IA Scholar Query: An additivity theorem for the genus of a graph.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 11 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Burnside groups and orbifold invariants of birational maps
https://scholar.archive.org/work/vkoo3txax5ennksz2kvxl575ha
We construct new invariants of equivariant birational isomorphisms taking values in equivariant Burnside groups.Andrew Kresch, Yuri Tschinkelwork_vkoo3txax5ennksz2kvxl575haThu, 11 Aug 2022 00:00:00 GMTThe hybrid number of a ploidy profile
https://scholar.archive.org/work/krjkc6mtbvgs7gzdosmcknoav4
Polyploidization, whereby an organism inherits multiple copies of the genome of their parents, is an important evolutionary event that has been observed in plants and animals. One way to study such events is in terms of the ploidy number of the species that make up a dataset of interest. It is therefore natural to ask: How much information about the evolutionary past of the set of species that form a dataset can be gleaned from the ploidy numbers of the species? To help answer this question, we introduce and study the novel concept of a ploidy profile which allows us to formalize it in terms of a multiplicity vector indexed by the species the dataset is comprised of. Using the framework of a phylogenetic network, we present a closed formula for computing the hybrid number (i.e. the minimal number of polyploidization events required to explain a ploidy profile) of a large class of ploidy profiles. This formula relies on the construction of a certain phylogenetic network from the simplification sequence of a ploidy profile and the hybrid number of the ploidy profile with which this construction is initialized. Both of them can be computed easily in case the ploidy numbers that make up the ploidy profile are not too large. To help illustrate the applicability of our approach, we apply it to a simplified version of a publicly available Viola dataset.Katharina T. Huber, Liam J. Maherwork_krjkc6mtbvgs7gzdosmcknoav4Thu, 11 Aug 2022 00:00:00 GMTVacuum energy in (2+1)-dimensional quantum field theory on curved spaces
https://scholar.archive.org/work/jpilhshm3zbrhem2mwc7efcxqa
Relativistic quantum degrees of freedom in their vacuum state endow geometric backgrounds with an energy, as demonstrated by the Casimir Effect. We explore the vacuum energy (or free energy at finite temperature) of (2+1)-dimensional ultrastatic relativistic quantum field theories as a functional of their spatial geometry. These theories have physical realisations as, for example, the low-energy effective description of the electronic structure of graphene: four free massless Dirac fermions. We define a UV-finite unambiguous measure of free energy for these setups: the free energy difference. We compute it for the free scalar with curvature coupling and free Dirac fermion using heat kernel methods, deriving analytic expressions for perturbative and long-wavelength deformations of maximally-symmetric two-spaces (namely the plane and the round sphere) and, using a novel numerical approach, highly-accurate estimates in the case of large (axisymmetric) deformations to the sphere. We find that for these theories, as with holographic conformal field theories (CFTs) dual to vacuum Einstein gravity with a negative cosmological constant, it is universally negative for non-trivial deformations of maximally-symmetric two-spaces and can be made arbitrarily negative as the geometry becomes singular. In fact, we find that the differenced heat kernel has a definite sign — a much stronger result. We also observe a qualitative similarity between the (appropriately normalised) vacuum energies of a conformal scalar, massless Dirac fermion and holographic CFT on deformations of the two-sphere, and a remarkably close quantitive agreement between the latter two — very dissimilar in nature — theories. Finally, we show vacuum energy negativity for all perturbative deformations to Poincaré-invariant, power-counting-renormalisable theories on the plane. Our results indicate that relativistic quantum degrees of freedom universally disfavour smooth geometries and we note this effect has the potential to be measured experimentally.Lucas Samuel Wallis, Toby Wiseman, Science And Technology Facilities Council (Great Britain)work_jpilhshm3zbrhem2mwc7efcxqaThu, 11 Aug 2022 00:00:00 GMTExploring the electric field around a loop of static charge: Rectangles, stadiums, ellipses, and knots
