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.
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).
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. 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.
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.
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.
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…
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…
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.
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.
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…
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.
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.
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…
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…
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…
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…
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…
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…
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.
Paper proves spectral filters can be transferred between graphs.
problem Proving spectral filters can be transferred between graphs.
method Introducing the Cayley smoothness space and proving filters in this space are linearly stable.
result Graph spectral filters are transferable if they are in the Cayley smoothness space.
This work prunes CNN filters based on their functionality, not just size.
problem Redundant filters in CNNs waste computation resources.
method Functionality-oriented filter pruning method.
result Pruning based on functionality optimizes computation and interprets filter importance.
A new SOHP filter improves trend estimation in economic time series.
problem Improving trend estimation in nonlinear economic time series.
method Recursive application of one-sided HP filter on updated cyclical components, combined with an incremental HP filtering algorithm.
result Better performance of SOHP filter compared to other HP-type filters on real economic data.
Deep density methods improve filtering in high-dimensional systems.
problem Nonlinear filtering in high-dimensional systems.
method Two deep density methods based on Feynman-Kac formulas and neural networks.
result Logarithmic deep backward stochastic differential equation filter outperforms classical methods in high dimensions.
The paper explores modifications to filter banks for speech recognition.
problem Improving speech recognition accuracy using modified filter banks.
method The authors investigate replacing triangular filters with Gabor or Gammatone filters, and rearranging filter bank computations to integrate features over smaller time scales.
result No significant improvements in phone error rate were observed with the modifications.
Pruning filters in CNNs improves interpretability, showing shape-selective filters are crucial for object recognition.
problem Interpreting the complex decision-making process of CNNs is challenging due to their large number of parameters.
method We developed a greedy structural compression scheme that prunes filters based on the classification accuracy reduction (CAR) index.
result Pruned filters in CNNs, especially those in the first and second layers, are more likely to be shape-selective, indicating their importance in object recognition.
Gradient filters track moving parameters under noisy data and misspecification.
problem Tracking multidimensional time-varying parameters under noisy observations and model misspecification.
method Gradient-based filters update parameters using the gradient of a postulated objective function, evaluated at either the predicted or updated parameters.
result Novel sufficient conditions for exponential stability of the filtered parameter path, and finite-sample and asymptotic mean squared error bounds.
We simplify Bayesian filtering by framing it as optimization, making it practical for high-dimensional systems.
problem Bayesian filtering struggles in high-dimensional state spaces like neural networks.
method We frame Bayesian filtering as optimization, using gradient descent for nonlinear cases.
result Our method results in effective, robust, and scalable filters for high-dimensional systems.
A new model optimizes Bloom filters using machine learning.
problem Improving the efficiency of Bloom filters for data sets.
method Modeling learned Bloom filters with machine learning, optimizing with sandwiching method.
result Optimized learned Bloom filters provide improved performance.
Develops an inverse particle filter for cognitive systems.
problem Tracking cognitive adversaries in counter-adversarial applications.
method Global filtering approach using Monte Carlo methods and differentiable I-PF.
result Demonstrates convergence to optimal inverse filter and improved estimation performance.
A novel method reduces dimensionality for filtering SRNs with observed variables.
problem Challenges in estimating hidden state variables in SRNs with limited observations.
method Filtered Markovian Projection (Filtered MP) for dimensionality reduction in filtering.
result Filtered MP guarantees consistency and superior computational efficiency in high dimensions.
Kernel learning FBSDE filter improves nonlinear filtering efficiency.
problem Nonlinear filtering problem in high-dimensional systems.
method Iterative and adaptive meshfree approach using forward backward SDE and KDE.
result Rigorous convergence analysis provided, supporting empirical results.
Paper develops a particle filter for rapid model parameter adaptation and change detection.
problem Rapidly adapting to changes in model parameters and distinguishing between regime shifts and stochastic volatility.
method Incorporates genetic algorithm elements into a particle filter for accelerated adaptation and change detection.
result The filter adapts to regime shifts extremely rapidly and provides a clear heuristic for distinguishing between regime shifts and stochastic volatility.
New method filters large networks from financial data to reveal key subnetworks.
problem Filtering large dimensional networks to isolate key constituents.
method Exploits spectral properties of high-dimensional data networks, tuning for sparsity and consistency.
result Shows method can interpolate between zero and maximal filtering, preserving spectral properties.
Many nonlinear extensions of the Kalman filter, e.g., the extended and the unscented Kalman filter, reduce the state densities to Gaussian densities. This approximation gives sufficient results in many cases. However, this filters only estimate states that are correlated with the observation. Therefore, sequential esti…
Constructs tangent groupoid for filtered manifolds without coordinates.
problem No specific problem stated; intrinsic construction of tangent groupoid.
method Intrinsic construction of tangent groupoid.
result Intrinsic construction of tangent groupoid for filtered manifolds.
This work analyzes the stability of graph filters under large perturbations.
problem Stability of graph filters under large edge rewires.
method Proves a bound on stability using frequency response and community structure.
result Graph filter stability depends on perturbation to community structure.