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Fil: Andrada, Adrian Marcelo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica; Argentina

Given a holomorphic family of pairs $\{(X_t,E_t)\}$, where each $E_t$ is holomorphic vector bundle over compact complex manifold $X_t$. For small enough $t$, we get a correspondence between the Dolbeault complex of $E_t$-valued $(p,q)$-forms on $X_t$ and the one of $E_0$-valued $(p,q)$-forms on $X_0$., Comment: 12 pages

A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system (GIFS), based on a combinatorial structure we call a pre-tree, is introduced. For each GIFS, a family of tilings is constructed indexed by a parameter. For what we call a commensurate GIFS, our method is used to define what we refer to as balanced tilings. Under mild conditions on the commensurate GIFS and the parameter, the resulting balanced tilings have a finite set of prototiles, are self-similar, and are quasiperiodic. A notion of rigidity is defined for a commensurate GIFS, and a necessary and sufficient condition for two rigid balanced tilings to be congruent is provided. For a given rigid GIFS, there are uncountably many balanced tilings, corresponding to uncountably many parameters. All rigid balanced tilings are non-periodic.

Given a pants decomposition $\mathcal{PC} = \{\gamma_1, \ldots, \gamma_{\xi}\}$ on a hyperbolizable surface $\Sigma$ and a vector $\underline{c} = (c_1, \ldots, c_{\xi}) \in \mathbb{R}_+^\xi$, we describe a plumbing construction which endows $\Sigma$ with a complex projective structure for which the associated holonomy representation $\rho$ is quasi-Fuchsian and for which $\ell_\rho(\gamma_i) = c_i$. When $\underline{c} \to \underline{0} = (0, \ldots, 0)$ this construction limits to Kra's plumbing construction. In addition, when $\Sigma = \Sigma_{1,1}$, the holonomy representations of these structures belong to the `linear slice' of quasi-Fuchsian space $\mathrm{QF}(\Sigma)$ defined by Komori and Parkonnen. We discuss some conjectures for these slices suggested by the pictures we created in joint work with Yamashita., Comment: 27 pages, 10 figures

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semi-simple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thus extend earlier rigidity results about quasi-isometric embeddings to the setting of semi-simple Lie groups. We also present some examples when the rigidity does not hold, including first examples in which every flat is mapped into multiple flats.

We study linear series on a general curve of genus g, whose images are exceptional with respect to their secant planes. Each such exceptional secant plane is algebraically encoded by an included linear series, whose number of base points computes the incidence degree of the corresponding secant plane. With enumerative applications in mind, we construct a moduli scheme of inclusions of limit linear series with base points over families of nodal curves of compact type, which we then use to compute combinatorial formulas for the number of secant-exceptional linear series when the spaces of linear series and of inclusions are finite., 37 pages, final version, to appear in Geometriae Dedicata

We study three different topologies on the moduli space $\mathscr{H}^{\rm loc}_m$ of equivariant isometry classes of $m$-dimensional locally homogeneous Riemannian spaces. As an application, we provide the first examples of locally homogeneous spaces converging to a limit space in the pointed $\mathcal{C}^{k,\alpha}$-topology, for some $k>1$, which do not admit any convergent subsequence in the pointed $\mathcal{C}^{k+1}$-topology., Comment: 17 pages. Some minor modifications. Accepted for publication in Geom. Dedicata

We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean line into the real euclidean plane. We consider these up to equivalence under the group of birational automorphisms of the real affine plane which are diffeomorphisms of its real locus. We show that in contrat with the situation in the categories of smooth manifolds with smooth maps and of real algebraic varieties with regular maps where there is only one equivalence class up to isomorphism, there are plenty of non-equivalent smooth rational closed embeddings up to such birational diffeomorphisms. Some of these are simply detected by the non-negativity of the real Kodaira dimension of the complement of their images. But we also introduce finer invariants derived from topological properties of suitable fake real planes associated to certain classes of such embeddings.

