arXiv research

We examine groups whose resonance varieties, characteristic varieties and Sigma-invariants have a natural arithmetic group symmetry, and we explore implications on various finiteness properties of subgroups. We compute resonance varieties, characteristic varieties and Alexander polynomials of Torelli groups, and we sho…

Study on pure virtual braids, their formality, and related ranks.

Study Alexander invariants and cohomology jump loci in group extensions with trivial monodromy.

Survey on hyperplane arrangements and their Milnor fibration topology.

The paper explores various braid-like groups and their properties.

Generalizes cohomology ring result for combinatorial line arrangements.

Advances in Koszul modules and syzygies of algebraic varieties.

Let $\mathcal{A}$ be a line arrangement in the complex projective plane $\mathbb{P}^2$, having the points of multiplicity $\geq 3$ situated on two lines in $\mathcal{A}$, say $H_0$ and $H_{\infty}$. Then we show that the non-local irreducible components of the first resonance variety $\mathcal{R}_1(\mathcal{A})$ are 2-…

Proves effective Chen ranks conjecture for Koszul modules.

Study cohomology jump loci in 3-manifolds, focusing on Alexander polynomial.

Study shows classical and quantum resonances match for hyperbolic surfaces.

New method combines simulations and data for anomaly detection.

We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. The jump loci of a space X come in two basic flavors: th…

Holomorphic vector bundles on Hopf manifolds admit flat connections.

New resonance theory for Anosov flows connects spectral properties to mixing measures.

We study the topology of the boundary manifold of a line arrangement in CP^2, with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial Delta(G), and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arbitrary complex representa…

We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by countting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspe…

This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative aspects. The unifying framework for this study is the stratification of the character …

Inverse problem solved for rotationally symmetric manifolds using eigenvalues and resonances.

We give a new algorithm computing local system cohomology groups for complexified real line arrangements. Using it, we obtain several conditions for the first local system cohomology to vanish and to be at most one-dimensional, which generalize a result by Cohen-Dimca-Orlik. The conditions are described in terms of dis…

Resonator Networks solve high-dimensional vector factorization better than optimization methods.

For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H^{<=2}(X), to the second nilpotent quotient, G/G_3. We define invariants of G/G_3 by counting normal subgroups of a fixed prime index p, according to their abelianization. We show how to compute this distribution fro…

Deep learning improves automated detection of epileptic seizures.

Study reveals a link between Ruelle-Pollicott resonances and cohomology eigenvalues for Anosov diffeomorphisms.

Proves new fixed point formulae for complex manifolds with boundary.

The study finds resonance points in polarised curves with polynomial conserved quantities.

Quantum resonances for tensors on hyperbolic spaces are studied.

We examine the Johnson filtration of the (outer) automorphism group of a finitely generated group. In the case of a free group, we find a surprising result: the first Betti number of the second subgroup in the Johnson filtration is finite. Moreover, the corresponding Alexander invariant is a non-trivial module over the…

Study of spectral properties of Lorentzian quasi-Fuchsian manifolds.

The resonant band is a useful notion for the computation of the nontrivial monodromy eigenspaces of the Milnor fiber of a real line arrangement. In this article, we develop the resonant band description for the cohomology of the Aomoto complex. As an application, we prove that real 4-nets do not exist.

We show that the resolvent of the Laplacian on SL(3,$\mathbb{R}$)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of $\mathbb{C}$. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue op…

Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.

We find a resonance free region polynomially close to the critical line on Conformally compact manifolds with polyhomogeneous metric.

For a conformally compact manifold that is hyperbolic near infinity and of dimension $n+1$, we complete the proof of the optimal $O(r^{n+1})$ upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we es…

SGDm with fixed step-size diverges under covariate shift, similar to a parametric oscillator.

Procedure maps quantum systems to curved spacetimes with resonant frequencies.

Study resonant forms for dissipative Anosov flows on 3-manifolds.

Proposes a method to improve skull stripping accuracy in MRI images.

The paper proves at least two non-contractible closed geodesics on Finsler RP^n.

The isoresidual fibration maps Riemann sphere strata to resonance arrangements.

Study geometric structures on LVM threefolds, focusing on resonant structures.

Study of resonances and residue operators for hyperbolic spaces.

We study the spectral theory of asymptotically hyperbolic manifolds with ends of warped product type. Our main result is an upper bound on the resonance counting function with a geometric constant expressed in terms of the respective Weyl constants for the core of the manifold and the base manifold defining the ends.

We prove the meromorphic extension to C for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the geometrically finite quotients with rational non-maximal rank cusps previous…

Resonant machine learning uses electrical network dynamics to optimize learning efficiently.

As a consequence of a result of Cardoso and Vodev, we show that the resolvent of the Laplacian on asymptotically hyperbolic manifolds is analytic in an exponential neighbourhood of the critical line. The case of non-trapping metrics with constant curvature near infinity is also considered: there exists a strip with at …

On an asymptotically hyperbolic manifold (X,g), we show that the resolvent resonances coincide, with multiplicities, with the poles of the renormalized scattering operator, except for the special points n/2-k (with k>0 integer) where an additional term appears: this is the dimension of the kernel of the k-conformal Lap…

We consider actions of Z^k, k \ge 2, by Anosov diffeomorphisms which are uniformly quasiconformal on each coarse Lyapunov distribution. These actions generalize Cartan actions for which coarse Lyapunov distributions are one-dimensional. We show that, under certain non-resonance assumptions on the Lyapunov exponents, a …