arXiv subjects

Riccardo Benedetti

Publications and source records attributed to Riccardo Benedetti.

At least 19 recordsLinked to original sources

arXiv2021

Assimilating X- and S-band Radar Data for a Heavy Precipitation Event in Italy

During the night between 9 and 10 September 2017, multiple flash floods associated to a heavy-precipitation event affected the town of Livorno, located in Tuscany, Italy. Accumulated precipitation exceeding 200 mm in two hours, associated with a return period higher than 200 years, caused all the largest streams of the Livorno municipality to flood several areas of the town. We used the limited-area Weather Research and Forecasting (WRF) model, in a convection-permitting setup, to reconstruct the extreme event leading to the flash floods. We evaluated possible forecasting improvements emerging from the assimilation of local ground stations and X- and S-band radar data into the WRF, using the configuration operational at the meteorological center of Tuscany region (LaMMA) at the time of the event. Simulations were verified against weather station observations, through an innovative method aimed at disentangling the positioning and intensity errors of precipitation forecasts. By providing more accurate descriptions of the low-level flow and a better assessment of the atmospheric water vapour, the results demonstrate that assimilating radar data improved the quantitative precipitation forecasts.

physics.ao-ph↗

arXiv2019

Lectures on Differential Topology

This text arises from teaching advanced undergraduate courses in differential topology for the master curriculum in Mathematics at the University of Pisa. So it is mainly addressed to motivated and collaborative master undergraduate students, having nevertheless a limited mathematical background. Overall this text is a collection of themes, in some cases advanced and of historical importance, with the common feature that they can be treated with "bare hands". This means by just combining certain cut-and-paste procedures and applications of transversality, mainly through the multiplicative structure of the cobordism rings of every smooth compact manifold; topics developed from scratch along with the text.

math.GT↗

arXiv2019

Ideal triangulations of 3-manifolds up to decorated transit equivalences

We consider 3-dimensional pseudo-manifolds M with a given set of marked point V such that M-V is the interior of a compact 3-manifold with boundary. An ideal triangulation T of (M, V ) has V as its set of vertices. A branching (T, b) enhances T to a Delta-complex. Branched triangulations of (M, V ) are considered up to the b-transit equivalence generated by isotopy and ideal branched moves which keep V pointwise fixed. We extend a well known connectivity result for naked triangulations by showing that branched ideal triangulations of (M, V) are equivalent to each other. A pre-branching is a system of transverse orientations at the 2-facets of T verifying a certain global constraint; pre-branchings are considered up to a natural pb-transit equivalence. If M is oriented, every branching b induces a pre-branching w(b) and every b-transit induces a pb-transit. The quotient set of pre-branchings up to transit equivalence is far to be trivial; we get some information about it and we characterize the pre-branchings of type w(b). Pre-branched and branched moves are naturally organized in subfamilies which give rise to restricted transit equivalences. In the branching setting we revisit early results about the sliding transit equivalence and outline a conceptually different approach to the branched connectivity and eventually also to the naked one. The basic idea is to point out some structures of differential topological nature which are carried by every branched ideal triangulation, are preserved by the sliding transits and can be modified by the whole branched transits. The non ambiguous transit equivalence already widely studied on pre-branchings lifts to a specialization of the sliding equivalence on branchings; we point out a few specific insights, again in terms of carried structures preserved by the non ambiguous and which can be modified by the whole sliding transits.

