My week as curator of the Twitter channel @realscientists has just ended. It was refreshing and a lot of fun. I had the chance to review my own work from a fresh perspective and to check out old problems I didn't read about for a while. For those who don't know what Real Scientists is, I will explain. It's a Twitter channel that is curated by a different scientist/researcher/science communicator each week. It's a brilliant way to put general public and other researchers in direct contact with science research fields. After all, it's the taxpayers who fund most of our work, so we own it to them to explain what we do, don't we?

During the week I shared many images, slides, topics. I decided to collect them all, so that they do not get lost in the feed and may turn up useful for myself or someone else in the future. I will not write complete explanations, it will be more of a list of resources. Feel free to use whatever, but please be polite and quote the source (I will indicate the author or the original website for everything).

### Mathematics and its applications

#### X-ray tomography

X-ray tomography is my area of research (see here for details about my current project). Here some resources:

- Wikipedia page on X-ray tomography
- X-ray tomography used in archeology: mummies and lost letters.
- Wikipedia pages on J. Radon and Radon transform.
- VT-cube, a device that reduces X-ray irradiation in dental imaging.
- Denoysing 4D cardiac tomography images.
- Lecture notes on tomography.
- How tomographic data is collected.
- Reference: Smith et al. 1977. "A finite set of radiographs tells nothing at all".
- Compare FBP with other reconstruction algorithms for sparse data.
- An application of dynamic CT: coronary angiography. About 10 million tests carried every year in US.
- Another research group working in dynamic CT.
- Public CT real dataset by University of Helsinki.

#### Other Inverse Problems

Wikipedia page on Inverse Problems.

*Image enhancement and inpainting*.

- Inpainting Wikipedia page.
- Image restoration: deblurring & co.
- Our group will soon start working in image enhancement.

*Invisibility cloak*. How Harry Potter meets mathematics: link 1, link 2.

*Creating synthetic voice*. I talked about this problem here. A video where my advisor explains the project. His slides: I love this talk, every time it has great success. Speech problems affect many people, 6 to 8 millions in US only.

*Seismic tomography*. Seismic tomography exploits seismic wave and advanced topology to infere about the inner structure of our planet.

*Electrical Impedance Tomography*. EIT is a widely used medical imaging technique, employed especially in breathing monitoring.

*Gravitational Lensing*. Wikipedia page: measuring space bodies' properties from their interaction with remote light.

*Forest monitoring*. The Tampere Inverse Problems group uses inverse problems techniques to monitor and model forests, contributing to fire prevention and paper industry estimations: video.

#### Other applications

(see also Computer Science section)

*Cryptography*. Internet transactions require sensible information to be properly masked. For instance, when you buy online with your credit card, data transmission is protected by cryptography techniques (ex. RSA). Here some links:

- RSA Wikipedia page.
- Prime numbers are fundamental for RSA.
- The standard protocol SSL uses RSA.

*Big Data*. Wikipedia page on Big Data.

*Rollercoaster engineering*. Rollercoaster loops are designed according to a mathematical curve called Euler spiral, to maximise the fun and minimise nausea. Euler spiral can be found also in highways, to make turning at high speed safer.

*Biomathematics*. A fun example of biomathematics modelling can be found in this zombie scenario paper. Biomathematics had a big role in containing the Ebola infection.

*Finance*. Likely the person dealing with your investments is a mathematician. Here a Wikipedia page.

### Computer Science

- Mathematics in Google algorithms: article on The Guardian.
- Mathematics in animation movies.
- P=NP problem: link 1, a simple explanation, an article on New Yorker,

### Women in Mathematics

- The Fields Medal was won by a woman for the first time in 2014.
- Agora, a movie about a female mathematician of antiquity, Hypatia.
- University of Nottingham playlist on women in mathematics.
- How to propose to a mathematician.
- Sofia Kovalevskaya. My post on her, Google's dedicate Doodle, a movie on her life.
- Book on Minorities in Mathematics.
- TV series: The Bletchey Circle.
- Sophie Germain, who studied with Lagrange.
- How gender stereotypes affect mathematics skills at early age.
- [not only about women] Childcare at conferences: a post on Tenure She Wrote, my vision, Geoscience conference that provides it, some family care grants: LMS - UK, Franklin Women, SMBE, AustMS.
- Representation of women in mathematics at conference: a practical guide.
- Some links on women in STEM.
- Emmy Noether, allowed to work (for free or under a male colleague's name!) in a time when women were not allowed in academic life. A video about her by Katy Mack.