https://scholar.archive.org/work/ucxeehflo5c2xp6c6welnonbvu
We study the electric field around a continuous one-dimensional loop of static charge, under the assumption that the charge is distributed uniformly along the loop. For rectangular or stadium-shaped loops in the plane, we find that the electric field can undergo a symmetry-breaking pitchfork bifurcation as the loop is elongated; the field can have either one or three zeros, depending on the loop's aspect ratio. For knotted charge distributions in three-dimensional space, we compute the electric field numerically and compare our results to previously published theoretical bounds on the number of equilibrium points around charged knots. Our computations reveal that the previous bounds are far from sharp. The numerics also suggest conjectures for the actual minimum number of equilibrium points for all charged knots with five or fewer crossings. In addition, we provide the first images of the equipotential surfaces around charged knots, and visualize their topological transitions as the level of the potential is varied.Max Lipton, Steven H Strogatz, Alex Townsendwork_ucxeehflo5c2xp6c6welnonbvuThu, 11 Aug 2022 00:00:00 GMTExplicit classification of isogeny graphs of rational elliptic curves
https://scholar.archive.org/work/rnopeof3wbceno3ysjadtjhy5u
Let n>1 be an integer such that X_0( n) has genus 0, and let K be a field of characteristic 0 or relatively prime to 6n. In this article, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a non-trivial isogeny over ℚ. We achieve this by introducing 56 parameterized families of elliptic curves 𝒞_n,i(t,d) defined over K(t,d), which have the following two properties for a fixed n: the elliptic curves 𝒞_n,i(t,d) are isogenous over K(t,d), and there are integers k_1 and k_2 such that the j-invariants of 𝒞_n,k_1(t,d) and 𝒞_n,k_2(t,d) are given by the Fricke parameterizations. As a consequence, we show that if E is an elliptic curve over a number field K with isogeny class degree divisible by n∈{4,6,9}, then there is a quadratic twist of E that is semistable at all primes 𝔭 of K such that 𝔭∤ n.Alexander J. Barrioswork_rnopeof3wbceno3ysjadtjhy5uThu, 11 Aug 2022 00:00:00 GMTA Gentle Introduction to the Non-Abelian Hodge Correspondence
https://scholar.archive.org/work/rgd6xcy7ezglbnaqkixmorna3q
We aim at giving a pedagogical introduction to the non-abelian Hodge correspondence, a bridge between algebra, geometric structures and complex geometry. The correspondence links representations of a fundamental group, the character variety, to the theory of holomorphic bundles. We focus on motivations, key ideas, links between the concepts and applications. Among others we discuss the Riemann--Hilbert correspondence, Goldman's symplectic structure via the Atiyah--Bott reduction, the Narasimhan--Seshadri theorem, Higgs bundles, harmonic bundles and hyperk\"ahler manifolds.Alexander Thomaswork_rgd6xcy7ezglbnaqkixmorna3qThu, 11 Aug 2022 00:00:00 GMTHigher Airy structures and topological recursion for singular spectral curves
https://scholar.archive.org/work/i63cmfdx5nbevefn3eqnwtu4vu
We give elements towards the classification of quantum Airy structures based on the W(𝔤𝔩_r)-algebras at self-dual level based on twisted modules of the Heisenberg VOA of 𝔤𝔩_r for twists by arbitrary elements of the Weyl group 𝔖_r. In particular, we construct a large class of such quantum Airy structures. We show that the system of linear ODEs forming the quantum Airy structure and determining uniquely its partition function is equivalent to a topological recursion à la Chekhov-Eynard-Orantin on singular spectral curves. In particular, our work extends the definition of the Bouchard-Eynard topological recursion (valid for smooth curves) to a large class of singular curves, and indicates impossibilities to extend naively the definition to other types of singularities. We also discuss relations to intersection theory on moduli spaces of curves, giving a general ELSV-type representation for the topological recursion amplitudes on smooth curves, and formulate precise conjectures for application in open r-spin intersection theory.Gaëtan Borot and Reinier Kramer and Yannik Schülerwork_i63cmfdx5nbevefn3eqnwtu4vuWed, 10 Aug 2022 00:00:00 GMTGenus zero transverse foliations for weakly convex Reeb flows on the tight 3-sphere
