### Tag Archives: Computer Science

## Mathematical Physics, Determinism, Game of Life (Hiking in Modern Math 6/7)

## Chaos Theory, Meteorology, Navier-Stokes, Wolfram (Hiking in Modern Math 5/7)

## Complexity Theory, P versus NP, RSA Cryptography (Hiking in Modern Math 2/7)

## Theory of Computation, Turing Machine (Hiking in Modern Math 1/7)

## Shannon’s Information Theory (Trek through Math 5/8)

## Univalent Foundations of Mathematics

In an effort to make mathematics more computable, a consortium of today's greatest mathematicians have laid out new foundations. Amazingly, they all lie upon one single axiom, called univalence. The goal of this axiom is to make formal mathematics more similar to informal mathematics. With univalence, our Arabic numbers aren't just like natural numbers; They are natural numbers. Univalence also has unforeseen and mesmerizing consequences.

## Homotopy Type Theory and Higher Inductive Types

In this article, we explore the possibilities allowed by higher inductive types. They enable a much more intuitive formalization of integers and new mind-blowing definitions of the (homotopical) circle and sphere.

## Type Theory: A Modern Computable Paradigm for Math

In 2013, three dozens of today's brightest minds have just laid out new foundation of mathematics after a year of collective effort. This new paradigm better fits both informal and computationally-checkable mathematics. There is little doubt that it will fundamentally change our perspective on rigorous knowledge, and it could be that, in a few decades, the book they published turns out to be the bedrock of all mathematics, and, by extension, all human knowledge! Have a primer of this upcoming revolution, with this article on type theory, the theory that the book builds upon!

## The Tortuous Geometry of the Flat Torus

Take a square sheet of paper. Can you glue opposite sides without ever folding the paper? This is a conundrum that many of the greatest modern mathematicians, like Gauss, Riemann, and Mandelbrot, couldn't figure out. While John Nash did answer yes, he couldn't say how. After 160 years of research, Vincent Borrelli and his collaborators have finally provided a revolutionary and breathtaking example of a bending of a square sheet of paper! And it is spectacularly beautiful!

## The New Big Fish Called Mean-Field Game Theory

In recent years, at the interface of game theory, control theory and statistical mechanics, a new baby of applied mathematics was given birth. Now named mean-field game theory, this new model represents a new active field of research with a huge range of applications! This is mathematics in the making!

## Numbers and Constructibility

Last summer, I got to discover Morellet's artwork on inclined grids. Amazingly, this artwork is a display of the irrationality of $\sqrt{2}$! It's also a strong argument for the existence of this number. In this article, after discussing that, I take readers further by discussing what numbers can be constructed geometrically, algebraically, analytically or set theoretically using the power of mathematics!

## Shannon’s Information Theory

Claude Shannon may be considered one of the most influential person of the 20th Century, as he laid out the foundation of the revolutionary information theory. Yet, unfortunately, he is virtually unknown to the public. This article is a tribute to him. And the best way I've found is to explain some of the brilliant ideas he had.

## Cryptography and Quantum Physics

Recent discoveries in the branch of physics known as quantum mechanics have powerful applications in the field of network security - they have the potential to break forms of internet security based on mathematics such as the RSA algorithm, and also present new ways to safely send information. In this article we’ll see how a physics-based method can be used to secure online information.

## Bayesian Games: Math Models for Poker

How to better understand Poker and card games in general? Bayesian games provide the right mathematical model just for that! These correspond to games with incomplete information and include probabilistic reasonings.

## Probabilistic Algorithms, Probably Better

Probabilities have been proven to be a great tool to understand some features of the world, such as what can happen in a dice game. Applied to programming, it has enabled plenty of amazing algorithms. In this article, we discuss its application to the primality test as well as to face detection. We'll also deal with quantum computers, as well as fundamental computer science open problems P=BPP and NP=BQP.

## Web Programming: From HTML to AJAX

The Internet is an unavoidable component of today's life, and will only become more and more important in the future. In this article, we"ll have an overview of its development in terms of coding languages. This article explains how webpages actually work. In particular, we'll discuss HTML, CSS, PHP, WordPress, MySQL, Javascript, XML and JSON, as well as their competitors.

## P versus NP: A Crucial Open Problem

P=NP is probably today's most crucial open problem. Not only is it a very theoretical question in computer science and mathematics, but it also has major implications to real world. Its resolution could revolutionize the world. This article gives the definition and major results of P=NP.

## From Divide and Conquer to Parallelization

Divide and conquer is a extremely powerful concept that is being used a lot in computer science, and that can also be applied in real life. We present its application to sorting algorithms. Then we'll talk about a major fundamental open mathematical problem, called P=NC.