The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
Classifies torus bundles bounding 4-manifolds with rational homology.
problem Classifying torus bundles over the circle that bound 4-manifolds with rational homology.
method Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
result Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
Classifies surgeries on torus knots and cables that bound rational homology balls.
problem Which surgeries on torus knots and cables bound rational homology balls?
method Classification based on integral surgeries and rational numbers q/p for cables.
result Set of rational numbers q/p for cables of a given knot K is bounded.
Classifies 3-manifolds bounding rational homology balls.
problem Identifying 3-manifolds that bound rational homology balls.
method Used constraints from Donaldson's diagonalization theorem and Heegaard Floer correction terms.
result Complete classification of spherical 3-manifolds bounding rational homology balls.
Lower bounds on rational slice genus using Heegaard Floer invariants.
problem Measuring complexity of homology classes in 4-manifolds.
method Introducing rational slice genus, bounding Heegaard Floer τ invariants, using satellite links and closed braids.
result Lower bounds on rational slice genus in terms of Heegaard Floer τ invariants.
New 3-manifolds bound rational 4-balls through specific operations.
problem Finding rational homology 3-spheres that bound rational homology 4-balls.
method Two operations that preserve lattice embedding obstruction to bounding rational homology balls.
result Explicit examples of rational surgeries on torus knots that bound rational homology balls.
New knot invariant λ bounds rational unknotting.
problem Bounding rational unknotting number.
method Extracted invariant λ from Khovanov homology.
result Invariant λ is lower bound for proper rational unknotting number.
Fintushel and Stern showed that the Brieskorn sphere Σ(2,3,7) bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to p…
New rational homology 3-spheres found that can't bound definite 4-manifolds.
problem Finding rational homology 3-spheres that can't bound simply connected definite 4-manifolds.
method Constructing infinite families of irreducible rational homology 3-spheres.
result Infinite families of rational homology 3-spheres that are homology cobordant to given examples but can't bound simply connected definite 4-manifolds.
In this paper, we introduce a rational τ invariant for rationally null-homologous knots in contact 3-manifolds with nontrivial Ozsváth-Szabó contact invariants. Such an invariant is an upper bound for the sum of rational Thurston-Bennequin invariant and the rational rotation number of the Legendrian representatives o…
Study shows Seifert fibered spaces don't bound rational homology balls.
problem Understanding when Seifert fibered spaces bound rational homology balls.
method Analyzes Seifert fibered spaces with different conditions and orientations.
result Characterizes conditions for Seifert fibered spaces to bound rational homology balls.
We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology cobordisms, and to determine the concordance order of each 2-bridge knot.
Local knots can't bound smaller surfaces in rational homology 3-spheres.
problem Understanding local knots and their bounds in rational homology 3-spheres.
method Using Heegaard Floer invariant ν+ and additivity results.
result Local knots from ν+ -sharp knots have rational slice genus equal to the slice genus of the original knot.
New knots bound rational homology balls, using Alexander polynomials.
problem Finding sliceness obstructions for knots.
method Computing twisted Alexander polynomials and simplifying their calculation.
result New non-slice knots with rational homology ball bounds.
New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.
problem Characterizing knots that bound PL-disks in integer homology balls.
method Involutive Heegaard Floer homology formal properties.
result Found infinitely many manifold-knot pairs (Y, J) where J does not bound a PL-disk in an integer homology ball but does in a rational homology ball.
New families of Brieskorn spheres bound rational homology balls.
problem Identifying new Brieskorn spheres that bound rational homology balls.
method Using techniques from Akbulut and Larson's work, we present new families of Brieskorn spheres.
result We discover new infinite families of Brieskorn spheres that non-trivially bound rational homology balls.
New findings on knots that are both topologically and rationally slice.
problem Understanding knots that are both topologically and rationally slice.
method Analyzing the concordance group of knots in S3. result There are infinitely many topologically slice knots that are strongly rationally slice but not slice.
Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine…
In this paper the theory of semi-bounded rationality is proposed as an extension of the theory of bounded rationality. In particular, it is proposed that a decision making process involves two components and these are the correlation machine, which estimates missing values, and the causal machine, which relates the cau…
Study shows surgeries on certain knots bound rational homology 4-balls.
problem Classifying surgeries on knots that bound rational homology 4-balls.
method Used lattice embedding obstruction and Donaldson's Theorem.
result Classified surgeries on specific knots that bound rational homology 4-balls.
The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.
problem Lowering the rational genus of knots in rational homology 3-spheres.
method Using Heegaard Floer homology and the d-invariant. result Same lower bound and minimizers as Ni and the first author's results.
The abstract discusses connections between K-stability, heights, and rational points on Fano varieties.
problem Understanding the density of rational points on Fano varieties.
method Exploring connections between height bounds, K-stability, and Peyre's conjecture.
result Established an analog of height inequalities for real points, related to Kähler-Einstein metrics.
The paper calculates bounds for unknotting rational tangles using knot Floer homology.
problem Calculating the minimum number of rational replacements to unknot a tangle.
method From the link Floer complex, extract a lower bound for the rational unknotting number using knot Floer homology.
result The torsion obstruction is a lower bound for the proper rational unknotting number.
New knots show linear independence in slice concordance.
problem Understanding the structure of rationally slice knots.
method Provided an infinite family of knots that are linearly independent.
result Found knots that are linearly independent and infinite order.
This work explains crises in markets without external news using bounded rational agents.
problem Inability to model out-of-equilibrium dynamics in economic markets.
method Modeling bounded rational strategic reasoning in multi-agent market games.
result Bounded rational strategic reasoning can lead to endogenously emerging crises.
