- Public Key Cryptography (29 January) in which I describe the famous work of HLF regulars Diffie and Hellman
- A puzzle ‘four’ the new year (11 January) in which I discuss number puzzles, which do and don’t involve the digits 2, 0, 1 and 9
- The Twelve Facts of Christmas: Pascal’s Triangle (20 December), a festive round-up of mathematical facts
- Right on the Money (4 December) in which I wonder if the new British £50 will have a mathematician on it
- Measuring the Change (21 November) on the forthcoming redefinition of the kilogram
- Permutations and Tribulations (13 November) on the superpermutations discovery made anonymously on the internet
- Katherine Johnson (7 September) on life of the mathematician
- Game, SET and maths (16 August) about the card game SET
- New shape discovered – right under your nose (6 August) in which I describe the newly-discovered scutoid shape, found in nose cells
- Flexagons and false advertising (27 July) in which I explain how ordinary paper can be much more impressive if you flex it, Martin Gardner-style
- Few tile attempts (5 July) in which I discuss tilings and tessellations, and then decide how to redecorate my bathroom
- Cake and Coincidence (21 June) in which a striking coincidence prompts a discussion on probability and birthdays
- The Bridges of Königsberg (29 May) discussing the famous problem in graph theory
- A truly special occasion (18 May) all about myself and my partner’s Golden (ratio) wedding anniversary
- Colouring in like a mathematician (3 May) covering the then-current Chromatic Number discovery

I’ll continue writing posts for HLF’s Spektrum blog as long as they want me to – keep checking their Twitter feed or the blog’s RSS feed to see them as they appear!

]]>Our official list of #Noethember facts has provided inspiration for many. We’ve of course had daily sketches from our ‘ring’-leader @Coni777 – who’s also shared some photos of the drawing process:

The making of #noethember day 1. Greetings from France!! pic.twitter.com/L6fifvVRBf

— Constanza Rojas-Molina (@Coni777) November 1, 2018

Beautiful fox-based illustrations from @Lele_Saa:

Excited to take part on #Noethember, this month's drawing challenge about Emmy Noether, the mathematician, designed by @Coni777 @aperiodical https://t.co/5QY38BMg3E pic.twitter.com/2FpwtwfUmD

— Lele Saa (@Lele_Saa) November 1, 2018

Pencil and biro sketches from @njj4 and @tkleinwalsh:

Amalie Emmy Noether. Born 23 March 1882 in Erlangen, daughter and eldest child of Max Noether and Ida Amalia Kaufmann. Max, a mathematician, worked on algebraic geometry. #noethember pic.twitter.com/h4bapYtdtu

— Dr Nicholas Jackson (@njj4) November 1, 2018

Emmy decided to attend university in 1900. Then she spent three years independently preparing for the entrance exam. NBD. #Noethember pic.twitter.com/ZwQgVXqS2f

— Tina Klein Walsh (@tkleinwalsh) November 6, 2018

Others have used the idea as a springboard to collect their own ideas, including @mathsbooks, who has compiled their own list to work from and used multiple media, including sketches, watercolours and collage; and @algebrasnotwar who’s been drawing lovely cartoons.

#Noethember my day 14: After their legendary walking tours, @uniGoettingen’s mathematicians enjoyed Emmy Noether's yummy „Pudding à la Noether“. They continued their maths conversation, which became known as „Pudding Seminars“.https://t.co/wxFbSc1P7Z pic.twitter.com/sBStkVpd7H

— Katharina H MathsBooks (@MathsBooks) November 14, 2018

My #Noethember so far. 10 of 30 – the first third.

In contrast to the contributions of the other #Noethember enthusiasts, my amateurish attempts occur as quite a jumbled mess of styles with no recognition value. pic.twitter.com/D1hwBtRvHP— Katharina H MathsBooks (@MathsBooks) November 11, 2018

Noether was invited to return to the University of Göttingen by David Hilbert and Felix Klein#noethember pic.twitter.com/5fJErE4mdF

— Robert Vandermolen (@algebrasnotwar) November 12, 2018

The project has been a chance for people to join in even if it’s not a daily commitment, and many have shared occasional images without posting every day, including myself (@stecks), @gravitywithhat and @christianp. Others are joining in with a single post – and it’s all good!

