1,990 research outputs found

    Symmetries of Riemann surfaces and magnetic monopoles

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    This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions

    Classical and quantum algorithms for scaling problems

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    This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Quantum ergodicity on the Bruhat-Tits building for PGL(3,F)\text{PGL}(3, F) in the Benjamini-Schramm limit

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    We study eigenfunctions of the spherical Hecke algebra acting on L2(Γn\G/K)L^2(\Gamma_n \backslash G / K) where G=PGL(3,F)G = \text{PGL}(3, F) with FF a non-archimedean local field of characteristic zero, K=PGL(3,O)K = \text{PGL}(3, \mathcal{O}) with O\mathcal{O} the ring of integers of FF, and (Γn)(\Gamma_n) is a sequence of cocompact torsionfree lattices. We prove a form of equidistribution on average for eigenfunctions whose spectral parameters lie in the tempered spectrum when the associated sequence of quotients of the Bruhat-Tits building Benjamini-Schramm converges to the building itself.Comment: 111 pages, 25 figures, 2 table

    Quantum traces for SLnSL_n-skein algebras

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    We establish the existence of several quantum trace maps. The simplest one is an algebra map between two quantizations of the algebra of regular functions on the SLnSL_n-character variety of a surface S\mathfrak{S} equipped with an ideal triangulation λ\lambda. The first is the (stated) SLnSL_n-skein algebra S(S)\mathscr{S}(\mathfrak{S}). The second X‾(S,λ)\overline{\mathcal{X}}(\mathfrak{S},\lambda) is the Fock and Goncharov's quantization of their XX-moduli space. The quantum trace is an algebra homomorphism trˉX:S‾(S)→X‾(S,λ)\bar{tr}^X:\overline{\mathscr{S}}(\mathfrak{S})\to\overline{\mathcal{X}}(\mathfrak{S},\lambda) where the reduced skein algebra S‾(S)\overline{\mathscr{S}}(\mathfrak{S}) is a quotient of S(S)\mathscr{S}(\mathfrak{S}). When the quantum parameter is 1, the quantum trace trˉX\bar{tr}^X coincides with the classical Fock-Goncharov homomorphism. This is a generalization of the Bonahon-Wong quantum trace map for the case n=2n=2. We then define the extended Fock-Goncharov algebra X(S,λ)\mathcal{X}(\mathfrak{S},\lambda) and show that trˉX\bar{tr}^X can be lifted to trX:S(S)→X(S,λ)tr^X:\mathscr{S}(\mathfrak{S})\to\mathcal{X}(\mathfrak{S},\lambda). We show that both trˉX\bar{tr}^X and trXtr^X are natural with respect to the change of triangulations. When each connected component of S\mathfrak{S} has non-empty boundary and no interior ideal point, we define a quantization of the Fock-Goncharov AA-moduli space A‾(S,λ)\overline{\mathcal{A}}(\mathfrak{S},\lambda) and its extension A(S,λ)\mathcal{A}(\mathfrak{S},\lambda). We then show that there exist quantum traces trˉA:S‾(S)→A‾(S,λ)\bar{tr}^A:\overline{\mathscr{S}}(\mathfrak{S})\to\overline{\mathcal{A}}(\mathfrak{S},\lambda) and trA:S(S)↪A(S,λ)tr^A:\mathscr{S}(\mathfrak{S})\hookrightarrow\mathcal{A}(\mathfrak{S},\lambda), where the second map is injective, while the first is injective at least when S\mathfrak{S} is a polygon. They are equivalent to the XX-versions but have better algebraic properties.Comment: 111 pages, 35 figure

