arXiv research

Experimental support for Neumann's conjecture on Bloch group.

Study on Neumann eigenvalues controlled by domain isoperimetric ratio.

Hot spots conjecture proven for small eigenvalue domains.

We prove a generalized version of the Strong Atiyah Conjecture for the infinite dihedral group W, replacing the group von Neumann algebra NW with the Hecke-von Neumann algebra N_qW.

For a given bounded domain $Ω\subset {\Bbb R}^n$ with $C^1$-smooth boundary, we prove the Pólya conjecture for the Neumann eigenvalues. In other words, we prove that \begin{eqnarray*} μ_{k+1}\le \frac{(2π)^2k^{2/n}}{(ω_n \cdot \mbox{vol}\, (Ω))^{2/n}} \quad \;\; \mbox{for all} \;\; k=0,1,2,3,\cdots,\end{eqnarray*} wher…

Researchers prove hot spots conjecture for Gaussian spaces.

In the article we prove the Casson Invariant Conjecture of Neumann--Wahl for splice type surface singularities. Namely, for such an isolated complete intersection, whose link is an integral homology sphere, we show that the Casson invariant of the link is one-eighth the signature of the Milnor fiber.

Local corner-factor conjecture for Neumann jump determinants supported by models.

The Hanna Neumann conjecture gives a bound on the intersection of finitely generated subgroups of free groups. We explore a natural extension of this result, which turns out to be true only in the finite index case, and provide counterexamples for the general case. We also see that the graph-based method of generating …

Proven isoperimetric inequality for Witten-Laplacian eigenvalues.

In this paper, we prove the existence of a classical solution to a Neumann boundary problem for Hessian equations in uniformly convex domain. The methods depend upon the established of a priori derivative estimates up to second order. So we give a affirmative answer to a conjecture of N. Trudinger in 1986.

The study proves constant-curvature analogues of hot spots conjecture for triangles.

The Milnor fiber conjecture is proven for splice type singularities.

Proves Pólya's conjecture for thin products and Riemannian manifolds.

The abstract proves a conjecture about geometric structures in Calabi-Yau orbifolds.

Let $Ω$ be an open, bounded domain in the plane with connected and smooth boundary, and $ω$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $μ> 0$. If the boundary value of $ω$ is a nonzero constant along the boundary, denoting $0 = μ_1(Ω) < μ_2(Ω) <= ...$ the set of all Neumann eigen…

Twisted Neumann--Zagier matrices for quantum invariants.

Extends Onsager's conjecture to Besov spaces on manifolds with boundary.

In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the $3$-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid con…

The paper proves an area inequality for metric balls in Riemannian manifolds.

The paper proves a Neumann eigenvalue sum inequality in non-Euclidean space forms.

Compactness proven for manifolds with nonnegative Ricci curvature and uniformly convex boundary.

We study quantum invariant Z(M) for cusped hyperbolic 3-manifold M. We construct this invariant based on oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in studies of the modular double of the quantum group. Following Thurston and Neuma…

Sharp inequality for Neumann eigenvalues on space forms.

Study shows knots with coprime polynomials can't be concordant.

Aitchison and Rubinstein constructed two knot complements that can be decomposed into two regular ideal dodecahedra. This paper shows that these knot complements are the only knot complements that decompose into n regular ideal dodecahedra, providing a partial solution to a conjecture of Neumann and Reid.

New mathematical invariants derived from polytopes of matrices over rings.

Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra. We prove…

New insights into mirror symmetry via Monge-Ampère domains and pre-Frobenius manifolds.

Researchers solve a complex problem about determining Riemannian manifolds.

This paper is devoted to a proof of a generalized Ray-Singer conjecture for a manifold with boundary (the Dirichlet and the Neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary). The Ray-Singer conjectur…

We establish the existence of new rigidity and rationality phenomena in the theory of nonabelian group actions on the circle, and introduce tools to translate questions about the existence of actions with prescribed dynamics into finite combinatorics. A special case of our theory gives a very short new proof of Naimi's…

The study restricts embedding of Seifert fibered spaces in $S^4$.

For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parametrization of the set of conjugacy classes of boundary-unipotent representations of the fundamental group of M into SL(n,C). Our parametrization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmueller spaces d…

The study sets lower bounds for eigenvalue sums of Laplacian on bounded domains and spheres.

New subdomains found on cylinder and sphere with nonconstant curvatures.

The paper solves the problem of fitting an ellipsoid to random points efficiently.

Neumann and Reid described in their paper "Rigidity of cusps in deformations of hyperbolic 3-orbifolds" (Math Ann. 295 (1993) no. 2, 223--237) a 2-cusped hyperbolic 3-orbifold in which the cusps are geometrically isolated. Based on numerical evidence provided by Jeff Weeks' snappea program, they conjectured that the cu…

New upper bound for Neumann Laplacian eigenvalues on convex domains.

Proves existence of classical Neumann problems for Hessian equations in uniformly convex domains.

For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-Müller the…

The paper characterizes Pólya's conjecture for spheres and hemispheres, deriving inequalities and bounds.

Sharp lower bound for first Neumann eigenvalue found in terms of diameter and width.

Direct proof of Neumann isoperimetric inequality for convex domains.

Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.

Study proves volume conjecture for specific 3-manifolds.

Proves long-term behavior of curvature flows with Neumann boundary.

The Quantum Modularity Conjecture of Zagier predicts the existence of a formal power series with arithmetically interesting coefficients that appears in the asymptotics of the Kashaev invariant at each root of unity. Our goal is to construct a power series from a Neumann-Zagier datum (i.e., an ideal triangulation of th…