arXiv research

We will show a theorem of a type of Cheeger and Muller for a noncompact complete hyperbolic threefold of finite vulume. As an application we will compute a special value of Ruelle L-function at the origin for a unitary local system which is cuspidal.

New $L$-functions for 3-manifolds connect to Witten invariants and relate to generalized Bernoulli polynomials.

The study introduces pseudo-arithmeticity for certain lattices in hyperbolic spaces.

For a unitary local system of rank one on a complete hyperbolic threefold of finite volume which has only one cusp, the Ruelle L-function is defined. We will show that if the first cohomology group of the local system vanishes its value at s=0 is equal to the square of the Franz-Reidemeister torsion.

We study the twisted knot module for the universal deformation of an ${\rm SL}_2$-representation of a knot group, and introduce an associated $L$-function, which may be seen as an analogue of the algebraic $p$-adic $L$-function associated to the Selmer module for the universal deformation of a Galois representation. We…

Study multiplicity of non-acyclic SL2-representations and L-functions of Whitehead links.

Study on quadratic L-functions using hyperelliptic curves and homology.

In this paper we present a major application of the l-function and the reduced volume of Perelman, namely their application to the analysis of the asymptotical limits of kappa solutions of the Ricci flow.

The main purpose of this paper is to present a number of analytic and geometric properties of the $l$-function and the reduced volume of Perelman, including in particular the monotonicity, the upper bound and the rigidities of the reduced volume.

Study non-acyclic SL2-representations of twist knots and their L-functions.

We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rati…

A simple equation explains standard model coupled to gravity.

We generalize work of Deligne and Gillet-Soulé on a Riemann-Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces $Γ\backslash\mathbb{H}$, for $Γ\subset PSL_{2}(\mathbb{R})$ a fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cus…

For a unitary local system of rank one on a complete hyperbolic threefold of finite volume which has only one cusp, we will compare the order of the Alexander invariant at t=1 and one of Ruelle-Selberg L-function at s=0. Our result may be considered as a geometric analog of the Iwasawa main conjecture in the algebraic …

For a local system on a compact hyperbolic threefold, under a cohomological assumption, we will show that the order of its twisted Alexander polynomial and of the Ruelle L function at $s=0$ coincide. Moreover we will show that their leading constant are also identical. These results may be considered as a solution of a…

We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that the expectation of their normalized L-distance is non-increasing. As an immediate corollary we obtain a new proof of a recent result of Topping (J. reine angew. Math. 636 (2009), …

Machine learning accurately distinguishes Sato-Tate groups for hyperelliptic curves.

Study geodesics on random hyperbolic surfaces, finding variance similar to prime number theory.

Researchers prove a spectral gap for Hecke covers of Schottky surfaces.

The paper counts primes in complex orthospectra and polynomial orbits.

Continues study on special Lagrangian graphs and flow solutions.

Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.

Machine learning predicts Shafarevich-Tate group orders of elliptic curves.

Study proves existence of solutions for a specific type of parabolic equations.

We show that any global solution to the special Lagrangian equations with the phase larger than a critical value must be quadratic.

The paper proves a Liouville theorem for special Lagrangian equations with convexity conditions.

We compute the eta function $η(s)$ and its corresponding $η$-invariant for the Atiyah-Patodi-Singer operator $\mathcal{D}$ acting on an orientable compact flat manifold of dimension $n =4h-1$, $h\ge 1$, and holonomy group $F\simeq \mathbb{Z}_{2^r}$, $r\in \mathbb{N}$. We show that $η(s)$ is a simple entire function tim…

We prove sharp inequalities for determinants of Toeplitz operators and twisted Laplace operators on the two-sphere, generalizing the Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures of Gillet-Soule and Fang motivated by Arakelov geometry is obtained; applications to SU(2)-invariant determ…

We derive a priori interior Hessian and gradient estimates for special Lagrangian equation of phase at least a critical value in dimension three.

We characterize the global maximizers of a certain non-local functional defined on the space of all positively curved metrics on an ample line bundle L over a Kahler manifold X. This functional is an adjoint version, introduced by Berndtsson, of Donaldson's L-functional and generalizes the Ding-Tian functional whose cr…

Study $G_2$-metrics from non-integrable special Lagrangian fibrations.

A special p-form is a p-form which, in some orthonormal basis {e_μ}, has components φ_{μ_1...μ_p} = φ(e_{μ_1},..., e_{μ_p}) taking values in {-1,0,1}. We discuss graphs which characterise such forms.

Paper doubles Hessian estimates for special Lagrangian equation with constraints.

We define geometric zeta functions for locally symmetric spaces as generalizations of the zeta functions of Ruelle and Selberg. As a special value at zero we obtain the Reidemeister torsion of the manifold. For hermitian spaces these zeta functions have as special value the quotient of the holomorphic torsion of Ray an…

Proves Hessian estimates for special Lagrangian equation with new proofs.

We derive generalized estimators for a number of spatial statistics that have been used in the analysis of spatially resolved omics data, such as Ripley's K, H and L functions, clustering index, and degree of clustering, which allow these statistics to be calculated on data modelled by arbitrary random measures (RMs). …

Study sharpens unlinking number bounds for special alternating links.

We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non…

Paper constructs special solutions for symplectic Dirac operator.

The paper connects special cycle heights to Siegel Eisenstein series.

We solve a century-old conjecture about Alexander polynomials of special alternating links.

New surfaces found in 5D space.

Set-valued risk measures on $L^p_d$ with $0 \leq p \leq \infty$ for conical market models are defined, primal and dual representation results are given. The collection of initial endowments which allow to super-hedge a multivariate claim are shown to form the values of a set-valued sublinear (coherent) risk measure. Sc…

In the theory of so called "Covariant Quantum Mechanics" a basic role is played by Hermitian vector fields on a complex line bundle in the frameworks of Galilei and Einstein spacetimes. In fact, it has been proved that the Lie algebra of Hermitian vector fields is naturally isomorphic to a Lie algebra of "special funct…

The holomorphic torsion of a hermitian locally symmetric space is expressed as a special value of a geometric zeta function.

In this paper, a class of fully nonlinear flows with nonlinear Neumann type boundary condition is considered. This problem was solved partly by the first author under the assumption that the flow is the parabolic type special Lagrangian equation in $\mathbb{R}^{2n}$. We show that the convexity is preserved for solution…

Complex-valued (p,q)-harmonic morphisms defined and studied.

A special spine of a three-manifold is said to be poor if it does not contain proper simple subpolyhedra. Using the Turaev-Viro invariants, we establish that every compact three-dimensional manifold M with connected nonempty boundary has a finite number of poor special spines. Moreover, all poor special spines of the m…