#noethember day 8 pic.twitter.com/XOnoqOLwGQ

— Christian Lawson-Perfect (@christianp) November 8, 2018

https://twitter.com/GravityWithHat/status/1062395396180328449

@aperiodical #Noethember Day 2: See Emily Play pic.twitter.com/tTgm4Fnoc5

— Soup Dragon (@soupie66) November 2, 2018

Charmed to learn that mathematicians are celebrating #Noethember this month in celebration of Emmy Noether with facts and #sciart! My portrait of her focuses on her impact on #physics – Noether’s theorem links symmetries and conservation laws, for exampl… https://t.co/dL5lmGBOe0 pic.twitter.com/Dk06OiQQNn

— Dr Ele Willoughby (@minouette) November 14, 2018

#noethember Emily applied to the Universitynof Erlangen to first observe mathematics lectures there.she could only do that because women where not allowed to enroll at the University @Coni777 pic.twitter.com/KKw6KfIfiH

— Andrea Troncoso (@aitroncoso) November 9, 2018

"Your work has changed the way we look at algebra, . . . you have left your name written indelibly across its pages . . . we will always cherish the legacy you left us" -Hermann Weyl #noethember pic.twitter.com/1B8YSHnz1b

— Rachel Quinlan (@rkquinlan) November 12, 2018

#noethember pic.twitter.com/WbpJEqyosj

— imperatrixmundi (@o_fortuna) November 9, 2018

Others are using the hashtag to share other types of Noether-related content – from archive photos and documents to videos and books.

Emily Noether's dismissal letter from Gottingen 1933. She along with every other Jewish Professor was barred from teaching. Unlike many contemporaries she luckily managed to escape the Nazis. #noethember pic.twitter.com/4zohKwOt6K

— Benjamin Leis (@benjamin_leis) November 13, 2018

#Noethember Letter by Emmy Noether to David Hilbert. Erlangen, 1 December 1914, digitized version via @subugoe: https://t.co/SZfbSu8i24 pic.twitter.com/4VjnG4bxXK

— Katharina H MathsBooks (@MathsBooks) November 9, 2018

#Noethember En 1932, Emmy Noether est la 1ère femme invitée à donner une conférence plénière au congrès international des mathématiciens, qui se tient à Zürich.

L'occasion de jouer au célèbre jeu "Où est Emmy ?" (bien qu'il ne doit pas garanti qu'elle soit sur la photo). pic.twitter.com/eirce5Wumq

— Albireo (@foretdesciences) November 14, 2018

Emmy Noether, une femme extraordinaire à découvrir. Il y a 100 ans, elle a publié un théorème très important qualifié de "monument de la pensée mathématique" par Albert Einstein.

Avec une magnifique miniature de Maïm Garnier.@maybeegreen #noethember https://t.co/6IR2JLNJD2— Scienticfiz (@Vidosscientifi1) November 14, 2018

A l'initiative de @Coni777, consacrons ce mois de novembre à Emmy Noether: #noethember

La couverture de ses œuvres choisies. pic.twitter.com/l7chTJdfmO

— Roger Mansuy (@roger_mansuy) November 10, 2018

#Noethember Letter by Emmy Noether to David Hilbert. Erlangen, 1 December 1914, digitized version via @subugoe: https://t.co/SZfbSu8i24 pic.twitter.com/4VjnG4bxXK

— Katharina H MathsBooks (@MathsBooks) November 9, 2018

Emmy Noether utilisait même les cartes postales pour parler mathématiques.#noethember @Coni777 @roger_mansuy @paljasn @GavageSylvie @mickaellaunay pic.twitter.com/cnODXTCfmo

— Scienticfiz (@Vidosscientifi1) November 3, 2018

There’s still half a month to go – remember, just because you haven’t drawn something every day so far, or if you’ve started and lapsed (like me… just too busy!) doesn’t mean you can’t join in now. And you don’t have to be amazing at drawing – it’s all about sharing ideas and stories. Join the #Noethember movement!

]]>However, since the Nerds are currently riding an exponential curve of silliness in their DVD release schedule (their last show Just for Graphs was released on VHS as well as DVD) this time the show has been released as an autographed limited edition set of 3.5″ floppy disks, one of which was sent to me in the post.

Each of the disks contains one 30-second segment of the show, and I’ve been sent disk number #211, containing seconds 6300 through to 6330 of science comedy content. I was apprehensive, as I know there are many different parts to the show, and I was hoping for one with some mathematical content that our Aperiodical readers here could really appreciate.

Probabilistically, this wasn’t much of a worry – even though the show covers a range of topics, there’s plenty of maths behind all of them. Singer and physicist Helen includes a song about the relative radioactive exposure from different daily activities, each given in terms of in the Banana Equivalent Dose (the amount of radioactivity you’re exposed to from eating a banana), while experiments guy Steve takes a break from exploding things to tell us about his attempts to find meaning in the sentence “the temperature outside a plane is six times colder than the temperature inside a freezer”, a struggle long-term Aperiodical readers will be familiar with, and Steve’s initial response to which is now immortalised in t-shirt form.