### Algebra and Number Theory

*Goldbach's conjecture* (Wikipedia page). Goldbach's conjecture is an open problem in number theory. It states that every even number greater than 2 is sum of two prime numbers. For instance: 8=5+3, 18=11+7, 22=11+11. This has been checked for huge numbers (), but to date there is no proof that it's true for all even numbers. Some links:

- Page with updates on the verification of the conjecture.
- Post on Terence Tao's blog on his progress.

*Collatz's conjecture* (Wikipedia page). The conjecture states: take any natural number n. If n is even, compute n/2. If n is odd, compute 3n+1. Iterate the procedure. You'll alway end up at 1.

*Fermat's Last Theorem* (Wikipedia page). Stated by Fermat in 1635 and proven only in 1995 by Andrew Wiles, it claims that, given any n greater than 2, there is no integer solution to the equation . For instance given a cube, you cannot split it in the sum of two other cubes. Fermat wrote on the margin of a book "I have discovered a truly remarkable proof of this theorem which this margin is too small to contain.". Here Wiles recalls the moment when he understood his proof was complete (after 7 years of work). If you find the topic interesting, don't miss this great popular book by Simon Singh.

*Induction*. A technique to prove statements for natural numbers is induction. It works this way: (1) I want to prove statement A is true for all natural numbers, (2) I check A's true for n=1, (3) I *assume* A's true for a generic n and based on that, I prove A's true for n+1.

*Proof of infinity of prime numbers*. Euclid proved that prime numbers are infinitely many by a reduction ad absurdum proof. One assumes something, through logic gets to something absurd and concludes the initial assumption must have been false in the first place.

*Perfect numbers*. A perfect number is an integer that is equal to the sum of its divisors (except the number itself). An example is 6, whose divisors are 1,2,3,6, and 1+2+3=6. A cool property is that the sum of the reciprocals of perfect numbers' divisors is always equal to 2: . So far only even perfect numbers have been found and the question if odd perfect numbers exist is open.

*Weird numbers*. Some weird names for numbers: sexy primes, friendly numbers, amicable numbers, sociable numbers, twin primes.

*Pi*. Pi is one the most celebrated numbers ever, even having its own anniversary. Even Star Trek talks about it and Kate Bush wrote a song! Pi is irrational, meaning it cannot be expressed as a fraction. An equivalent formulation is that its infinitely many digits have no pattern, so in principle you can easily find any string of numbers in there. Another property of Pi is that it cannot be expressed as zero of an polynomial with rational coefficients ( is irrational but a zero of instead), making it trascendental. The most decimal places of Pi memorised is 70,000, and was achieved by Rajveer Meena in 2015.

### Geometry and Topology

*Art and mathematics*.

[add Mobius scarf and bacon: downloaded. Add Bathsheba Grossman.]

I mentioned two weird geometry constructions (manifolds) in particular: Möbius' strip and Klein's bottle. They are non-orientable. Möbius's strip has not "over" or "below", Klein's bottle has no "inside" or "outside.

*Non-euclidean geometries*. Eucliden geometry, the "classical one", is based on some axioms. At some point some mathematicians wondered if they were all necessary and experimented with creating new geometry settings.

Here the Wikipedia page on the topic. Also, non-euclidean geometries relate to Einstein's General Relativity Theory.

*The Seven Bridged of Königsberg*. This problem was solved by Euler in 1736 and led to the birth of graph theory. Graph theory is used when solving or creating mazes. Another related famous problems is the Four Colour Theorem: given any map, you can use only four colours to fill all countries so that any two adjacent countries have different color. The latter was proved by Kenneth Appel and Wolfgang Haken in 1976. They reduced to 1'936 possible cases and then checked them all with a computer. It was the first time computing was used in a formal proof.

*The happy ending problem*. Below I state the problem visually. George Szekeres went even further and proved you can always draw the polygon you want, if you draw enough initial points (for a convex pentagon: you need 9 points). There is a conjecture stating you need points to be sure to find a convex polygon with sides. The question was named this way because it led to the long-lasting love of Esther and George Szekeres. They were married 70 years and died within one hour from each other. Mathematics can make you fall in love sometime...

*Homeomorphisms*. One of the funniest tools in topology. Here a video showing why donuts are mugs after all. Below another weird transformation: from a square to a (almost) torus.