math.GT↗

arXiv2019

On ideal triangulations of surfaces up to branched transit equivalences

We consider triangulations of closed surfaces S with a given set of vertices V; every triangulation can be branched that is enhanced to a Delta-complex. Branched triangulations are considered up to the b-transit equivalence generated by b-flips (i.e. branched diagonal exchanges) and isotopy keeping V point-wise fixed. We extend a well known connectivity result for `naked' triangulations; in particular in the generic case when S has negative Euler-Poincare' characteristic c(S), we show that branched triangulations are equivalent to each other if c(S) is even, while this holds also for odd c(S) possibly after the complete inversion of one of the two branchings. Moreover we show that under a mild assumption, two branchings on a same triangulation are connected via a sequence of inversions of ambiguous edges (and possibly the total inversion of one of them). A natural organization of the b-flips in subfamilies gives rise to restricted transit equivalences with non trivial (even infinite) quotient sets. We analyze them in terms of certain preserved structures of differential topological nature carried by any branched triangulations; in particular a pair of transverse foliations with determined singular sets contained in V, including as particular cases the configuration of the vertical and horizontal foliations of the square of an Abelian differential on a Riemann surface.

math.GT↗

arXiv2018

Framing 3-manifolds with bare hands

After surveying existing proofs that every closed, orientable 3-manifold is parallelizable, we give three proofs using minimal background. In particular, our proofs use neither spin structures nor the theory of Stiefel-Whitney classes.

math.GT↗

arXiv2016

Non ambiguous structures on 3-manifolds and quantum symmetry defects

The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic oriented cusped $3$-manifolds can be split in a "symmetrization" factor and a "reduced" state sum. We show that these factors are invariants on their own, that we call "symmetry defects" and "reduced QHI", provided the manifolds are endowed with an additional "non ambiguous structure", a new type of combinatorial structure that we introduce in this paper. A suitably normalized version of the symmetry defects applies to compact $3$-manifolds endowed with $PSL_2(\mathbb{C})$-characters, beyond the case of cusped manifolds. Given a manifold $M$ with non empty boundary, we provide a partial "holographic" description of the non-ambiguous structures in terms of the intrinsic geometric topology of $\partial M$. Special instances of non ambiguous structures can be defined by means of taut triangulations, and the symmetry defects have a particularly nice behaviour on such "taut structures". Natural examples of taut structures are carried by any mapping torus with punctured fibre of negative Euler characteristic, or by sutured manifold hierarchies. For a cusped hyperbolic $3$-manifold $M$ which fibres over $S^1$, we address the question of determining whether the fibrations over a same fibered face of the Thurston ball define the same taut structure. We describe a few examples in detail. In particular, they show that the symmetry defects or the reduced QHI can distinguish taut structures associated to different fibrations of $M$. To support the guess that all this is an instance of a general behaviour of state sum invariants of 3-manifolds based on some theory of 6j-symbols, finally we describe similar results about reduced Turaev-Viro invariants.

math.GT↗

arXiv2015

Analytic families of quantum hyperbolic invariants

We organize the quantum hyperbolic invariants (QHI) of $3$-manifolds into sequences of rational functions indexed by the odd integers $N\geq 3$ and defined on moduli spaces of geometric structures refining the character varieties. In the case of one-cusped hyperbolic $3$-manifolds $M$ we generalize the QHI and get rational functions $\mathcal{H}_N^{h_f,h_c,k_c}$ depending on a finite set of cohomological data $(h_f,h_c,k_c)$ called {\it weights}. These functions are regular on a determined Abelian covering of degree $N^2$ of a Zariski open subset, canonically associated to $M$, of the geometric component of the variety of augmented $PSL(2,\mathbb{C})$-characters of $M$. New combinatorial ingredients are a weak version of branchings which exists on every triangulation, and state sums over weakly branched triangulations, including a sign correction which eventually fixes the sign ambiguity of the QHI. We describe in detail the invariants of three cusped manifolds, and present the results of numerical computations showing that the functions $\mathcal{H}_N^{h_f,h_c,k_c}$ depend on the weights as $N\rightarrow + \infty$, and recover the volume for some specific choices of the weights.

math.GT↗

arXiv2013

Spin structures on 3-manifolds via arbitrary triangulations

Let M be an oriented compact 3-manifold and let T be a (loose) triangulation of M, with ideal vertices at the components of the boundary of M and possibly internal vertices. We show that any spin structure s on M can be encoded by extra combinatorial structures on T. We then analyze how to change these extra structures on T, and T itself, without changing s, thereby getting a combinatorial realization, in the usual "objects/moves" sense, of the set of all pairs (M,s). Our moves have a local nature, except one, that has a global flavour but is explicitly described anyway. We also provide an alternative approach where the global move is replaced by simultaneous local ones.

