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Let G be a group, and H a G-group defined by an imbedding map $G\rightarrow H$; in [12] we have defined a topology on a subset of normal subgroups of $H$, the so-called prime ideals. In this work, we generalize this topology to other…
We extend Greenberg's strong approximation theorem to schemes of finite presentation over valuation rings with arbitrary value group, using the ultraproduct method of Becker, Denef, Lipshitz and van den Dries. As an application, we prove a…
We argue that the complex numbers are an irreducible object of quantum probability. This can be seen in the measurements of geometric phases that have no classical probabilistic analogue. Having complex phases as primitive ingredient…
In a "naive" attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups T and F for any subfactor. The Thompson group elements are the "local scale transformations"…
This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice…
We introduce a notion of topological property (T) for \'etale groupoids. This simultaneously generalizes Kazhdan's property (T) for groups and geometric property (T) for coarse spaces. One main goal is to use this property (T) to prove the…
We present a precise definition of extended homotopy quantum field theories and develop an orbifold construction for these theories when the target space is the classifying space of a finite group $G$, i.e. for $G$-equivariant topological…
Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened "homotopy type theory". In this…
The paper contains an exposition of part of topology using partitions of unity. The main idea is to create variants of the Tietze Extension Theorem and use them to derive classical theorems. This idea leads to a new result generalizing…
The language of homotopy type theory has proved to be appropriate as an internal language for various higher toposes, for example with Synthetic Algebraic Geometry for the Zariski topos. In this paper we apply such techniques to the higher…
It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of…
These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum…
Group field theories are particular quantum field theories defined on D copies of a group which reproduce spin foam amplitudes on a space-time of dimension D. In these lecture notes, we present the general construction of group field…
Recent work suggests that topological features of certain quantum gravity theories can be interpreted as particles, matching the known fermions and bosons of the first generation in the Standard Model. This is achieved by identifying…
One of the main motivations behind so-called topos physics, as developed by Chris Isham and Andreas Doering [4-7], is to provide a framework for new theories of quantum gravity. In this article we do not search for such theories, but ask…
We study groups, exponential groups and ordered groups equipped with valuations. We investigate algebraic and topological features of such valued structures, and apply our findings in order to solve regular equations over groups using…
Motivated by the recent rapid development of complexity theory applied to quantum mechanical processes we present the complete derivation of Nielsen's complexity of unitaries belonging to the representations of oscillator group. Our…
Inspired by classical ("actual") Quantum Theory over $\mathbb{C}$ and Modal Quantum Theory (MQT), which is a model of Quantum Theory over certain finite fields, we introduce General Quantum Theory as a Quantum Theory -- in the K{\o}benhavn…
In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets S, T and U of a group G satisfying the Triple Product Property (TPP) as a means to bound the exponent $\omega$…
Tame geometry originated in mathematical logic and implements strong finiteness properties by defining the notion of tame sets and functions. In part I we argued that observables in a wide class of quantum field theories are tame functions…