Paper discusses recent advancements in Anosov actions and Zimmer's conjecture.
problem Rigidity of Anosov actions and Zimmer's conjecture.
method Analyzes recent progress and connects different developments.
result Surprising connections and future research ideas.
This paper is a survey on the {\em Zimmer program}. In it's broadest form, this program seeks an understanding of actions of large groups on compact manifolds. The goals of this survey are (1) to put in context the original questions and conjectures of Zimmer and Gromov that motivated the program, (2) to indicate t…
Surveying matrix group actions on manifolds.
problem Topological Zimmer's conjecture on matrix group actions on manifolds.
method Surveying existing research and literature.
result Status update on matrix group actions on manifolds.
We classify compact Kähler manifolds M of dimension n≥3 on which acts a lattice of an almost simple real Lie group of rank ≥n−1. This provides a new line in the so-called Zimmer program, and characterizes certain type of complex tori by a property of their automorphisms groups.
Finite groups cannot effectively act on closed flat manifolds, except trivially.
problem Finite group actions on closed flat manifolds.
method Sufficient and necessary conditions for effective actions.
result Every group action of G on a closed flat manifold Mk (k<n) by homeomorphisms is trivial when n≥3. We consider Zimmer's program of lattice actions on surfaces by PL homomorphisms. It is proved that when the surface is not the torus or Klein bottle the action of any finite-index subgroup of SL(n,Z), n>4, (more generally for any 2-big lattice), factors through a finite group action. The proof is based on an establishm…
The paper explores actions on metric spaces similar to 3D manifolds, proving rigidity results.
problem Rigidity of actions on metric spaces similar to 3D manifolds.
method Reexamined isometry groups of geometric 3-manifolds, considered homomorphisms to them, established a dichotomy.
result Established a dichotomy between finite image or infinite volume of quotient spaces.
Let SL(n,Z) be the special linear group over integers and M=S1r×S2r,T1r×S2r , or T0r×S1r×S2r, products of spheres and tori. We prove that any group action of SL(n,Z) on Mr by diffeomorphims or piecewise linear homeomorphisms is trivial if r<n−1. This confirms a conjec…
Automorphism groups of free groups act trivially on certain manifolds.
problem Understanding actions of automorphism groups on manifolds.
method Analyzing the unique subgroup of index two in the automorphism group of a free group.
result Group actions of the automorphism group on manifolds are trivial.
We classify all holomorphic actions of higher rank lattices on compact Kaehler manifolds of dimension 3. This provides a complete answer to Zimmer's program for holomorphic actions on compact Kaehler manifolds of dimension at most 3.
The study examines group actions of SL_n(Z) on aspherical manifolds and their implications.
problem Understanding group actions of SL_n(Z) on aspherical manifolds.
method Analyzing the induced group homomorphisms and holonomy groups.
result The group SL_n(Z) cannot act nontrivially on aspherical manifolds when the dimension is less than the group's rank.
Let M be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group G. If m_0, n_0 are the dimensions of the maximal lightlike subspaces tangent to M and G, respectively, where G carries any bi-invariant metric, then we have n_0 \leq …
The paper proves Zimmer's conjecture for non-uniform lattices by controlling mass escape and Lyapunov exponents.
problem Proving Zimmer's conjecture for non-uniform lattices in higher-rank semisimple Lie groups.
method Establishes finiteness of low-dimensional actions, introduces novel techniques to control mass escape and Lyapunov exponents.
result Proves Zimmer's conjecture for many non-uniform lattices, improving previous results.
Proves Zimmer's conjecture for certain actions of SL(m, Z) subgroups.
problem Proving Zimmer's conjecture for actions of finite-index subgroups of SL(m, Z) with m>3.
method Combines earlier proof techniques with new ideas for non-compact spaces, using algebraic, geometric, and dynamical tools.
result Proves Zimmer's conjecture for C2 actions by finite-index subgroups of SL(m, Z) for m>3. Study bounds on lattice actions on compact pseudo-Riemannian manifolds.
problem Bounding real-rank of lattices acting on compact pseudo-Riemannian manifolds.
method Investigates conformal actions of cocompact lattices in higher-rank Lie groups on compact pseudo-Riemannian manifolds.
result Proves a general bound on the real-rank of the lattice and shows manifold conformally flat when real-rank is maximal.
Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.
problem Characterizing Lie foliations with symmetric leaves.
method Analyzing the rigidity of Lie foliations with locally symmetric leaves.
result Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.
In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued co…
Maximal measurable cocycles of complex hyperbolic lattices are cohomologous to representations.
problem Characterizing maximal measurable cocycles of complex hyperbolic lattices.
method Utilizing Zimmer's Superrigidity Theorem and proving the existence of a boundary map.
result Maximal measurable cocycles are cohomologous to representations of PU(p,1) into SU(m,n).
We prove several cases of Zimmer's conjecture for actions of higher-rank cocompact lattices on low dimensional manifolds. For example, if Γ is a cocompact lattice in Sl(n,R), M is a compact manifold, and ω a volume form on M we show that any homomorphism $ρ\colon Γ\rightarrow \mathrm{Diff}(M…
In this paper, we investigate the ergodic and rigidity properties of weakly hyperbolic group actions. Motivated by classical theorems describing Anosov diffeomorphisms, we obtain two main results: First, all C^2 volume preserving weakly hyperbolic actions on closed manifolds are ergodic. This result generalizes Anosov'…
We prove a generalization of Livsic's Theorem on the vanishing of the cohomology of certain types of dynamical systems. As a consequence, we strengthen a result due to Zimmer concerning algebraic hulls of Anosov actions of semisimple Lie groups. Combining this with Topological Superrigidity, we find a Holder geometric …
We announce a generalization of Zimmer's cocycle superrigidity theorem proven using harmonic map techniques. This allows us to generalize many results concerning higher rank lattices to all lattices in semisimple groups with property (T). In particular, our results apply to SP(1,n) and F4−20 and lattices in tho…
This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, technique…
R. Zimmer proved that, on a compact manifold, a foliation with a dense leaf, a suitable leafwise Riemannian symmetric metric and a transverse Lie structure has arithmetic holonomy group. In this work we improve such result for totally geodesic foliations by showing that the manifold itself is arithmetic. This also give…
Embedding theorem for tractor bundles applied to conformal geometry.
problem Embedding theorem for tractor bundles in Cartan geometries.
method Extension of Gromov-Zimmer embedding theorem to tractor bundles.
result Rigidity result for conformal actions of special pseudo-unitary groups.
In this article we consider asymptotically harmonic manifolds which are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature h. We prove the following equivalences for asymptotically harmonic manifolds X under the additional assumpti…
Study shows only one type of proper domain in certain spaces.
problem Classifying proper domains in Hermitian symmetric spaces.
method Analyzing Shilov boundaries and automorphism groups.
result Classification of closed proper manifolds locally modeled on Shilov boundaries.
The article discusses invariant measures outside homogeneous dynamics.
problem Understanding invariant measures in non-homogeneous settings.
method Comparing two recent results on rigidity and invariant measures.
result The importance of invariant and partially invariant measures in rigidity questions.
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.
Paper proves finite BMS measure for SPR groups in higher rank Lie groups.
problem Finite measure for certain groups in higher rank Lie groups.
method Developed SPR property and proved finite BMS measure.
result Finite Bowen-Margulis-Sullivan measure for SPR groups in higher rank Lie groups.
New criterion for smoothing actions with singularities.
problem Smoothability of actions with singular diffeomorphisms.
method Criterion for conjugacy of actions to smooth ones.
result Aperiodic actions by diffeomorphisms with countably many singularities are conjugate to smooth actions.
New volume invariant for cocycles of hyperbolic lattices, proving rigidity results.
problem Volume calculation for cocycles of hyperbolic lattices.
method Introducing a new volume invariant and proving Milnor-Wood type inequalities.
result Characterization of maximal cocycles and proof of rigidity results.
