Logical and algebraic structures from Quantum Computation

The main motivation for this thesis is given by the open problems regarding the axiomatisation of quantum computational logics. This thesis will be structured as follows: in Chapter 2 we will review some basics of universal algebra and functional analysis. In Chapters 3 through 6 the fundamentals of quantum gate theory will be produced. In Chapter 7 we will introduce quasi-MV algebras, a formal study of a suitable selection of algebraic operations associated with quantum gates. In Chapter 8 quasi-MV algebras will be expanded by a unary operation hereby dubbed square root of the inverse, formalising a quantum gate which allows to induce entanglement states. In Chapter 9 we will investigate some categorial dualities for the classes of algebras introduced in Chapters 7 and 8. In Chapter 10 the discriminator variety of linear Heyting quantum computational structures, an algebraic counterpart of the strong quantum computational logic, will be considered. In Chapter 11, we will list some open problems and, at the same time, draw some tentative conclusions. Lastly, in Chapter 12 we will provide a few examples of the previously investigated structures

On pp-adic LL-functions for Hilbert modular forms

We construct $p$-adic $L$-functions associated with $p$-refined cohomological cuspidal Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in $p$-adic families, and does not require any small slope or non-criticality assumptions on the $p$-refinement. The main new ingredients are an adelic definition of a canonical map from overconvergent cohomology to a space of locally analytic distributions on the relevant Galois group and a smoothness theorem for certain eigenvarieties at critically refined points.Comment: 88 page

On pp-adic LL-functions for Hilbert modular forms

We construct $p$-adic $L$-functions associated with $p$-refined cohomological cuspidal Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in $p$-adic families, and does not require any small slope or non-criticality assumptions on the $p$-refinement. The main new ingredients are an adelic definition of a canonical map from overconvergent cohomology to a space of locally analytic distributions on the relevant Galois group and a smoothness theorem for certain eigenvarieties at critically refined points

An axiomatic approach to virtual chains

We introduce a category of Kuranishi presentations, whose objects are a variant of the Kuranishi structures introduced by Fukaya and Ono, and which can be seen as a refinement of the version studied by Pardon. We then formulate the notion of virtual chains categorically as a natural transformation between two functors from this category to the category of chain complexes; we call such a datum 'a theory of virtual counts'. To show that this definition carries non-trivial content, we then construct a multicategory whose objects are Kuranishi flow categories, and show that a theory of virtual counts determines a multifunctor to the multicategory of chain complexes. We then implement this construction in the setting of Hamiltonian Floer theory, borrowing from some joint work with Groman and Varolgunes, yielding a construction of Hamiltonian Floer groups (and operations on them) as an output of this machine. We plan to provide a similar account for Lagrangian Floer theory in subsequent joint work.Comment: 99 pages, 5 figures. Revised to modify the discussion of Hamiltonian Floer to conform more closely to the framework in arXiv:2210.1102

BPS cohomology for 2-Calabi—Yau categories

The integrality conjecture predicts how the refined Donaldson—Thomas invariants of Abelian categories are determined by smaller BPS invari- ants. In this thesis, we prove the cohomological integrality conjecture for 2-Calabi—Yau categories, that is, we prove that the underlying mixed Hodge structure of the Borel—Moore homology of the moduli stacks of objects is isomorphic to a symmetric algebra generated by BPS cohomology and the C*-equivariant cohomology of a point. 2-Calabi—Yau (2CY) categories are homological dimension 2 categories together with bifunctorial pairing on the shifted extension groups. This pairing can roughly be thought of as a symplectic form on the category. The ubiquity of 2CY categories is illustrated by the following list of examples: categories of coherent sheaves on symplec- tic surfaces, local systems on Riemann surfaces, categories of Higgs bundles, and representations of preprojective algebras of quivers. The cohomological Hall algebra (CoHA) A of a 2CY category plays a prominent role. Using the CoHA we identify the BPS algebra as the universal enveloping algebra of a generalised Kac—Moody (GKM) Lie algebra which we define to be the BPS Lie algebra of the 2CY category. Thus we realise the BPS cohomology as the underlying mixed Hodge structure of the GKM Lie algebra; we prove a Poincare ́—Birkhoff—Witt- type isomorphism for the CoHA in terms of the BPS Lie algebra and the C*-equivariant-equivariant cohomology of a point. The explicit description of the BPS Lie algebra as a GKM algebra reveals a core aspect of our result: the BPS Lie algebra is generated in terms of the intersection cohomology of good moduli spaces of objects in the 2CY category. In the case of totally negative 2CY categories (pairs of nonzero objects of the category have negative Euler pairing) we find the BPS Lie algebra to be the free Lie algebra generated by the intersection cohomology. The main results of the thesis are based on joint work with Ben Davison and Lucien Hennecart

Hodge and Gelfand theory in Clifford analysis and tomography

2022 Summer.Includes bibliographical references.There is an interesting inverse boundary value problem for Riemannian manifolds called the Calderón problem which asks if it is possible to determine a manifold and metric from the Dirichlet-to-Neumann (DN) operator. Work on this problem has been dominated by complex analysis and Hodge theory and Clifford analysis is a natural synthesis of the two. Clifford analysis analyzes multivector fields, their even-graded (spinor) components, and the vector-valued Hodge–Dirac operator whose square is the Laplace–Beltrami operator. Elements in the kernel of the Hodge–Dirac operator are called monogenic and since multivectors are multi-graded, we are able to capture the harmonic fields of Hodge theory and copies of complex holomorphic functions inside the space of monogenic fields simultaneously. We show that the space of multivector fields has a Hodge–Morrey-like decomposition into monogenic fields and the image of the Hodge–Dirac operator. Using the multivector formulation of electromagnetism, we generalize the electric and magnetic DN operators and find that they extract the absolute and relative cohomologies. Furthermore, those operators are the scalar components of the spinor DN operator whose kernel consists of the boundary traces of monogenic fields. We define a higher dimensional version of the Gelfand spectrum called the spinor spectrum which may be used in a higher dimensional version of the boundary control method. For compact regions of Euclidean space, the spinor spectrum is homeomorphic to the region itself. Lastly, we show that the monogenic fields form a sheaf that is locally homeomorphic to the underlying manifold which is a prime candidate for solving the Calderón problem using analytic continuation