Finite image of mapping class group representations proved using graph embeddings.
problem Finiteness of images of mapping class group representations in twisted Dijkgraaf-Witten theory.
method Translation of problem into graph manipulation, using TVBW representations and spherical fusion categories.
result Finiteness of images of mapping class group representations in twisted Dijkgraaf-Witten theory is proven.
Proves a conjecture about quantum invariants of three-manifolds from Hopf algebras.
problem Proves a 20-year-old conjecture about two quantum invariants of three-manifolds.
method Uses Hopf algebras and special Heegaard diagrams to handle non-semisimple invariants.
result Affirmative answer to the open question about the relation between Kuperberg and Hennings-Kauffman-Radford invariants.
New representations on surfaces with positive cross ratios.
problem Understanding representations of surfaces with specific geometric properties.
method Using geodesic currents and Anosov representations, proving systolic inequalities.
result Systolic inequalities hold for all positively ratioed representations.
Classifies SL(2;C) representations of a Brieskorn homology 3-sphere.
problem Classifying SL(2;C) representations of a Brieskorn homology 3-sphere.
method Shows irreducible representations are conjugate to SU(2) or SL(2;R) representations. Constructs SL(2;R) representations from PSL(2;R) representations of the orbifold.
result Classifies SL(2;C) representations and provides a construction method.
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.
This article reviews statistical methods for learning data representations.
problem Learning meaningful representations of data.
method Statistical perspective on unsupervised and supervised representation learning.
result Recent advances in representation learning from a statistical viewpoint.
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. New representation connects two link invariants.
problem Link invariants of different types.
method Augmentation representation of link group.
result Connects two types of link invariants.
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.
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.
Characterizes groups for specific types of representations into SL(d,R).
problem Understanding groups admitting certain types of representations into SL(d,R).
method Characterization and bounds on cohomological dimension.
result Bounds on cohomological dimension and characterizations of representations.
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…
Introduces representations of surface groups into Lie groups preserving order.
problem Representations of surface groups into Lie groups that preserve the group order.
method Relates order preserving representations to weakly maximal representations and uses geometric and causal structure.
result Order preserving representations into Lie groups of Hermitian type are faithful with discrete image and form a closed set.
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.
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…
Paper defines weak representations and their equivalence to VB-groupoids.
problem Understanding representations of Lie groupoids and VB-groupoids.
method Introduces weak representations and shows equivalences between categories.
result Equivalence between 2-term representations up to homotopy and weak representations of G.
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.
Method learns state representations without supervision for Atari games.
problem Learning state representations without supervision.
method Maximizes mutual information across features of neural encoder.
result New benchmark for evaluating representation learning models.
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. Investigates latent variable models for useful generative concept representations.
problem Creating latent representations that support various concepts and attributes.
method Latent variable modeling, including latent variable models, latent representations, and latent spaces.
result Hierarchical latent representations and latent space vectors and geometry are effective for generative concept representations.
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.
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.
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.
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.
New representations for framed braid groups defined.
problem Understanding framed braid groups through new representations.
method Homological and monodromy representations introduced.
result Conjectural equivalence between homological and monodromy representations.
This study compares global vs local observation and action representations for DRL in RTS games.
problem Improving Deep Reinforcement Learning performance in RTS games.
method Comparing two observation and action representations in μRTS.
result Local representation outperforms global representation in resource harvesting tasks.
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.
Robots learn state representation from demonstrations.
problem Robots need a compact state representation for efficient interaction.
method Imitation learning using a multi-head neural network.
result Trained representation improves performance and efficiency in reinforcement learning.
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 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 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.
Anosov representations uniquely determined by boundary maps for surface groups to SL(3,R).
problem Uniqueness of Anosov representations determined by boundary maps.
method Analysis of boundary maps and representation properties.
result Anosov representations uniquely determined by boundary maps if not reducible.
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.
ICP separates and competes feature representations to learn diverse information.
problem Learning representations with diversified information.
method Information Competing Process (ICP) separates representations into parts with different mutual information constraints, forcing them to learn independently in a competitive environment.
result ICP facilitates obtaining diversified representations with rich information.
The paper proposes a model to learn disentangled representations using mutual information.
problem Learning disentangled representations from shared and exclusive attributes.
method Mutual information maximization for shared attributes and minimization for disentanglement.
result The proposed model outperforms state-of-the-art models in representation disentanglement.
New representations of Lie algebras via monoidal category actions.
problem Constructing representations of Lie algebras using monoidal categories.
method Using crossed homomorphisms and monoidal categories to generate representations.
result Established new bifunctor for weak and admissible representations of Lie-Rinehart algebras.
Proved conjecture about knot representations.
problem Number of SL(2,R) representations of 2-bridge knots.
method Proved conjecture about knot representations.
result Proved Riley's conjecture on 2-bridge knots.
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). This paper learns actionable representations for reinforcement learning.
problem Learning comprehensive representations in reinforcement learning.
method Focuses on goal-conditioned policies to learn salient, actionable representations.
result Actionable representations improve exploration and hierarchical reinforcement learning.
This paper explores the complexity of learning representations in contextual linear bandits.
problem Understanding the complexity of representation learning in contextual linear bandits.
method Systematic approach to representation learning in contextual linear bandits, focusing on instance-dependent perspective.
result Representation learning is fundamentally more complex than linear bandits, with some cases being arbitrarily harder.
This paper improves few-shot learning by reducing sample complexity using representation learning.
problem Reducing sample complexity for target tasks with limited data.
method Representation learning to pool all source task samples for target task learning.
result Representation learning can achieve substantial sample size reduction, bypassing the $Ω( rac{1}{T})$ barrier.
This work improves understanding of reinforcement learning state representations.
problem Lack of precise characterization of how and when state representations generalize.
method Developed a bound on the generalization error based on effective dimension.
result Bound quantifies the tension between generalization and approximation.