1,140 research outputs found

    A Unifying Theory for Graph Transformation

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    The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO

    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

    Composition and Cobordism Maps

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    We study the relationship between the algebra of module homomorphisms under composition and 4-dimensional cobordisms in the context of bordered Heegaard Floer homology. In particular, we prove that composition of module homomorphisms of type-DD structures induces the pair of pants cobordism map on Heegaard Floer homology in the morphism spaces formulation of the latter, due to Lipshitz--Ozsv\'{a}th--Thurston. Along the way, we prove a gluing result for cornered 4-manifolds constructed from bordered Heegaard triples. As applications, we present a new algorithm for computing arbitrary cobordism maps on Heegaard Floer homology and construct new nontrivial A∞A_\infty-deformations of Khovanov's arc algebras. Motivated by this last result and a K\"{u}nneth theorem for Heegaard Floer complexes of connected sums, we also prove the existence of a tensor product decomposition for arc algebras in characteristic 2 and show that there cannot be such a splitting over Z\Z

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    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

    Locality and Exceptional Points in Pseudo-Hermitian Physics

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    Pseudo-Hermitian operators generalize the concept of Hermiticity. Included in this class of operators are the quasi-Hermitian operators, which define a generalization of quantum theory with real-valued measurement outcomes and unitary time evolution. This thesis is devoted to the study of locality in quasi-Hermitian theory, the symmetries and conserved quantities associated with non-Hermitian operators, and the perturbative features of pseudo-Hermitian matrices. An implicit assumption of the tensor product model of locality is that the inner product factorizes with the tensor product. Quasi-Hermitian quantum theory generalizes the tensor product model by modifying the Born rule via a metric operator with nontrivial Schmidt rank. Local observable algebras and expectation values are examined in chapter 5. Observable algebras of two one-dimensional fermionic quasi-Hermitian chains are explicitly constructed. Notably, there can be spatial subsystems with no nontrivial observables. Despite devising a new framework for local quantum theory, I show that expectation values of local quasi-Hermitian observables can be equivalently computed as expectation values of Hermitian observables. Thus, quasi-Hermitian theories do not increase the values of nonlocal games set by Hermitian theories. Furthermore, Bell's inequality violations in quasi-Hermitian theories never exceed the Tsirelson bound of Hermitian quantum theory. A perturbative feature present in pseudo-Hermitian curves which has no Hermitian counterpart is the exceptional point, a branch point in the set of eigenvalues. An original finding presented in section 2.6.3 is a correspondence between cusp singularities of algebraic curves and higher-order exceptional points. Eigensystems of one-dimensional lattice models admit closed-form expressions that can be used to explore the new features of non-Hermitian physics. One-dimensional lattice models with a pair of non Hermitian defect potentials with balanced gain and loss, Δ±iγ, are investigated in chapter 3. Conserved quantities and positive-definite metric operators are examined. When the defects are nearest neighbour, the entire spectrum simultaneously becomes complex when γ increases beyond a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the PT-phase transition is dictated by the topological phase of the system. In the thermodynamic limit, PT-symmetry spontaneously breaks in the topologically non-trivial phase due to the presence of edge states. Chiral symmetry and representation theory are utilized in chapter 4 to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators. These intertwining operators include positive-definite metric operators in the quasi-Hermitian case. The PT-phase transition is explicitly determined in a special case

    SQISignHD: New Dimensions in Cryptography

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    We introduce SQISignHD, a new post-quantum digital signature scheme inspired by SQISign. SQISignHD exploits the recent algorithmic breakthrough underlying the attack on SIDH, which allows to efficiently represent isogenies of arbitrary degrees as components of a higher dimensional isogeny. SQISignHD overcomes the main drawbacks of SQISign. First, it scales well to high security levels, since the public parameters for SQISignHD are easy to generate: the characteristic of the underlying field needs only be of the form 2f3f2˘7−12^{f}3^{f\u27}-1. Second, the signing procedure is simpler and more efficient. Third, the scheme is easier to analyse, allowing for a much more compelling security reduction. Finally, the signature sizes are even more compact than (the already record-breaking) SQISign, with compressed signatures as small as 116 bytes for the post-quantum NIST-1 level of security. These advantages may come at the expense of the verification, which now requires the computation of an isogeny in dimension 44, a task whose optimised cost is still uncertain, as it has been the focus of very little attention

    Categorical structures for deduction

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    We begin by introducing categorized judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and first-order logic as special kinds of categorized judgemental theories. We believe our analysis sheds light on both the topics, providing a new point of view. In the case of type theory, we provide an abstract definition of type constructor featuring the usual formation, introduction, elimination and computation rules. For first-order logic we offer a deep analysis of structural rules, describing some of their properties, and putting them into context. We then put one of the main constructions introduced, namely that of categorized judgemental dependent type theories, to the test: we frame it in the general context of categorical models for dependent types, describe a few examples, study its properties, and use it to model subtyping and as a tool to prove intrinsic properties hidden in other models. Somehow orthogonally, then, we show a different side as to how categories can help the study of deductive systems: we transport a known model from set-based categories to enriched categories, and use the information naturally encoded into it to describe a theory of fuzzy types. We recover structural rules, observe new phenomena, and study different possible enrichments and their interpretation. We open the discussion to include different takes on the topic of definitional equality

    On the post-quantum future of Elliptic Curve Cryptography

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    This thesis is a literature study on current published quantum-resistant isogeny-based key exchange protocols. Here we cover the topic from foundations. Chapters 1 and 2 discuss classical computation models, algorithm complexity, and how these concepts support the security of modern elliptic curve cryptography methods, such as ECDH and ECDSA. Next, in Chapters 3 to 5, we present quantum computation models, and how Shor's algorithm on quantum computers presents a threat to the future security of classical asymmetric cryptography. We explore the foundations of isogeny-based cryptography, and two key exchange protocols of this kind: SIDH and CSIDH. Appendices A and B are provided for readers wanting more in-depth background explanations on the algebraic geometry of elliptic curves, and quantum mechanics respectively

    The Distribution of Sandpile Groups of Random Graphs with their Pairings

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    We determine the distribution of the sandpile group (also known as the Jacobian) of the Erd\H{o}s-R\'{e}nyi random graph G(n,q)G(n,q) along with its canonical duality pairing as nn tends to infinity, fully resolving a conjecture from 2015 due to Clancy, Leake, and Payne and generalizing the result by Wood on the groups. In particular, we show that a finite abelian pp-group GG equipped with a perfect symmetric pairing δ\delta appears as the Sylow pp-part of the sandpile group and its pairing with frequency inversely proportional to ∣G∣∣Aut(G,δ)∣|G||\mathrm{Aut}(G,\delta)|, where Aut(G,δ)\mathrm{Aut}(G,\delta) is the set of automorphisms of GG preserving the pairing δ\delta. While this distribution is related to the Cohen-Lenstra distribution, the two distributions are not the same on account of the additional algebraic data of the pairing. The proof utilizes the moment method: we first compute a complete set of moments for our random variable (the average number of epimorphisms from our random object to a fixed object in the category of interest) and then show the moments determine the distribution. To obtain the moments, we prove a universality result for the moments of cokernels of random symmetric integral matrices whose dual groups are equipped with symmetric pairings that is strong enough to handle both the dependence in the diagonal entries and the additional data of the pairing. We then apply results due to Sawin and Wood to show that these moments determine a unique distribution.Comment: 43 page
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