math.GT↗

arXiv2011

Alexander quandle lower bounds for link genera

We denote by Q_F the family of the Alexander quandle structures supported by finite fields. For every k-component oriented link L, every partition P of L into h:=|P| sublinks, and every labelling z of such a partition by the natural numbers z_1,...,z_n, the number of X-colorings of any diagram of (L,z) is a well-defined invariant of (L,P), of the form q^(a_X(L,P,z)+1) for some natural number a_X(L,P,z). Letting X and z vary in Q_F and among the labellings of P, we define a derived invariant A_Q(L,P)=sup a_X(L,P,z). If P_M is such that |P_M|=k, we show that A_Q(L,P_M) is a lower bound for t(L), where t(L) is the tunnel number of L. If P is a "boundary partition" of L and g(L,P) denotes the infimum among the sums of the genera of a system of disjoint Seifert surfaces for the L_j's, then we show that A_Q(L,P) is at most 2g(L,P)+2k-|P|-1. We set A_Q(L):=A_Q(L,P_m), where |P_m|=1. By elaborating on a suitable version of a result by Inoue, we show that when L=K is a knot then A_Q(K) is bounded above by A(K), where A(K) is the breadth of the Alexander polynomial of K. However, for every g we exhibit examples of genus-g knots having the same Alexander polynomial but different quandle invariants A_Q. Moreover, in such examples A_Q provides sharp lower bounds for the genera of the knots. On the other hand, A_Q(L) can give better lower bounds on the genus than A(L), when L has at least two components. We show that in order to compute A_Q(L) it is enough to consider only colorings with respect to the constant labelling z=1. In the case when L=K is a knot, if either A_Q(K)=A(K) or A_Q(K) provides a sharp lower bound for the knot genus, or if A_Q(K)=1, then A_Q(K) can be realized by means of the proper subfamily of quandles X=(F_p,*), where p varies among the odd prime numbers.

math.GT↗

arXiv2011

Levels of knotting of spatial handlebodies

Given a (genus 2) cube-with-holes M, i.e. the complement in S^3 of a handlebody H, we relate intrinsic properties of M (like its cut number) with extrinsic features depending on the way the handlebody H is knotted in S^3. Starting from a first level of knotting that requires the non-existence of a planar spine for H, we define several instances of knotting of H in terms of the non-existence of spines with special properties. Some of these instances are implied by an intrinsic counterpart in terms of the non-existence of special cut-systems for M. We study a natural partial order on these instances of knotting, as well as its intrinsic counterpart, and the relations between them. To this aim, we recognize a few effective "obstructions" based on recent quandle-coloring invariants for spatial handlebodies, on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links, on the analysis of appropriate coverings of M, and on the very classical use of Alexander elementary ideals of the fundamental group of M. Our treatment of the matter also allows us to revisit a few old-fashioned beautiful themes of 3-dimensional geometric topology.

math.GT↗

arXiv2011

The Kashaev and quantum hyperbolic link invariants

We show that the link invariants derived from 3-dimensional quantum hyperbolic geometry can be defined by means of planar state sums based on link diagrams and a new family of enhanced Yang-Baxteroperators (YBO) that we compute explicitly. By a local comparison of the respective YBO's we show that these invariants coincide with the Kashaev specializations of the colored Jones polynomials. As a further application we disprove a conjecture about the semi-classical limits of quantum hyperbolic partition functions, by showing that it conflicts with the existence of hyperbolic links that verify the volume conjecture.