The show also contains a wonderful thread of purely maths content from in-house number ninja Matt Parker, who tells us of his campaign to rid the UK’s signage of geometrically inaccurate football symbols – just one of a range of bits on geometrical topics: from his struggles tiling his own bathroom, to playing with stereographic projections and spherical camera footage.

The 30-second clip on my disk is from a section on recursive spirals, inspired by the MC Escher lithograph of a person looking at a painting of a city in which there’s an art gallery through the window of which can be seen that same person, and in the show Matt takes this idea and runs with it… maybe slightly too far.

My impression of this 30-second clip was that it was a little out of context and could have done with a better explanation to introduce what Matt was talking about (he also seemed to just start talking mid-sentence, and didn’t really round off what he was saying at the end of the clip either – it’s a bit of a cliffhanger); we also didn’t get to see much of Steve or Helen, apart from them being over on the left of the stage listening to Matt.

You’ll be pleased to hear that each floppy disk edition also comes with a code to download and watch the whole show, not just the 30 seconds on the disk. It can be ordered from the FOTSN online shop, along with a version of the show on DVD for normal people, which includes a ‘safer for schools’ soundtrack, behind-the-scenes DVD extras and an infinitely long DVD unboxing video extra (of course). There’s also a standalone download edition, for if you don’t want to be encumbered by messy real-world objects – or if, like the nerds themselves, you prefer the purity of buying just the ones and zeroes.

]]>Earlier this year, one of our colleagues made an interesting observation about the Wikiquote page for Mathematics . Wikiquote collects interesting and pithy statements, with references, and categorises them by subject. What our friend Colin had noticed that was almost all of the quotes on the page about mathematics were from men.

This reflects on existing gender imbalance in mathematics, which will hopefully improve. But mathematics is done by all kinds of people all over the world, and many of them have occasion to say interesting and/or profound things about their subject – even the ones who aren’t men. Since Wikiquote is part of the Wikimedia network and can be edited and contributed to, we held a small editing day back in May where we added a number of quotes from female mathematicians to the page, to try to balance out the score.

Walking around the Women in Mathematics featured here at the HLF this week, I was featured by the amazing quotes taken from the interviews with each of the mathematicians. I’ve picked some of my favorites – and yes, I’ve added them to the Wikiquote page . Enjoy!

“In doing mathematics, I express something personal. It is a source of joy to know that, despite this personal aspect, the fruit of my work can be of interest to other mathematicians. “- **Nalini Anantharaman**

“Mathematics offers a common language across borders. It’s a real joy. “- **Alice Fialowski**

“My earliest mathematical memory is my father’s explanation of the theorem that three angles in a triangle add up to 180 degrees. The idea that something could have proved to be true was very appealing to me. “- **Frances Kirwan**

“I used to fear I was on the right track. You need to develop a personal conviction that you *are a mathematician* , and that’s what you are doing. “- **Katarzyna Rejzner**

“I enjoy being surprised by mathematics and its intrinsic difficulty. The moment I enjoy the fall of one coherent whole. “- **Katrin Wendland**

“You should not choose to do mathematics if you want to make money; your salary as a mathematician will never correspond to the amount of time invested in your work. “- **Margarida Mendes Lopes
**

“I like to find out as much as I can about a mathematical object as possible, just as you would like to understand a person as well as possible.” – **Oksana Yakimova**

For the Wednesday afternoon of HLF, the entire conference gets on a (very large) boat and heads off for a gentle cruise down the river, drink in hand and ready to enjoy the scenery. The young researchers, along with the Laureates and the rest of us, are effectively trapped on the boat for a few hours – so just like last year, we took the opportunity to corner some of the PhD and postgrad students and ask them about their research – and the numbers they find central to their work.

*Mathematician, Functional Analysis*

Becky’s research is in pure mathematics, working with objects known as *C*-algebras* – originally developed to provide a mathematical framework for quantum physics. Using cohomological data, she ‘twists’ these algebras with 2-cocycles (giving the 2 Becky chose as the number to represent her work), and studies how this changes the fundamental properties of the objects, such as *commutativity* – whether two operations give the same result if performed in either order. This contributes to an overall effort to completely classify C*-algebras.

