mathematics

2017 LMS Winners

The London Mathematics Society (LMS) has announced the winners for their various prizes and medals this year. In this blogpost I will give a breakdown of who won.

  • Polya Prize: This was awarded to Professor Alex Wilkie FRS from the University of Oxford due to his contributions to model theory and its connections to real analytic geometry.
  • Senior Whitehead Prize: Professor Peter Cameron, from the University of St Andrews, was awarded this prize for his research on combinatorics and group theory.
  • Senior Anne Bennett Prize: Awarded to Professor Alison Etheridge FRS from the University of Oxford for her “research on measure-valued stochastic processes and applications to population biology” as well as outstanding leadership.
  • Naylor Prize and Lectureship: This was given to Professor John Robert King from the University of Nottingham due to profound contributions to the theory of non-linear PDEs and applied mathematical modelling.
  • Berwick Prize: This was awarded to Kevin Costello of the Perimeter Institute in Canada for his paper entitled The partition function of a topological field theory (published in the Journal of Topology in 2009). In this paper Costello “characterises the function as the unique solution of a master equation in a Fock space.”
  • Whitehead Prize:
    • Dr Julia Gog (University of Cambridge) for her research on the mathematical understanding of disease dynamics, in particular influenza.
    • Dr András Máthé (Univeristy of Warwick) due to his insights into problems in the fields of geometric measure theory, combinatorics and real analysis.
    • Ashley Montanaro (University of Bristol) for her contributions to quantum computation and quantum information theory.
    • Dr Oscar Randal-Williams (University of Cambridge) due to his contributions to algebraic topology, in particular the study of moduli spaces of manifolds.
    • Dr Jack Thorne (University of Cambridge) for research in number theory, in particular the Langlands program.
    • Professor Michael Wemyss (University of Glasgow) for his “applications of algebraic and homological techniques to algebraic geometry.”

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Pictures of Math VII

This post was inspired by an article I read on sciencealert.com.

Simon Beck’s Snow Art

(source)

Simon Beck is the world’s first snow artist. Each pattern takes him 11 hours, and he uses nothing more than a compass and his snowshoes. He chooses to draw maths due to the simplicity of the patterns.

Fabergé Fractals

UK physicist Tom Beddard decided to create digital renderings of 3D Fabergé eggs covered in fractal patterns.

“The formulae effectively fold, scale, rotate or flip space. They are truly fractal in the fact that more and more detail can be revealed the closer to the surface you travel.” – Beddard

For more read here.

3D Models by Henry Segerman

625x465 1480313 6762325 1446759854

Henry Segerman is an Australian mathematician who creates 3D printed models that express mathematical formulae and concepts.  This allows his students to better understand them.

“The language of mathematics is often less accessible than the language of art, but I can try to translate from one to the other, producing a picture or sculpture that expresses a mathematical idea.” 

Hevea Project

A French team of mathematicians called the Hevea Project, have created digital constructions of isometric embeddings.

“Take a sphere – say the surface of a tennis ball – and imagine shrinking it down to have a nanometre radius,” writes Daniel Matthews about isometric embedding. “Nash and Kuiper show that by ‘ruffling’ the surface sufficiently (but always smoothly; no creasing or folding or ripping or tearing allowed!) you can have an isometric copy of your original tennis ball, all contained within this nanometre radius.”

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News: Mirzakhani has died at 40

When I read the news that Maryam Mirzakhani had sadly passed away with breast cancer aged 40 I was honestly shocked. I remember finding out that she was the first women to win the Fields Medal in 2014 and feeling a huge sense of pride that we had achieved such a big milestone in mathematics – a mostly male dominated subject.

Professor at Stanford Universtiy, Mirzakhani was awarded the notorious Field’s Medal for her work on complex geometry and dynamic systems. She specialised in areas of theoretical mathematics that “read like a foreign language by those outside of mathematics” such as moduli spaces, Teichmüller thoery, hyperbolic geometry, Ergodic theory and symplectic geometry. By mastering these fields, Mirzakhani could describe the geometric and dynamic complexities of curved surfaces, spheres, donut shapes and even amoebas in a huge amount of detail. Furthermore, her work had implications in a vast amount of fields, ranging from cryptography to the physics of how the universe was created.


Moduli Spaces

Moduli Spaces can be thought of as geometric solutions to geometric classification problems. In broad terms, a moduli problem consists of three main categories:

  1. Objects: which geometric objects do we want to parametrise?
  2. Equivalences: when do we identify two objects as being isomorphic?
  3. Families: how do we allow our objects to vary?

Read more here.

Teichmüller Theory

Teichmüller theory, which brings together an array of fundamental ideas from different mathematical fields (including complex analysis, hyperbolic geometry, differential geometry, etc), is concerned with the Teichmüller space.