math.GT↗

arXiv2010

The topology of Helmholtz domains

The goal of this paper is to describe and clarify as much as possible the 3-dimensional topology underlying the Helmholtz cuts method, which occurs in a wide theoretic and applied literature about Electromagnetism, Fluid dynamics and Elasticity on domains of the ordinary space. We consider two classes of bounded domains that satisfy mild boundary conditions and that become "simple" after a finite number of disjoint cuts along properly embedded surfaces. For the first class (Helmholtz), "simple" means that every curl-free smooth vector field admits a potential. For the second (weakly-Helmholtz), we only require that a potential exists for the restriction of every curl-free smooth vector field defined on the whole initial domain. By means of classical and rather elementary facts of 3-dimensional geometric and algebraic topology, we give an exhaustive description of Helmholtz domains, realizing that their topology is forced to be quite elementary (in particular, Helmholtz domains with connected boundary are just possibly knotted handlebodies, and the complement of any non-trivial link is not Helmholtz). The discussion about weakly-Helmholtz domains is a bit more advanced, and their classification appears to be a quite difficult issue. Nevertheless, we provide several interesting characterizations of them and, in particular, we point out that the class of links with weakly-Helmholtz complements eventually coincides with the class of the so-called homology boundary links, that have been widely studied in Knot Theory.

math.GT↗

arXiv2007

Notes on a paper of Mess

These notes are a companion to the article "Lorentz spacetimes of constant curvature" by Geoffrey Mess, which was first written in 1990 but never published. Mess' paper will appear together with these notes in a forthcoming issue of Geometriae Dedicata.

math.DG↗

arXiv2007

(2+1)-Einstein spacetimes of finite type

The aim of this survey is to give an overview on the geometry of Einstein maximal globally hyperbolic 2+1 spacetimes of arbitrary curvature, conatining a complete Cauchy surface of finite type. In particular a specialization to the finite type case of the canonicla Wick rotation-rescaling theory, previously developed by the authors, is provided. This includes, for arbitrary curvatures, parameterizations in terms of suitable measured geodesic laminations on open hyperbolic surfaces of finite type. The same geometric objects also parameterize complex projective structures on the surfaces. The coincidence of such parameter space is explained by means of geometric correlations between spacetimes of different curvatures and projective surfaces realized via canonical WR-rescaling along the cosmological times. We also specialize on AdS case mostly referring to recent results achieved by other authors. In particular we describe maximal causal extensions of AdS globally hyperbolic spacetimes and an AdS approach to the theory of earthquakes for hyperbolic surfaces of finite type. A general earthquake theorem is proved for the so called enhanced Teichmuller space. The case of spacetimes with conical timelike singularities is also treated.

math.DG↗

arXiv2006

Quantum hyperbolic geometry

We construct a new family, indexed by the odd integers $N\geq 1$, of $(2+1)$-dimensional quantum field theories called {\it quantum hyperbolic field theories} (QHFT), and we study its main structural properties. The QHFT are defined for (marked) $(2+1)$-bordisms supported by compact oriented 3-manifolds $Y$ with a properly embedded framed tangle $L_\Ff$ and an {\it arbitrary} $PSL(2,\C)$-character $ρ$ of $Y \setminus L_\Ff$ (covering, for example, the case of hyperbolic cone manifolds). The marking of QHFT bordisms includes a specific set of parameters for the space of pleated hyperbolic structures on punctured surfaces. Each QHFT associates in a constructive way to any triple $(Y,L_\Ff,ρ)$ with marked boundary components a tensor built on the matrix dilogarithms, which is holomorphic in the boundary parameters. We establish {\it surgery formulas} for QHFT partitions functions and describe their relations with the {\it quantum hyperbolic invariants} of \cite{BB1,BB2} (either defined for unframed links in closed manifolds and characters trivial at the link meridians, or hyperbolic {\it cusped} 3-manifolds). For every $PSL(2,\mc)$-character of a punctured surface, we produce new families of conjugacy classes of "moderately projective" representations of the mapping class groups.

