks are robust to adversarial attacks.
method Topological Data Analysis via persistence diagrams, Lipschitz stability.
result Certified -robustness on ORBIT5K dataset.
On the Optimal Representation Efficiency of Barlow Twins: An Information-Geometric Interpretationcs.LG
This paper analyzes Barlow Twins' representation efficiency using information-geometric methods.
problem Understanding and comparing the efficiency of self-supervised learning methods.
method Introduces an information-geometric framework to quantify representation efficiency and applies it to Barlow Twins.
result Proves that Barlow Twins achieves optimal representation efficiency (η=1).
We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent bundle of some smooth manifold.
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.
Proof shows volumes of certain geometric representations are always integers.
problem Integrality of volumes of specific geometric representations.
method Elementary, combinatorial-geometrical proof.
result Volumes of representations are integers when .
Defines new representations for hyperbolic groups, unifying existing definitions.
problem Geometrically finite behavior in higher rank groups.
method Introduces a new family of discrete representations for relatively hyperbolic groups.
result Stability of these representations under certain deformations.
GSAN learns adaptive node representations using geometric scattering and attention.
problem Oversmoothing in node representation learning.
method Attention-based architecture integrating geometric scattering and GCN channels.
result GSAN outperforms previous networks in semi-supervised node classification.
New construction provides non-trivial representations for geometric quantisation.
problem Geometric quantisation of non-integral symplectic structures.
method Construction from Noncommutative Differential Geometry adapted to diffeology.
result The construction provides non-trivial representations.
A new geometric metric identifies true data changes from parametrization artifacts in high-dimensional representations.
problem Quantifying representation drift in high-dimensional data using Euclidean or cosine distances can misattribute changes due to arbitrary parametrizations.
method Introducing the Fubini Study metric to identify representations that differ only by gauge transformations.
result The Fubini Study metric isolates intrinsic evolution by remaining invariant under gauge-induced fluctuations, providing a diagnostic for meaningful structural changes.
We describe recent links between two topics: geometric structures on manifolds in the sense of Ehresmann and Thurston, and dynamics "at infinity" for representations of discrete groups into Lie groups.
Geometric stability measures neural network robustness, distinguishing from similarity metrics.
problem Lack of robustness in neural network representations.
method Introduces geometric stability, quantified by Shesha metric measuring self-consistency.
result Stability and similarity are uncorrelated, revealing distinct properties of neural network robustness.
Geometric structures on 5-manifolds from surface group representations of G2'.
problem Constructing geometric structures on 5-manifolds from G2'-surface group representations.
method Using Higgs bundles and partial flag manifolds of G2' to construct geometric structures.
result Developing maps of geometric structures are the domain of discontinuity.
Geometric properties near singular points of surfaces given by certain representation formulaemath.DG
Study geometric properties of surfaces with specific formulae.
problem Calculate curvature of surfaces with singular points.
method Investigate singular curvature and limiting normal curvature of surfaces.
result Determine geometric properties of surfaces with singular points.
New method learns disentangled representations using Gromov-Monge maps.
problem Learning disentangled representations from unlabelled data.
method Introduces a novel approach based on Gromov-Monge maps to preserve geometric features while aligning data distributions.
result Demonstrates effectiveness on four benchmarks, outperforming other methods.
Proves critical exponent for positive representations in discrete subgroups.
problem Determining the critical exponent for positive representations.
method Analyzes discrete subgroups and their geometric properties.
result Equality of critical exponent holds if and only if is a lattice for geometrically finite .
3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.
problem Understanding which 3-manifold groups can have convex co-compact representations.
method Analyzing representations of 3-manifold groups into projective general linear group, focusing on convex co-compactness.
result Fundamental groups of closed irreducible orientable 3-manifolds can only admit convex co-compact representations if they are geometric or hyperbolic.
This note summarizes in an informal way some geometric properties of Anosov representations into the symplectic group, which were presented in a talk at the conference What is Next. The mathematical legacy of Bill Thurston, held in June 2014 in Cornell.
Researchers extend geometric quantization to complex Abelian Lie supergroups.
problem Quantization of super Kähler structures on complex Abelian Lie supergroups.
method Extended geometric quantization scheme to super Kähler setting, constructed unitary representation.
result Irreducible subrepresentations of the constructed representation are determined by the moment map.
A novel geometric algebra-based KG embedding framework improves link prediction.
problem KG embedding to model entities and relations in a low-dimensional space.
method Utilizes multivector representations and geometric product in geometric algebra.
result Outperforms state-of-the-art models in link prediction experiments.
Study on projective structures linked to Hitchin representations.
problem Understanding the topology and geometric properties of projective structures.
method Using gauge theory, flat bundles, Higgs bundles, and geometric constructions.
result Some projective structures are fibered in a standard way.
Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.
problem Understanding Fock-Goncharov positivity and its geometric implications.
method Geometric interpretation and bending deformations of Fuchsian representations.
result Stabilization of a uniform Finsler quasi-convex disk in the symmetric space.