To get an short introduction to Teichmüller theory, click here.

Hyperbolic Geometry

Hyperbolic geometry is a non-Euclidean geometry, where the parallel postulate of Euclidean geometry is replaced with:

For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R.” – Wikipedia

Source: debate.org

Ergodic Theory

Ergodic theory was initially developed to solve problems in statistical physics and is a branch of mathematics that studies “dynamical systems with an invariant measure”. An invariant measure is a measure that is preserved by some function.

Symplectic Geometry

Symplectic Geometry is a branch of differential geometry and differential topology that studies symplectic manifolds. These are differentiable manifolds that have a closed, non-degenerate 2-form.


 

“Mirzakhani once described her work as ‘like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out’.”

– Guardian

Mirzakhani will be remembered not only for her extraordinary work, but also as being an inspiration to thousands of women to pursue maths and science.

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Goodstein Theorem

On reading a magazine on Gödel’s Incompleteness Theorems, I came across a family of sequences of non-negative integers called Goodstein sequences and the Goodstein Theorem involving these sequences.

Goodstein’s thoerem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence converges to 0.

What is a Goodstein Sequence?

To understand what a Goodstein sequence, first we must understand what hereditary base-n notation is. Whilst this notation is very similar to the usual base-n notation, base-n notation is not sufficient for Goodstein’s theorem.

To convert a base-n representation to a hereditary base-n notation, first rewrite all of the exponents in base-n notation. Then rewrite any exponents inside the exponents, and continue this way until every number in the expression has been converted to base-n notation.

For example, 35 = 25 + 2 + 1 in ordinary base-2 notation but Screen Shot 2017-07-11 at 5.37.45 PM.png in hereditary base-2 notation.

Now we can define the Goodstein sequence G(m) of a natural number m. In general the (n+1)-st term G(m)(n+1) of the Goodstein sequence of m is given as follows (taken from Wikipedia):

  • Take the hereditary base-n + 1 representation of G(m)(n).
  • Replace each occurrence of the base-n + 1 with n + 2.
  • Subtract one. (Note that the next term depends both on the previous term and on the index n.)
  • Continue until the result is zero, at which point the sequence terminates.

In spite of the rapid growth of the terms in the sequence, Goodstein’s theorem states that every Goodstein sequence eventually terminates at 0, no matter what the starting value is.

Mathematicians Laurie Kirby and Jeff Paris proved in 1982 that Goodstein’s theorem is not provable in ordinary Peano arithmetic. In other words, this is the type of theorem described in 1931 by Gödel’s incompleteness theorem.

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Eiffel Tower

“Is it not true that the very conditions which give strength also conform to the hidden rules of harmony ? … Now to what phenomenon did I have to give primary concern in designing the Tower? It was wind resistance. Well then ! I hold that the curvature of the monument’s four outer edges, which is as mathematical calculation dictated it should be … will give a great impression of strength and beauty, for it will reveal to the eyes of the observer the boldness of the design as a whole.”

-Eiffel in an interview in Le Temps of February 14th 1887

Gustave Eiffel was proud of the good-looking Tower he created whose shape resulted from mathematical calculation: at any height on the Tower, the moment of the weight of the higher part of the Tower, up to the top, is equal to the moment of the strongest wind on this same part. By writing the differential equation of this equilibrium, we can find the harmonious equation that describes the shape of the Tower.

The Equation

Let A be a point on the edge of the tower, and x be the distance between the top of the Tower and A. Let P(x) be the weight of the part of the Tower above A up to the top of the tower. If f(x) is half the width of the Tower at A, then the moment of the weight of the Tower relative to A = P(x)f(x).

Consider a piece of the Eiffel Tower at a distance t from the top of the Tower, with its thickness equal to dt. Then, viewed from the top, this looks like a square with width 2f(t).

The forces on this piece are:

  • Weight dP(t):  proportional to its volume – dP(t) = 4kf(t)^2dt
  • horizontal wind dV(t): proportional to the part of the surface in direction of the wind – dV(t) = 2Kf(t)dt

Differential Equation

The absolute value of the moment of dP(t) relative to A = dP(t)f(x) and the absolute value of the moment of dV(t) relative to A = dV(t)(x-t). These should be equal so

Screen Shot 2017-07-10 at 4.39.08 PM.png

 

Let a = 2k/K. Then f, the function which give the width of the Eiffel Tower as a function of the distance from the top, is a solution to

Screen Shot 2017-07-10 at 4.40.22 PM.png

 

Hence giving the equation for the shape of the Eiffel Tower.