https://scholar.archive.org/work/7t5rs7hgujbadjzwocx4ncraj4
A contact form on the tight 3-sphere (S^3,ξ_0) is called weakly convex if the Conley-Zehnder index of every Reeb orbit is at least 2. In this article, we study Reeb flows of weakly convex contact forms on (S^3,ξ_0) admitting a prescribed finite set of index-2 Reeb orbits, which are all hyperbolic and mutually unlinked. We present conditions so that these index-2 orbits are binding orbits of a genus zero transverse foliation. In addition, we show in the real-analytic case that the topological entropy of the Reeb flow is positive if the branches of the stable/unstable manifolds of the index-2 orbits are mutually non-coincident.Naiara V. de Paulo, Umberto Hryniewicz, Seongchan Kim, Pedro A. S. Salomãowork_7t5rs7hgujbadjzwocx4ncraj4Wed, 10 Aug 2022 00:00:00 GMTBrieskorn spheres, cyclic group actions and the Milnor conjecture
https://scholar.archive.org/work/k65da2v7argrjh2uojz7u6prj4
In this paper we further develop the theory of equivariant Seiberg-Witten-Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain the following applications. First, we show that the knot concordance invariants θ^(c) defined by the first author satisfy θ^(c)(T_a,b) = (a-1)(b-1)/2 for torus knots, whenever c is a prime not dividing ab. Since θ^(c) is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture of a similar flavour to the proofs using the Ozsváth-Szabó τ-invariant or Rasmussen s-invariant. Second, we prove that a free cyclic group action on a Brieskorn homology 3-sphere Y = Σ(a_1 , ... , a_r) does not extend smoothly to any contractible smooth 4-manifold bounding Y. This generalises to arbitrary r the result of Anvari-Hambleton in the case r=3. Third, given a finite subgroup of the Seifert circle action on Y = Σ(a_1 , ... , a_r) of prime order p acting non-freely on Y, we prove that if the rank of HF_red^+(Y) is greater than p times the rank of HF_red^+(Y/ℤ_p), then the ℤ_p-action on Y does not extend smoothly to any contractible smooth 4-manifold bounding Y. We also prove a similar non-extension result for equivariant connected sums of Brieskorn homology spheres.David Baraglia, Pedram Hekmatiwork_k65da2v7argrjh2uojz7u6prj4Wed, 10 Aug 2022 00:00:00 GMTWonderful compactifications and rational curves with cyclic action
https://scholar.archive.org/work/5fca6zkqavbjjgfy26mqhs5gvi
We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows.Emily Clader, Chiara Damiolini, Shiyue Li, Rohini Ramadaswork_5fca6zkqavbjjgfy26mqhs5gviWed, 10 Aug 2022 00:00:00 GMTOn symmetric simplicial (super)string backgrounds, (super-)WZW defect fusion and the Chern-Simons theory
https://scholar.archive.org/work/ypb62pe4grgbjair3skxeu7oti
The super-σ-model of dynamics of the super-charged loop in an ambient supermanifold in the presence of worldsheet defects of arbitrary topology is formalised within Gawȩdzki's higher-cohomological approach, drawing inspiration from the precursor arXiv:0808.1419 [hep-th]. A distinguished class of the corresponding backgrounds (supertargets with additional bicategorial supergeometric data), organised into simplicial hierarchies, is considered. To these, configurational (super)symmetry of the bulk field theory is lifted coherently, whereby the notion of a maximally (super)symmetric background, and in particular that of a simplicial Lie background, arises as the target structure requisite for the definition of the super-σ-model with defects fully transmissive to the currents of the bulk (super)symmetry. The formal concepts are illustrated in two settings of physical relevance: that of the WZW σ-model of the bosonic string in a compact simple 1-connected Lie group and that of the GS super-σ-model of the superstring in the Minkowski super-space. In the former setting, the structure of the background is fixed through a combination of simplicial, symmetry(-reducibility) and cohomological arguments, and a novel link between fusion of the maximally symmetric WZW defects of Fuchs et al. and the 3d CS theory with timelike Wilson lines with fixed holonomy is established. Moreover, a purely geometric interpretation of the Verlinde fusion rules is proposed. In the latter setting, a multiplicative structure compatible with supersymmetry is shown to exist on the GS super-1-gerbe of arXiv:1706.05682 [hep-th], and subsequently used in a novel construction of a class of maximally (rigidly) supersymmetric bi-branes whose elementary fusion is also studied.Rafał R. Suszekwork_ypb62pe4grgbjair3skxeu7otiWed, 10 Aug 2022 00:00:00 GMTRay structures on Teichmüller Space
