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…
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.
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.
Study on flat metrics on 3D and 4D manifolds, focusing on topology and algebra.
problem Topology and algebraic structure of flat metrics on manifolds.
method Algebraic and topological descriptions of moduli spaces.
result Algebraic description and topology of moduli spaces for 4D manifolds with a single holonomy generator.
Explains how knots relate to 4D shapes.
problem Understanding 4D shapes through knot theory.
method Combines knot theory with 4D manifold topology.
result Connects 4D shapes to knot theory and other geometries.
Topological twists for 4d N=2 theories depend on spacetime type, gerbe connections, and generalized spin-c structures.
problem Defining topologically twisted partition functions for 4d N=2 theories.
method Topological twisting for general 4d N=2 theories, introducing generalized spin-c structures.
result Topological partition functions depend on spacetime type, gerbe connections, and generalized spin-c structures.
Proof outlined for 4D smooth Poincaré conjecture.
problem 4-dimensional smooth Poincaré conjecture.
method Outline of proof.
result Proof of 4D smooth Poincaré conjecture.
New minimal hypersurfaces in 4D sphere found.
problem Constructing embedded minimal hypersurfaces in S4. method Equivariant min-max theory and suspended Hopf action.
result Infinitely many topological S1-bundles and Seifert fibered manifolds found. The paper explores gauge theory invariants and their duals via topological-holomorphic twist.
problem Understanding gauge theory invariants and their duals in 4d and 2d.
method Topological-holomorphic twist of N=4 supersymmetric gauge theory.
result Derived novel topological and holomorphic invariants and their Langlands duals.
Research on the least complex surface in certain 4D shapes.
problem Finding the simplest surface in specific four-dimensional shapes.
method Analyzing smooth four-manifolds to determine minimal genus.
result Results on the minimal genus for studied four-manifolds.
This paper confirms volumes of geodesic balls can identify 4D space forms.
problem Determining if a 4D manifold is a space form using geodesic ball volumes.
method Tensor calculus and classical theorems, not topological characterizations.
result Similar results for 4D manifold space forms confirmed.
We construct new classes of exact solutions of the 4D vacuum Einstein equations which describe ellipsoidal black holes, black tori and combined black hole -- black tori configurations. The solutions can be static or with anisotropic polarizations and running constants. They are defined by off--diagonal metric ansatz wh…
Study Murasugi sum in 4D for knotted surfaces, defining arborescent surfaces.
problem Defining and understanding Murasugi sum in 4D for knotted surfaces.
method Introduced a 4D Murasugi sum to define arborescent knotted surfaces.
result Defined and studied arborescent knotted surfaces using 4D Murasugi sum.
Classifies theories with eight supercharges using pseudo-periodic maps and Riemann surfaces.
problem Classifying theories with eight supercharges using mathematical tools.
method Assumes theories are given by genus g fibrations of Riemann surfaces, uses pseudo-periodic maps of negative type in mapping class group.
result Identifies dual graphs and 3d mirror quivers, unifies various SCFTs in combinatorial framework.
We explain how, starting with a stack of D4-branes ending on an NS5-brane in type IIA string theory, one can, via T-duality and the topological-holomorphic nature of the relevant worldvolume theories, relate (i) the lattice models realized by Costello's 4d Chern-Simons theory, (ii) links in 3d analytically-continued Ch…
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. The paper studies 4D Ricci flow manifolds with curvature constraints.
problem Investigating 4D Ricci flow manifolds with specific curvature conditions.
method Analyzing 4D manifolds with curvature constraints via Ricci flow.
result Proves topological and geometric gap theorems for maximal volume growth.
The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.
problem Understanding spectral networks in symplectic topology and their role in Lagrangian fillings.
method Analytic results on adiabatic degeneration of Floer trajectories and explicit computation of continuation strips.
result Established equivalence between Family Floer functor and non-abelianization functor for Lagrangian fillings with spectral networks.
We propose a way to define and compute invariants of general smooth 4-manifolds based on topological twists of non-Lagrangian 4d N=2 and N=3 theories in which the problem is reduced to a fairly standard computation in topological A-model, albeit with rather unusual targets, such as compact and non-compact Gepner models…
A new tensorial metric describes geometry in 4D space.
problem Understanding the structure of hypercomplex space.
method Developed a new geometry group in R^4 with a tensorial metric.
result Riemannian and Euclidean distances are special cases of the Alpha Group's metric.
Survey on geometric, analytic, and topological aspects of 4D equations.
problem No specific problem stated in abstract.
method Geometric, analytic, and topological discussions.
result New solution of the Cauchy problem over null hypersurfaces.