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New Books in Maths: June 2017

Scale

Author: Geoffery West

Geoffery West, one of the most influential scientists of today, has written Scale to explore the hidden laws that govern the life cycle of a myriad of different things, from plants and animals to the cities we live in.

Although this book is more biological rather than mathematical, Scale promises to be an exciting adventure story about the laws that bind us together in simple but profound ways. This book aims to show the reader  how cities, companies and biological life are ‘dancing to the same simple powerful tune

“On one level, ‘Scale’ is a book about Mr West’s peculiar career path. But on another, it is about the hidden mathematical patterns underlying life, cities and commerce. Many things that appear unrelated are actually linked, he says.”

– The Economist

It’s All a Game

Author: Tristan Donovan

Games expert and British journalist Tristan Donovan has recently released a book explaining the incredible and often surprising history and psychology of board games.

By describing the evolution of the game across different cultures, times and continents, Donovan recollects compellingstories and characters, such as the paranoid Chicago toy genius behind Operation and Mouse Trap or the role of Monopoly in helping prisoners of war escape the Nazis, revealing why board games ‘have captured hearts and minds all over the world for generations‘.

“[A] timely book…It’s All a Game provides a wonderfully entertaining trip around the board, through 4,000 years of game history.”

― The Wall Street Journal

The Calculus of Happiness

Author: Oscar E. Fernandez

IThe Calculus of Happiness, Oscar Fernandez shows us that math allows us to gain interesting insights into health, wealth and love. This book is great for anybody that doesn’t have a lot of mathematical understanding as Fernandez uses only high-school-level math to guide the reader through many surprising results, such as an easy rule of thumb for choosing foods that lower the risk of developing diabetes.

“A nutrition, personal finance, and relationship how-to guide all in one, ‘The Calculus of Happiness’ invites you to discover how empowering mathematics can be.

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Seven Statistical Sins

Inspired by an article on phys.org I decided to compile a list of the seven statistical sins. Statistics is a vital tool to understanding the patterns in the world around us, however our intuition often lets us down when it comes to interpreting these patterns.

1.Assuming small differences are meaningful

Examples of this include small fluctuations in the stock market, or differences in polls where one party is ahead by one point or two. These represent chance rather than anything meaningful.

To avoid drawing any false conclusions that may arise due to this statistical noise we must consider the margin of error related to the numbers. If the difference is smaller than the margin of error, there is likely no meaningful difference and is probably due to random fluctuations.

2. Equating statistical significance to real-world significance

Statistical data may not represent real-world generalisations, for example stereotypically women are more nurturing while men are physically stronger. However, given a pile of data, if you were to pick two men at random there is likely to be quite a lot of difference in their physical strength; if you pick one man and one women they may end up being very similar in terms of nurturing or the man may be more nurturing than the woman.

This error can be avoided by analysing the effect size of the differences between groups, which is a measure of how the average of one group differs from the average of another. Then if the effect size is small, the two groups are very similar. Even if the effect size is large, each group will still have a lot of variation so not all members of one group will be different from all members of the other (hence giving rise to the error described above).

3. Neglecting to look at the extremes

This is relevant when looking at normal distributions.

normal-distrubution-large.gif

Source: mathsisfun.com

In these cases, when there is a small change in performance for the group, whilst there is no effect on the average person the character of the extremes changes more drastically. To avoid this, we have to reflect on whether we’re dealing with the extreme cases or not. If we are, these small differences can radically affect the data.

4. Trusting coincidence

If we look hard enough, we can find patterns and correlations between the strangest things, which may be merely due to coincidence. So, when analysing data we have to ask ourselves how reliable the observed association is. Is it a one-off? Can future associations be predicted? If it has only been seen once, then it is probably only due to chance.

5. Getting causation backwards

When we find a correlation between two things, for example unemployment and mental health, it may be tempting to see a causal path in one direction: mental health problems lead to unemployment. However, sometimes the causal path goes in the other direction: unemployment leads to mental health problems.

To get the direction of the causal path correct, think about reverse causality when you see an association. Could it go in the other direction? Could it even go in both ways (called a feedback loop)?

6. Forgetting outside cases

Failure to consider a third factor that may create an association between two things may lead to an incorrect conclusion. For example, there may be an association between eating at restaurants and high cardiovascular strength. However, this may be due to the fact that those who can afford to eat at restaurants regularly are in high socioeconomic bracket, which in turn means they can also afford better health care.

Therefore, it is crucial to think about possible third factors when you observe a correlation.

7. Deceptive Graphs

A lot of deception can arise from the way that the axis are labeled (specifically the vertical axis) on graphs. The labels should show a meaningful range for the data given. For example, by choosing a narrower range a small difference looks more impactful (and vice versa).

Source: phys.org

In fact, check out this blog filled with bad graphs.

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