https://scholar.archive.org/work/k5t45d6gqbhw3o6rdxhgduhdoi
While there may be many Thurston metric geodesics between a pair of points in Teichmüller space, we find that by imposing an additional energy minimization constraint on the geodesics, thought of as limits of harmonic map rays, we select a unique Thurston geodesic through those points. Extending the target surface to the Thurston boundary yields, for each point Y in Teichmüller space, an "exponential map" of rays from that point Y onto Teichmüller space with visual boundary the Thurston boundary of Teichmüller space. We first depict harmonic map ray structures on Teichmüller space as a geometric transition between Teichmüller ray structures and Thurston geodesic ray structures. In particular, by appropriately degenerating the source of a harmonic map between hyperbolic surfaces (along "harmonic map dual rays"), the harmonic map rays through the target converge to a Thurston geodesic; by appropriately degenerating the target of the harmonic map, those harmonic map dual rays through the domain converge to Teichmüller geodesics. We then extend this transition to one from Teichmüller disks through Hopf differential disks to stretch-earthquake disks. These results apply to surfaces with boundary, resolving a question on stretch maps between such surfaces.Huiping Pan, Michael Wolfwork_k5t45d6gqbhw3o6rdxhgduhdoiWed, 10 Aug 2022 00:00:00 GMTGenericity of sublinearly Morse directions in CAT(0) spaces and the Teichmüller space
https://scholar.archive.org/work/bv652d3wpjdafeo5dnafjfqlri
We show that the sublinearly Morse directions in the visual boundary of a rank-1 CAT(0) space with a geometric group action are generic in several commonly studied senses of the word, namely with respect to Patterson-Sullivan measures and stationary measures for random walks. We deduce that the sublinearly Morse boundary is a model of the Poisson boundary for finitely supported random walks on groups acting geometrically on rank-1 CAT (0) spaces. We prove an analogous result for mapping class group actions on Teichm\"uller space. Our main technical tool is a criterion, valid in any unique geodesic metric space, that says that any geodesic ray with sufficiently many (in a statistical sense) strongly contracting segments is sublinearly contracting.Ilya Gekhtman, Yulan Qing, Kasra Rafiwork_bv652d3wpjdafeo5dnafjfqlriTue, 09 Aug 2022 00:00:00 GMTOn moduli spaces of roots in algebraic and tropical geometry
https://scholar.archive.org/work/nr2q3enuwjhhrcagrvhzsrhrta
In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that the combinatorics of the moduli space of tropical roots is governed by the poset of flows, a poset parametrizing certain flows on graphs.Alex Abreu, Marco Pacini, Matheus Seccowork_nr2q3enuwjhhrcagrvhzsrhrtaTue, 09 Aug 2022 00:00:00 GMTConformal dynamics at infinity for groups with contracting elements
https://scholar.archive.org/work/tm2cesqd4fd3zjtl6xhwuonchi
This paper develops a theory of conformal density at the infinity for groups with contracting elements. We start by introducing a class of contractive boundary encompassing many known hyperbolic-like boundaries, on which a detailed study of conical points and Myrberg points is carried out. The basic theory of conformal density is then established, including the Sullivan shadow lemma and Hopf-Tsuji-Sullivan theorem. This gives a unification of the theory of conformal density on the Gromov and Floyd boundary for (relatively) hyperbolic groups, the visual boundary for rank-1 CAT(0) groups, and Thurston boundary for mapping class groups. Besides that, the conformal density on the horofunction boundary provides a new important example of our general theory. At last, we present applications towards the co-growth problem of divergent groups, measure theoretical results for CAT(0) groups and mapping class groups.Wenyuan Yangwork_tm2cesqd4fd3zjtl6xhwuonchiTue, 09 Aug 2022 00:00:00 GMTAn Upsilon torsion function for knot Floer homology
https://scholar.archive.org/work/otyrcmfckvbvvlmzv5atjgoq2m