math.GT↗

arXiv2006

Canonical Wick rotations in 3-dimensional gravity

We develop a ``canonical Wick rotation-rescaling theory in 3-dimensional gravity''. This includes: (a) A simultaneous classification that shows how generic maximal globally hyperbolic spacetimes of constant curvature, which admit a complete Cauchy surface (in particular a compact one), as well as complex projective structures on arbitrary surfaces, are all encoded by pairs (H,L), H being a ``straight convex sets'' in the hyperbolic plane, and L a ``measured geodesic laminations'' suitably defined on H. (b) Canonical geometric correlations: spacetimes of different curvature, that share a same encoding pair (H,L), are related to each other by ``canonical rescaling''; they can be transformed by ``canonical Wick rotations'' in hyperbolic 3-manifolds, that carry asymptotically the corresponding projective structures. Both Wick rotations and rescalings act along the "canonical cosmological time" and have ``universal rescaling functions''. These correlations are functorial with respect to isomorphisms of the respective geometric categories. We analyze the behaviour along a ray of measured laminations, (broken) T-symmetry by spacetimes of negative curvature, the relationship with ``earthquake theory'', beyond the case of compact Cauchy surface. WR-rescaling does apply on the ``ends'' of geometrically finite hyperbolic 3-manifolds, that hence realize concrete interactions of their globally hyperbolic ``ending spacetimes'' of constant curvature. It also provides further "classical amplitudes" of these interactions, beyond the volume of the hyperbolic convex cores.

math.DG↗

arXiv2005

Classical and quantum dilogarithmic invariants of flat PSL(2,C)-bundles over 3-manifolds

We introduce a family of matrix dilogarithms, which are automorphisms of C^N tensor C^N, N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy fundamental five-term identities that correspond to decorated versions of the 2 --> 3 move on 3-dimensional triangulations. Together with the decoration, they arise from the solution we give of a symmetrization problem for a specific family of basic matrix dilogarithms, the classical (N=1) one being the Rogers dilogarithm, which only satisfy one special instance of five-term identity. We use the matrix dilogarithms to construct invariant state sums for closed oriented 3-manifolds $W$ endowed with a flat principal PSL(2,C)-bundle rho, and a fixed non empty link L if N>1, and for (possibly "marked") cusped hyperbolic 3-manifolds M. When N=1 the state sums recover known simplicial formulas for the volume and the Chern-Simons invariant. When N>$, the invariants for M are new; those for triples (W,L,rho) coincide with the quantum hyperbolic invariants defined in [Topology 43 (2004) 1373-1423], though our present approach clarifies substantially their nature. We analyse the structural coincidences versus discrepancies between the cases N=1 and N>1, and we formulate "Volume Conjectures", having geometric motivations, about the asymptotic behaviour of the invariants when N tends to infinity.

math.GT↗

arXiv2004

Wick rotations in 3D gravity: ML(H2)-spacetimes

"Ends of hyperbolic 3-manifolds should support canonical Wick Rotations, so they realize effective interactions of their ending globally hyperbolic spacetimes of constant curvature." We develop a consistent sector of WR-rescaling theory in 3D gravity, that, in particular, concretizes the above guess for many geometrically finite manifolds. ML(H2)-spacetimes are solutions of pure Lorentzian 3D gravity encoded by measured geodesic laminations of the hyperbolic plane H2, possibly invariant by any given torsion-free discrete isometry group G. The rescalings which correlate spacetimes of different curvature, as well as the conformal Wick rotations towards hyperbolic structures, are directed by the gradient of the respective canonical cosmological times, and have universal rescaling functions that only depend on their value. We get an insight into the WR-rescaling mechanism by studying rays of ML(H2)-spacetimes emanating from the static case. In particular, we determine the "derivatives" at the starting point of each ray. We point out the tamest behaviour of the cocompact G case against the different general one, even when G is of cofinite area, but non-compact. We analyze brocken T-symmetry of AdS ML(H2)-spacetimes and related earthquake failure. This helps us to figure out the main lines of development in order to achieve a complete WR rescaling theory.

math.DG↗