New constraints found for algebro-geometric subgroups of mapping class groups.
problem Constraints for algebro-geometric subgroups of mapping class groups.
method Using deep work of Gibney, Keel, and Morrison, constraints on the Shafarevich morphism are derived to prove the infinite restriction of certain representations.
result Most Reshetikhin-Turaev representations of the mapping class group restrict to infinite representations on algebro-geometric subgroups when the genus is at least 3.
Geometrically represents L-homology classes using normal maps.
problem Representing L-homology classes geometrically.
method Establishing a relationship between normal cobordism classes and L-homology.
result Existence of a map Γ representing L-homology classes by normal degree one maps.
Anderson and Canary have shown that if the algebraic limit of a sequence of discrete, faithful representations of a finitely generated group into PSL(2,C) does not contain parabolics, then it is also the sequence's geometric limit. We construct examples that demonstrate the failure of this theorem for certain sequences…
This work introduces a geometric approach to probability representation and option pricing.
problem Representing probability distributions geometrically for better understanding and approximation.
method Introducing a geometric representation of probability using implied volatility and geometric transformations.
result Any probability distribution on positive reals can be represented by a planar curve, facilitating approximation and analysis.
Constructs constant mean curvature surfaces using geometric flow.
problem Creating surfaces with constant mean curvature in 3D spaces.
method Using a generalized Weierstrass representation and geometric flow on potentials.
result Describes construction of CMC surfaces with symmetries.
Li-Bland's correspondence between linear Courant algebroids and Lie -algebroids is explained and shown to be an equivalence of categories. Decomposed VB-Courant algebroids are shown to be equivalent to split Lie 2-algebroids in the same manner as decomposed VB-algebroids are equivalent to 2-term representations up t…
Anosov deformations created for surfaces with specific properties.
problem Creating proper slices in character varieties for specific surface groups.
method Constructing proper slices in character varieties for surface groups into SL(3,R).
result Representations are Borel Anosov.
In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these repr…
Let be a connected, linear, real reductive Lie group with compact centre. Let be compact. Under a condition on , which holds in particular if is maximal compact, we give a geometric expression for the multiplicities of the -types of any tempered representation (in fact, any standard representation) …
Unified proof of Teichmüller components using robust submanifolds.
problem Existence of higher Teichmüller components in representation spaces.
method Introducing robust families of submanifolds for a linear Lie group.
result Unified short proof of Teichmüller components for specific groups.
Researchers describe character varieties for Hopf links, proving geometric properties.
problem Character variety geometry of Hopf links with twists.
method Geometric descriptions of irreducible and totally reducible representations.
result Complete geometric description of -character variety for .
In this article we give a geometric interpretation of the Hitchin component for PSL(4,R) in the representation variety of a closed oriented surface of higher genus. We show that representations in the Hitchin component are precisely the holonomy representations of properly convex foliated projective structures on the u…
Novel representer theorem for metric and preference learning in RKHSs.
problem Metric and preference learning problems in Hilbert spaces.
method Regularization with respect to task structure norm, RKHS representation, and novel algorithm.
result Significant performance improvement over baseline methods in real-world rank inference benchmarks.
New framework explains neural network behavior through geometric postulates.
problem Understanding neural network mechanisms and making them more transparent.
method Introducing the Pursuit of Subspaces (PoS) hypothesis as an axiomatic framework.
result Unified geometric perspective on neural network representation, computation, and generalization.
This note removes technical assumptions and characterizes relatively dominated representations.
problem Geometrically finiteness and Anosov conditions in higher-rank settings.
method Characterization using eigenvalue gaps and limit maps.
result Relatively dominated representations are characterized using eigenvalue gaps and limit maps.
Research classifies geometric structures on manifolds using surface group representations.
problem Classifying geometric structures on manifolds related to surface group actions on character varieties.
method Surveying results on surface groups, focusing on compact and non-compact target groups, discussing various representations and their dynamics.
result Dichotomy in dynamics of character varieties based on compactness of target groups.
New method for high-fidelity shape representations from raw data.
problem Creating accurate shape representations from raw data.
method A simple loss function encouraging neural network to vanish on input point cloud and have unit norm gradient.
result Our method produces high-fidelity, smooth, and natural zero level set surfaces.
Researchers present and compare different representations of dissipative Hamiltonian DAE systems.
problem Understanding and transforming dissipative Hamiltonian DAE systems.
method Global geometric and algebraic points of view, translations between representations, characterizations, and numerical methods for computing structural information.
result A general DAE system can be transformed into a dissipative Hamiltonian or port-Hamiltonian DAE system.
Higher Vector Bundlesmath.CT
Geometric models for representations up to homotopy using simplicial vector bundles.
problem Geometric models for representations up to homotopy of Lie groupoids.
method Application of higher analogs of cleavages in simplicial fibrations to geometric models.
result An equivalence between representations up to homotopy and simplicial vector bundles endowed with a cleavage.