Heegaard Floer theory produces chain complexes associated to knots. Viewed as modules over polynomial rings, such complexes yield torsion invariants that offer constraints on cobordisms between knots. For instance, Juhasz, Miller and Zemke used torsion invariants to bound the number of local maxima and minima in cobordisms between pairs of knots. Gong and Marengon defined a related torsion invariant and used it to study nonorientable knot cobordisms. In this paper we define a one parameter family of Heegaard Floer torsion invariants that yields a piecewise linear function defined on the interval [0,2]. We call this the Upsilon torsion function; it is closely related to the Heegaard Floer Upsilon function defined by Ozsvath, Stipsicz and Szabo. In a natural way, this Upsilon torsion function interpolates between the Juhasz-Miller-Zemke invariant and the Gong-Marengon invariant. In addition to bounding the number of local maxima and minima in knot cobordisms, the Upsilon torsion function provides new obstructions related to the Gordian distance between knots.Samantha Allen, Charles Livingstonwork_otyrcmfckvbvvlmzv5atjgoq2mTue, 09 Aug 2022 00:00:00 GMTThere is no Enriques surface over the integers
https://scholar.archive.org/work/tszxcxin2bepdctndael74j43e
We show that there is no family of Enriques surfaces over the ring of integers. This extends non-existence results of Minkowski for families of finite \'etale schemes, of Tate and Ogg for families of elliptic curves, and of Fontaine and Abrashkin for families of abelian varieties and more general smooth proper schemes with certain restrictions on Hodge numbers. Our main idea is to study the local system of numerical classes of invertible sheaves. Among other things, our result also hinges on counting rational points, Lang's classification of rational elliptic surfaces in characteristic two, the theory of exceptional Enriques surfaces due to Ekedahl and Shepherd-Barron, some recent results on the base of their versal deformation, Shioda's theory of Mordell--Weil lattices, and an extensive combinatorial study for the pairwise interaction of genus-one fibrations.Stefan Schröerwork_tszxcxin2bepdctndael74j43eTue, 09 Aug 2022 00:00:00 GMTUnstable minimal surfaces in symmetric spaces of non-compact type
https://scholar.archive.org/work/26jxlqdztjc23jbcxuuxevk5v4
We prove that if Σ is a closed surface of genus at least 3 and G is a split real semisimple Lie group of rank at least 3 acting faithfully by isometries on a symmetric space N, then there exists a Hitchin representation ρ:π_1(Σ)→ G and a ρ-equivariant unstable minimal map from the universal cover of Σ to N. This follows from a new lower bound on the index of high energy minimal maps into an arbitrary symmetric space of non-compact type. Taking G=PSL(n,ℝ), n≥ 4, this disproves the Labourie conjecture.Nathaniel Sagman, Peter Smilliework_26jxlqdztjc23jbcxuuxevk5v4Tue, 09 Aug 2022 00:00:00 GMTPoint Counting on Igusa Varieties for function fields
https://scholar.archive.org/work/3x6nro6jpvborebbslrij7i63e
Igusa varieties over the special fibre of Shimura varieties have demonstrated many applications to the Langlands program via Mantovan's formula and Shin's point counting method. In this paper we study Igusa varieties over the moduli space of global 𝒢-shtukas and (under certain conditions) calculate the Hecke action on its cohomology. As part of their construction we prove novel results about local G-shtukas in both equal and unequal characteristic and also discuss application of these results to Barsotti-Tate groups and Shimura varieties.Paul Hamacher, Wansu Kimwork_3x6nro6jpvborebbslrij7i63eTue, 09 Aug 2022 00:00:00 GMTFundamental Exact Sequence for the Pro-Étale Fundamental Group
https://scholar.archive.org/work/n6pxz2rhlbcwrlzknxjv3usv5a
The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups – the usual étale fundamental group π_1^et defined in SGA1 and the more general group defined in SGA3. It controls local systems in the pro-étale topology and leads to an interesting class of "geometric covers" of schemes, generalizing finite étale covers. We prove the homotopy exact sequence over a field for the pro-étale fundamental group of a geometrically connected scheme X of finite type over a field k, i.e. that the sequence 1 →π_1^proet(X_k̅) →π_1^proet(X) →Gal_k → 1 is exact as abstract groups and the map π_1^proet(X_k̅) →π_1^proet(X) is a topological embedding. On the way, we prove a general van Kampen theorem and the Künneth formula for the pro-étale fundamental group.Marcin Larawork_n6pxz2rhlbcwrlzknxjv3usv5aMon, 08 Aug 2022 00:00:00 GMT