New 2-links created in 4D spaces, topologically unknotted but not smoothly.
problem Creating exotic 2-links in 4D spaces.
method A construction procedure to generate infinite sets of 2-links.
result First examples of 2-links that are topologically unknotted but not smoothly.
Recently we introduced T-duality in the study of topological insulators, and used it to show that T-duality trivialises the bulk-boundary correspondence in 2 dimensions. In this paper, we partially generalise these results to higher dimensions and briefly discuss the 4D quantum Hall effect.
We construct infinitely many smooth oriented 4-manifolds containing pairs of homotopic, smoothly embedded 2-spheres that are not topologically isotopic, but that are equivalent by an ambient diffeomorphism inducing the identity on homology. These examples show that Gabai's recent "Generalized" 4D Lightbulb Theorem does…
Explains peculiarities of 4D scalar curvature via Yamabe invariant.
problem Interplay between scalar curvature and differential topology in 4-manifolds.
method Careful discussion of Yamabe invariant (sigma constant).
result Proves new results and identifies open problems.
New stable shrinking Ricci soliton found in 4D.
problem Stability of 4D Ricci shrinkers.
method Explicit metric description, tensor harmonic analysis, Kähler structure, selfduality insights.
result First non-cylindrical linearly stable shrinking Ricci soliton in 4D.
Study links between surface germs and knot theory in 4D.
problem Understanding the relationship between surface germs and knot theory in R4. method Constructing surface germs XK linked to knots K in S3 and studying their Lipschitz geometry. result Ambient bi-Lipschitz equivalence of surface germs is related to isotopy of knots, and Jones polynomial can recognize non-equivalent germs.
New spheres can split a 4D link in ways not possible in 3D.
problem Exploring how splitting spheres behave in 4D space.
method Constructing specific 2-component surface-links in S4. result Found non-isotopic splitting spheres in S4∖Lm,n. The paper characterizes and contrasts knots with high 4D clasp numbers.
problem Characterizing knots with high 4-dimensional clasp numbers.
method Topological category analysis and construction of counterexamples.
result Characterization and contrast of knots with high 4D clasp numbers.
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.
We study the twisted index of 4d N = 2 class S theories on a closed hyperbolic 3-manifold M3. Via 6d picture, the index can be written in terms of topological invariants called analytic torsions twisted by irreducible flat connections on the 3-manifold. Using the topological expression, we determine the …
Algorithm trisects 4D book into three chapters.
problem Trisecting 4D book into chapters.
method Algorithm for open book decompositions of 4-manifolds.
result Algorithm works for various types of manifolds.
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.
The 'anholonomic frame' method (see gr-qc/0005025, gr-qc/0001060 and hep-th/0110250) is applied for constructing new classes of exact solutions of vacuum Einstein equations with off-diagonal metrics in 4D and 5D gravity. We examine several black tori solutions generated by anholonomic transforms with non-trivial topolo…
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. Study sharp geometric and topological properties of pinched 4D submanifolds.
problem Pinched submanifolds in space forms.
method Four-dimensional geometry, Riemannian manifolds with nonnegative isotropic curvature, Bochner technique.
result Sharp results extend previous work without additional assumptions.
Four-dimensional scanning transmission electron microscopy (4D-STEM) of local atomic diffraction patterns is emerging as a powerful technique for probing intricate details of atomic structure and atomic electric fields. However, efficient processing and interpretation of large volumes of data remain challenging, especi…
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.
Classifies 4D toric Hermitian ALF metrics with conical singularities.
problem Classifying specific types of 4D Riemannian metrics.
method Explicit formulas provided for classification.
result Examples of metrics with conical singularities have infinitely many distinct topologies.
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.
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. Unified framework for 3D and 4D manifold and knot theory.
problem Unified understanding of manifold and knot theory.
method Unified correspondence between different subfields of low-dimensional topology.
result Unified algebraic manifestations of 3D and 4D manifold and knot theory.
In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-gr…
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.
Geometric interpretation of 2d-4d wall-crossing formulas.
problem Understanding wall-crossing phenomena in coupled 2d-4d systems.
method Deformation theory of holomorphic pairs and relation to scattering diagrams.
result Geometric interpretation of wall-crossing formulas.
Proves local maximizers for higher Ekeland-Hofer capacities in 4D star-shaped domains.
problem Finding local maximizers for higher Ekeland-Hofer capacities in specific domains.
method Analogous to 4D local Viterbo conjecture, proving maximizers for rational ellipsoids.
result Local maximizers of the k-th Ekeland-Hofer capacities are symplectomorphic to rational ellipsoids.