The paper offers new methods to determine if certain 3D links can be formed by intersecting spheres in 4D space.
problem Determining if a 3D link can be formed by intersecting spheres in 4D space.
method Using obstructions from multivariable signature, Blanchfield form, and generalised Seifert matrices.
result Provides lower bounds on the doubly slice genus of links.
Lecture notes on link homologies and knotted surfaces, focusing on 4D obstructions.
problem Understanding knotted surfaces using link homology theories.
method Overview of link homology theories, introduction to Khovanov homology, and computational techniques.
result New insights into the homomorphisms assigned to link cobordisms for knotted surfaces.
New method finds non-orientable knotted surfaces in 4D.
problem Reducibility of knotted surfaces in 4D.
method Elementary obstruction to reducibility.
result Construction of stably irreducible non-orientable surfaces.
The paper develops obstructions for embedding 2D complexes into 4D space.
problem Embedding 2D complexes into 4D space and understanding obstructions.
method Uses Goodwillie-Weiss calculus and intersections of Whitney disks.
result Two approaches to obstructions lead to the same result.
New method shows nonorientable surfaces in 4D are topologically unknotted.
problem Tackles isotopy classes of nonorientable surfaces in D4. method Calculations implemented in Sage to show ambient isotopy.
result Closed, nonorientable surfaces in S4 are topologically unknotted. Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.
problem Finding topological obstructions to compact gradient shrinking Ricci solitons in dimension four.
method Discussion of background material, introduction of new problem, exploration of limitations of current results.
result Introduction of new problem and limitations of current results in extending Hitchin-Thorpe inequality.
Enhances Hantzsche's theorem for 3-manifolds in 4D.
problem Embedding 3-manifolds in 4-dimensional space.
method Using Heegaard diagrams and a property called doubly unlinked (DU).
result Enhanced Hantzsche's embedding obstruction.
The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.
problem Existence and uniqueness of almost complex blow-ups on almost complex manifolds.
method Definition and construction of almost complex blow-ups, proving their existence and uniqueness.
result Existence and uniqueness of almost complex blow-ups on 4D almost complex manifolds.
Study proves rigidity and gap theorems for specific metrics.
problem Existence and properties of self-dual and even Poincaré-Einstein metrics in 4D.
method Rigorous mathematical proofs, including gap theorems and rigidity results.
result Obtained new scalar conformal invariants and identified obstructions to metric existence.
2D complexes can be almost-embedded in 4D space without self-intersections.
problem Understanding the embedding properties of 2-dimensional complexes in 4-dimensional space.
method Analyzing specific 2-complexes constructed by Freedman-Krushkal-Teichner and showing they can be PL immersed in R4 without self-intersections. result Many 2-complexes can be PL almost-embedded in R4 with singularities only as self-intersections of some 2-cells. The study proves that certain minimal hypersurfaces in 4D space must be planes.
problem The extension of the half-space theorem to higher dimensions is obstructed.
method Analyzes topological properties of minimal hypersurfaces in R4. result Complete, properly embedded minimal hypersurfaces in R4 with bounded curvature and diffeomorphic to R3 must be planes. A 3D catenoid in 4D space is a minimal hypersurface that cannot be extended to a higher-dimensional half-space.
problem Extending the half-space theorem to higher dimensions in R4. method Analyzing the topological constraints on minimal hypersurfaces in R4. result A complete, properly embedded minimal hypersurface in a slab in R4 must be a hyperplane. Symmetrizes 4d and 3d BPS quivers for Argyres-Douglas theories.
problem Understanding the relationship between 4d and 3d BPS quivers.
method Analyzes geometric backgrounds and uses skein modules to derive quiver partition functions.
result Proves isomorphism between 4d wall-crossing and unlinking of symmetric quivers.
We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose a…
New 1-parameter family of ovals identified in 4d Ricci flow classification.
problem Classifying κ-solutions in 4d Ricci flow. method Introducing conjectures and constructing new examples.
result Established canonical neighborhood theorem for 4d Ricci flow.
Study of 4D Ricci solitons with symmetry, finding precise geometric asymptotics.
problem Classifying 4D gradient steady Ricci solitons and understanding their geometric properties.
method Analysis of 4D gradient steady Ricci solitons with O(3)-symmetry under a weak curvature decay condition.
result Find precise geometric asymptotics similar to 3D compact κ-solutions.