*Computer Scientist, Medical monitoring*

Edward is working on a system that can monitor haemoglobin levels in a person’s blood, using only the camera and flash in an everyday smartphone camera. Putting your finger over the camera lens, the system can see the colour of the blood each time a heartbeat pushes it through the fingertip. Using devices that are already prevalent brings obvious advantages in terms of cost and ease of distribution. The number Edward is most familiar with is the resting heart rate of his most frequent test subject – himself.

*Computer Scientist, Hardware Architecture*

Sandhya works in hardware architecture, designing integrated circuits for applications such as machine learning – her research happens somewhere in the overlap between computer science and electrical engineering. To represent her research, Sandhya chose a binary sequence of 0s and 1s, since she works right down at the level of the individual bits shuttling around the hardware. (For the picture, we chose an arbitrary sequence, despite our previous thoughts on this issue.)

*Computer Scientist, Approximation Algorithms*

Eric’s research concerns approximation algorithms – routines for finding good solutions to mathematically difficult optimisation problems (for example the famous ‘Travelling Salesman’ problem). He works specifically with a class of problems computing things known as *submodular functions*. For this class of problems there is a mathematical limit on how well they can be approximated, given by $*1-(\frac{1}{e})$*, or about $0.632$. This forms a kind of ‘holy grail’ for these algorithms, so naturally Eric chose it as his number.

*Computer Scientist, Healthcare Informatics*

Moumita develops machine learning systems, using patient information and electronic health records to predict outcomes like drug interactions, or hospitalisation rates for patients with chronic illnesses. Accurate predictions allow for more effective clinical decision-making, in turn leading to improved outcomes, that is to say, saving lives. Her chosen number – 67% – provides a kind of informal benchmark when assessing these models for possible adoption.

]]>Alongside the HLF this year, an exhibit celebrates female mathematicians from around Europe (written about by Gina in a post earlier this week) and includes photographs, interview quotes and beautiful mathematical equations, which come together to create an illuminating and at times poetic snapshot of the life and work of a mathematician.

The equations in particular caught my eye, each splashed across the poster in a different bright colour. They’re used more as an illustration of the type of maths each person works on, rather than giving any mathematical background – but I was intrigued, and thought I’d investigate a few of them, and share some of the mathematics represented.

Subject area: Cluster algebras, cluster categories, representation theory, categorification

Karin’s work involves **algebras** – structures defined using a basic collection of objects, with an underlying set of numbers used to scale them. An example might be the **complex numbers**, discussed in the post on the Riemann Hypothesis, in which the objects are just the number *i*, and the underlying set is the real numbers. You can combine multiples of $*i$* with real numbers, to get complex numbers of the form $*a + ib$*, and these can be combined by addition, or multiplication to give other complex numbers. Some algebras have more than one object that can be combined in different multiplicities – for example, the **quaternion numbers** are defined in a similar way but using $i, j$ and $k$ – each of which can be multiplied by scalars, but also interact with each other in specified ways (in this case, $i^2 = j^2 = k^2 = ijk = -1$). In particular, Baur’s work is on **Cluster Algebras**, whose collection of objects is generated from a seed set using particular rules.

The equation relates A, an algebra, to End_{B}, which is a collection of endomorphisms. Endomorphisms are maps from a particular object to itself – for example, given the set of numbers {1, 2, 3} you could consider all the maps taking this set to itself, by reordering the numbers:

{1, 2, 3} → {1, 2, 3}

{1, 2, 3} → {1, 3, 2}

{1, 2, 3} → {2, 1, 3}

{1, 2, 3} → {2, 3, 1}

{1, 2, 3} → {3, 1, 2}

{1, 2, 3} → {3, 2, 1}

Once you have defined the endomorphisms of an object, they can then be considered as a structure in themselves. In the same way as with the objects in the algebra, you can combine one endomorphism with another to get a different one:

e.g. {1, 2, 3} → {1, 3, 2} combined with {1, 2, 3} → {2, 1, 3}

This would swap the second and third elements, then swap the first and second elements, giving the result {1, 2, 3} → {3, 1, 2}. Studying all the possible endomorphisms of a given algebraic object tells you something about its structure, and the collection of endomorphisms can also form an object to be studied.

The equation is from Karin’s paper “Dimer Models and Cluster Categories of Grassmannians”, with Alastair King and Robert J. Marsh. In this paper, they show that the particular algebra they are studying, A, is isomorphic to (the same as) a particular endomorphism algebra, giving them a better understanding of its structure.

Subject area: Geometric analysis, minimal surfaces, Dirichlet problem, geometric maximum principle

This equation is from Nelli’s paper, Minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$, written with Harold Rosenberg, in which they prove various results about the space $\mathbb{H}^2 \times \mathbb{R}$.