*Manifolds*. Another powerful maps are diffeomorphisms, a level on top of homeomorphisms. Manifolds are geometrical objects that are locally similar to for some n. For instance a sphere is a manifold of dimension 2, small regions are comparable to planar regions. If you consider a sphere, you imagine it embedded in tridimensional space, but truth is you don't know the third dimension. All you know is your 2-dimensional "curved" life on the sphere. This is why you need new tools to define differentiation, that is usually linear and "flat". Manifolds find many real-life applications: for instance seismic tomography (see above) or robotics.

### Analysis and Calculus

*Harmonic series*. Their name comes from music. The basic question is: can we get infinity by summing up infinitely many infinitesimal quantities? Turns out it depends, as you can see from the figures. Relating to harmonic series is the problem of book stacking.

*Integration*. The theory of integration started several centuries ago, when people questioned how to calculate "difficult areas", but it was rigorously stated by Riemann in the XIX century. Below you find my explanation of his theory and a visual solution about integrability of monotone functions.

*Pompeiu problem*. Formulated in the past century, it's still open. Here's my favourite reference on the topic. See the problem below.

*Banach-Tarski paradox*. With a very formal proof, Banach and Tarski showed it is in principle possible to cut a sphere in pieces and recombine the pieces to get two spheres identical to the first. The trick is to decompose the sphere in non-measurable sets, that is something really artificial and odd.

### History of Mathematics

*Math duels* ("disfide"). Around 1500, mathematicians used to challenge each other to math duels, to prove who was smarter and more able to solve problems. A famous series of such episodes concerned the formula to solve cubic equations, that is equations of the form . It all started with Tartaglia, an Italian mathematician nicknamed like that for his stammering. Born in a poor family, he had been a soldier, a topographer, and was a talented self-taught mathematician. At the time, mathematicians rarely published their results and only shared them with their students and maybe family, so that they would have an ace in their hole when it came to being hired by some university. At some point, a mathematician contemporary to Tartaglia, Antonio Maria del Fiore, started bragging about knowing the formula to solve cubic equations, which he learned from his teacher Scipione Del Ferro. Tartaglia was an ambitious researcher and found the formula independently. He then accepted a "math duel" from del Fiore. Tartaglia annihilated del Fiore and became immediately famous. Cardano, famous mathematician of the time, invited him to Milan. Tartaglia told him the "secret formula", under the oath he would not reveal it. However, he still hesitated to publish it. Years later, Cardano found out that Del Ferro - then long gone - had discovered the formula previously and independently, thus he felt relieved from his promise to Tartaglia. He then allowed his student Ferrari to publish their research and improvements on Tartaglia's formula. The result was a duel, in which Tartaglia lost due to his stammering and lack of confidence in public speaking. Luckily history gave him credit and the solving formula is named also after him.

*Evariste Galois*. One of the most brilliant mathematicians ever existed, he died very young in a duel (a real one!). Here his biography. His results were crucial to Wiles to prove Fermat Last Theorem.

*The stolen theorem*. One of the most popular calculus theorems is believed to have been commissioned by De L'Hopital to Johann Bernoulli.

*Riemann's hypothesis*. A great popular book to know more about prime numbers and a legendary problem in analytical number theory is The Music of Primes by Marcus Du Sautoy.

*The Newton-Leibniz controversy*. The two illustrious contemporary scientists fought to be recognised as inventor of calculus. Historians now believe they both were right and invented calculus independently, but at the time Newton's influence allowed him to win the argument and Leibniz died in disgrace. This rivalry was quoted in the popular show The Big Bang Theory.

*Paul Erdos*. Some say he was the greatest mathematician of the past century. He collaborated with more than 500 people, writing more than 1,525 papers, in many different areas and topics! Erdos belonged to an Hungarian Jewish family that lived during Nazism. His father died in the Holocaust and his mother survived in hiding. He didn't have a home: he kept travelling around the world, stopping at conferences or hosted by colleagues. He would knock in the middle of the night at the door of some colleague and say he was ready to solve some problem. No one would dare to turn him down and throw the chance away, 'cause his productivity and genius were out of ordinary. He received 15 honorary doctorates.

*Alexander Grothendieck*. I wrote about him when he passed last year. Someone translated some of his lectures in English. You can read more on his life here.

### Riddles

*The hanging picture problem*. Hang a picture to the wall using two nails and a string. Fix the string in such a way that if one (*any*) of the two nails is removed, the picture falls down.

