Improved upper bound for discrete isometric filling of cycles.
problem Finding the minimum number of vertices in a discrete isometric filling of cycle graphs.
method Explicit construction of isometric fillings using concentric annular structures.
result Explicit construction of isometric fillings with \( |V(K_n)| \le \left(\frac{1}{6} + o(1)
ight)n^2 \), improving the upper bound to \( D^* \le \frac{1}{6} \).
This paper tackles Gromov's filling area conjecture using discrete graph theory.
problem Finding the smallest surface area for isometrically filling a circle.
method Using graph-theoretic tools like Menger's theorem to derive bounds.
result Discrete bounds translate to a lower bound of 1.36π for the hemisphere's surface area.
The paper proposes a new way to approximate Riemannian metrics using discrete wall systems.
problem Approximating Riemannian metrics and proving geometric conjectures.
method Discretization of metrics using walls and triangulations.
result The discrete filling area conjecture is equivalent to Gromov's original conjecture.
We prove that, for a hyperbolic two bridge knot, infinitely many Dehn fillings are rigid in SO0(4,1). Here rigidity means that any discrete and faithful representation in SO0(4,1) is conjugate to the holonomy representation in SO0(3,1). We also show local rigidity for almost all Dehn fillings.
Proves minimization for Kähler manifolds with automorphisms.
problem Minimizing CM degree for Kähler manifolds with automorphisms.
method Equivariant CM minimization conjecture, asymptotic filtration Chow stability.
result Strict minimization by extremal filling.
This paper identifies a negative profit effect in limit order fills.
problem Profit drag in limit order fills due to adverse price movements.
method Discrete market model, empirical simulation of US Treasury Bond futures.
result Existence of negative drift in limit order fills.
Determining the space of free discrete two generator groups of Möbius transformations is an old and difficult problem. In this paper we show how to construct large balls of full dimension in this space. To do this, we begin with a marked discrete group of non-separating disjoint circle type. Such a group determines thr…
A method for efficient approximate inference on discrete distributions.
problem Applying SVGD to discrete distributions.
method Transforming discrete distributions to piecewise continuous distributions for SVGD application.
result Outperforms traditional algorithms and ensemble methods on discrete graphical models.
This research explores how different discrete diffusion kernels affect graph generation quality.
problem The impact of different discrete diffusion kernels on graph generation quality.
method Developed a family of discrete diffusion kernels that converge to different Bernoulli priors.
result The quality of generated graphs is sensitive to the prior used, challenging previous intuitions.
We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit example…
Let X=S×E×B be the metric product of a symmetric space S of noncompact type, a Euclidean space E and a product B of Euclidean buildings. Let Γ be a discrete group acting isometrically and cocompactly on X. We determine a family of quasi-isometry invariants for such Γ, namely the k-dimension…
Proves hardness of semi-discrete optimal transport and proposes regularization methods.
problem Computing Wasserstein distance between discrete and non-discrete probability measures.
method Proves hardness, introduces distributionally robust dual optimal transport, regularizes primal objective, uses stochastic gradient descent.
result Regularization schemes and improved convergence guarantees for semi-discrete optimal transport problems.
Gerbes encode spectral gaps in topological insulators.
problem Capturing spectral gaps in topological insulators.
method Geometric encoding through 'Real' gerbes.
result Gerbes precisely capture spectral gaps.
Deep Reinforcement Learning (DRL) has been applied to address a variety of cooperative multi-agent problems with either discrete action spaces or continuous action spaces. However, to the best of our knowledge, no previous work has ever succeeded in applying DRL to multi-agent problems with discrete-continuous hybrid (…
In this research, we develop a trading strategy for the discrete-time optimal liquidation problem of large order trading with different market microstructures in an illiquid market. In this framework, the flow of orders can be viewed as a point process with stochastic intensity. We model the price impact as a linear fu…
Proposes ABC method for discrete data, improving likelihood-free inference.
problem Discrete data likelihood-free inference problems.
method Population-based MCMC ABC framework with a new Markov kernel inspired by Differential Evolution.
result High potential and superiority of the new Markov kernel demonstrated.
Study Stein and Milnor fillings of links from surface singularities.
problem Comparing Stein and Milnor fillings of links from surface singularities.
method Analyzing the topology and obstructions of Stein fillings and Milnor fillings.
result Milnor fillings have bounded topology, while Stein fillings can be more varied.