$\mathbb{H}^2$ here represents the **hyperbolic plane** – a surface in which the space curves away from itself at every point. This sounds confusing, but it’s difficult to picture as it doesn’t fit in our normal 3D universe. Compared to our usual geometry (called Euclidean geometry), it has major differences. One of the fundamental ideas in geometry is if you have a point near a line, there’s only one line that passes through that point and is parallel to the original line – this is known as Playfair’s axiom. In hyperbolic space, there can be more than one such line – all of which pass through the same point, but don’t intersect each other, or cross the original line, anywhere else. This is because of the way the space is curved, and it’s a fundamental property of so-called Hyperbolic geometry.

Hyperbolic space can be visualised using the **Poincaré Disc** – a circle in which distance behaves strangely near the edges. Distances in the centre of the disc behave just as normal in the regular 2D space we’re used to, but as you get nearer the edges, distances get shorter – until, on the boundary of the disc, points which look close together are actually far apart, and straight lines become curved.

A nice way to think about it is to imagine you’re creating a circular piece of crochet, but each line of stitches as you go further out from the centre has more stitches in it. Such a crochet piece might start off flat, but as you get further out it’ll need to ripple up and down to accommodate the extra stitches – but in the hyperbolic plane, this extra surface is just squashed in to the same space, and the definition of distance is difference there.

The space$\mathbb{H}^2 \times \mathbb{R}$ represents the result of taking the **direct product** of the hyperbolic plane with the real line $\mathbb{R}$. Direct products are analogous to putting two things at right angles – for example, the direct product of a circle and the infinite real line $\mathbb{R}$ is an infinite cylinder, and the direct product of two real lines $(\mathbb{R}$ \times\mathbb{R})$ is $\mathbb{R}^2$, the 2-dimensional real plane.

Nelli’s work here concerns the **Dirichlet Problem**, which involves solving differential equations, and this equation describes the divergence (a property of the related vector field) of a function defined on $\mathbb{H}$^n, n-dimensional hyperbolic space.

Subject area: Algebraic groups and Lie algebras, Poisson structures, harmonic analysis on Gelfand pairs

This equation is known as **Weyl’s character formula**, and it’s from a branch of maths called **representation theory**, which is connected to the idea of **groups** – collections of objects, much like algebras, but without the underlying set of scalars. Group theory is concerned with how the objects making up the group behave when combined in different ways, and group structures often underlie other mathematical ideas – the group describing all possible moves on a Rubik’s cube is a nice example.

Representation theory involves representing elements of a group or other algebraic structure in terms of linear transformations of vector spaces. Results from this common ground can then be used to learn about the original structures. Representation theory often pops up in other areas of research, as the underlying groups the structures are based on show the connections between different areas.

As with numbers and primes, representations can be ‘broken down’ into products of smaller representations (transformations of spaces with fewer dimensions). Those representations which cannot be broken down are called irreducible representations. Much like prime numbers, properties of irreducible representations are particularly important.

The **character of a representation** condenses this further, assigning to elements of the group a single ‘number’ which captures much of the information in the representation. Weyl’s character formula gives information on the characters of the irreducible representations of a certain type of group, known as a compact Lie group. Lie groups, which Oksana works with, are groups that can also be considered as topological shapes.

The HLF included a talk from 2018 Fields medalist Caucher Birkar. His subject area, algebraic geometry, is one of the largest fields of research within pure mathematics (over a quarter of the 60 Fields medals awarded since 1936 have been to people working in algebraic geometry), and it has connections to many other fields of maths including topology, algebra and number theory. But what exactly is algebraic geometry? Well, if you’ve studied maths at school, there’s a pretty good chance you’ve already done some.

Algebraic geometry considers spaces of polynomials – equations which use one or more variables raised to powers. A simple example might be:

$x^3 + 3x^2 – 4x + 5 = 0$

Each of the terms in the polynomial has a coefficient in front of it, which can be 1 (in which case it’s not written), and can be 0 (in which case that term doesn’t appear) or can be another number. Depending on what kind of polynomial you’re working with (more strictly, over which field you’re defining your space of polynomials), you might use whole numbers, fractions or even complex numbers in front of each term. In our example above, there’s an $x^3$ term, an $x^2$ term, an $x$ term (which in this case has a negative coefficient) and a constant term.