Let $X$ be a smooth complex projective curve, and let $x\in X$ be a point. We compute the automorphism group of the moduli space of framed vector bundles on $X$ of rank $r \geq 2$ with a framing over $x$. It is shown that this automorphism group is generated by the following three: (1) pullbacks using automorphisms of the curve $X$ that fix the marked point $x$, (2) tensorization with certain line bundles over $X$ and (3) the action of $\operatorname{PGL}_r(\mathbb{C})$ through composition with the framing., 33 pages

Let G be a compact semisimple linear Lie group. We study the action of $$\text {Aut}(F_r)$$ on the space $$H_*(G^r; {\mathbb {Q}})$$ . We compute the image of this representation and prove that it only depends on the rank of $${\mathfrak {g}}$$ . We show that the kernel of this representation is always the Torrelli subgroup $$\text {IA}_r$$ of $$\text {Aut}(F_r)$$ .

In this paper, we sketch some constructions providing a link between complex-analytic geometry and nonstandard algebraic geometry via a categorical point of view. The analytic category is seen as a completed fiber of a family of nonstandard algebraic geometries by applying a standard part functor. We indicate how various notions of analytic objects fit into this context (as for example banachanalytic spaces, Kahler spaces etc.)

A collection $ \Delta $ of simple closed curves on an orientable surface is an algebraic $ k $-system if the algebraic intersection number $\langle \alpha,\beta \rangle$ is equal to $k $ in absolute value for every $ \alpha , \beta \in \Delta $ distinct. Generalizing a theorem of [MRT14] we compute that the maximum size of an algebraic $k$-system of curves on a surface of genus $g$ is $2g+1$ when $g\ge 3$ or $k$ is odd, and $2g$ otherwise. To illustrate the tightness in our assumptions, we present a construction of curves pairwise geometrically intersecting twice whose size grows as $g^2$., Comment: 7 pages, 2 figures

This article is concerned with the study of the holonomy group of flat solvmanifolds. It is known that the holonomy group of a flat solvmanifold is abelian; we give an elementary proof of this fact and moreover we prove that any finite abelian group is the holonomy group of a flat solvmanifold. Furthermore, we show that the minimal dimension of a flat solvmanifold with holonomy group $\mathbb{Z}_n$ coincides with the minimal dimension of a compact flat manifold with holonomy group $\mathbb{Z}_n$. Finally, we give the possible holonomy groups of flat solvmanifolds in dimensions 3, 4, 5 and 6; exhibiting in the latter case a general construction to show examples of non cyclic holonomy groups., 25 pages

An immersed surface in $${{\mathbb {R}}}^4$$ is said to has constant Jordan angles (CJA) if the angles between its tangent planes and a fixed plane do not depend on the choice of the point. The constant Jordan angles surfaces in $${{\mathbb {R}}}^4$$ has been proved to exist, Bayard et al. (Geom Dedicata 162:153–176, 2013), but there are only explicit examples of non planar surfaces for the extremal angles 0 and $$\frac{\pi }{2}$$ as the Clifford torus. In this work, an alternative proof of the existence has been obtained that is based on the solution of a hyperbolic partial differential equation. Finally, after a study of the known solutions of the hyperbolic equation, an explicit expression with arbitrary CJA is provided for a family of immersions with an additional geometric property written in terms of the local invariants.

We give a condition for a function to produce a M\"obius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of M\"obius invariant knot energies can produce M\"obius invariant and parametrization invariant weighted inner products. They would give a natural way to study the evolution of knots in the framework of M\"obius geometry., Comment: 14 pages

Let g be an arbitrary Jordan loop and let G denote the space of rectangles R which are inscribed in g in such a way that the cyclic order of the vertices of R is the same whether it is induced by R or by g. We prove that G contains a connected set S satisfying one of three properties: 1. S consists of rectangles of uniformly large area, including a square, and every point of g is the vertex of a rectangle in S. 2. S consists of rectangles having all possible aspect ratios, and all but at most 4 points of g are vertices of rectangles in S. 3. S contains rectangles of every sufficiently small diameter, and all but at most 2 points of g are vertices of rectangles in S., Comment: 32 pages. This paper is a revision of the first version, inspired in part by comments from an anonymous referee. The current version of the paper will probably appear in Geometriae Dedicata

