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. 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.
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). Survey on stated skein algebras and their representations.
problem None explicitly stated in the abstract.
method None explicitly stated in the abstract.
result None explicitly stated in the abstract.
Develops theory of relatively Anosov representations using flow examples.
problem Understanding relatively Anosov representations.
method Uses a contracting flow on a bundle to define Anosov representations and builds examples.
result Builds families of examples of relatively Anosov representations.
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.
New CR representations are found and shown to be redundant.
problem Identifying and classifying CR representations of 3-manifolds.
method Experimental computation of limit sets and exact computations of triangle groups.
result Many CR representations are redundant and conjugate.
In this paper, I introduce weak representations of a Lie groupoid G. I also show that there is an equivalence of categories between the categories of 2-term representations up to homotopy and weak representations of G. Furthermore, I show that any VB-groupoid is isomorphic to an action groupoid associated to a weak…
We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants of different types. We also study properties of the augmentation representation…
The paper proposes a learning-theoretic perspective on representation alignment.
problem Understanding how AI models' representations become aligned as they scale.
method Reviewing and connecting different notions of alignment, focusing on stitching.
result Relating properties of stitching to kernel alignment of representations.
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.
Extending braid group representations to singular braid monoids and groups.
problem Extending braid group representations to singular braid monoids and groups.
method Investigating the extension of representations from braid groups to singular braid monoids and groups, and computing defects.
result Constructing a linear representation of the singular braid group that is an extension of the Lawrence-Krammer-Bigelow representation and computing its defect.
Theory of Θ-positive representations for real closed fields.
problem Generalizing positive representations to real closed fields.
method Developing theory for Fuchsian groups to linear groups over real closed fields.
result Theory encompasses many generalizations of positive or Anosov representations.
New framework improves reliability of learned representations by modeling uncertainty and structural constraints.
problem Uncertainty in learned representations treated as deterministic, leading to unreliable models.
method Proposes a principled framework for reliable representation learning with uncertainty-aware regularization and structural constraints.
result Improves stability, calibration, and robustness of learned representations.
Characterizes totally elliptic surface group representations into Lie groups.
problem Understanding totally elliptic surface group representations into Lie groups.
method Characterization of representations into PSL2R and PSL2C by their mapping properties. result They are either into a compact subgroup or Deroin--Tholozan representations.
Linear disentangled representations improve unsupervised action estimation.
problem Learning linear disentangled representations for unsupervised action estimation.
method Developed a method to induce irreducible representations in VAE models without labeled action sequences.
result Linear disentangled representations are a desirable property for unsupervised action estimation.
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.
Study Anosov representations of reducible suspensions of hyperbolic groups.
problem Characterize dynamical properties of reducible suspensions of Anosov representations.
method Analyzing linear representations of non-elementary hyperbolic groups, focusing on weak unipotent actions on subspaces.
result Characterize when reducible suspensions are discrete and faithful, quasi-isometrically embedded, and Anosov.
Study Heisenberg homology on surface configurations, revealing new representations of mapping class groups.
problem Homology of surface configurations with Heisenberg group representations.
method Analysis of unordered configurations in a surface, using Heisenberg group actions and representations.
result Obtained genuine and projective representations of mapping class groups from Heisenberg group actions.
Optimal transport metric transfers deep network representations efficiently.
problem Efficiently transfer deep network representations for new tasks.
method Use optimal transport to quantify representation similarity and regularize student network.
result Optimal transport distance promotes similarity between teacher and student representations.
The paper introduces cataclysm deformations for Anosov representations.
problem Deforming Anosov representations in Lie groups.
method Constructing cataclysm deformations for θ-Anosov representations into semisimple Lie groups. result Cataclysm deformations are injective for Hitchin representations but not for all θ-Anosov representations. In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…
Robots could learn their own state and world representation from perception and experience without supervision. This desirable goal is the main focus of our field of interest, state representation learning (SRL). Indeed, a compact representation of such a state is beneficial to help robots grasp onto their environment …