This kind of polynomial equation (*poly* = many, *nomos* = part; a thing with many parts) can be used to describe all kinds of systems and objects. Simple example polynomials might not have terms with high powers – for example, a polynomial with just an $x$ term and a constant term is called linear, so-called because it describes a straight line if you draw it on a graph.

This line represents all the points for which $y = 2x + 3$: at $x = 0, y = 3$ and at $x = 1, y = 5$ as you would expect. Many of you will be familiar with this kind of graphical representation of an equation, and might have also seen what happens to the graph of a polynomial as you include higher powers as terms.

A graph with a squared term (left) is a quadratic, and will have a u-shaped (or n-shaped, depending on whether the $x^2$ term has a positive or negative coefficient) curve. A cubic polynomial (centre) will usually have a wiggle in the middle, but will have the ends going in opposite directions, and a polynomial with an $x^4$term (a quartic, right) will look like a quadratic but with an extra bump. The more terms, the more bumps you can get.

Some of you may be wondering why this area of research is known as Algebraic Geometry – the algebra is clearly there, in the form of equations – but where’s the geometry?

Well, there are two problems here. Firstly, the type of algebra referred to in the name isn’t the kind of algebra you’ll remember from school, where you do the same thing to both sides of an equation and move terms around to get the answer. At university and research level, algebra is the branch of pure maths concerned with structures like groups and fields. Such objects are abstractly defined, and are essentially sets with rules. The set of all real numbers forms a field, because it’s a collection of things (numbers) you can combine together using addition and multiplication, and if you do this you always get another thing that lives in the same field (a number). There are a few other conditions to specify the different types of objects – called groups, rings, fields, and even algebras, and these can be used to describe real objects or structures – the group associated with moves on the Rubik’s cube being a nice example.

So algebraic geometry involves considering not just individual polynomials, but the ring of all possible polynomials, given the variables you’re putting in (sometimes just $x$, or sometimes a whole set of variables, usually labelled something like $x_1, x_2\ldots x_n$) and the field you’re using to pick coefficients (such as the real numbers, the set of all fractions, or the complex numbers).

The geometry aspect is slightly less obvious – but it has actually already started making itself apparent. By defining the linear polynomial* $y = 2x + 3$*, we’ve already seen how this corresponds to an object on a 2D graph. If your polynomial has more than one variable, this shape will exist in more than 2 dimensions. Each polynomial defines an object in space – a line, a plane or a solid, or anything else, and by studying the polynomials you can examine the geometry of these shapes and the connections between the abstract numerical object and the slightly more tangible (at least in lower dimensions) shapes. Algebraic curves (shapes defined by polynomials) range from simple lines and circles to beautiful complicated shapes with exciting names like the cycloid, the quadrifolium and the Conchoid of Nicomedes.

Birkar’s work in algebraic geometry concerns the structures you can construct from the rings of polynomials, how these generalise to higher dimensions, and how to deal with singularities in the space (points where the polynomial doesn’t look like a simple curve). His Fields medal was awarded for work on morphisms, maps used to ‘patch’ varieties (collections of polynomials) at points where they have singularities, so that they can be studied more easily.

Algebraic geometry connects to the study of shapes in topology, the use of polynomials to model mathematical problems, and it has connections to cryptography through the polynomials used in encryption, and to higher level physics concepts such as string theory. Since you’ve certainly already done a very small amount yourself – whenever you’ve sketched a graph of $x^2$ – you can maybe appreciate the importance of Birkar’s achievement and the beauty of this kind of mathematics, even if you’re not quite at Fields medal level yet.

]]>

At last year’s HLF, Turing Award Leslie Lamport gave us his (not wholly complimentary) thoughts on the state of proof-writing in mathematics. Since he has worked in both maths and computer science, members of the latter discipline may have felt they got off quite lightly. Perhaps to redress the balance, this year we found out what he thinks is wrong with most people’s code and algorithms, in a talk titled *If You’re Not Writing a Program, Don’t Use a Programming Language*.

Lamport made a distinction between programs and algorithms: programs are the real-world code written in programming languages, while an algorithm is the underlying abstract concept. He paraphrases fellow Turing laureate Tony Hoare to summarise the situation: “inside every large program is an algorithm trying to get out”. The problem as Lamport sees it is that too often, people try to verify their algorithms at the level of the program, where the algorithm is obscured by the messy details of the code: the variable types, the attention to edge cases and so on.

In an article for Wired, he analogises the situation to another field: “Architects don’t make their blueprints out of bricks.” – and they certainly don’t press on without making a blueprint at all.

