arXiv research

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

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

This paper classifies quadratic form parameters over integers and computes their Witt groups.

Proof of wall-crossing formula using spectral networks.

Abstract: Mapping 3-manifold bordisms to topological orders and domain walls.

This paper computes the quadratic Witt groups (the Wall L-groups) of the polynomial ring Z[t] and the integral group ring of the infinite dihedral group, with various involutions. We show that some of these groups are infinite direct sums of cyclic groups of order 2 and 4. The techniques used are quadratic linking form…

Let P be a closed smooth (4j-2)-connected 8j-manifold. We complete Wilkens' classification of the manifolds P for j = 1,2 and give an alternative proof to Wall's classification of the manifolds for j > 2. The Hopf-invariant-one dimensions (j=1,2) are characteristed by the fact that the quadratic linking functions which…

We investigate classification results for general quadratic functions on torsion abelian groups. Unlike the previously studied situations, general quadratic functions are allowed to be inhomogeneous or degenerate. We study the discriminant construction which assigns, to an integral lattice with a distinguished characte…

We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special…

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

Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex.

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…

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

The paper classifies biharmonic quadratic maps between spheres, proving their energy density properties.

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.

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…

A tubular group is a group that acts on a tree with $\mathbb{Z}^2$ vertex stabilizers and $\mathbb{Z}$ edge stabilizers. This paper develops further a criterion of Wise and determines when a tubular group acts freely on a finite dimensional CAT(0) cube complex. As a consequence we offer a unified explanation of the fai…

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

Convolutional networks predict turbulence from wall quantities.

A correspondence between different $Pin$-type structures on a compact surface and quadratic (linear) forms on its homology is constructed. Addition of structures is defined and expressed in terms of these quadratic forms.

The paper proposes a new way to approximate Riemannian metrics using discrete wall systems.

New method uses binary quadratic forms to classify Seifert surfaces in 4-ball.

Enhances power of covariance matrix tests for high-dimensional data.

Study on veering triangulations and their flow graphs, proving new applications.

Extends index theorem to domain walls with discontinuous Riemannian connections.

The paper characterizes when numerical criteria for PDE solvability fail and provides effective criteria for existence.

Study wall singularities in spaces with upper curvature bounds.

We introduce and study a canonical quadratic form, called the torsion quadratic form, of the determinant line of a flat vector bundle over a closed oriented odd-dimensional manifold. This quadratic form caries less information than the refined analytic torsion, introduced in our previous work, but is easier to construc…

Novel link classification connects quadratic forms and knot theory.

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

Study quadratic one-forms on logarithmic Higgs bundles on pointed curves.

We study the moduli space of SU(3) structure manifolds X that form the internal compact spaces in four-dimensional N=1/2 domain wall solutions of heterotic supergravity with flux. Together with the direction perpendicular to the four-dimensional domain wall, X forms a non-compact 7-manifold Y with torsionful G2 structu…

Neural network predicts turbulence from wall shear stress.

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

Geometric interpretation of 2d-4d wall-crossing formulas.

New quadratic forms expand and rotate linear endomorphisms in geometric theory.

Modeling aortic wall inhomogeneities to predict dissection risks.

Study shows K-moduli spaces of curves on quadrics and K3 surfaces match with VGIT quotients.

Study centers of mapping-torus groups to define knot and mapping class invariants.

When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on $R^3 \times S^1$ are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these…

Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.

Hurwitz transformations are defined as specific automorphisms of a Cayley-Dickson algebra. These transformations generate quadratic and nonquadratic forms. We investigate here the Hurwitz transformations corresponding to Cayley-Dickson algebras of dimensions 2m = 2, 4 and 8. The Hurwitz transformations which lead to qu…

Reformulates mod-two APS index using domain-wall fermion.

We study commensurating actions of groups and the associated properties FW and PW, in connection with wallings, median graphs, CAT(0) cubings and multi-ended Schreier graphs.