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This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on…

A program schema defines a class of programs, all of which have identical statement structure, but whose functions and predicates may differ. A schema thus defines an entire class of programs according to how its symbols are interpreted. A…

Let $N$ be a normal subgroup of a group $G$. An $N$-module $Q$ is $G$-stable provided that $Q$ is equivalent to the twist $Q^g$ of $Q$ by $g$, for every $g\in G$. If the action of $N$ on $Q$ extends to an action of $G$ on $Q$, $Q$ is…

This paper describes a way to improve the scalability of program synthesis by exploiting modularity: larger programs are synthesized from smaller programs. The key issue is to make each "larger-created-from-smaller" synthesis sub-problem be…

In this note, we introduce the notion of support graph to define explanations for any model of a logic program. An explanation is an acyclic support graph that, for each true atom in the model, induces a proof in terms of program rules…

We construct the derived scheme of stable sheaves on a smooth projective variety via derived moduli of finite graded modules over a graded ring. We do this by dividing the derived scheme of actions of Ciocan-Fontanine and Kapranov by a…

A stable pair on a projective variety consists of a sheaf and a global section subject to stability conditions parameterized by rational polynomials. We will show that for a smooth projective threefold and a class of a rank 2 sheaf, there…

While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is known to be unstable in a certain precise sense. Until now, it has not…

Relaxed highest-weight modules play a central role in the study of many important vertex operator (super)algebras and their associated (logarithmic) conformal field theories, including the admissible-level affine models. Indeed, their…

Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the…

We present an exploration of the rich theoretical connections between several classes of regularized models, network flows, and recent results in submodular function theory. This work unifies key aspects of these problems under a common…

We study modules over stacks of deformation quantization algebroids on complex Poisson manifolds. We prove finiteness and duality theorems in the relative case and construct the Hochschild class of coherent modules. We prove that this class…

In a Systems Engineering setting, various models are produced using a variety of methods and tools. Focusing on a type of models -- called descriptive models -- which we shall describe, we argue that, while the clarity and precision of…

A logic programming paradigm which expresses solutions to problems as stable models has recently been promoted as a declarative approach to solving various combinatorial and search problems, including planning problems. In this paradigm,…

We present a new type system with support for proofs of programs in a call-by-value language with control operators. The proof mechanism relies on observational equivalence of (untyped) programs. It appears in two type constructors, which…

In this paper, we define the asymptotic stable division property for submodules of the Bergman module. We show that under a mild condition, a submodule with the asymptotic stable division property is p-essentially normal for all p>n. A new…

Recently, symbolic structures were proposed as finite representations of potentially infinite first-order structures, where Linear Integer Arithmetic terms and formulas define the domain and interpretations of a structure. We generalize…

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module…

The theory of spin models intersects with condensed matter physics, complex systems, graph theory, combinatorial optimization, computational complexity and neural networks. Many ensuing applications rely on the fact that complicated spin…

While neural models show remarkable accuracy on individual predictions, their internal beliefs can be inconsistent across examples. In this paper, we formalize such inconsistency as a generalization of prediction error. We propose a…