We derive a lower bound on the size of finite non-cyclic quotients of the braid group that is superexponential in the number of strands. We also derive a similar lower bound for nontrivial finite quotients of the commutator subgroup of the braid group., Comment: 7 pages, final version; to appear in Geometriae Dedicata

In this paper we extend the notion of the Brauer–Clifford group to the case of an Azumaya–Poisson (S, H)-algebra, when H is a cocommutative Hopf algebra and S is an H-module Poisson algebra. This is the situation that arises in applications with connections to differential geometry, algebraic geometry and quantum groups. These Brauer–Clifford groups turn out to be an example of the Brauer group of a symmetric monoidal category.

In this short note, we use the Bochner technique and the Hodge theory in complex differential geometry to prove several injectivity results for the cohomology of holomorphic vector bundles on compact Kahler manifolds, which generalize Enoki’s original injectivity theorem.

In this paper we translate the necessary and sufficient conditions of Tanaka’s theorem on the finiteness of effective prolongations of a fundamental graded Lie algebras into computationally effective criteria, involving the rank of some matrices that can be explicitly constructed. Our results would apply to geometries, which are defined by assigning a structure algebra on the contact distribution.

Let $\Sigma$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $\Sigma$. A geodesic curve $\gamma$ in $\Sigma$ is filling, if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word in some generators of $\pi_1(\Sigma)$, is filling. In the process, we find an explicit bound for the combinatorial length of a curve given by its Dehn-Thurston coordinate, in terms of the hyperbolic length. This gives us an efficient method for producing a collection which is guaranteed to contain all words corresponding to simple geodesics of bounded hyperbolic length., Comment: Minor revisions; to appear in Geometriae Dedicata

The purpose of this paper is to investigate relationship between the automorphism group of a rational surface and that of its Hilbert scheme of n points.

In this survey, written for the proceedings of the conference K-theory in algebra, analysis and topology, Buenos-Aires, Argentina (satellite event of the ICM 2018), we give a rigorous overview of the noncommutative counterparts of some celebrated conjectures of Grothendieck, Voevodsky, Beilinson, Weil, Tate, Parshin, Kimura, Schur, and others., Survey article; 23 pages. Revised version. arXiv admin note: text overlap with arXiv:1808.00950

We explore hybrid subgroups of certain non-arithmetic lattices in $${\text {PU}}(2,1)$$ . In particular, we show that all of Mostow’s lattices are virtually hybrids; moreover, we show that some of these non-arithmetic lattices are virtually hybrids of two non-commensurable arithmetic lattices in $${\text {PU}}(1,1)$$ .

A Morse 2-function is a generic smooth map f from a manifold M of arbitrary finite dimension to a surface B. Its critical set maps to an immersed collection of cusped arcs in B. The aim of this paper is to explain exactly when it is possible to move these arcs around in B by a homotopy of f and to give a library of examples when M is a closed 4-manifold. The last two sections give applications to the theory of crown diagrams of smooth 4-manifolds.

A Ricci flow (M, g(t)) on an n-dimensional Riemannian manifold M is an intrinsic geometric flow. A family of smoothly embedded submanifolds \((S(t), g_E)\) of a fixed Euclidean space \(E = \mathbb {R}^{n+k}\) is called an extrinsic representation in \(\mathbb {R}^{n+k}\) of (M, g(t)) if there exists a smooth one-parameter family of isometries \((S(t), g_E) \rightarrow (M, g(t))\). When does such a representation exist? We formulate a new way of framing this question for Ricci flows on surfaces of revolution immersed in \(\mathbb {R}^3\). This framework allows us to construct extrinsic representations for the Ricci flow initialized by any compact surface of revolution immersed in \(\mathbb {R}^3\). In particular, we exhibit the first explicit extrinsic representations in \(\mathbb {R}^4\) of the Ricci flows initialized by toroidal surfaces of revolution immersed in \(\mathbb {R}^3\).

We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom–Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of curves.

