arXiv research

New method uses broken scattering to uniquely identify Finsler manifolds.

Consider a broken geodesics $α([0,l])$ on a compact Riemannian manifold $(M,g)$ with boundary of dimension $n\geq 3$. The broken geodesics are unions of two geodesics with the property that they have a common end point. Assume that for every broken geodesic $α([0,l])$ starting at and ending to the boundary $\partial M$…

Samplets and multiwavelets constructed from scattered data converge to specific densities in the limit.

New proofs confirm travel time data determine simple metrics on a disc.

Billiard trajectories (broken generalised geodesics) are considered in the exterior of an obstacle $K$ with smooth boundary on an arbitrary Riemannian manifold. We prove a generalisation of the well-known Santalo's formula. As a consequence, it is established that if the set of trapped points has positive measure, then…

Survey on broken ray transforms and open problems.

The broken genera are orientation preserving diffeomorphism invariants of closed oriented 4-manifolds, defined via broken Lefschetz fibrations. We study the properties of the broken genera invariants, and calculate them for various 4-manifolds, while showing that the invariants are sensitive to exotic smooth structures…

We consider the broken ray transform on Riemann surfaces in the presence of an obstacle, following earlier work of Mukhometov. If the surface has nonpositive curvature and the obstacle is strictly convex, we show that a function is determined by its integrals over broken geodesic rays that reflect on the boundary of th…

We reduce the broken ray transform on some Riemannian manifolds (with corners) to the geodesic ray transform on another manifold, which is obtained from the original one by reflection. We give examples of this idea and present injectivity results for the broken ray transform using corresponding earlier results for the …

The space of broken hyperbolic structures generalizes the Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations (equivalently, the space of projectivized affine foliations) generalizes the space of projectivized measured foliations. Just as projectivized measured foliations…

It is known that an arbitrary smooth, oriented 4-manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. Th…

The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles, and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefsch…

In this article, we generalize the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classify genus one Lefschetz fibrations over the 2-disk with certain non-trivial global monodromies using chart descriptions, and identify…

We show that there exists a non-trivial simplified broken Lefschetz fibration which has infinitely many homotopy classes of sections. We also construct a non-trivial simplified broken Lefschetz fibration which has a section with non-negative square. It is known that no Lefschetz fibration satisfies either of the above …

The purpose of this note is to explain a combinatorial description of closed smooth oriented 4-manifolds in terms of positive Dehn twist factorizations of surface mapping classes, and further explore these connections. This is obtained via monodromy representations of simplified broken Lefschetz fibrations on 4-manifol…

This PhD thesis studies the broken ray transform, a generalization of the geodesic X-ray transform where geodesics are replaced with broken rays that reflect on a part of the boundary. The fundamental question is whether this transform is injective. We employ four different methods to approach this question, and each o…

A hyperelliptic broken Lefschetz fibration is a generalization of a hyperelliptic Lefschetz fibration. We construct and compute a local signature of hyperelliptic directed broken Lefschetz fibrations by generalizing Endo's local signature of hyperelliptic Lefschetz fibrations. It is described by his local signature and…

Given a closed Riemannian manifold of dimension $n$ and a Morse-Smale function, there are finitely many $n$-part broken trajectories of the negative gradient flow. We show that if the manifold admits a hyperbolic metric, then the number of $n$-part broken trajectories is always at least the hyperbolic volume. The proof…

Study proves uniqueness for ray transform on surfaces with obstacles.

We prove a lower bound on the number of maximally broken trajectories of the negative gradient flow of a Morse-Smale function on a closed aspherical manifold in terms of integral (torsion) homology.

We present explicit algorithms for simplifying the topology of indefinite fibrations on 4-manifolds, which include broken Lefschetz fibrations and indefinite Morse 2-functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1-parameter families. In particular, given an arbitr…

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism type…

Much work has been done on the existence and uniqueness of broken Lefschetz fibrations such as those by Auroux et al., Gay and Kirby, Lekili, Akbulut and Karakurt, Baykur, and Williams, but there has been a lack of explicit examples. A theorem of Gay and Kirby suggests the existence of a broken Lefschetz fibration of S…

This paper shows how to create surface-links with many triple points.

We study the integral transform over a general family of broken rays in $\mathbb{R}^2$. It is natural for broken rays to have conjugate points, for example, when they are reflected from a curved boundary. If there are conjugate points, we show that the singularities cannot be recovered from local data and therefore art…

We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. This paper will demonstrate the potential of the scattering symplect…

We introduce general scattering transforms as mathematical models of deep neural networks with l2 pooling. Scattering networks iteratively apply complex valued unitary operators, and the pooling is performed by a complex modulus. An expected scattering defines a contractive representation of a high-dimensional probabil…

Paper explains scattering diagrams' role in mirror symmetry.

We prove that the topology, smooth structure, and metric of a compact Lorentzian manifold with boundary is uniquely determined by data at the boundary. The data consists of the lengths and directions of future-directed once-broken geodesics connecting points on the boundary, which are first timelike and then lightlike.…

GSAN learns adaptive node representations using geometric scattering and attention.

New method learns soliton dynamics from scattering data without assuming known equations.

Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, i.e., manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and broken Lefschetz fibrations on them. We present a set of four moves which allow …

Scattering representations simplify SBI for images without extra compression.

This note presents the handlebody argument for modifying achiral Lefschetz singularities into broken Lefschetz fibrations, yielding a handlebody proof of the existence of broken Lefschetz fibrations on arbitrary closed smooth oriented 4-manifolds based on the earlier work of Gay and Kirby. Appeared in Geometry and Topo…

The paper establishes scattering theory for wave equations on Schwarzschild spacetime.

Paper develops formulas for shape derivatives in wave scattering.

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of the success of convolutional neural networks (ConvNets) in image data analysis and other tasks. Inspired by recent interest in geometric deep learning, which aims to generalize ConvNets to manifold and gra…

Study scattering rigidity on stationary manifolds using geodesics.

MODWST improves classification tasks with wavelet scattering.

Study on scattering geodesics on modular surface and their sojourn times.

We prove that every closed oriented smooth 4-manifold X admits a broken Lefschetz fibration (aka singular Lefschetz fibration) over the 2-sphere. Given any closed orientable surface F of square zero in X, we can choose the fibration so that F is a fiber. Moreover, we can arrange it so that there is only one Lefschetz c…

Here we show that every compact smooth 4-manifold X has a structure of a Broken Lefschetz Fibration (BLF in short). Furthermore, if b_{2}^{+}(X)> 0 then it also has a Broken Lefschetz Pencil structure (BLP) with nonempty base locus. This imroves a Theorem of Auroux, Donaldson and Katzarkov, and our proof is topological…

Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.

Study on recovering Lorentzian metrics from scattering data.

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric…

Bayesian Scattering offers a simple baseline for image data uncertainty.

Scattering theory for linearised gravity on Schwarzschild black hole exterior.

The scattering transform is a multilayered wavelet-based deep learning architecture that acts as a model of convolutional neural networks. Recently, several works have introduced generalizations of the scattering transform for non-Euclidean settings such as graphs. Our work builds upon these constructions by introducin…