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Probabilistic models learned as density estimators can be exploited in representation learning beside being toolboxes used to answer inference queries only. However, how to extract useful representations highly depends on the particular…
We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer…
For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect…
We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length $\omega$ to computations on tapes of arbitrary ordinal length. We show that a set of ordinals is ordinal computable from a finite…
A well-known result on Lie Theory states that every finite-dimensional complex solvable Lie algebra can be represented as a matrix Lie algebra, with upper-triangular square matrices as elements. However, this result does not specify which…
Computational problems are classified into computable and uncomputable problems. If there exists an effective procedure (algorithm) to compute a problem then the problem is computable otherwise it is uncomputable. Turing machines can…
We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and…
We prove new Brown representability theorems for triangulated categories using metric techniques as introduced in the work of Neeman. In the setting of algebraic geometry, this gives us new representability theorems for homological and…
The notion of weak truth-table reducibility plays an important role in recursion theory. In this paper, we introduce an elaboration of this notion, where a computable bound on the use function is explicitly specified. This elaboration…
This paper concerns algorithms that give correct answers with (asymptotic) density $1$. A dense description of a function $g : \omega \to \omega$ is a partial function $f$ on $\omega$ such that $\left\{n : f(n) = g(n)\right\}$ has density…
We discuss computability and computational complexity of conformal mappings and their boundary extensions. As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures and…
In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame…
Let $G$ be a finite solvable group. Then $G$ always has a useful presentation, which we call a "long presentation". Using a "long presentation" of $G$, we present an inductive method of constructing the irreducible representations of $G$…
We introduce a uniform representation of general objects that captures the regularities with respect to their structure. It allows a representation of a general class of objects including geometric patterns and images in a sparse, modular,…
A relatively new topic in computability theory is the study of notions of computation that are robust against mistakes on some kind of small set. However, despite the recent popularity of this topic relatively foundational questions about…
We propose the representation principle to study physical systems with a given symmetry. In the context of symmetry enriched topological orders, we give the appropriate representation category, the category of SET orders, which include SPT…
We make two theoretical contributions to disentanglement learning by (a) defining precise semantics of disentangled representations, and (b) establishing robust metrics for evaluation. First, we characterize the concept "disentangled…
We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model $\mathcal A$ of a computably enumerable, model complete theory, the…
As suggested by the title, this paper is a survey of recent results and questions on the collection of computably enumerable sets under inclusion. This is not a broad survey but one focused on the author's and a few others' current…
We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of…