Generic Hitchin representations generate dense subgroups.
problem Understanding dense subgroups in SL_n(R) representations.
method Using a theorem by Rapinchuk, Benyash-Krivetz, and Chernousov.
result Generic Hitchin representations are strongly dense.
New lattices in higher dimensions have dense surface subgroups.
problem Finding dense subgroups in higher-dimensional arithmetic lattices.
method Exhibited nonuniform arithmetic lattices in SO(n,1).
result Contain Zariski-dense surface subgroups.
We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…
Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.
problem Characterizing dense subgroups of algebraic groups.
method Bi-Lipschitz rigidity theorem for Zariski dense discrete subgroups.
result No C 1 C^1 C 1 -smooth slim limit set for higher rank semisimple algebraic groups. The study finds conditions for certain groups to be dense in a specific mathematical space.
problem Conditions for linear reflection groups to be dense in a projective space.
method Analyzes necessary and sufficient conditions for Zariski-density, applies to Coxeter groups and surface subgroups.
result Establishes conditions for Zariski-dense subgroups in S L n ( Z ) \mathrm{SL}_n(\mathbb{Z}) SL n ( Z ) for various n n n . New representations of hyperbolic 3-manifold groups into larger groups.
problem Finding representations of hyperbolic 3-manifold groups into larger matrix groups.
method Holonomy representations from projective deformations of hyperbolic structures.
result First examples of strongly dense representations into S L ( 4 , R ) SL(4,\mathbb{R}) S L ( 4 , R ) and S U ( 3 , 1 ) SU(3,1) S U ( 3 , 1 ) . Detects dense subhypergraphs in heterogeneous random hypergraphs.
problem Testing for the existence of a dense subhypergraph in heterogeneous random hypergraphs.
method Established detection boundaries and constructed asymptotically powerful and adaptive tests.
result Developed tests for distinguishing between null and alternative hypotheses.
Dense neural networks can't approximate all functions.
problem Approximation capabilities of dense neural networks.
method Model compression approach combining weak regularity lemma and graph neural networks.
result Existence of Lipschitz continuous functions not approximable by dense neural networks.
The key idea of current deep learning methods for dense prediction is to apply a model on a regular patch centered on each pixel to make pixel-wise predictions. These methods are limited in the sense that the patches are determined by network architecture instead of learned from data. In this work, we propose the dense…
Odd-dimensional SL(n,Q) contains dense surface subgroups.
problem Finding dense subgroups in SL(n,Q) for odd n.
method Constructing a continuous path of representations.
result Existence of dense surface subgroups in SL(n,Q) for odd n.
Classifies manifolds with dense conjugacy classes in their mapping class groups.
problem Classifying manifolds based on conjugacy classes in their mapping class groups.
method Analyzing connected orientable 2-manifolds and their mapping class groups.
result Mapping class groups of certain manifolds have dense conjugacy classes.
The paper finds dense subgroups in certain Lie groups.
problem Finding dense subgroups in Lie groups.
method Constructing dense surface subgroups in specific Lie groups.
result Uniform lattices contain infinitely many dense Hitchin representations.
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
Sharp boundaries for detecting dense subhypergraphs established.
problem Detecting dense subhypergraphs in random hypergraphs.
method Established sharp detection boundaries for known and unknown edge probabilities.
result Sharp detectable regions differ significantly from graph counterparts.
Method generates dense fields from sparse measurements without needing spatial statistics or examples.
problem Generating dense physical fields from sparse measurements.
method Introduces a differentiable numerical simulator into neural network training.
result Superior results on fluid mechanics problems compared to statistical and neural network methods.
Deforms surface groups to be Zariski dense in SL(n,R)
problem Finding Zariski dense surface groups in SL(n,R)
method Deforming K K K -integral representations of surface groups result Generalizes Long and Thistlethwaite's method to SL(n,R)
Paper tackles dense subgraph discovery with noisy feedback.
problem Discover dense subgraphs in edge-weighted graphs with noisy feedback.
method Proposes polynomial-time and scalable algorithms for dense subgraph discovery.
result Polynomial-time algorithm obtains nearly-optimal solution with high probability.