In my days as a PhD student working on problems related to computational group theory, I grappled with the problem of demonstrating the correctness of an algorithm, while at the same time giving a comprehensible overview of how it works in the event that anyone might one day want to actually implement it. So, while by no stretch of the imagination being a computer scientist or even a serious coder, I can appreciate that this problem is real and non-trivial to solve.

So what is the solution? Lamport says we should describe our algorithms not with code – real or pseudo – but with mathematics. Lamport shows us how to encode an algorithm in a set of predicates: logical statements describing the initial state of the algorithm, the transitions between subsequent states, and the required outputs and conditions for the algorithm to terminate. Treating algorithms as mathematics allows us to more easily appreciate their overall structure, and to successfully ‘debug’ them. He’s involved with a system called TLA which implements this idea.

Via a brief detour into topology, Lamport backs up the credentials for his system with quotes from some of the big guns – Microsoft and Amazon Web Services. While I suspect that his desired revolution in mathematical proof may be a long time coming, it seems like his ideas on improving algorithms have some friends in high places.

You can watch the talk on the HLF website.

]]>This year at the HLF there are multiple sessions in the program concerning the Riemann Hypothesis, including a talk from one of the laureates, and one of the young-researcher-led workshop sessions. But what exactly is the Riemann Hypothesis, and what is its place in mathematics?

The Riemann Hypothesis was conjectured in 1859 by Bernhard Riemann, a mathematician working in analysis and number theory. It concerns a function called the Riemann Zeta function, which is defined as follows:Given an ‘input’ number $s$, to calculate the value of the function, you add together the numbers $\frac{1}{1^s}$, $\frac{1}{2^s}$ and so on. For some choices of $s$, as you add more numbers the sum approaches a limit – for example if $s = 2$ the sum converges to $\frac{π^2}{6}$. But if $s$ is 1 or less, the sum diverges: it just keeps getting bigger, exceeding any possible upper bound. However, a technique known as ‘analytic continuation’ can be used to extend the function and assign finite values even in these cases. The Riemann zeta function is the version of this function extended across the entire set of numbers, including – and this is where the real interest lies – the complex numbers.

A complex number is one of the form $a + ib$, where $i$ is the square root of $-1$. Such numbers are considered to exist in a 2-dimensional plane, called the complex plane, with the “real” part a extending left-to-right like a normal number line, and another perpendicular axis extending vertically for the value of $b$ – the “imaginary” part of the number.

Numbers in the complex plane can also be inputs to the Riemann Zeta function – for each point in this 2D plane you can evaluate what the function would be equal to for that input.

The thing analysts are often interested in when studying a function is where it’s equal to zero – these are points that give a nice insight into how the function behaves. For the Zeta function, there are plenty of known points that evaluate to zero, known as ‘zeroes of the function’. Infinitely many, in fact – every negative even integer will give the value zero, meaning there’s an infinite line of points spaced along the negative horizontal axis which all give zero when put through this function.

The only other points that have been found to make Zeta equal zero are on a different straight line in this diagram, called the ‘critical line’ – it’s the line where all the numbers have real part ($a$) equal to $\frac{1}{2}$, so it’s a vertical line running just to the right of the imaginary axis. The first one of these (above the horizontal, going upwards) is at $\frac{1}{2} + 14.134725\ldots i$, and they continue upwards, with identical zeroes at points below the line with negative equivalents of these values ($\frac{1}{2} – 14.134725\ldots i$ and so on).

It’s been proven that infinitely many zeroes lie on this critical line, all with real value $\frac{1}{2}$ and with no discernible pattern in their imaginary values, beyond the obvious vertical symmetry. Over 10,000,000,000,000 such zeroes have been computed numerically (initially by hand, and following work by Alan Turing in 1953, by computer), and billions of others have also been checked in the region around $\frac{1}{2} + 10^{24} i$. We also know that at least 40% of the zeroes of this function are on the critical line, and all the ones known so far are at irrational values of b – numbers that can’t be expressed as a fraction.

But what we don’t know is that there aren’t any others anywhere else – it can’t be proved that all the zeroes are either on the negative axis at even whole number points (these are know as the ‘trivial’ zeroes) or lie somewhere on the critical line. The Riemann Hypothesis is exactly this – the statement that all non-trivial zeroes of the Riemann Zeta function lie on the line $a =\frac{1}{2}$.

People have been working on this problem for over 150 years, and as you would expect, some progress has been made towards a proof – other functions which have properties similar to the Zeta function can be studied, and a number of approaches including self-adjoint operators, random matrix theory, non-commutative geometry, statistical mechanics and even classification of quasicrystals have all been employed in attempts at a proof.

