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Both lectures focus on the first part of the so-called 'mathematical part' of Plato's Theaetetus. In this passage, the young Theaetetus briefly recounts the mathematical lesson given by the geometer Theodorus. The first lecture delves into…
These lectures notes were written for a summer school on Mathematics for post-quantum cryptography in Thi\`es, Senegal. They try to provide a guide for Masters' students to get through the vast literature on elliptic curves, without getting…
Shortly after the middle of the past century, a comprehensive presentation of Continuum Mechanics was written under supervision of Clifford Ambrose Truesdell III in two volumes of Siegfried Fluegge's Handbuch der Physik, a first in 1960…
The problem of drawing geographical maps is the one of mapping a subset of the sphere, representing a country or some other region on the surface of the Earth, into the Euclidean plane, minimising certain distortion properties that are…
Canonical algebras, introduced by C.M. Ringel in 1984, play an important role in the representation theory of finite dimensional algebras. They are equipped with a large contact surface to many further mathematical subjects like function…
Diophantine approximation explores how well irrational numbers can be approximated by rationals, with foundational results by Dirichlet, Hurwitz, and Liouville culminating in Roth's theorem. Schmidt's subspace theorem extends Roth's results…
The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues…
Since the beginning of the quest of hypercomplex numbers in the late eighteenth century, many hypercomplex number systems have been proposed but none of them succeeded in extending the concept of complex numbers to higher dimensions. This…
Recently, in a letter to Nature, del Rio et al.8 exploited the quantum viewpoint of the old but well-known thought experiment of Maxwell's demon, a tiny "man-machine" that processes only a single unit of information. In their work, they…
In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic…
This book invites readers to see mathematics not just as formulas and rules, but as the deepest expression of human thought. It begins by exploring the timeless idea of mathematics as a universal language, contrasting its precision with the…
The Lorenz attractor was introduced in 1963 by E. N. Lorenz as one of the first examples of \emph{strange attractors}. However Lorenz' research was mainly based on (non-rigourous) numerical simulations and, until recently, the proof of the…
A perplexing problem in understanding physical reality is why the universe seems comprehensible, and correspondingly why there should exist physical systems capable of comprehending it. In this essay I explore the possibility that rather…
Parametric geometry of numbers is a new theory, recently created by Schmidt and Summerer, which unifies and simplifies many aspects of classical Diophantine approximations, providing a handle on problems which previously seemed out of…
In this paper we discuss how teaching of mathematics for middle school and high school students can be improved dramatically when motivation of concepts and ideas is done through the classical problems and the history of mathematics. This…
In this work we investigate the topological information captured by the Euler obstruction of a map, $f:(X,0)\to (\mathbb{C}^{2},0)$, with $(X,0)$ a germ of a complex $d$-equidimensional singular space, with $d > 2$, and its relation with…
Repeatedly adding or subtracting the digital reversal to or from an integer, depending on which one is larger, can be treated as a dynamical system. On one hand, a three-digit version of this map running only two steps is the 1089…
Our aim is to emphasize the role of mathematical models in physics, especially models of geometry and probability. We briefly compare developments of geometry and probability by pointing to similarities and differences: from Euclid to…
Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [A.…
These notes are based on a talk given at the Summer School "Infinite-dimensional representations of finite-dimensional algebras" held at the University of Manchester in September 2015. They intend to provide a brief introduction to the…