Wave fronts on certain surfaces become dense.
problem Density of wave fronts on surfaces.
method Proof of density for specific surfaces.
result Wave fronts become dense on flat torus, square billiard, Klein bottle, and cube surface.
Example shows dense subgroup of SL5(Z) not finitely presented.
problem Finding dense subgroups of SL5(Z) that are not finitely presented.
method Discussing an example of a Zariski-dense finitely generated subgroup of SL5(Z).
result Example shows a subgroup that is dense but not finitely presented.
The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.
problem Finding free semigroups with critical exponents arbitrarily close to a subgroup's in dense subgroups of Lie groups.
method Analyzing Zariski dense discrete subgroups of Lie groups, showing the existence of free semigroups with critical exponents arbitrarily close to the subgroup's.
result The existence of free semigroups with critical exponents arbitrarily close to the subgroup's in dense subgroups of Lie groups.
We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety.
A dense amalgam connects boundaries of groups split by finite subgroups.
problem Understanding boundaries of groups split by finite subgroups.
method Introducing dense amalgam and applying it to E Z E\mathcal{Z} E Z -boundaries. result Boundaries of groups split by finite subgroups have a dense amalgam structure.
The study finds dense orbits and absolute period leaves for complex flows.
problem Existence of dense orbits for real Rel flows on holomorphic 1-forms.
method Established a density criterion for m S L ( 2 , R ) {
m SL}(2,\mathbb{R}) m S L ( 2 , R ) -orbit closures, verified using explicit constructions. result Found dense leaves and examples of absolute period foliation.
RDL-Net improves speech enhancement with fewer parameters and better performance.
problem Improving speech enhancement with fewer parameters and better performance.
method Proposes RDL-Net, a CNN combining residual and dense aggregations without over-allocating parameters.
result RDL-Net achieves higher speech enhancement performance with fewer parameters and lower computational requirements.
Study on detecting and recovering hidden dense cycles in random graphs.
problem Detecting and recovering hidden dense cycles in random graphs.
method Information-theoretic analysis of thresholds for detection and recovery.
result Characterization of information-theoretic thresholds for detection and recovery.
We address feature interpretation and reproducibility issues in dense nets, proposing a modified loss function.
problem Feature interpretation and reproducibility issues in dense nets.
method Proposed a modified loss function to circumvent basis collapse.
result Substantially concise nets with 100x fewer parameters and lower MSE loss.
Generative model captures hubs and dense communities in social networks.
problem Capturing both hubs and dense communities in social networks.
method Graphon mixture model with a new condition on sparse graphs.
result Estimation of hub normalized degree and graphon for sparse components.
Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.
problem Characterizing neural network approximations to Lipschitz functions.
method Analyzing L L L -Lipschitz neural networks and their density in L L L -Lipschitz functions. result One layer neural networks are dense in the set of all L L L -Lipschitz functions. In this paper are given examples of tori T^2 embedded in S^3 with all their asymptotic lines dense.
We prove that a dense subgroup of H o m e o + ( I ) \mathrm{Homeo}_{+}(I) Homeo + ( I ) is not elementary amenable. We also show that the topological group H o m e o + ( I ) \mathrm{Homeo}_{+}(I) Homeo + ( I ) does not satisfy the Stability of the Generators Property, moreover, any finitely generated subgroup of H o m e o + ( I ) \mathrm{Homeo}_{+}(I) Homeo + ( I ) admits a faithful discrete representation …
We investigate several situations where the local homogeneity of a geometric structure on a dense open subset of a manifold implies the local homogeneity everywhere. This results in a strengthening of the conclusions in Gromov's open-dense orbit theorem. In particular, we show that any smooth closed 3-dimensional Loren…
In each Menger manifold M M M we construct: (i) a closed nowhere dense subset M 0 M_0 M 0 which is homeomorphic to M M M and is universal nowhere dense in the sense that for each nowhere dense set A ⊂ M A\subset M A ⊂ M there is a homeomorphism h h h of M M M such that h ( A ) ⊂ M 0 h(A)\subset M_0 h ( A ) ⊂ M 0 ; (ii) a meager F σ F_σ F σ -set Σ 0 ⊂ M Σ_0\subset M Σ 0 ⊂ M which is univers…
The paper classifies dense conjugacy classes in mapping class groups of locally finite graphs.
problem Identifying which mapping class groups have dense conjugacy classes.
method Developed flux homomorphisms and combinatorial criteria for stability.
result A complete classification for self-similar locally finite graphs and a criterion for stability.
