Alternative definition of cobordism map in ECH theory.
problem Defining and proving equivalence of cobordism map in ECH theory.
method Reformulating and proving independence of filtered ECH cobordism map using Seiberg-Witten theory.
result The new definition of cobordism map is equivalent to existing definitions.
New bound on ECH sub-leading asymptotics.
problem Understanding ECH capacities in detail.
method Analyzing sub-leading asymptotics of ECH spectrum.
result New bound on sub-leading asymptotics of ECH capacities.
Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the…
Proves existence of elliptic Reeb orbit on real projective 3-space using ECH.
problem Existence of elliptic Reeb orbits on real projective 3-space.
method Use of ECH (Embedded Contact Homology) to find distinguished pseudoholomorphic curves.
result Existence of elliptic Reeb orbit proven for some contact forms on RP3. The embedded contact homology (ECH) of a 3-manifold with a contact form is a variant of Eliashberg-Givental-Hofer's symplectic field theory, which counts certain embedded J-holomorphic curves in the symplectization. We show that the ECH of T^3 is computed by a combinatorial chain complex which is generated by labeled c…
We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic k-torsion in the context of ech, a variant of ECH used in a proof of the isomorphism between Heegaard Floer and Seiberg-Witten Floer homologies; and we explain how it translates into Heegaard Floer homology.
Researchers describe ECC of concave lens spaces and compute symplectic capacities.
problem Computing symplectic capacities of concave toric domains.
method Combinatorial description of ECC, ECH capacities computation, packing of symplectic manifolds.
result Computed ECH capacities of certain concave toric domains.
New invariant shows no homotopy 4-sphere difference.
problem Determining homotopy 4-spheres using ECH and SWF.
method Embedded contact homology (ECH) and Seiberg-Witten theory (SWF).
result ECH and SWF are ineffective for distinguishing homotopy 4-spheres.
The study connects ECH capacities to Anosov flows, proving infinite capacities and obstructions.
problem Understanding ECH capacities and their relation to Anosov flows.
method Relating ECH capacities to Anosov flows dynamics, proving infinite capacities and obstructions.
result ECH capacities are infinite for many symplectic 4-manifolds, including cotangent disk bundles over surfaces of genus at least two.
This paper improves energy estimates for Seiberg-Witten Floer generators.
problem Recovering the volume of contact 3-manifolds using ECH capacities.
method Stronger estimates on the energy of min-max Seiberg-Witten Floer generators.
result Directly proves the ECH capacities recover volume theorem.
Paper proves ellipticity of certain Reeb orbits and estimates ECH spectrum on lens spaces.
problem Proving ellipticity of Reeb orbits in lens spaces and estimating ECH spectrum.
method Using rational self-linking number, Conley-Zehnder index, and ECH computations.
result First ECH spectrum on dynamically convex L(3,1) is estimated and shown to be equal to contact area infimum.
This paper finds a global surface of section in dynamically convex L(p,p-1) using ECH.
problem Finding a global surface of section in dynamically convex L(p,p-1).
method Using Embedded Contact Homology (ECH).
result Relates periods of the surface of section to the first ECH spectrum.
In this paper, we prove (1): for any closed contact three-manifold with a C∞-generic contact form, the union of periodic Reeb orbits is dense, (2): for any closed surface with a C∞-generic Riemannian metric, the union of closed geodesics is dense. The key observation is C∞-closing lemma for 3D R…
Established equivalence of Atiyah classes for generalized holomorphic vector bundles.
problem Defining and comparing Atiyah classes for generalized holomorphic vector bundles.
method Used three approaches: \(\check{C}\)ech cohomology, first jet short exact sequence, and Lie algebroid pairs.
result Equivalence of Atiyah classes defined by different methods.
This is the third of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an au…
This is the second of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an a…
Computes knot filtered ECH for torus knots on tight 3-sphere.
problem Computing knot filtered ECH for torus knots.
method Generalized knot filtered ECH definition, Morse-Bott methods, energy filtered Seiberg-Witten theory.
result Computed knot filtered ECH for T(2,q) knots (q odd, positive).