Study Gromov hyperbolic domains in Minkowski space, proving equivalence to boundary properties.
problem Investigate Gromov hyperbolic domains in Minkowski space.
method Explicit comparisons between metrics, dynamical arguments, and quasi-hyperbolic metric.
result Gromov hyperbolicity of convex, future complete domains is equivalent to stable acausality of the boundary.
SED integrates synthesis, execution, and debugging for neural program synthesis.
problem Challenges in synthesizing complex programs that match specifications.
method SED combines synthesis, execution, and debugging to improve neural program generation.
result SED reduces error rates and outperforms standard decoding methods.
The paper proves that certain spaces have injective balls of any radius.
problem The injectivity radius of certain geometric spaces is infinite.
method Analyzes higher rank simple and semisimple Lie groups with specific properties.
result The locally symmetric spaces have injective balls of any radius.
Stochastic programs simplify complex models with noise and nondeterminism.
problem Handling models with nuisance parameters, noise, and nondeterminism.
method Developed a reference implementation for stochastic probabilistic programs and inference.
result Efficient inference in models with noise and nondeterminism is possible.
Deployable probabilistic programming for Go and other languages.
problem Adding probabilistic programming to mainstream languages.
method Design guidelines and Infergo implementation for Go.
result Infergo demonstrates performance and applicability in various use cases.
Extends program induction for probabilistic programming.
problem Automatic probabilistic program synthesis for diverse data types.
method Further steps to extend previous work on program induction.
result Generalization over various data types (text, image, video).
Improves probabilistic programming by analyzing program structure.
problem Inefficiency and limitations of single inference algorithms in probabilistic programming.
method Three novel techniques: static and dynamic analyses to adapt programs for more efficient inference.
result Improves probabilistic programming by making inference more efficient.
Study shows exact dimensionality and regularity of manifolds for specific groups.
problem Exact dimensionality and regularity of manifolds for relatively Anosov groups.
method Dynamical methods, including finite and mixing of Bowen–Margulis–Sullivan measures.
result Manifolds are C1-regular and growth indicator is strictly concave. A neural program synthesis method with iterative fix operations.
problem Creating correct programs from input-output examples.
method Combines encoder-decoder synthesis with a differentiable fixer.
result Improves synthesis accuracy by reducing discrepancies between outputs and desired outputs.
Paper introduces techniques to learn higher-order programs, improving predictive accuracy and reducing learning times.
problem Expressing and learning complex programs in ILP.
method Extending meta-interpretive learning to support higher-order definitions as background knowledge.
result Learning higher-order programs reduces hypothesis space and sample complexity, improving predictive accuracy and reducing learning times.
Graph-based approach repairs programs from diagnostic feedback.
problem Learning to repair programs from limited labeled data and compiler error messages.
method Introduces program-feedback graph and graph neural network for reasoning, and self-supervised learning with unlabeled programs.
result DrRepair significantly outperforms prior work, achieving high repair rates.
Improves neural program synthesis by addressing aliasing and syntax issues.
problem Ignoring program aliasing and syntax in neural program synthesis.
method Reinforcement learning and direct syntax maximization training.
result Improved accuracy, especially with limited training data.
Neural model guides PBE problem solving in programming.
problem Synthesizing programs from example inputs/outputs.
method Uses a neural model to guide miniKanren's constraint logic programming system.
result Synthesizes programs faster and generalizes to larger problems.
COSET benchmarks neural program embeddings using diverse source-code datasets.
problem Evaluating neural program embeddings is challenging due to lack of straightforward metrics.
method COSET framework with labeled programs, transformations, and a pilot study.
result COSET identifies strengths and weaknesses of neural models and program characteristics.
SPoC uses search to translate pseudocode into correct programs with error localization.
problem Mapping pseudocode to functionally correct long programs.
method Search-based approach guided by compilation errors for credit assignment.
result Search improves synthesis success rate from 25.6% to 44.7%.
Dataset of human-written problem statements and solutions for program synthesis.
problem Creating programs from natural language problem descriptions.
method Crowdsourced problem statements and solutions from programming competitions.
result Best model achieved 8.8% accuracy, indicating high complexity.