We show that, in dimension at least $4$, the set of locally finite simplicial volumes of oriented connected open manifolds is $[0, \infty]$. Moreover, we consider the case of tame open manifolds and some low-dimensional examples., Comment: 12 pages, 1 figure

We briefly overview how, historically, string theory led theoretical physics first to precise problems in algebraic and differential geometry, and thence to computational geometry in the last decade or so, and now, in the last few years, to data science. Using the Calabi-Yau landscape -- accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades -- as a starting-point and concrete playground, we review some recent progress in machine-learning applied to the sifting through of possible universes from compactification, as well as wider problems in geometrical engineering of quantum field theories. In parallel, we discuss the programme in machine-learning mathematical structures and address the tantalizing question of how it helps doing mathematics, ranging from mathematical physics, to geometry, to representation theory, to combinatorics, and to number theory., Comment: 32 pages, 3 figures. Invited review for IJMPA, based on various colloquia, seminars and conference talks in the 2019-2020 academic year

We study the asymptotics of Ohsawa-Takegoshi extension operator and orthogonal Bergman projector associated with high tensor powers of a positive line bundle. More precisely, for a fixed complex submanifold in a complex manifold, we consider the operator which associates to a given holomorphic section of a positive line bundle over the submanifold the holomorphic extension of it to the ambient manifold with the minimal $L^2$-norm. When the tensor power of the line bundle tends to infinity, we obtain an explicit asymptotic expansion of this operator. This is done by proving an exponential estimate for the Schwartz kernel of the extension operator, and showing that this Schwartz kernel admits a full asymptotic expansion in powers of the line bundle for close parameters. Along the way we prove similar results for the projection onto the holomorphic sections orthogonal to those which vanish along the submanifold. Motivated by possible applications to the invariance of plurigenera problem, our results are proved for manifolds and embeddings of bounded geometry. As an application, we obtain asymptotically optimal $L^{\infty}$-bounds on the holomoprhic extensions from submanifolds, refining previous results of Zhang and Bost., Comment: A version of the extension theorem with respect to the supremum norm is added, see Theorem 1.10. References to earlier works have been added. A minor gap in the proof from Section 4.3 is corrected by a simpler argument

In recent years, it has been shown that some classical inequalities follow from a local stochastic dominance for naturally associated random polytopes. We strengthen planar isoperimetric inequalities by attaching a stochastic model to some classical inequalities, such as Mahler’s Theorem, and a reverse Lutwak–Zhang inequality, the polar for $$L_p$$ centroid bodies. In particular, we obtain the dual counterpart to a result of Bisztriczky–Boroczky.

Motivated by calculations of motivic homotopy groups, we give widely attained conditions under which operadic algebras and modules thereof are preserved under (co)localization functors, Comment: 29 pages

Several topological methods have been used successfully in the study of the hypercomplex analysis; for example, in the theory of functions of several complex variables (Grauert et al., in: Encyclopaedia of mathematical science, vol. 74, Springer, 1991, Hirzebruch, in: Topological methods in algebraic geometry, Classics in Mathematics. Reprint of the 1978 Edition. Springer, Berlin, 1995, Krantz, in Function theory of several complex variables, 2nd edn. American Mathematical Society, Providence, 2001), in the Clifford analysis (Sabadini et al. in Adv. Appl. Clifford Algebras 24:1131–1143, 2014), and in the theory of slice regular functions (Colombo et al. in Math Nachr 285:949–958, 2012). Particularly, the fiber bundle is one of these topological subjects that had an intensive development in a number of papers (Bernstein and Philips in Sci Am 245(1):122–137, 1981, Bleecker, in: Guage Theory and Variational Principles. Dover Books on physics Dover Books on mathematics. Courier Corporation, North Chelmsford, 2005, Bredon in Topology and geometry, Springer, Berlin, 1913, Cohen in The topology of fiber bundles, Stanford University, Stanford, 1998, Hatcher in Algebraic-Topology, Cambridge University Press, Cambridge, 2002, Husemoller in fibre bundles, 3rd edn, Springer, Berlin, 1993, Steenrod in The topology of fibre bundles, Princeton University Press, Princeton, 1951, Walschap in Metric structures in differential geometry, Springer, New York, 2004, Weatherall in Synthese 193:2389–2425, 2016). The aim of this work is to show how the Splitting Lemma and the Representation Formula intrinsically determine a fiber bundle over the space of quaternionic slice regular functions and as a consequence, several properties of this function space are interpreted in terms of sections, pullbacks and isomorphism of fiber bundles.

