Linear differential equations form the central topic of this volume, Galois theory being the unifying theme.
A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used.
This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

Galois theory of linear differential equations

Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified complement. Under suitable hypotheses, the maximum likelihood degree equals the top Chern class of a sheaf of logarithmic differential forms. Exact formulae in terms of degrees and Newton polytopes are given for polynomials with generic coefficients.

The maximum likelihood degree

We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic cohomology. These constructions yield new applications in symplectic topology relating to the Arnol'd chord conjecture and to exact contact embeddings. We prove that if a Liouville domain M admits an exact embedding into an exact convex symplectic manifold X, and the boundary of M is displaceable in X, then the symplectic cohomology of M vanishes and the chord conjecture holds for any Lagrangianly fillable Legendrian lying in the boundary. The TQFT respects the isomorphism between the symplectic cohomology of a cotangent bundle and the homology of the free loop space, so it recovers the TQFT of string topology. Finally, we use the TQFT to prove that symplectic cohomology vanishes iff Rabinowitz Floer cohomology vanishes.

Topological quantum field theory structure on symplectic cohomology

Radical ontic structural realism (ROSR) asserts an ontological commitment to 'free-standing' physical structures understood solely in terms of fundamental relations, without any recourse to relata which stand in these relations. Bain (2011) has recently defended ROSR against the common charge of incoherence by arguing that a reformulation of fundamental physical theories in category-theoretic terms (rather than the usual set-theoretic ones) offers a coherent and precise articulation of the commitments accepted by ROSR. In this essay, we argue that category theory does not offer a more hospitable environment to ROSR than set theory. We also show that the application of category-theoretic tools to topological quantum field theory and to algebraic generalisations of general relativity do not warrant the claim that these theories describe 'object-free' structures. We conclude that category theory offers little if any comfort to ROSR.

/pdf/no-categorial-support-for-radical-ontic-structural-realism-407at0p0om.pdf

No categorial support for radical ontic structural realism

The main objective of this thesis is to clarify concepts of generalised symmetries in noncommutative geometry (i.e., the noncommutative analogue of groupoids and Lie algebroids) and their associated (co)homologies. These ideas are incorporated by the notion of Hopf algebroids and Hopf-cyclic (co)homology. Among the various existing approaches, we focus on the definition of Hopf algebroids by Boehm-Szlachanyi, which understands a Hopf algebroid as consisting of left and right bialgebroid structures and an antipode intertwining these structures. Examples of bialgebroid and Hopf algebroid structures we discuss in detail include universal enveloping algebras of Lie-Rinehart algebras, jet spaces, and convolution and function algebras over etale groupoids. Generalised Connes-Moscovici algebras, i.e., spaces of transverse differential operators on arbitrary etale groupoids are also treated: we give a general description of the background procedure of the constructions by Connes-Moscovici and Moscovici-Rangipour, introducing the concept of matched pairs of bialgebroids and bicrossed product bialgebroids. The Connes-Moscovici algebras can then be shown to arise in such a way. Further general constructions we provide include for instance a categorical equivalence between left bialgebroid comodules and modules over its duals in the sense of Kadison-Szlachanyi. Central in this thesis, we indicate that Hopf-cyclic cohomology is naturally defined when using the concept of Hopf algebroids from Boehm-Szlachanyi: we explain how the Hopf-cyclic cohomology fits into the monoidal category of (Hopf algebroid) modules and show that it descends in a canonical way from the cyclic cohomology of corings. This generalises an analogous approach for Hopf algebras by Crainic. We also develop a dual cyclic homology theory for Hopf algebroids, obtained by cyclic duality in the sense of Connes and a generalised Hopf-Galois map (in the sense of Schauenburg) canonically associated to the Hopf algebroid. Such a map is required to mediate between the involved bialgebroid module and comodule categories. Ramifications of the theory we discuss comprise the identification of the Hochschild theory as certain derived functors. Also, we give general structure theorems for the cyclic theory of commutative and cocommutative Hopf algebroids in terms of their respective Hochschild groups. This generalises similar considerations for Hopf algebras in the work of Khalkhali-Rangipour. We then calculate Hopf-cyclic cohomology and dual Hopf-cyclic homology in concrete examples of Hopf algebroids, such as universal enveloping algebras of Lie-Rinehart algebras, jet spaces and convolution algebras over etale groupoids. The results of these computations are establishing Hopf-cyclic (co)homology as a noncommutative extension of both Lie-Rinehart (co)homology and groupoid homology. Moreover, a special method to obtain dual Hopf-cyclic homology for convolution algebras is constructed. This is achieved by a method which shows how in particular cases the dual theory fits into the monoidal category of (left and right bialgebroid) comodules. Finally, we prove a theorem that intimates that $\times_A$-Hopf algebras (in the sense of Schauenburg) are a key concept for multiplicative structures (such as cup, cap and Yoneda products) and certain duality isomorphisms in algebraic (co)homology theories. In particular, results on Hochschild (co)homology by Van den Bergh and Lie-Rinehart (co)homology by Huebschmann are obtained this way.

/pdf/hopf-algebroids-and-their-cyclic-theory-5b17bt48by.pdf

Hopf Algebroids and Their Cyclic Theory

The“Gel’fand sheaf” of a topological algebra is endowed with auniform structure, this being complete if and only if, the spectrum of the algebra considered is complete. Examples are also provided.

On the sheaf theory

We examine some topological algebras with ascending or descending chain condition. We prove that a commutative noetherian F-algebra is necessarily a Q-algebra. We characterize noetherian F-algebras which are Q-algebras among those whose left ideals are closed. We show that any commutative artinian m-convex algebra is finite dimensional.

Topological algebras with ascending or descending chain condition

/pdf/algebres-topologiques-a-ideaux-a-gauche-fermes-8067xd9eki.pdf

Algèbres topologiques à idéaux à gauche fermés

/pdf/fonctions-entieres-et-m-convexite-2ewulmvh6u.pdf

Fonctions entières et {$m$}-convexité

We show that a regular von Neumann Q-m-convex Frechet algebra is of finite dimension. We also show that a regular von Neumann m-convex Frechet algebra is a projective limit of finite dimensional algebras. Finally, we prove that a bilateral Q-F-algebra is a regular von Neumann algebra if and only if it is isomorphic to a finite product of algebras which are also fields.

/pdf/on-some-von-neumann-topological-algebras-35dln2bjz4.pdf

On some von neumann topological algebras

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