This paper is the sequel to "The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I" and is devoted to proving some of the technical parts of the HF=ECH isomorphism.
Given a closed oriented 3-manifold M, we establish an isomorphism between the Heegaard Floer homology group HF^+(-M) and the embedded contact homology group ECH(M). Starting from an open book decomposition (S,h) of M, we construct a chain map Φ^+ from a Heegaard Floer chain complex associated to (S,h) to an embedded co…
Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding…
In joint work with Yang Huang, we defined a canonical absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields. A similar grading was defined on embedded contact homology by Michael Hutchings. In this paper we show that the isomorphism between these homology theories defined by Colin-G…
A new version of ECK for rationally null-homologous knots, with a surgery formula.
problem Defining and computing ECK for rationally null-homologous knots.
method Generalized ECK to rational open book decompositions, proving independence of choices, and proving a surgery formula.
result Large negative n-surgery formula for ECK, supporting equivalence with knot Floer homology.
We show that sutured embedded contact homology is a natural invariant of sutured contact 3-manifolds which can potentially detect some of the topology of the space of contact structures on a 3-manifold with boundary. The appendix, by C. H. Taubes, proves a compactness result for the completion of a sutured contact 3-ma…
Investigates the rotating Kepler problem for energy values ≤ -3/2.
problem Understanding periodic orbits and symplectic structures in rotating celestial mechanics.
method Ligon-Schaaf and Levi-Civita symplectic regularizations, special concave toric domain construction.
result Identification of a special concave toric domain (SCTD) for the RKP phase space.
Constructs invariants from Lagrangian tori in symplectic 4-manifolds.
problem Tackles invariants of Lagrangian tori in symplectic 4-manifolds.
method Uses ECH and monopole Floer homology to construct distinguished elements.
result Repackages Gromov and Seiberg-Witten invariants of torus surgeries.
The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the h…
We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit va…
The paper studies automorphisms of Weyl manifolds and constructs modified contact Weyl diffeomorphisms.
problem Analyzing the automorphisms of Weyl manifolds associated with symplectic structures.
method Investigates the automorphisms of Weyl manifolds corresponding to Poincaré-Cartan classes and constructs modified contact Weyl diffeomorphisms.
result Construction of modified contact Weyl diffeomorphisms and analysis of automorphisms of Weyl manifolds.
This is the fourth of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an a…
Algorithm finds connected components on Lie groups for multi-orientation image analysis.
problem Finding connected components in complex, multi-orientation images.
method Iterative algorithm using morphological dilations and Hamilton-Jacobi-Bellman kernels on Lie groups.
result Algorithm converges in finitely many steps and can differentiate between crossing and aligned structures.
The aim of this paper is to review and discuss in detail local aspects of principal bundles with groupoid structure. Many results, in particular from the second and third section, are already known to some extents, but, due to the lack of a ``unified'' point of view on the subject, I decided nonetheless to (re)define a…
This paper classifies symplectic and Stein fillings of contact 3-manifolds with spinal open book decompositions.
problem Classifying symplectic and Stein fillings of contact 3-manifolds with spinal open book decompositions.
method Using holomorphic curves and Lefschetz fibrations to classify fillings.
result Symplectic and Stein fillings of contact 3-manifolds with spinal open book decompositions can be classified up to deformation equivalence.
We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that any aspherical subcomplex of a random 2-complex satisfies…
Random complexes can be embedded linearly if certain conditions on parameters are met.
problem Embedding random simplicial complexes linearly in Euclidean space.
method Established strict inequalities on parameters for linear embedding into R^(2d).
result Necessary and sufficient conditions for linear embedding of random complexes.
Random walks on cell complexes link to Laplacians and Novikov-Shubin invariants.
problem Computing Novikov-Shubin invariants for complex cell structures.
method Construct random walks on cell complexes, relate to Laplacians, and use return probabilities.
result Novikov-Shubin invariants can be recovered from random walk return probabilities.
