Related papers: Counting points and acquiring flesh
These notes discuss, in a style intended for physicists, how to average data and fit it to some functional form. I try to make clear what is being calculated, what assumptions are being made, and to give a derivation of results rather than…
These are lecture notes prepared for London Mathematical Society Symposium "Quantum Groups" held in Durham 19-29 July 1999. We give a survey on cyclotomic Hecke algebras. These algebras have been studied by Dipper, James, Malle, Mathas and…
Group theory is a particularly fertile field for the design of practical algorithms. Algorithms have been developed across the various branches of the subject and they find wide application. Because of its relative maturity, computational…
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication…
Even though flt is a number theoretic result we prove that the result depends on the topological as well as the field structure of the underlying space.
After the first heuristic ideas about `the field of one element' F_1 and `geometry in characteristics 1' (J.~Tits, C.~Deninger, M.~Kapranov, A.~Smirnov et al.), there were developed several general approaches to the construction of…
This is a draft of an article to appear in the October 2022 issue of the Notices of the AMS. In this survey article we explore a fascinating area called descriptive combinatorics and its recently discovered connections to distributed…
This book is based on notes compiled over the many years I have been teaching the course "Applied Functional Analysis" in the first year of the Master programme at Delft University of Technology, for students with previous exposure to the…
This manuscript synthesizes almost fifteen years of research in algebraic combinatorics, in order to highlight, theme by theme, its perspectives. In part one, building on my thesis work, I use tools from commutative algebra, and in…
We model the field $F_1$ of one element as a lambda ring $\bf Z$ with the canonical lambda structure. We show that then we can calculate the Riemann zeta function of integers in two ways: the first, geometrical, as a zeta function of the…
The recent hardware trend towards reduced precision computing has reignited the interest in numerical techniques that can be used to enhance the accuracy of floating point operations beyond what is natively supported for basic arithmetic…
This is an expository paper, in which we give a summary of some of the joint work of John Luecke and the author on Dehn surgery. We consider the situation where we have two Dehn fillings $M(\alpha)$ and $M(\beta)$ on a given 3-manifold $M$,…
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…
This paper collects and extends the lectures I gave at the "XXIV International Fall Workshop on Geometry and Physics" held in Zaragoza (Spain) August 31 - September 4, 2015. Within these lectures I review the formulation of Quantum…
This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Winter term 2000-2001, to undergraduate Mathematics and Physics students. Contents: Chapter 1 - Examples of Dynamical Systems Chapter 2 -…
This presentation is a concise teaser for the Effective Field Theory (EFT) of two and three nucleon systems as it emerged in the last three years, using a lot of words and figures, and a few cheats.
In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in…
This article is a rough, quirky overview of both the history and present state of the art of density functional theory. The field is so huge that no attempt to be comprehensive is made. We focus on the underlying exact theory, the origin of…
In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset…
We establish the equivalence of three notions of $\mathbb{F}_q$-rational points on weighted projective spaces $\mathbb{P}_{\mathbf{w}}^n$ and derive explicit combinatorial formulas for their enumeration, leveraging Burnside's lemma and gcd…