6,040 research outputs found

    Profinite Galois descent in K(h)-local homotopy theory

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    We investigate the category of K(h)-local spectra through the action of the Morava stabiliser group. Using condensed mathematics, we give a model for the continuous action of this profinite group on the ∞-category of K(h)-local modules over Morava E-theory, and explain how this gives rise to descent spectral sequences computing the Picard and Brauer groups of K(h)-local spectra. In the second part, we focus on the computation of these spectral sequences at height one, showing that they recover the Hopkins-Mahowald-Sadofsky computation of the Picard group, and giving a complete computation of the Brauer group relative to p-completed complex K-theory

    Hidden in the Cloud : Advanced Cryptographic Techniques for Untrusted Cloud Environments

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    In the contemporary digital age, the ability to search and perform operations on encrypted data has become increasingly important. This significance is primarily due to the exponential growth of data, often referred to as the "new oil," and the corresponding rise in data privacy concerns. As more and more data is stored in the cloud, the need for robust security measures to protect this data from unauthorized access and misuse has become paramount. One of the key challenges in this context is the ability to perform meaningful operations on the data while it remains encrypted. Traditional encryption techniques, while providing a high level of security, render the data unusable for any practical purpose other than storage. This is where advanced cryptographic protocols like Symmetric Searchable Encryption (SSE), Functional Encryption (FE), Homomorphic Encryption (HE), and Hybrid Homomorphic Encryption (HHE) come into play. These protocols not only ensure the confidentiality of data but also allow computations on encrypted data, thereby offering a higher level of security and privacy. The ability to search and perform operations on encrypted data has several practical implications. For instance, it enables efficient Boolean queries on encrypted databases, which is crucial for many "big data" applications. It also allows for the execution of phrase searches, which are important for many machine learning applications, such as intelligent medical data analytics. Moreover, these capabilities are particularly relevant in the context of sensitive data, such as health records or financial information, where the privacy and security of user data are of utmost importance. Furthermore, these capabilities can help build trust in digital systems. Trust is a critical factor in the adoption and use of digital services. By ensuring the confidentiality, integrity, and availability of data, these protocols can help build user trust in cloud services. This trust, in turn, can drive the wider adoption of digital services, leading to a more inclusive digital society. However, it is important to note that while these capabilities offer significant advantages, they also present certain challenges. For instance, the computational overhead of these protocols can be substantial, making them less suitable for scenarios where efficiency is a critical requirement. Moreover, these protocols often require sophisticated key management mechanisms, which can be challenging to implement in practice. Therefore, there is a need for ongoing research to address these challenges and make these protocols more efficient and practical for real-world applications. The research publications included in this thesis offer a deep dive into the intricacies and advancements in the realm of cryptographic protocols, particularly in the context of the challenges and needs highlighted above. Publication I presents a novel approach to hybrid encryption, combining the strengths of ABE and SSE. This fusion aims to overcome the inherent limitations of both techniques, offering a more secure and efficient solution for key sharing and access control in cloud-based systems. Publication II further expands on SSE, showcasing a dynamic scheme that emphasizes forward and backward privacy, crucial for ensuring data integrity and confidentiality. Publication III and Publication IV delve into the potential of MIFE, demonstrating its applicability in real-world scenarios, such as designing encrypted private databases and additive reputation systems. These publications highlight the transformative potential of MIFE in bridging the gap between theoretical cryptographic concepts and practical applications. Lastly, Publication V underscores the significance of HE and HHE as a foundational element for secure protocols, emphasizing its potential in devices with limited computational capabilities. In essence, these publications not only validate the importance of searching and performing operations on encrypted data but also provide innovative solutions to the challenges mentioned. They collectively underscore the transformative potential of advanced cryptographic protocols in enhancing data security and privacy, paving the way for a more secure digital future

    Lie pairs

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    Extending the theory of systems, we introduce a theory of Lie semialgebra ``pairs'' which parallels the classical theory of Lie algebras, but with a ``null set'' replacing 0. A selection of examples is given. These Lie pairs comprise two categories in addition to the universal algebraic definition, one with ``weak Lie morphisms'' preserving null sums, and the other with ``⪯-morphisms'' preserving a surpassing relation ⪯ that replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt) Theorem in these three categories