Knots generating infinite subgroup bound rational homology balls.
problem Understanding knots that bound rational homology balls.
method Cyclic branched covers and rational homology balls.
result Infinite two-torsion subgroup in knot concordance group.
Determines surgeries on chain links bounding rational homology balls using lattice-theoretic methods.
problem Integral surgeries on chain links bounding rational homology balls.
method Lattice-theoretic cubiquity obstruction and practical computation methods.
result Proves slice-ribbon conjecture for quasi-alternating 3-braid links, extending previous results.
Using the Heegaard Floer homology of Ozsvath and Szabo we investigate obstructions to definite intersection pairings bounded by rational homology spheres. As an application we obtain new lower bounds for the four-ball genus of Montesinos links.
The paper proves that rational concordance of double twist knots is reciprocal.
problem Determining when double twist knots are rationally slice.
method Using Donaldson's diagonalization theorem, the paper constructs rational ball obstructions for non-rational sliceness.
result Infinitely many prime power-fold cyclic branched covers do not bound a rational ball, proving the converse of rational concordance.
Paper finds new 3D shapes that can be inside a 4D space.
problem Finding new 3D shapes with specific properties.
method Examined arithmetic hyperbolic 3-manifolds and their homology.
result Discovered infinitely many 3D shapes that are rational homology spheres and can bound geometrically.
We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links. In particular we consider the problem of which rational homology S1×S2's bound rational homology S1×D3'…
New 4-manifold accounts for rationally slice knots.
problem Characterize rationally slice knots.
method Show independence of construction of rational homology ball VK. result Single 4-manifold accounts for all known rationally slice knots.
An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed, simply connected, rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivariant rational homotopy type.
New rational curvature measures for 2-complexes.
problem Measuring curvature in 2-dimensional cell complexes.
method Defined and proved rational curvature invariants.
result Computable rational curvature bounds for 2-complexes.
The study sets constraints on 4-manifold forms linked to specific invariants.
problem Understanding the intersection forms of spin 4-manifolds bounded by Seifert rational homology 3-spheres.
method Analyzes constraints using the μ-bar and κ invariants.
result The difference between κ and -μ-bar for a Seifert rational homology 3-sphere is at most 2, and under certain conditions, it is 0.
Rational neural networks approximate functions more efficiently with less depth.
problem Choosing optimal nonlinear activation functions in neural networks.
method Rational activation functions with optimal bounds and efficiency proofs.
result Rational neural networks approximate smooth functions more efficiently than ReLU networks with exponentially smaller depth.
The article classifies cubiquitous sublattices and applies them to branched covers.
problem Understanding cubiquitous sublattices as obstructions to rational homology 4-balls.
method Developed a geometric Wu obstruction to classify cubiquitous sublattices and applied it to branched covers.
result Completely classified which sublattices with orthogonal bases are cubiquitous.
Modeling market dynamics with a bounded-rational agent learning from alpha signals and market impact.
problem Understanding and predicting market dynamics with a bounded-rational agent.
method Formulating a neuroscience-inspired Bounded Rational Information Theoretic Inverse Reinforcement Learning (BRIT-IRL) model to simulate market dynamics.
result Effective asset price dynamics with non-linear mean reversion generated dynamically.
Alternating links bound rational homology balls if their chessboard lattice is cubiquitous.
problem When do alternating links bound rational homology balls?
method Heegaard Floer homology and flows on planar graphs.
result The normalized determinant of the link's chessboard lattice is a necessary and sufficient condition for the link to bound a rational homology ball.
Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.
problem Understanding which Seifert fibered spaces can be boundaries of symplectic rational homology balls.
method Analyzes convex boundaries and Lagrangian disk fillings of Legendrian knots.
result Strong restrictions on which Seifert fibered spaces can bound symplectic rational homology balls.
The study proves rationality of complex projective varieties with holomorphic vector fields.
problem Rationality of complex projective varieties with holomorphic vector fields.
method Key technique by Harvey-Lawson on finite volume flows.
result Uniform upper bound on Betti numbers for varieties with holomorphic vector fields.
Bounded rationality, that is, decision-making and planning under resource limitations, is widely regarded as an important open problem in artificial intelligence, reinforcement learning, computational neuroscience and economics. This paper offers a consolidated presentation of a theory of bounded rationality based on i…
We show that for rational surface singularities with odd determinant the mu-bar invariant defined by W. Neumann is an obstruction for the link of the singularity to bound a rational homology 4-ball. We identify the mu-bar invariant with the corresponding correction term in Heegaard Floer theory.
New model for rational tropical points using sp4-webs and measures.
problem Understanding rational tropical points of Fock-Goncharov moduli space.
method Introducing rational bounded sp4-laminations and defining tropical coordinate systems. result Established a bijection between rational tropical points and sp4-webs. We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsváth and Szabó's correction terms in Heegaard Floer homology to obtain general constraints on the surgery coefficients. We then turn our attention to the case of integral surgeries, with particular emphasis on p…
It is known that Garside groups are strongly translation discrete. In this paper, we show that the translation numbers in a Garside group are rational with uniformly bounded denominators and can be computed in finite time. As an application, we give solutions to some group-theoretic problems.
This work uses neural networks to optimize decision-making processes constrained by limited information.
problem Decision-making under resource constraints.
method Adaptive neural network priors integrated with anytime MCMC optimization.
result Neural network priors improve decision-making efficiency over fixed priors.
In this note we study the Seifert rational homology spheres with two complementary legs, i.e. with a pair of invariants whose fractions add up to one. We give a complete classification of the Seifert manifolds with 3 exceptional fibers and two complementary legs which bound rational homology balls. The result translate…