    On the use of senders for minimal Ramsey theory

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    This thesis investigates problems related to extremal and probabilistic graph theory. Our focus lies on the highly dynamic field of Ramsey theory. The foundational result of this field was proved in 1930 by Franck P. Ramsey. It implies that for every integer t and every sufficiently large complete graph Kn, every colouring of the edges of Kn with colours red and blue contains a red copy or a blue copy of Kt. Let q ⩾ 2 represent a number of colours, and let H1,..., Hq be graphs. A graph G is said to be q-Ramsey for the tuple (H1,...,Hq) if, for every colouring of the edges of G with colours {1, . . . , q}, there exists a colour i and a monochromatic copy of Hi in colour i. As we often want to understand the structural properties of the collection of graphs that are q-Ramsey for (H1,..., Hq), we restrict our attention to the graphs that are minimal for this property, with respect to subgraph inclusion. Such graphs are said to be q-Ramsey-minimal for (H1,..., Hq). In 1976, Burr, Erdős, and Lovász determined, for every s, t ⩾ 3, the smallest minimum degree of a graph G that is 2-Ramsey-minimal for (Ks, Kt). Significant efforts have been dedicated to generalising this result to a higher number of colours, q⩾3, starting with the ‘symmetric’ q-tuple (Kt,..., Kt). In this thesis, we improve on the best known bounds for this parameter, providing state-of-the-art bounds in different (q, t)-regimes. These improvements rely on constructions based on finite geometry, which are then used to prove the existence of extremal graphs with certain key properties. Another crucial ingredient in these proofs is the existence of gadget graphs, called signal senders, that were initially developed by Burr, Erdős, and Lovász in 1976 for pairs of complete graphs. Until now, these senders have been shown to exist only in the two-colour setting, when q = 2, or in the symmetric multicolour setting, when H1,..., Hq are pairwise isomorphic. In this thesis, we then construct similar gadgets for all tuples of complete graphs, providing the first known constructions of these tools in the multicolour asymmetric setting. We use these new senders to prove far-reaching generalisations of several classical results in the area

    The galaxy of Coxeter groups

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    In this paper we introduce the galaxy of Coxeter groups -- an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures. Along the way we show profinite rigidity of triangle Coxeter groups -- a result which is possibly of independent interest.Comment: 30 pages, 6 figures. v2: Incorporated referee's suggestions; Corrected a mistake in the proof of Theorem 4.25 (formerly 4.24), improved other proofs and text; Final version, to appear in the Journal of Algebr

    Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders

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    Orientations of supersingular elliptic curves encode the information of an endomorphism of the curve. Computing the full endomorphism ring is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is O\mathfrak{O}-orientable for a fixed imaginary quadratic order O\mathfrak{O} provides non-trivial information towards computing an endomorphism corresponding to the O\mathfrak{O}-orientation. We provide explicit algorithms and in-depth complexity analysis. We also consider the question in terms of quaternion algebras. We provide algorithms which compute an embedding of a fixed imaginary quadratic order into a maximal order of the quaternion algebra ramified at pp and ∞\infty. We provide code implementations in Sagemath which is efficient for finding embeddings of imaginary quadratic orders of discriminants up to O(p)O(p), even for cryptographically sized pp

    Measured foliations at infinity and constant mean curvature surface in quasi-Fuchsian manifolds close to the Fuchsian locus

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    Given an orientied, closed hyperbolic surface SS, we study quasi-Fuchsian hyperbolic manifolds homeomorphic to S×RS\times \mathbb{R}. We study two questions regarding them: one is on \textit{measured foliations at infinity} and the other is on \textit{foliation by constant mean curvature surfaces}. Measured foliations at infinity of quasi-Fuchsian manifolds are a natural analog at infinity to the measured bending laminations on the boundary of its convex core. Given a pair of measured foliations (F+,F−)(\mathsf{F}_{+},\mathsf{F}_{-}) which fill a closed hyperbolic surface SS and are {arational}, we prove that for t>0t>0 sufficiently small tF+t\mathsf{F}_{+} and tF−t\mathsf{F}_{-} can be uniquely realised as the measured foliations at infinity of a quasi-Fuchsian manifold homeomorphic to S×RS\times \mathbb{R}, which is sufficiently close to the Fuchsian locus. The proof is based on that of Bonahon in \cite{bonahon05} which shows that a quasi-Fuchsian manifold close to the Fuchsian locus can be uniquely determined by the data of filling measured bending laminations on the boundary of its convex core. Finally, we interpret the result in half-pipe geometry. For the second part of the thesis we deal with a conjecture due to Thurston asks if almost-Fuchsian manifolds admit a foliation by CMC surfaces. Here, almost-Fuchsian manifolds are defined as quasi-Fuchsian manifolds which contain a unique minimal surface with principal curvatures in (−1,1)(-1,1) and it is known that in general, quasi-Fuchsian manifolds are not foliated by surfaces of constant mean curvature (CMC) although their ends are. However, we prove that almost-Fuchsian manifolds which are sufficiently close to being Fuchsian are indeed monotonically foliated by surfaces of constant mean curvature. This work is in collaboration with Filippo Mazzoli and Andrea Seppi.Comment: Ph.D. thesis, combination of the papers arXiv:2111.01614 and arXiv:2204.0573
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