    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

    Quantum-Classical hybrid systems and their quasifree transformations

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    The focus of this work is the description of a framework for quantum-classical hybrid systems. The main emphasis lies on continuous variable systems described by canonical commutation relations and, more precisely, the quasifree case. Here, we are going to solve two main tasks: The first is to rigorously define spaces of states and observables, which are naturally connected within the general structure. Secondly, we want to describe quasifree channels for which both the Schrödinger picture and the Heisenberg picture are well defined. We start with a general introduction to operator algebras and algebraic quantum theory. Thereby, we highlight some of the mathematical details that are often taken for granted while working with purely quantum systems. Consequently, we discuss several possibilities and their advantages respectively disadvantages in describing classical systems analogously to the quantum formalism. The key takeaway is that there is no candidate for a classical state space or observable algebra that can be put easily alongside a quantum system to form a hybrid and simultaneously fulfills all of our requirements for such a partially quantum and partially classical system. Although these straightforward hybrid systems are not sufficient enough to represent a general approach, we use one of the candidates to prove an intermediate result, which showcases the advantages of a consequent hybrid ansatz: We provide a hybrid generalization of classical diffusion generators where the exchange of information between the classical and the quantum side is controlled by the induced noise on the quantum system. Then, we present solutions for our initial tasks. We start with a CCR-algebra where some variables may commute with all others and hence generate a classical subsystem. After clarifying the necessary representations, our hybrid states are given by continuous characteristic functions, and the according state space is equal to the state space of a non-unital C*-algebra. While this C*-algebra is not a suitable candidate for an observable algebra itself, we describe several possible subsets in its bidual which can serve this purpose. They can be more easily characterized and will also allow for a straightforward definition of a proper Heisenberg picture. The subsets are given by operator-valued functions on the classical phase space with varying degrees of regularity, such as universal measurability or strong*-continuity. We describe quasifree channels and their properties, including a state-channel correspondence, a factorization theorem, and some basic physical operations. All this works solely on the assumption of a quasifree system, but we also show that the more famous subclass of Gaussian systems fits well within this formulation and behaves as expected

    On the Global Topology of Moduli Spaces of Riemannian Metrics with Holonomy Sp(n)\operatorname{Sp}(n)

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    We discuss aspects of the global topology of moduli spaces of hyperkähler metrics. If the second Betti number is larger than 44, we show that each connected component of these moduli spaces is not contractible. Moreover, in certain cases, we show that the components are simply connected and determine the second rational homotopy group. By that, we prove that the rank of the second homotopy group is bounded from below by the number of orbits of MBM-classes in the integral cohomology. \\ An explicit description of the moduli space of these hyperkähler metrics in terms of Torelli theorems will be given. We also provide such a description for the moduli space of Einstein metrics on the Enriques manifold. For the Enriques manifold, we also give an example of a desingularization process similar to the Kummer construction of Ricci-flat metrics on a Kummer K3K3 surface.\\ We will use these theorems to provide topological statements for moduli spaces of Ricci-flat and Einstein metrics in any dimension larger than 33. For a compact simply connected manifold NN we show that the moduli space of Ricci flat metrics on N×TkN\times T^k splits homeomorphically into a product of the moduli space of Ricci flat metrics on NN and the moduli of sectional curvature flat metrics on the torus TkT^k

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Algebraic solutions of linear differential equations: an arithmetic approach

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    Given a linear differential equation with coefficients in Q(x)\mathbb{Q}(x), an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz pp-curvature conjecture.Comment: 47 page

    On the essential torsion finiteness of abelian varieties over torsion fields

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    The classical Mordell-Weil theorem implies that an abelian variety AA over a number field KK has only finitely many KK-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension Kcyc=KQabK^{\rm cyc}=K\mathbb{Q}^{\mathrm{ab}} by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety AA over the infinite algebraic extension KBK_B obtained by adjoining the coordinates of all torsion points of an abelian variety BB. Assuming the Mumford-Tate conjecture, and up to a finite extension of the base field KK, we give a necessary and sufficient condition for the finiteness of A(KB)torsA(K_B)_{\rm tors} in terms of Mumford--Tate groups. We give a complete answer when both abelian varieties have dimension both three, or when both have complex multiplication.Comment: 35 page

    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
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