Liouville’s theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally equivalent if they are combinatorially equivalent and the lengths of corresponding edges are related by scale factors associated with the vertices.

In this paper, we generalize a result of Satoh to show that for any odd natural $n$, the connected sum of the $n$-twist spun sphere of a knot $K$ and an unknotted projective plane in the 4-sphere is equivalent to the same unknotted projective plane. We additionally provide a fix to a small error in Satoh's proof of the case that $K$ is a 2-bridge knot.

Let $$S_{g,n}$$ be a surface of genus $$g > 1$$ with $$n>0$$ punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new variational proof of this result. We also give an alternative proof of the existence and uniqueness of a hyperbolic polyhedral metric with prescribed curvature in a given conformal class.

The Orlicz Minkowski problem is a generalization of the $$L_p$$ Minkowski problem. For a class of appropriate functions and discrete measures that have no essential subspaces, the existence is demonstrated for the discrete Orlicz Minkowski problem. This is a non-trivial extension of the discrete $$L_p$$ Minkowski problem for $$p

The aim of this paper is to investigate the existence of proper, weakly biharmonic maps within a family of rotationally symmetric maps $$u_a : B^n \rightarrow {\mathbb {S}}^n$$, where $$B^n$$ and $${\mathbb {S}}^n$$ denote the Euclidean n-dimensional unit ball and sphere respectively. We prove that there exists a proper, weakly biharmonic map $$u_a$$ of this type if and only if $$n=5$$ or $$n=6$$. We shall also prove that these critical points are unstable.

Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on $\mathbb{R}^n$ and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of $\mathrm{SU}(3)$-structures in dimension $6$ whose associated $3$-form is Killing, we then show that every Killing $3$-form on a compact $n$-dimensional Riemannian manifold with negative sectional curvature vanishes if $n\ge 4$., 16 pages

We describe how to parameterize a Poincare section for horocycle flow on the moduli space, $${\mathcal {H}}(\alpha )$$ of translation surfaces. We give an explicit description of this Poincare section in the case when $$\alpha = 2$$ and bound the return time under horocycle flow.

General integral geometric invariants for convex bodies are introduced and two integral geometric inequalities for them are established. The equality cases for the inequalities are kinematic formulas, which are characterizations of integral geometric valuations. Those characterizations are analogues of Hadwiger’s characterization theorem.

We recall the notion of partial presimplicial set and its geometric realization. We show that any semiadequate diagram yields a partial presimplicial set leading to a geometric realization of the almost-extreme Khovanov homology of the diagram. We give a concrete formula for the homotopy type of this geometric realization, involving wedge of spheres and a suspension of the projective plane.

We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier in [1] using Toeplitz operators of compact K\"ahler manifolds and in [10] using skein theory., Comment: Geometriae Dedicata (online), 10 pages, minor changes

We construct the symplectic resolution of a symplectic orbifold whose isotropy locus consists of disjoint submanifolds with homogeneous isotropy, that is, all its points have the same isotropy groups.

We construct new classes of self-similar groups: S-arithmetic groups, affine groups and metabelian groups. Most of the soluble ones are finitely presented and of type $$FP_n$$ for appropriate n.

In this paper we produce a sequence of Riemannian manifolds $M_j^m$, $m \ge 2$, which converge in the intrinsic flat sense to the unit $m$-sphere with the restricted Euclidean distance. This limit space has no geodesics achieving the distances between points, exhibiting previously unknown behavior of intrinsic flat limits. In contrast, any compact Gromov-Hausdorff limit of a sequence of Riemannian manifolds is a geodesic space. Moreover, if $m\geq3$, the manifolds $M_j^m$ may be chosen to have positive scalar curvature., Comment: 21 pages, 3 figures, comments welcome