Study 4D solitons with specific curvature properties, proving curvature bounds and classifying solutions.
problem Investigate 4D gradient solitons with specific curvature properties.
method Analyze 4D gradient steady and shrinking solitons with nonnegative or half nonnegative isotropic curvature.
result Prove 2-nonnegativity of Ricci curvature and bound the curvature tensor for ancient solutions.
Study classifies special 4D shapes with certain curvature.
problem Classifying specific types of 4D shapes.
method Classifying compact almost-Kähler four manifolds with nonnegative biorthogonal curvature.
result Classified compact almost-Kähler four manifolds with nonnegative biorthogonal curvature.
Perelman's proof confirmed, new method uses 4D topology.
problem Confirming the classical Poincaré conjecture.
method 4D topology, spun torus-knots, ribbonness, disk-chord system, Bing's result.
result Homotopy 3-sphere is diffeomorphic to the 3-sphere.
A class of 3d N=2 supersymmetric gauge theories are constructed and shown to encode the simplicial geometries in 4-dimensions. The gauge theories are defined by applying the Dimofte-Gaiotto-Gukov construction in 3d/3d correspondence to certain graph complement 3-manifolds. Given a gauge theory in this class…
We consider compactification of type IIA supergravity on nearly Kaehler manifolds. These represent a simple class of SU(3) structure manifolds which includes S^6 and CP^3. We exhibit for the first time an explicit reduction ansatz in this context, obtaining an N=2 gauged supergravity in 4d with a single vector and hype…
We explore 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at θ=π (SU(2)θ=π YM), by turning on backg…
New surfaces generalize Dini surfaces in 4D.
problem None explicitly stated; focuses on surface generalization.
method Introducing a new family of surfaces in 4D.
result Generalized Dini surfaces exist in 4D.
New ribbon disks in 4D space, non-isotopic to each other.
problem Non-isotopic ribbon disks in 4D.
method Using corks to construct diffeomorphic ribbon disks.
result Non-isotopic ribbon disks constructed in 4D.
The study calculates harmonic functions and 1-forms on specific 4D spaces.
problem Computing harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
method Computed the expansion of harmonic functions and 1-forms.
result Computed the expansion of harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
Study of symmetries in 4D Lie groups.
problem Understanding symmetries in specific Lie groups.
method Analyzing isometry groups of left-invariant metrics.
result Full description of isometry groups for 4D Lie groups.
A registration-free framework monitors shape and color in 4D point clouds.
problem Monitoring shape and color changes in complex parts without registration.
method Laplace-Beltrami operator spectral properties for geometric and color feature capture; combined monitoring scheme for shape and color anomalies.
result Effective detection of shape deformations and color anomalies without registration or mesh reconstruction.
Minimal moves for surfaces in 4D identified.
problem Classifying surfaces embedded in 4D space.
method Derived minimal generating set of planar moves.
result Identified minimal moves for surfaces in 4D.
Method resolves 4D symplectic orbifolds using complex geometry.
problem Resolving symplectic orbifolds in 4 dimensions.
method Combining complex geometry techniques with symplectic form gluing.
result Examples of 4D symplectic orbifolds successfully resolved.
Study describes flat metric moduli spaces on 4D manifolds.
problem Understanding flat metrics on 4D closed manifolds.
method Algebraic and topological description of moduli spaces.
result Algebraic and topological description of moduli spaces of flat metrics.
The paper classifies singularities of line congruences in 4D space.
problem Classifying singularities of line congruences in 4D space.
method Generic classification approach for 3-parameter line congruences and Blaschke normal congruences.
result Generic classification of singularities of 3-parameter line congruences in R4. We study second-order PDEs in 4D for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge-Ampere property. Since half-flatness of the conformal structure is equivalent to the exi…
We present a simple explicit construction of hyper-Kaehler and hyper-symplectic (also known as neutral hyper-Kaehler or hyper-parakaehler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kaehler and hyper-symplectic structures in dimension four.
