## Archive for the ‘**MATHEMATICS**’ Category

## Solid angles in perspective

**Paul Quincey**

The specialised uses of solid angles mean that they are quite unfamiliar quantities. This article, apart from making solid angles a little more familiar, brings out several topics of general interest, such as how units are interrelated and how equations depend on the choice of units. Although the steradian is commonly used as the unit for solid angle, another unit, the square degree, is used in astronomy, and a unit introduced here, the solid degree (with 360 solid degrees in a hemisphere) could be used with benefits that are similar to those of the degree when it is used as the unit for plane angle. The article, which is suitable for students at A-level and introductory undergraduate level, also shows how solid angles can provide a gentle introduction to crystal structure, spherical trigonometry and non-Euclidean geometry.

read more at arxiv.org/abs/2108.05226

## Applying physics to mathematics

**by Tadashi Tokieda**

abstract : Humans tend to be better at physics than at mathematics. When an apple falls from a tree, there are more people who can catch it—we know physically how the apple moves—than people who can compute its trajectory from a differential equation. Applying physical ideas to discover and establish mathematical results is therefore natural, even if it has seldom been tried in the history of science. (The exceptions include Archimedes, some old Russian sources, a recent book by Mark Levi, as well as my articles and lectures.) This TMC Distinguished Lecture presents a diversity of examples, and tries to make them easy for imaginative beginners and difficult for seasoned researchers.

## From Ramanujan to renormalization: the art of doing away with divergences and arriving at physical results

Wolfgang Bietenholz

A century ago Srinivasa Ramanujan – the great self-taught Indian genius of mathematics – died, shortly after returning from Cambridge, UK, where he had collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding results to different branches of mathematics, like analysis and number theory, with a focus on special functions and series. Here we refer to apparently weird values which he assigned to two simple divergent series, and . These values are sensible, however, as analytic continuations, which correspond to Riemann’s ζ-function. Moreover, they have applications in physics: we discuss the vacuum energy of the photon field, from which one can derive the Casimir force, which has been experimentally measured. We also discuss its interpretation, which remains controversial. This is a simple way to illustrate the concept of renormalization, which is vital in quantum field theory.

Read more at

## Packing Moons Inside the Earth

**Sunil K. Chebolu**

Using ideas of sphere packing problem we estimate the number of solid moons that can be packed inside the Earth, assuming that both the Moon and the Earth are perfect sphere.

Read more at https://arxiv.org/abs/2006.00603

## Joe Polchinski Memorial Lecture: A Brief History of Branes

**Paul Townsend (University of Cambridge, Department of Applied Mathematics and Theoretical Physics, UK)**

Abstract of the memorial lecture “A Brief History of Branes”: Joe Polchinski made many groundbreaking discoveries in theoretical physics. This talk will focus on his contributions to the circle of ideas that led to M-theory in the late 1990s, especially his work of the 1980s on supermembranes (’86) and D-branes and T-duality (’89). This will be part of a survey of the changing role of branes in physics, with personal commentary on various related topics (such as M-branes, U-dualities, black branes) in supergravity and string theory.

Polchinski was a professor at the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara. His great contributions to theoretical physics, including the discovery of D-branes –– a type of membrane in string theory –– have led to advances in the understanding of string theory and quantum gravity. In 2008, he shared ICTP’s Dirac Medal with Juan Maldacena and Cumrun Vafa for their fundamental contributions to superstring theory. The three scientists’ profound achievements have helped to address outstanding questions like confinement of quarks and QCD mass spectrum from a new perspective and have found applications in practical calculations. In addition to the Dirac Medal, Polchinski was awarded the American Physical Society’s 2007 Dannie Heineman Prize for Mathematical Physics, the Milner Foundation’s Physics Frontiers Prize in 2013 and 2014, as well as the 2017 Breakthrough Prize in Fundamental Physics. His work touched the lives of many ICTP scientists, from the hundreds who attended his lectures to those who worked directly with him.

## Is The Starry Night Turbulent?

**James Beattie, Neco Kriel**

Vincent van Gogh’s painting, The Starry Night, is an iconic piece of art and cultural history. The painting portrays a night sky full of stars, with eddies (spirals) both large and small. Kolmogorov1941’s description of subsonic, incompressible turbulence gives a model for turbulence that involves eddies interacting on many length scales, and so the question has been asked: is The Starry Night turbulent? To answer this question, we calculate the azimuthally averaged power spectrum of a square region (1165×1165 pixels) of night sky in The Starry Night. We find a power spectrum, P(k), where k is the wavevector, that shares the same features as supersonic turbulence. It has a power-law P(k)∝k

^{2.1±0.3 }in the scaling range, 34≤k≤80. We identify a driving scale, k

_{D}=3, dissipation scale, kν=220 and a bottleneck. This leads us to believe that van Gogh’s depiction of the starry night closely resembles the turbulence found in real molecular clouds, the birthplace of stars in the Universe.

Read more at https://arxiv.org/pdf/1902.03381.pdf