arXiv research

Eta invariant of (2,3,5) nilmanifolds vanishes but eta function is nontrivial.

On a spin manifold with conformal cusps, we prove under an invertibility condition at infinity that the eta function of the twisted Dirac operator has at most simple poles and is regular at the origin. For hyperbolic manifolds of finite volume, the eta function of the Dirac operator twisted by any homogeneous vector bu…

Let $\mathcal{A}_0$ and $\mathcal{A}_1$ be two self-adjoint Fredholm Dirac-type operators defined on two non-compact manifolds. If they coincide at infinity so that the relative heat operator is trace-class, one can define their relative eta function as in the compact case. The regular value of this function at the zer…

We study the eta invariants of Dirac operators and the regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner M"uller and use relative traces to define these spectral invariants. We show the regularity of eta and zeta functions at s=0. The Selberg trace formula and the detai…

Numerical discovery matches eta invariant on Berger spheres with conformal anomaly on round spheres.

The eta invariant appears regularly in index theorems but is known to be directly computable from the spectrum only in certain examples of locally symmetric spaces of compact type. In this work, we derive some general formulas useful for calculating the eta invariant on closed manifolds. Specifically, we study the eta …

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 extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $δ$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values $η(δ_t,0)$ and $tζ(δ_t,0)$ are smooth functions of …

Road Network Metric Learning improves ETA prediction accuracy by addressing data sparsity.

We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a uniq…

Let P be a selfadjoint elliptic operator of order m>0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form s=k/m, where k ranges over all non-zero integers less than or equa…

We give a useful classification of the metabelian unitary representations of pi_1(M_K), where M_K is the result of zero-surgery along a knot K in S^3. We show that certain eta invariants associated to metabelian representations pi_1(M_K) --> U(k) vanish for slice knots and that even more eta invariants vanish for ribbo…

The zeta and eta-functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are stressed, as the intrisic ambiguity present in the definition of the associated fermion …

Cochran, Orr and Teichner introduced $L^2$--eta--invariants to detect highly non--trivial examples of non slice knots. Using a recent theorem by Lück and Schick we show that their metabelian $L^2$--eta--invariants can be viewed as the limit of finite dimensional unitary representations. We recall a ribbon obstruction t…

Eta-Einstein and $(κ,μ)$-structures studied in dimension 3.

The integral of the top dimensional term of the multiplicative sequence of Pontryagin forms associated to an even formal power series is calculated for special Riemannian metrics on the unit ball of a hermitean vector space. Using this result we calculate the generating function of the reduced Dirac and signature eta-i…

Using adiabatic limits of Eta invariants, Rho invariants of the total space of a fiber bundle are investigated. One concern is to formulate the aspects of local index theory for families of Dirac operator in terms of the odd signature operator, and place known results in a context which permits the treatment of Rho inv…

FMA-ETA predicts travel time using FFN with attention.

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{Γ,Σ}^{\rm o}(λ)$ associated to the spinor bundle $Σ$ on odd dimensional convex co-compact hyperbolic manifolds $X:=Γ\backslash\hh^{2n+1}$. We define a natural eta invariant $η(D)$ associated to the Dirac operator $D$ on $X…

For an acyclic representation of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to a unitary representation, we define a refinement of the Ray-Singer torsion associated to this representation. This new invariant can be viewed as an analytic counterpart of the refined combina…

Accurate expected time of arrival (ETA) information is crucial in maintaining the quality of service of public transit. Recent advances in artificial intelligence (AI) has led to more effective models for ETA estimation that rely heavily on a large GPS datasets. More importantly, these are mainly cabs based datasets wh…

In this paper, we define the equivariant eta form of Bismut-Cheeger for a compact Lie group and establish a formula about the functoriality of equivariant eta forms with respect to the composition of two submersions.

Using H. Donnelly result from the article "Eta Invariants for G-Spaces" we calculate the eta invariants of the signature operator for almost all 7-dimensional flat manifolds with cyclic holonomy group. In all cases this eta invariants are an integer numbers. The article was motivated by D. D. Long and A. Reid article "…

Self-exciting Hawkes processes are used to model events which cluster in time and space, and have been widely studied in seismology under the name of the Epidemic Type Aftershock Sequence (ETAS) model. In the ETAS framework, the occurrence of the mainshock earthquakes in a geographical region is assumed to follow an in…

We extend the theory of the universal eta-invariant to the case of relative bordism groups of manifolds with boundaries. This allows the construction of secondary descendants of the universal eta-invariant. We obtain an interpretation of Laures' f-invariant as an example of this general construction. As an aside we imp…

Paper establishes a new formula for Atiyah-Patodi-Singer index using eta invariants.

We study the eta-invariants of links and show that in many cases they form link concordance invariants, in particular that many eta-invariants vanish for slice links. This result contains and generalizes previous invariants by Smolinsky and Cha--Ko. We give a formula for the eta-invariant for boundary links. In several…

New method calculates eta invariant without analytic continuation.

Paper combines multiple ETA models into a stacked ensemble for better ETA predictions.

Constructs weight 1/2 multiplier systems for a specific group and relates to geometric edge paths.

Paper generalizes spectral flow formulas for compact Lie group actions.

We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold. In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.

Eta invariant computed for circle bundles over Fano manifolds.

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group cyclic of odd prime order p. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p=n=3. We use the expressions ob…

Researchers compute Floer homotopy types and eta invariants for Seifert 3-manifolds.

Study eta invariant remainder on contact manifolds, improving previous results.

We calculate the eta-invariant for the odd signature operator relative to a specific submersion metric on the Milnor fibration of a quasihomogeneous hypersurface singularity using certain global boundary conditions in terms of the data of its fibre intersection form, monodromy and variation mapping resp. the monomial d…

Paper extends variational formula for Bismut-Cheeger eta form, proving key theorems in K-theory.

We give a survey on eta invariants including methods of computation and applications in differential topology.

We consider families of Dirac operators on the unit interval which depend on parameters via boundary conditions. We study the associated eta forms and Maslov cocyles. With this simple example we show how previous results of Lesch/Woiciechowski and the first author on the eta invariant of cylinders generalize to the fam…

We prove two geometric index theorems for a family of first-order elliptic operators over a manifold with boundary by computing eta form representatives for the Chern character classes of the index bundle. The eta forms occur as relative and regularized traces on infinite-dimensional vector bundles realized as the limi…

Study eta invariant on non-compact manifolds with positive scalar curvature.

We study eta-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem for Sasakian geometry to prove the existence of eta-Einstein structures on many different compact manifolds, including exotic spheres. We also relate…

Study of twisted Ruelle zeta function on hyperbolic manifolds and its relation to analytic torsion.

Let B_n be the Artin braid group on n strings with standard generators sigma_1, ..., sigma_{n-1}, and let SB_n be the singular braid monoid with generators sigma_1^{+-1}, ..., sigma_{n-1}^{+-1}, tau_1, ..., tau_{n-1}. The desingularization map is the multiplicative homomorphism eta: SB_n --> Z[B_n] defined by eta(sigma…

In the previous work ([14]) we introduced the well-posed boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$ and ${\mathcal P}_{+, {\mathcal L}_{1}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined…

Analytic surgery and gluing formula for torsion forms in fiber bundles.

In this note, we prove the regularity of eta forms by the Clifford asymptotics. Then we generalize this result to the equivariant case.