Solves partial assignment problems using random clique complexes.
problem Partial assignment problems, especially with severe occlusions and distortions.
method Formulate as matching random clique complexes, analyze k-skeletons, match adjacency matrices, consider geometric neighbourhoods.
result Outperforms diverse matching algorithms significantly.
Enhances random forest consistency and introduces DMRF for improved performance.
problem Improving the consistency and efficiency of random forest algorithms.
method Strengthened proof methods and propose DMRF algorithm.
result DMRF achieves better theoretical and experimental performance than previous variants.
Random branched covers of groups are homotopy equivalent to geometrically small cancellation complexes.
problem Understanding the topological properties of random branched covers of groups.
method Constructing a random model for branched covers and showing asymptotic homotopy equivalence to geometrically small cancellation complexes.
result The fundamental group of a random branched cover is Gromov hyperbolic and has small cohomological dimension.
Study on vanishing cohomology groups in random simplicial complexes.
problem Determining when cohomology groups vanish in random simplicial complexes.
method Analysis of binomial random (k+1)-uniform hypergraphs and their downward-closure. result Vanishing of cohomology groups with coefficients in F2 has a sharp threshold. Study on expected topology of random subcomplexes in subdivided finite simplicial complexes.
problem Understanding the expected topology of random subcomplexes in subdivided finite simplicial complexes.
method Analysis of successive barycentric subdivisions and study of expected Betti numbers, average Morse inequalities, and Euler characteristic.
result Asymptotic upper and lower bounds for the expected Betti numbers of random subcomplexes.
Proposes random Euler filters for efficient complex-valued nonlinear signal processing.
problem Efficiently processing complex-valued nonlinear signals with reduced computational cost.
method Introduces linear and widely-linear random Euler complex-valued filters with fixed network structures.
result Analytical minimum mean square error and optimum step-size derived for transient and steady-state performances.
The paper studies random Čech complexes on Riemannian manifolds and phase transitions in homological connectivity.
problem Understanding homological connectivity in random Čech complexes on Riemannian manifolds.
method Study of a random Čech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M, focusing on phase transitions for homological connectivity.
result The homology of the complex becomes isomorphic to that of M at a phase transition.
Random polynomial dynamical systems often have negative Lyapunov exponents.
problem Understanding the behavior of random polynomial dynamical systems.
method Investigation of i.i.d. random complex dynamical systems generated by probability measures.
result For a generic system, the Lyapunov exponent of almost every sequence of maps is negative for most initial values.
Random groups in the square model have Haagerup property and are residually finite.
problem Understanding properties of random groups in the square model.
method Generalized Isoperimetric Inequality and introduced modified hypergraphs to structure Cayley complex.
result Random groups in the square model have the Haagerup property and are residually finite.
Lower bounds show linear complexity for linear regression.
problem Computational complexity of linear regression.
method Reduction to estimating the least eigenvalue of a random Wishart matrix.
result Θ(d) calls to the oracle are necessary and sufficient for polynomial accuracy.
Gradient boosting with randomized trees reduces discontinuities and complexity.
problem Discontinuities in regression functions due to sparse training data.
method Gradient boosting machine with partially randomized decision trees.
result Improves robustness and computational efficiency of gradient boosting.
Randomness is crucial for stability in learning and statistics, especially for differential privacy.
problem Quantifying the amount of randomness needed for algorithmic stability.
method Weak-to-strong boosting theorem for stability, characterizing randomness complexity of PAC Learning.
result Randomness complexity is tightly controlled by the best replication probability of any deterministic algorithm solving the task.
Low-complexity spiking networks learn complex tasks with minimal trainable parameters.
problem Training complex reinforcement learning tasks with minimal resources.
method Reinforcement learning on simple networks of spiking neurons with random connections.
result Small random spiking networks achieve learning efficiency similar to humans on complex tasks.