So what would happen if the Riemann Hypothesis was proved? The most straightforward consequence, if the least mathematically interesting, would be that the person to do it would be awarded one million dollars. The Riemann Hypothesis is one of the Millennium Prize Problems, a list of seven then unsolved questions in maths produced by the Clay Institute in 2000. To date, only one of these has been solved: the Poincaré Conjecture, by Grigori Perelman in 2003. Its inclusion on the list gives an indication of how important the result is considered among mathematicians.

The Riemann Hypothesis is closely related to the prime numbers – alternative formulations of the Zeta function involve stating it in terms of the prime numbers, and while these can’t be used to evaluate the function, it means that fully understanding the Zeta function could unlock some aspect of the as-yet-unknown patterns in the primes. Many results about primes have been proved on the condition that the Riemann Hypothesis is true. So a proof would deliver a large number of further important results ‘for free’.

Since modern encryption methods are based on prime numbers, there’s a popular idea that a proof of the Riemann Hypothesis would compromise this security – that it would literally (as it surely would figuratively) “break the internet”. An episode of the mid-2000s US TV series *Numb3rs* referred to this in an episode where the daughter of a mathematician who claimed to have a proof was kidnapped. Of course, the bare knowledge of the truth value of the hypothesis would have no such effect: any committed hacker would be perfectly happy with a method for compromising encryption that might be based on a false assumption, if it worked anyway. But if the techniques employed by a proof delivered a method for factorising large numbers quickly, such consequences might not be totally beyond the bounds of possibility.

Tomorrow at the HLF, mathematician and double Laureate Sir Michael Atiyah has announced he’ll be presenting “a simple proof using a radically new approach” of the Riemann Hypothesis. Such a proof would be huge news, and a big step forward for mathematics. We’re all looking forward to hearing what he has to say!

]]>We spoke to Nat Alison (@tesseralis), creator of the amazing Polyhedra Viewer.

The Polyhedra Viewer is a web app that lets you explore the relationships and transformations between convex polyhedra. You can push different buttons to rectify a tetrahedron into an octahedron, or augment a pentagonal prism with a pyramid, or gyrate the components of a rhombicosidodecahedron. You can also change the colors of each polyhedron and look at their geometric information. It’s a passion project I’ve been working on for the past eight months while I’ve been “funemployed”.

The bilunabirotunda and the triangular hebesphenorotunda. They’re “elementary” Johnson solids, in that they can’t be made from gluing Platonic or Archimedean solids together. But they’re still intricately related to them! The bilunabirotunda forms a honeycomb with the cube and dodecahedron, and the triangular hebesphenorotunda, if you put two of them together, shares coordinates with the icosidodecahedron!

Try typing in the name of a polyhedron that the app doesn’t know about (e.g. polyhedra.tessera.li/

If you click on the animation on the top of the page or type polyhedra.tessera.li/

Also, if you go to the info tab for each solid you can download a 3d model of it!

I always thought the polyhedra were beautiful, especially the way that you could construct them from a few simple shapes and operations. I wanted to find a way to convey that beauty and connectedness to other people. I loved learning about polyhedra from websites like George Hart’s Virtual Polyhedra, but I often found that those sites were not well-designed or too complex for people who only have passing knowledge of math and polyhedra. I wanted to make something accessible, something tangible that encourages people to explore even if they didn’t know the basic concepts.

You can actually transform between everything except the section labelled “Elementary Johnson Solids”! The proof of that will be left as an exercise to the reader. :)

To display the 3D polyhedron models, I used X3DOM, a library that lets you encode 3D graphics in an XML format (like SVG is for 2D graphics). Each type of operation was coded by hand. For example, for truncation, I needed to write an algorithm that says “replace each vertex in the polyhedron with $n$ new vertices and push those new vertices in along the edges”.

Because I knew what the results would be each time, I could do a lot of “cheating”! For example, when doing an expansion, because I knew what the result was going to be and had models of all the polyhedra, I didn’t need to calculate the amount to pull out the faces by. I could just rely on the model of the result and measure it from there!

Yes! I’ve gone through several iterations of this project. I also did another visualization showing the relationships between polyominoes. And while the work isn’t public, a lot of the knowledge on how to design and create an interactive web app was from my previous work as a software engineer.

That’s still up in the air, and depends a lot on what I can do to support myself! I’m thinking about using the mechanics of the polyhedra viewer into a puzzle game. I’d also like to do more creative coding work, visualizing other complex concepts in a way that is easy for people to understand. If you’d like to see more work like this, you can buy me a coffee at ko-fi.com/