Proves upper bound on systolic ratio for circle fillings.
problem Bounding systolic ratio for circle fillings.
method Proved upper bound on systolic ratio depending on genus.
result Filling Area Conjecture holds for large genus.
The paper constructs minimal coherent filling pairs on surfaces.
problem Finding minimal intersecting coherent filling pairs on surfaces.
method Geometric procedure starting from a torus filling pair.
result Construction of minimal intersecting coherent filling pairs on Sg for g≥3. The paper classifies symplectic fillings of lens spaces and constructs cobordisms.
problem Classifying symplectic fillings of lens spaces and constructing cobordisms.
method Analyzing tight and universally tight contact structures, using plumbing of disk bundles, and constructing cobordisms.
result Maximal second homology Stein fillings of lens spaces are given by specific plumbing.
Computes A-polynomials of knots from Whitehead sister link fillings.
problem Computing A-polynomials for knots from Whitehead sister link fillings.
method Using results on A-polynomials of Dehn fillings, formulas are derived for the A-polynomials of knots.
result Formulas to compute A-polynomials for knots from Whitehead sister link fillings.
If a simple 3-manifold M admits a reducible and a toroidal Dehn filling, the distance between the filling slopes is known to be bounded by three. In this paper, we classify all manifolds which admit a reducible Dehn filling and a toroidal Dehn filling with distance 3.
We give three infinite families of examples of nonhyperbolic Dehn fillings on hyperbolic manifolds. A manifold in the first family admits two Dehn fillings of distance two apart, one of which is toroidal and annular, and the other is reducible and ∂-reducible. A manifold in the second family has boundary consi…
Study negative definite spin fillings of knot covers.
problem Existence of negative definite spin fillings in branched double covers.
method Derive obstructions and characterize special knots.
result Characterization of special alternating knots.
The paper proves finiteness of cosmetic fillings on a specific type of 3-manifold.
problem Cosmetic fillings on specific 3-manifolds.
method Analysis of Dehn fillings on compact, orientable 3-manifolds with incompressible torus boundaries.
result There are only finitely many pairs of purely cosmetic fillings.
Study generalizes Lebesgue curves to new space-filling and fractal sets.
problem Generating space-filling curves from planar substitutions.
method Generalized Lebesgue's construction to new curves and fractal sets.
result Some substitutions create relatively dense fractal-like sets.
We use Menke's JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces. For virtually overtwisted structures on elliptic or parabolic torus bundles, this gives a complete …
Let M be a simple 3-manifold with a toral boundary component partial_0 M. If Dehn filling M along partial_0 M one way produces a toroidal manifold and Dehn filling M along partial_0 M another way produces a boundary-reducible manifold, then we show that the absolute value of the intersection number on partial_0 M of th…
If a hyperbolic 3-manifold M admits a reducible and a finite Dehn filling, the distance between the filling slopes is known to be 1. This has been proved recently by Boyer, Gordon and Zhang. The first example of a manifold with two such fillings was given by Boyer and Zhang. In this paper, we give examples of hyperboli…
Affirmative proof that rank 3 3-manifolds have filling links.
problem Existence of filling links in rank 3 3-manifolds.
method Introduced and used the concept of filling links, proved for rank 3 manifolds.
result Every rank 3 closed orientable 3-manifold contains a filling link.
Study filling links in 3-manifolds to understand their topological properties.
problem Understanding the topological properties of 3-manifolds through filling links.
method Introduce and study filling links, prove their existence and injectivity properties.
result Existence of k-filling links in the 3-torus for each k>1.
New method fills cluster seeds with exact Lagrangian structures.
problem Surjectivity of map from Lagrangian fillings to cluster seeds.
method Construction of quiver with potential and Lagrangian disk surgeries.
result Trivial deformation space of quiver allows manipulation of fillings.
Study shows connectedness of Bowditch boundary persists in long Dehn fillings.
problem Persistence of Bowditch boundary connectedness in Dehn fillings.
method Analysis of relatively hyperbolic group pairs and peripheral subgroups.
result Connectedness of Bowditch boundary persists in sufficiently long Dehn fillings without needing restrictions.
