17,345 research outputs found
A Note on Derived Geometric Interpretation of Classical Field Theories
In this note, we would like to provide a conceptional introduction to the
interaction between derived geometry and physics based on the formalism that
has been heavily studied by Kevin Costello. Main motivations of our current
attempt are as follows: (i) to provide a brief introduction to derived
algebraic geometry, which can be, roughly speaking, thought of as a higher
categorical refinement of an ordinary algebraic geometry, (ii) to understand
how certain derived objects naturally appear in a theory describing a
particular physical phenomenon and give rise to a formal mathematical
treatment, such as redefining a perturbative classical field theory (or its
quantum counterpart) by using the language of derived algebraic geometry, and
(iii) how the notion of factorization algebra together with certain higher
categorical structures come into play to encode the structure of so-called
observables in those theories by employing certain cohomological/homotopical
methods. Adopting such a heavy and relatively enriched language allows us to
formalize the notion of quantization and observables in quantum field theory as
well.Comment: 14 pages. This note serves as an introductory survey on certain
mathematical structures encoding the essence of Costello's approach to
derived-geometric formulation of field theories and the structure of
observables in an expository manne
Sheaves and schemes: an introduction to algebraic geometry
Master of ScienceDepartment of MathematicsRoman FedorovThe purpose of this report is to serve as an introduction to the language of sheaves and schemes via algebraic geometry. The main objective is to use examples from algebraic geometry to motivate the utility of the perspective from sheaf and scheme theory. Basic facts and definitions will be provided, and a categorical approach will be frequently incorporated when appropriate
A Higher-Order Calculus for Categories
A calculus for a fragment of category theory is presented. The types in the language denote categories and the expressions functors. The judgements of the calculus systematise categorical arguments such as: an expression is functorial in its free variables; two expressions are naturally isomorphic in their free variables. There are special binders for limits and more general ends. The rules for limits and ends support an algebraic manipulation of universal constructions as opposed to a more traditional diagrammatic approach. Duality within the calculus and applications in proving continuity are discussed with examples. The calculus gives a basis for mechanising a theory of categories in a generic theorem prover like Isabelle
Algebraic groups in non-commutative probability theory revisited
The role of coalgebras as well as algebraic groups in non-commutative
probability has long been advocated by the school of von Waldenfels and
Sch\"urmann. Another algebraic approach was introduced more recently, based on
shuffle and pre-Lie calculus, and resulting in another construction of groups
of characters encoding the behaviour of states. Comparing the two, the first
approach, recast recently in a general categorical language by Manzel and
Sch\"urmann, can be seen as largely driven by the theory of universal products,
whereas the second construction builds on Hopf algebras and a suitable
algebraization of the combinatorics of noncrossing set partitions. Although
both address the same phenomena, moving between the two viewpoints is not
obvious. We present here an attempt to unify the two approaches by making
explicit the Hopf algebraic connections between them. Our presentation,
although relying largely on classical ideas as well as results closely related
to Manzel and Sch\"urmann's aforementioned work, is nevertheless original on
several points and fills a gap in the free probability literature. In
particular, we systematically use the language and techniques of algebraic
groups together with shuffle group techniques to prove that two notions of
algebraic groups naturally associated with free, respectively Boolean and
monotone, probability theories identify. We also obtain explicit formulas for
various Hopf algebraic structures and detail arguments that had been left
implicit in the literature.Comment: 24 page
On Multi-Language Semantics: Semantic Models, Equational Logic, and Abstract Interpretation of Multi-Language Code
Modern software development rarely takes place within a single programming language. Often, programmers appeal to cross-language interoperability. Benefits are two-fold: exploitation of novel features of one language within another, and cross-language code reuse. For instance, HTML, CSS, and JavaScript yield a form of interoperability, working in conjunction to render webpages. Some object oriented languages have interoperability via a virtual machine host (.NET CLI compliant languages in the Common Language Runtime, and JVM compliant languages in the Java Virtual Machine). A high-level language can interact with a lower level one (Apple's Swift and Objective-C). Whilst this approach enables developers to benefit from the strengths of each base language, it comes at the price of a lack of clarity of formal properties of the new multi-language, mainly semantic specifications. Developing such properties is a key focus of this thesis. Indeed, while there has been some research exploring the interoperability mechanisms, there is little development of theoretical foundations. In this thesis, we broaden the boundary functions-based approach Ă la Matthews and Findler to propose an algebraic framework that provides systematic and more general ways to define multi-languages, regardless of the inherent nature of the underlying languages. The aim of this strand of research is to overcome the lack of a formal model in which to design the combination of languages. Main contributions are an initial algebra semantics and a categorical semantics for multi-languages. We then give ways in which interoperability can be reasoned about using equations over the blended language. Formally, multi-language equational logic is defined, within which one may deduce valid equations starting from a collection of axioms that postulate properties of the combined language. Thus, we have the notion of a multi-language theory and part of the thesis is devoted to exploring the properties of these theories. This is accomplished by way of both universal algebra and category theory, giving us a very general and flexible semantics, and hence a wide collection of models. Classifying categories are constructed, and hence equational theories furnish each categorical model with an internal language. From this we establish soundness and completeness of the multi-language equational logic. As regards static analysis, the heterogeneity of the multi-language context opens up new and unexplored scenarios. In this thesis, we provide a general theory for the combination of abstract interpretations of existing languages in order to gain an abstract semantics of multi-language programs. As a part of this general theory, we show that formal properties of interest of multi-language abstractions (e.g., soundness and completeness) boil down to the features of the interoperability mechanism that binds the underlying languages together. We extend many of the standard concepts of abstract interpretation to the framework of multi-languages. Finally, a minor contribution of the thesis concerns language specification formalisms. We prove that longstanding syntactical transformations between context-free grammars and algebraic signatures give rise to adjoint equivalences that preserve the abstract syntax of the generated terms. Thus, we have methods to move from context-free languages to the algebraic signature formalisms employed in the thesis
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