Low entropy hypersurfaces in 4D are isotopic to a sphere.
problem Characterizing hypersurfaces with low entropy in 4D.
method Proving isotopy to the standard 3-sphere for hypersurfaces with entropy ≤ cylinder entropy.
result Closed hypersurfaces with low entropy are isotopic to the standard 3-sphere.
Sharp inequality proven for symmetric functions on a 4D sphere.
problem Proving a sharp Beckner's inequality for axially symmetric functions on S4. method Utilized pointwise properties of Gegenbauer polynomials.
result Sharp Beckner's inequality established for axially symmetric functions on S4. M-theory compactified on G2-holonomy manifolds results in 4d N=1 supersymmetric gauge theories coupled to gravity. In this paper we focus on the gauge sector of such compactifications by studying the Higgs bundle obtained from a partially twisted 7d super Yang-Mills theory on a supersymmetric three-cycle…
The study examines 4D steady gradient Ricci solitons with nonnegative curvature away from a compact set.
problem Analyzing noncompact steady gradient Ricci solitons with nonnegative curvature operator.
method Examining the asymptotic behavior of noncompact κ-noncollapsed steady gradient Ricci solitons with nonnegative curvature operator away from a compact set.
result 4D noncompact κ-noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature must be a Bryant Ricci soliton up to scaling.
Study Lefschetz fibrations with 4D fibers using Seiberg-Witten theory.
problem Constraints on the topology of Lefschetz fibrations with 4D fibers.
method Using the family Bauer-Furuta invariant and framed bordism class of 1D Seiberg-Witten moduli spaces.
result New obstructions to the smooth isotopy of compositions of Dehn twists.
Study uses superalgebra homology to classify 4D Engel-like Lie algebras.
problem Classifying 4-dimensional Engel-like Lie algebras.
method Applied homology groups of Lie superalgebras.
result Distinguished and classified 4D Engel-like Lie algebras.
Method synthesizes 4D CMR images from XCAT model using GAN and SPADE.
problem Synthesizing realistic 4D CMR images with annotations and adaptable styles.
method Hybrid GAN approach with XCAT anatomical ground truth and SPADE for semantic preservation.
result Synthesized images with modality-specific features learned from real CMR data.
The study examines Lipschitz normally embedded Hölder triangles in 4D space.
problem Comparing ambient and outer Lipschitz geometry of Hölder triangles.
method Analyzes Lipschitz normally embedded Hölder triangles in \(\mathbb{R}^4\).
result Infinitely many equivalence classes of microknots.
New theorem for 4D links simplifies characterisation problem.
problem Long-standing open problem in link characterisation.
method Reidemeister Theorem for solid ribbon torus links.
result Complete characterisation of a related class of links.
It is formulated a new 'anholonomic frame' method of constructing exact solutions of Einstein equations with off--diagonal metrics in 4D and 5D gravity. The previous approaches and results are summarized and generalized as three theorems which state the conditions when two types of ansatz result in integrable gravitati…
Extends spectral Einstein functionals computation to 4D spin manifolds with boundary.
problem Computing spectral Einstein functionals for 4D spin manifolds with boundary.
method Generalizes Dabrowski's results to 4D spin manifolds with boundary using noncommutative residue.
result Generalized spectral Einstein functionals computation for 4D spin manifolds with boundary.
Global regularity proved for 4D Ricci flow with scalar curvature integral bound.
problem Global regularity of 4D Ricci flow with integral scalar curvature bound.
method Extended Ge-Jiang's result to include integral bound on scalar curvature.
result Global ε-regularity for 4D Ricci flow with integral scalar curvature bound. New deep learning method improves 4D Flow MRI super-resolution under domain shift.
problem Domain shift in low-resolution 4D Flow MRI data.
method Distributional deep learning framework for domain generalization.
result Framework significantly outperforms traditional methods in real data applications.
New infinite family of 2-complexes intrinsically linked in 4D.
problem Intrinsic linking of 2-complexes in 4D.
method Examining suspensions of graphs containing K6 as a minor.
result Embeddings of suspensions contain intrinsically linked cycles.
Minimal hypertori found in 4D sphere, solving Bernstein conjecture.
problem Finding minimal embedded hypertori in 4D sphere.
method Analyzing minimally embedded and immersed hypertori and hyperspheres.
result Infinitely many non-isometric minimally embedded hypertori and hyperspheres found.