This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.
problem Classifying and determining the length of the shortest filling pairs on a specific type of surface.
method Classifying and determining the length of the shortest filling pairs on a specific type of surface.
result The paper classifies and determines the length of the shortest minimal filling pairs on a genus two surface.
Positive braids have endless filling possibilities.
problem Understanding exact Lagrangian fillings of positive braid links.
method Analyzing positive braid Legendrian links and their fillings.
result Positive braid Legendrian links have infinitely many exact Lagrangian fillings.
Let M be an irreducible, compact, connected, orientable 3-manifold whose boundary is a torus. We show that if M is hyperbolic, then it admits at most six finite/cyclic fillings of maximal distance 5. Further, the distance of a finite/cyclic filling to a cyclic filling is at most 2. If M has a non-boundary-paralle…
Four-dimensional Einstein Dehn filling is impossible.
problem Complex-hyperbolic Einstein Dehn filling in four dimensions.
method Proof of impossibility.
result Complex-hyperbolic Einstein Dehn filling cannot be performed in dimension four.
Study Stein fillings of planar contact 3-manifolds with relative trisection genus 2.
problem Classify Stein fillings of planar contact 3-manifolds under constraints on their relative trisections.
method Partial classification of diffeomorphism types of fillings with relative trisections of genus at most 2.
result Partially classify the diffeomorphism types of Stein fillings with relative trisections of genus at most 2.
Study on Legendrian knots and their non-orientable Lagrangian fillings.
problem Conditions for Legendrian knots to have non-orientable exact Lagrangian fillings.
method Developed combinatorial obstructions and classified fillability for various knot families.
result Completely determined decomposably non-orientable fillability for alternating and plus-adequate knots.
Study shows no large mean curvature fill-ins for nonnegative scalar curvature.
problem Existence of fill-ins with nonnegative scalar curvature and large mean curvature.
method Analyzes closed Riemannian manifolds and scalar curvature properties.
result No closed Riemannian manifold admits a fill-in with nonnegative scalar curvature and large mean curvature.
We extend Thurston's metric to projective filling currents, embedding Teichmüller space into the larger space.
problem Embedding Teichmüller space into the space of projectivized filling currents.
method Extending the symmetrized Thurston metric to PCfill(S) and studying its geometry. result There is no quasi-isometric projection back from PCfill(S) to T(S). We give an algorithm which produces infinitely many pairwise exotic Stein fillings of the same contact 3-manifolds, applying positive allowable Lefschetz fibrations over the disk. As a corollary, for a large class of Stein fillings, we realize the topological invariants (i.e. fundamental group, homology group, homology…
Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.
problem Characterizing symplectic fillings of prequantization bundles with finite capacities.
method Analysis of symplectic capacities and diffeomorphisms.
result Symplectic fillings of prequantization bundles are diffeomorphic to disk bundles under finite capacity conditions.
Filling length measures the length of the contracting closed loops in a null-homotopy. The filling length function of Gromov for a finitely presented group measures the filling length as a function of length of edge-loops in the Cayley 2-complex. We give a bound on the filling length function in terms of the log of an …
Market makers face a trade-off between fill probability and post-fill returns, requiring contrarian strategies.
problem Navigating the trade-off between fill probability and post-fill returns in market making.
method Analysis of live trading data from Binance Bitcoin perpetual.
result A negative correlation between maker fill likelihood and post-fill returns, necessitating contrarian strategies.
Develops a diagrammatic method for symplectic filling classifications.
problem Classifying exact/weak symplectic fillings of 3D contact manifolds.
method Symplectic JSJ decomposition applied to contact surgery diagrams.
result Recover symplectic fillings for certain lens spaces and torus bundles, and classify fillings for a large class of plumbed 3-manifolds.
The study of symplectic fillings for rational cuspidal curves.
problem Understanding symplectic fillings of contact manifolds associated with rational cuspidal curves.
method Exploration through Stein handlebodies and rational blow-downs.
result Examples of contact manifolds that are links of normal surface singularities, and those that do not admit symplectic fillings.
Study finds many Lagrangian fillings for certain Legendrian links.
problem Understanding Lagrangian fillings for Legendrian links of finite type.
method Use of N-graphs and combinatorics of seed patterns.
result Proves existence of at least seeds many exact embedded Lagrangian fillings for Legendrian links of type ADE.