Currents on cusped hyperbolic surfaces have a denseness property similar to compact surfaces.
problem Proving denseness of rational currents on cusped hyperbolic surfaces.
method Using geodesic currents and subset currents, proving denseness through examples and continuous extension.
result Denseness of rational currents on cusped hyperbolic surfaces, including geodesics connecting cusps.
Busemann points are sparse in Teichmüller spaces.
problem Characterizing the limits of geodesic rays in Teichmüller spaces.
method Proof of nowhere density of Busemann points in horoboundaries.
result Teichmüller metrics lack non-positive curvature.
Integral points are potentially dense in character varieties of quasi-projective varieties.
problem Density of integral points in character varieties of quasi-projective varieties.
method Reduction to Riemann surfaces and use of Corlette-Simpson work.
result Integral points have Zariski-dense orbit under the mapping class group.
The paper finds bounds on shortest dense curves on surfaces.
problem Finding shortest dense curves on surfaces.
method Quantitative density of closed geodesics and orthogeodesics.
result Upper bounds on shortest dense curves.
Detection of dense cycles in graphs reveals a gap between easy detection and hard recovery.
problem Detecting and recovering dense cycles in Erdős-Rényi graphs.
method Characterization of computational thresholds for detection and recovery using low-degree polynomial algorithms.
result A gap exists between the detection and recovery thresholds for certain parameter regimes.
New framework for dense weighted networks with community-specific patterns.
problem Dense networks with varying edge weights across communities.
method Proposes a new model with functions mapping node characteristics to edge weights, requiring few parameters.
result Developed a bootstrap methodology for generating new networks.
We are raising questions on discrete and dense subgroups of Diff(I). Most of the questions are around the problems discussed in [A1]-[A4].
New proof shows almost all surface group actions are dense.
problem Transitivity of normal subgroups on character varieties.
method Proved almost minimal action of non-trivial normal subgroups.
result Almost all points in character variety have dense orbits.
Polynomial-time test for detecting dense subgraphs in heterogeneous networks.
problem Detecting a planted community in heterogeneous networks.
method Proposes a polynomial-time test with a standard normal distribution null limiting distribution.
result The test is efficient and performs well in both simulations and real data.
This paper explores loss landscapes of sparse neural networks, finding unique characteristics compared to dense networks.
problem Understanding the loss landscape of sparse neural networks, especially one-hidden-layer networks.
method Analyzes sparse networks with dense and sparse final layers, focusing on linear and non-linear models.
result Sparse networks can have no spurious valleys under certain conditions, but spurious valleys and minima can exist for wide sparse networks.
We show that the bounded Borel class of any dense representation $ρ: G\to \PSL_n\bC$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group G G G . When n = 2 n=2 n = 2 , the Borel class is equal to the 3 3 3 -dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, …
We introduce and study the operation, called dense amalgam, which to any tuple X_1,...,X_k of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted ⊔ ~ ( X 1 , . . . , X k ) \widetilde\sqcup(X_1,...,X_k) ⊔ ( X 1 , ... , X k ) , in which there are many appropriately distributed copies of the spaces X_1,...,X_k. We then sh…
We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…
In this paper, we prove (1): for any closed contact three-manifold with a C ∞ C^\infty C ∞ -generic contact form, the union of periodic Reeb orbits is dense, (2): for any closed surface with a C ∞ C^\infty C ∞ -generic Riemannian metric, the union of closed geodesics is dense. The key observation is C ∞ C^\infty C ∞ -closing lemma for 3D R…
The paper bounds crossing numbers of dense graphs on surfaces.
problem Estimating the minimum number of edge crossings for dense graphs on surfaces.
method Proved lower and upper bounds on crossing numbers, providing explicit families of surfaces.
result Upper and lower bounds on crossing numbers match up to constant factors.