arXiv research

We show that a four-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to $S^1 \times S^3\# n \overline{\mathbb{C} P^2}$, $\# m\mathbb{C} P^2 \#n\overline{\mathbb{C} P^2}$ or $\# m (S^2 \times S^2)$. Given the relation between boundary Lefschetz fibrations and stable general…

Study calculates homotopy groups and derivatives for disc diffeomorphisms.

Ancient curve shortening flow in a disc with mixed boundary conditions is solved.

The paper finds conditions for slicing knots in 4-manifolds.

Study the topology of stable vector fields and Lyapunov functions on R^n.

For $L \hookrightarrow X$ a Lagrangian embedding associated with a real homogeneous space, we construct the moduli space of stable holomorphic discs mapping to $(X,L)$ as an orbifold with corners equipped with a group action. Some essential constructions involving orbifolds with corners are also discussed, including th…

Metric spaces with upper curvature bounds have controlled Dehn functions.

Constructs a functor for equivariant smooth h-cobordisms.

Classifies stable diffeomorphism of spin 4-manifolds with specific fundamental groups.

Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.

The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.

Study shows almost all Arnold stable solutions have no conjugate points.

Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.

New knots found with tough, unsliceable discs.

Classifies ancient flows in a disc with boundary.

We consider components of Hurwitz moduli space of G-Galois covers and set up a powerful algebraic framework to study the set of corresponding equivalence classes of monodromy maps. Within that we study geometric stabilisation by various G-covers branched over the disc. Our results addresses the problem to decide equiva…

The paper explores isometric models and Busemann functions for Funk and Hilbert discs.

Let $Γ$ be either the infinite cyclic group $\mathbb{Z}$ or the Baumslag-Solitar group $\mathbb{Z} \ltimes \mathbb{Z}[\frac{1}{2}]$. Let $K$ be a slice knot admitting a slice disc $D$ in the 4-ball whose exterior has fundamental group $Γ$. We classify the $Γ$-homotopy ribbon slice discs for $K$ up to topological ambien…

The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.

Study smooth manifolds using disc-presheaves.

This note characterizes monohedral tilings of regular polygons with up to three tiles.

Study horizontal discs in fat distributions, proving their existence.

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.

Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.

The paper classifies homotopy ribbon discs with specific groups.

It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two …

Study of rotation angles in a rotating disc model.

The study finds conditions for free boundary Hamiltonian stationary discs in complex 2-space.

Algebraic treatment of connection reduction over a special disc.

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used …

Holomorphic discs converge to maximal surfaces under specific flows.

The paper finds the Finsler structure of Apollonian weak metric on unit disc.

This paper compares two methods for training neural ODEs in time-series regression and CNFs.

We show that a relatively hyperbolic graph with uniformly hyperbolic peripheral subgraphs is hyperbolic. As an application, we show that the disc graph and the electrified disc graph of a handlebody H of genus g>1 are hyperbolic, and we determine their Gromov boundaries.

In this note we prove that any monohedral tiling of the closed circular unit disc with $k \leq 3$ topological discs as tiles has a $k$-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain t…

Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.

New phenomena in 4-manifolds show discs with special properties.

We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying c…

We reduce a question of Eskin-Kontsevich-Zorich and Forni-Matheus-Zorich, which asks for a classification of all $\text{SL}_2(\mathbb{R})$-invariant ergodic probability measures with completely degenerate Kontsevich-Zorich spectrum, to a conjecture of Möller's. Let $\mathcal{D}_g (1)$ be the subset of the moduli space …

Curvature bounds preserved in length-minimizing disks.

In this paper we describe the homology and cohomology of some natural bimodules over the little discs operad, whose components are configurations of non-$k$-overlapping discs. At the end we briefly explain how this algebraic structure intervenes in the study of spaces of non-$k$-equal immersions.

We construct knotted proper holomorphic embeddings of the unit disc in C^2.

New minimal discs and annuli found in ellipsoids.

Modified proof constructs dual spheres for 4-manifolds.

The article proves Randers Poincaré disc satisfies isoperimetric equality.

Glaucoma is the second leading cause of blindness all over the world, with approximately 60 million cases reported worldwide in 2010. If undiagnosed in time, glaucoma causes irreversible damage to the optic nerve leading to blindness. The optic nerve head examination, which involves measurement of cup-to-disc ratio, is…

The paper finds linked periodic orbits in disc homeomorphisms using braids.

Study detects if a circuit bounds a disc using curve intersections.