arXiv research

The study finds infinite nodal solutions for equations on positive Ricci curvature manifolds.

The paper studies elliptic surfaces and proves unique fibered structures.

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus $2$ fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic …

The Seiberg-Witten family of elliptic curves defines a Jacobian rational elliptic surface $\Z$ over $\mathbb{C}\mathrm{P}^1$. We show that for the $\bar{\partial}$-operator along the fiber the logarithm of the regularized determinant $-1/2 \log \det' (\bar\partial^* \bar\partial)$ satisfies the anomaly equation of the …

We study the asymptotic properties of the conormal cycle of nodal sets associated to a random superposition of eigenfunctions of the Laplacian on a smooth compact Riemannian manifold without boundary. In the case where the dimension is odd, we show that the expectation of the corresponding current of integration equidi…

The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.

Paper finds infinite pairs of fiber-type curves with same topology but different embeddings.

The Weil-Petersson and Takhtajan-Zograf metrics on the Riemann moduli spaces of complex structures for an $n$-fold punctured oriented surface of genus $g,$ in the stable range $g+2n>2,$ are shown here to have complete asymptotic expansions in terms of Fenchel-Nielsen coordinates at the exceptional divisors of the Knuds…

Eigenfunction maxima inside high-d nodal domains.

New bounds found for nodal sets on special manifolds.

In 2009 Gaiotto, Moore and Neitzke presented a new construction of hyperkähler metrics on the total spaces of certain complex integrable systems, represented as a torus fibration $\mathcal{M}$ over a base space $\mathcal{B}$, except for a divisor $D$ in $\mathcal{B}$, in which the torus fiber degenerates into a nodal t…

New nodal domain theorems for symmetric matrices via signed graphs.

Study how nodal domains change on surfaces under perturbations.

We consider a Laplace eigenfunction $\varphi_λ$ on a smooth closed Riemannian manifold, that is, satisfying $-Δ\varphi_λ= λ\varphi_λ$. We introduce several observations about the geometry of its vanishing (nodal) set and corresponding nodal domains. First, we give asymptotic upper and lower bounds on the volume of a tu…

The paper studies eigenfunctions and nodal sets of the Witten-Laplacian.

We prove by Hilbert-Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves, and of Hasset on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solved a questi…

We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any nodal domain $Ω_λ$ almost fully contains a ball of radius $\sim \frac{1}{\sqrtλ}$. …

Abstract: Shows nonexistence of Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.

The paper studies nodal sets of solutions to parabolic equations, proving finiteness and monotonicity properties.

We use layer potential to establish that the boundary biharmonic Steklov operators are elliptic pseudo-differential operators. Thus we are able to establish lower bounds on both the measure of boundary nodal sets and interior nodal sets for biharmonic Steklov eigenfunctions.

New estimates for nodal and singular sets of parabolic inequalities.

We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami operator …

Given a Laplace eigenfunction on a surface, we study the distribution of its extrema on the nodal domains. It is classically known that the absolute value of the eigenfunction is asymptotically bounded by the 4-th root of the eigenvalue. It turns out that the number of nodal domains where the eigenfunction has an extre…

We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.

Study compares nodal sets of solutions to the Allen-Cahn equation.

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we stud…

We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.

Study on the nodal set of Dirac equation solutions on manifolds.

Sharp estimate for nodal domains intersecting a ball on a Riemannian manifold.

Study small perturbations on low energy Laplace eigenfunctions.

New harmonic functions show nodal sets can be topologically complex despite frequency and regularity constraints.

We prove existence results for nodal solutions of the Yamabe equation that are constant along the level sets of an isoparametric function.

We show that on a compact Riemmanian manifold $(M,g)$, nodal sets of linear combinations of any $p+1$ smooth functions form an admissible $p-$sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a resul…

We prove a natural inequality which implies the known lower bounds for the $(n-1)$-dimensional Hausdorff measure of nodal sets for smooth compact manifolds.

Recently, Sogge-Zelditch and Colding-Minicozzi gave new power law lower bounds on the size of the nodal sets of eigenfunctions. The purpose of this short note is to point out a third method to obtain a power law lower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growth bound for …

The family Blow Up formula is recalled. Certain combinatoric graphs are introduced for the discussion of the counting of nodal curves on an Kahler surface.

Study ALE spaces via nodal curves and compactifications.

Continuing the program of math.SG/0012067 and math.SG/0310450, we introduce refinements of the Donaldson-Smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components. In cases where a corresponding analogue of the Gromov-Tau…

This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic …

For generic torus-invariant metrics, eigenspaces are 2D and nodal sets are connected hypersurfaces.

Characterizes nodal volumes of Gaussian fields on manifolds, extending previous work.

Study bounds the measure of zero sets of Neumann Laplace eigenfunctions.

In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy (1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets.This paper is a …

We study the nodal sets of non-degenerate eigenfunctions of the Laplacian on fibre bundles $π{:}\, M\to B$ in the adiabatic limit. This limit consists in considering a family $G_\varepsilon$ of Riemannian metrics, that are close to Riemannian submersions, for which the ratio of the diameter of the fibres to that of the…

In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface Σin a compact n-manifold M, there is a Riemannian metric on M such that the nodal set of its first nontrivial eigenfunction is Σ. We present…

Researchers extend topological classification to knotted semimetals in 3D.

Generalizing Courant's nodal domain theorem, the "Extended Courant property" is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. In a previous paper (Documenta Mathematica, 2018, Vol. 23, pp. 1561--1585), we gave simple counterexamples to this property, including co…

The paper estimates the measure of nodal sets for solutions to a specific type of Schrödinger equation.