680 research outputs found

    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

    New techniques for integrable spin chains and their application to gauge theories

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    In this thesis we study integrable systems known as spin chains and their applications to the study of the AdS/CFT duality, and in particular to N “ 4 supersymmetric Yang-Mills theory (SYM) in four dimensions.First, we introduce the necessary tools for the study of integrable periodic spin chains, which are based on algebraic and functional relations. From these tools, we derive in detail a technique that can be used to compute all the observables in these spin chains, known as Functional Separation of Variables. Then, we generalise our methods and results to a class of integrable spin chains with more general boundary conditions, known as open integrable spin chains.In the second part, we study a cusped Maldacena-Wilson line in N “ 4 SYM with insertions of scalar fields at the cusp, in a simplifying limit called the ladders limit. We derive a rigorous duality between this observable and an open integrable spin chain, the open Fishchain. We solve the Baxter TQ relation for the spin chain to obtain the exact spectrum of scaling dimensions of this observable involving cusped Maldacena-Wilson line.The open Fishchain and the application of Functional Separation of Variables to it form a very promising road for the study of the three-point functions of non-local operators in N “ 4 SYM via integrability

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Views from a peak:Generalisations and descriptive set theory

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    This dissertation has two major threads, one is mathematical, namely descriptive set theory, the other is philosophical, namely generalisation in mathematics. Descriptive set theory is the study of the behaviour of definable subsets of a given structure such as the real numbers. In the core mathematical chapters, we provide mathematical results connecting descriptive set theory and generalised descriptive set theory. Using these, we give a philosophical account of the motivations for, and the nature of, generalisation in mathematics.In Chapter 3, we stratify set theories based on this descriptive complexity. The axiom of countable choice for reals is one of the most basic fragments of the axiom of choice needed in many parts of mathematics. Descriptive choice principles are a further stratification of this fragment by the descriptive complexity of the sets. We provide a separation technique for descriptive choice principles based on Jensen forcing. Our results generalise a theorem by Kanovei.Chapter 4 gives the essentials of a generalised real analysis, that is a real analysis on generalisations of the real numbers to higher infinities. This builds on work by Galeotti and his coauthors. We generalise classical theorems of real analysis to certain sets of functions, strengthening continuity, and disprove other classical theorems. We also show that a certain cardinal property, the tree property, is equivalent to the Extreme Value Theorem for a set of functions which generalize the continuous functions.The question of Chapter 5 is whether a robust notion of infinite sums can be developed on generalisations of the real numbers to higher infinities. We state some incompatibility results, which suggest not. We analyse several candidate notions of infinite sum, both from the literature and more novel, and show which of the expected properties of a notion of sum they fail.In Chapter 6, we study the descriptive set theory arising from a generalization of topology, κ-topology, which is used in the previous two chapters. We show that the theory is quite different from that of the standard (full) topology. Differences include a collapsing Borel hierarchy, a lack of universal or complete sets, Lebesgue’s ‘great mistake’ holds (projections do not increase complexity), a strict hierarchy of notions of analyticity, and a failure of Suslin’s theorem.Lastly, in Chapter 7, we give a philosophical account of the nature of generalisation in mathematics, and describe the methodological reasons that mathematicians generalise. In so doing, we distinguish generalisation from other processes of change in mathematics, such as abstraction and domain expansion. We suggest a semantic account of generalisation, where two pieces of mathematics constitute a generalisation if they have a certain relation of content, along with an increased level of generality

    Computation and Physics in Algebraic Geometry

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    Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry

    Generalised Braiding of Anyonic Excitations and Topological Quantum Computation.

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    This thesis investigates various topological phases of matter in two-dimensional and quasi one-dimensional systems. These exotic states of matter have applications in topological quantum computation; which is an inherently fault-tolerant quantum computation scheme. In these schemes, computations are implemented by braiding anyonic excitations. In this thesis we examine three aspects of braiding: braiding of anyonic excitations on graphs, topological lattice models, and non-adiabatic perturbations of a qubit constructed from Majorana bound states. In the first part of the thesis we introduce a universal framework to discuss the braiding of anyonic excitations on graphs as a model of a quantum wire network. We show that many features of the planar algebraic theory of anyons may be extended to graphs. In this direction, we introduce graph hexagon equations, a generalisation of the planar hexagon equations and demonstrate that this framework has several similarities and differences from its planar counterpart. Notably, depending on the graph, we find solutions that do not exist in the planar theory. We study this framework on a variety of graphs and tabulate solutions. In the second part, we investigated non-adiabatic perturbations of a topological memory that consists of two p-wave superconducting wires separated by a nontopological junction. We consider noise in the potential creating the non-trivial topological phase and also the effect of shuttling the Majoranas, a necessary step in braiding. We examine a mechanism for bit and phase flip errors where excitations from one wire tunnel through a junction into another wire, we also outline a scheme that utilises disorder to minimise such situations. In the final part of the thesis, we construct a modified toric code from Hopf algebra gauge theory. We find that introducing a non-trivial quasitriangular structure on the gauge group changes the identification of braiding statistics in the quantum double, although it is of the same topological order as the toric code. In particular, when the gauge group is CZN we can interpret this as a form of flux attachment, where under exchange, the electric charges behave as if they have fluxes attached

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Projected Langevin dynamics and a gradient flow for entropic optimal transport

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    The classical (overdamped) Langevin dynamics provide a natural algorithm for sampling from its invariant measure, which uniquely minimizes an energy functional over the space of probability measures, and which concentrates around the minimizer(s) of the associated potential when the noise parameter is small. We introduce analogous diffusion dynamics that sample from an entropy-regularized optimal transport, which uniquely minimizes the same energy functional but constrained to the set Π(μ,ν)\Pi(\mu,\nu) of couplings of two given marginal probability measures μ\mu and ν\nu on Rd\mathbb{R}^d, and which concentrates around the optimal transport coupling(s) for small regularization parameter. More specifically, our process satisfies two key properties: First, the law of the solution at each time stays in Π(μ,ν)\Pi(\mu,\nu) if it is initialized there. Second, the long-time limit is the unique solution of an entropic optimal transport problem. In addition, we show by means of a new log-Sobolev-type inequality that the convergence holds exponentially fast, for sufficiently large regularization parameter and for a class of marginals which strictly includes all strongly log-concave measures. By studying the induced Wasserstein geometry of the submanifold Π(μ,ν)\Pi(\mu,\nu), we argue that the SDE can be viewed as a Wasserstein gradient flow on this space of couplings, at least when d=1d=1, and we identify a conjectural gradient flow for d≥2d \ge 2. The main technical difficulties stems from the appearance of conditional expectation terms which serve to constrain the dynamics to Π(μ,ν)\Pi(\mu,\nu)
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