arXiv research

Geometrically, a new obstruction is found for 4-manifold realizations.

It is shown that Schneiderman and Teichner's invariant $τ$ detects the example due to Kirk of a link map $f:S^2_+\cup S^2_-\to S^4$ for which the restricted map $f|S^2_+:S^2_+\to S^4-f(S^2_-)$ has vanishing Wall self-intersection but is not homotopic to an embedding.

The study explores deep and shallow slice knots in 4-manifolds, linking them to conjectures and proving existence and nonexistence results.

Study self-intersections of arcs on a pair of pants, proving natural number spectrum.

This is the beginning of an obstruction theory for deciding whether a map f:S^2 --> X^4 is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall's self-intersection number mu(f) which tells the whole story in higher dimensions…

The paper calculates self-intersections on a pair of pants using Bowen and Series' coding.

Given a hyperbolic surface $§$, a classic result of Birman and Series states that for each $K$, all complete geodesics with at most $K$ self-intersections can only pass through a certain nowhere dense, Hausdorff dimension 1 subset of $§$. We define a self-intersection function for each complete geodesic, which bounds t…

The study counts curves on a once-punctured torus with self-intersections.

Paper proves curves can be smoothed to reduce self-intersection by exactly 1.

Sharp lower bound on fold singularities self-intersections.

The study examines elastic curves with self-intersections and their properties.

Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.

Suppose a smooth planar curve $γ$ is $2π$-periodic in the $x$ direction and the length of one period is $\ell$. It is shown that if $γ$ self-intersects, then it has a segment of length $\ell- 2π$ on which it self-intersects and somewhere its curvature is at least $2π/(\ell - 2π)$. The proof involves the projection $Γ$ …

The entropy of geodesic currents on hyperbolic surfaces is bounded by their self-intersection number.

Study uses neural networks to predict wall quantities in turbulent flows.

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class…

The study finds that most minimal surfaces in generic 4D manifolds intersect in complex ways.

Generic potential primes have no self-intersections or intersections.

This paper studies the interplay between the N=2 gauge theories in three and four dimensions that have a geometric description in terms of twisted compactification of the six-dimensional (2,0) SCFT. Our main goal is to construct the three-dimensional domain walls associated to any three-dimensional cobordism. We find t…

New $G_2$-holonomy manifolds from 5d N=1 theories domain walls.

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We pr…

Study connects Morse theory with cluster variables for wall-crossing in Cerf diagrams.

We explain how to adapt a construction of M. Sageev's to construct a proper action on a CAT(0) cube complex starting from a proper action on a wall space, and use this to deduce that if G is a group containing an amenable subgroup H of super-polynomial growth and G acts properly on a space with walls then there are arb…

Neural network predicts turbulence near-wall regions efficiently.

Farrell and Hsiang noticed that the geometric surgery groups defined By Wall, Chapter 9, do not have the naturality Wall claims for them. They were able to fix the problem by augmenting Wall's definitions to keep track of a line bundle. The definition of geometric Wall groups involves homology with local coefficients a…

The index theorem connects anomalies on a domain wall to global integrals.

Geodesics with bounded angles have zero Hausdorff dimension.

Convolutional networks predict turbulence from wall quantities.

In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is the number of letters of the corresponding word. The self-intersection of a free…

We find the minimal number of self-intersections of the boundary of a surface of genus g generically immersed in the plane.

Analyzes how quadratic differential trajectories change with variation, proving a wall-crossing formula.

We address the problem of computing bounds for the self-intersection number (the minimum number of self-intersection points) of members of a free homotopy class of curves in the doubly-punctured plane as a function of their combinatorial length L; this is the number of letters required for a minimal description of the …

Given an orientable surface with boundary and a free homotopy class, we present a purely combinatorial algorithm which produces a representative of that homotopy class with minimal self intersection.

Wall's result extended to 4-manifolds with definite intersection forms.

Extends index theorem to domain walls with discontinuous Riemannian connections.

The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.

Regular homotopy classes of immersions of a 3-sphere in 5-space constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an embedding is described in terms of geometric invariants of its self intersection. Ge…

Study wall singularities in spaces with upper curvature bounds.

We describe a correspondence between spaces with walls and CAT(0) cube complexes.

The purpose of this note is to give a self contained description of Walls finiteness obstruction.

In a previous paper, we defined an operation $μ$ that generalizes Turaev's cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding que…

A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may have odd Euler characteristic if and only if k is congruent to 1 modulo 4 or k+…

Modeling aortic wall inhomogeneities to predict dissection risks.

Proof of wall-crossing formula using spectral networks.

Study of Hamiltonian flows on character varieties for self-intersecting curves.

Minimal geodesics on hyperbolic surfaces are long.

We review our recent work on solitons in the Higgs phase. We use U(N_C) gauge theory with N_F Higgs scalar fields in the fundamental representation, which can be extended to possess eight supercharges. We propose the moduli matrix as a fundamental tool to exhaust all BPS solutions, and to characterize all possible modu…

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity. We also consider the minimal genus of a subsurface tha…