Solution: name by variables the following actions: means "to pass the string over nail 1 clockwise", means "to pass the string over nail 1 counterclockwise". Similarly for and nail 2. We can combine actions through a sort of multiplications. Passing the string over nail 1 in one sense and then the other would be , so we understand that corresponds to "nothing is done", meaning "the picture falls down". Also, observe (physically) that the actions are not commutative: try and check that it is not equivalent to do only . Also, removing for instance nail 1 corresponds to ignore all actions as and . Hence our problem translates to: write a product so that by removing all occurrences of $x, x^{-1}$ or you get 1. Since our product is not commutative, one solution will be: . Through abstraction you can easily generalise this trick to any number of nails!

A reference link on the topology behind this.

*The prisoners*. Three prisoners meet a guard in a room. The guard says: "I have hats with me, two of which are black and the others are white". The prisoners ask: "How many hats there are all together?", the guard replies: "It's a secret!". The guard picks three of the hats and puts them on the prisoners' heads. All prisoners see the others' hats color, but not their own. The guard asks them, one after another: "What is the color of your hat?". The first prisoner looks around and replies: "I don't know". The second prisoner looks around and replies: "I don't know". What does the third prisoner reply and what is his hat's color?

Solution: denote B=black and W=white. Because of the replies of prisoner 1 and 2, we can rule out the combinations: W B B and B W B. We are left with the following cases: W W B, B W W, W W W, B B W, W B W. All cases except one show that the third prisoner's hat must be W. Consider the case W W B. The second prisoner hears the first's reply, so he understand he cannot see two black hats, leaving only the possibilities of him seeing two white hats or one black and one white. When his turn comes, he sees the third wearing black, so he concludes his own must be white and replies so. Since he says he doesn't know, we must rule this case out and conclude the third prisoner has white hat and says so.

*Palindromes*. Find the smallest 3-digit palindrome number that is divisible by 18.

Solution: our number, denote by **ABA**, must be divisible by 2 and 9. A number is divisible by 9 if the sum of its digits is divisible by 9, hence A+B+A=2A+B is divisible by 9. ABA must be also even, that leaves us the following possibilities:

A=2 -> 2A+B=4+B -> B must be 5

A=4 -> B=1

A=6 -> B=6

A=8 -> B=2

The smallest is 252.

*The Monty Hall problem*. This is a counter-intuitive problem of probability. Even Paul Erdos, considered the best mathematician of past century, did not believe the solution until he saw computer simulations! The problem asks: you are a guest of a TV programme. The presenter shows you three closed doors: behind one there's the car of your dreams, behind the other two there are goats. He lets you pick a door (say, n.1), but before opening it, he opens one of the other two (say, n.3) and shows a goat behind it. He then asks you: "Do you want to stick with your choice or change to door n.2?". What's the best strategy? The solution says the best strategy is changing your initial choice. Here some material to understand this: link 1, link 2, link 3.

*True statements*. How many statements are true?

At most 0 statements in this block are true.

At most 1 statement in this block is true.

At most 2 statements in this block are true.

At most 3 statements in this block are true.

Solution: only the last two statements are true. If you assume no statement is true, then the first is true, which is a contradiction. If you assume only one is true, the last three are true, again a contradiction.

*The 12 ball problem*. You have 12 balls, looking exactly the same, but one is an odd weight (you do not know if it's lighter or heavier). You have a scale (see figure below) and at most three chances to use it. How do you find the odd ball? Solution is in the gallery below.

*Sticks*. Use 6 identical sticks to build 4 identical triangles.

Solution: click here.

### Miscellanea

- How to draw a perfect ellipse.
- The golden ratio to be found in architecture and nature.
- Visualising Pythagoras' theorem.
- Communication of science: SoapBoxScience.
- The Mathematics of Sudoku. 17 is the minimum number of clues to have a unique solution. The "most difficult scheme" was created by a Finn.
- Magic square. In antiquity, they were believed to hold mystical power. Euler's work on magic squares (translated to English).
- Bayesian approach to Inverse Problems: link 1, link 2.
- Gödel incompleteness theorems. Proved by a young Gödel in 1931, they threw mathematicians in panic (still do). A funny look into the topic.
- Mathematical models for traffic jams.
- Free version of the book "A Mathematician's Apology" by the great number theorist G. H. Hardy.