Tanya Khovanova's Math Blog

This is my second crocheting project to help finance our new program, Yulia’s Dream, in honor of Yulia Zdanovska, a young Ukrainian math talent killed in the war.

I made four Whitehead links in the colors of the Ukrainian flag. You can read about my other crocheting project in my previous post: Hyperbolic Surfaces for Ukraine.

Fun trivia about the Whitehead links.

• Why are these links so famous? They are the simplest non-trivial links with the linking number zero.
• What’s the linking number? The linking number is an invariant of a link. If two loops are not linked (they are called an unlink), their linking number is zero. If they are linked, then their linking number is usually not zero. Here the loops are linked, but the linking number is nevertheless zero. Thus, the linking number can’t differentiate this link from an unlink. To explain how to calculate the linking number, I need to explain another simpler invariant: the crossing number.
• What’s the crossing number? The crossing number is the smallest number of crossings when projecting the link on a plane. The top two pictures have 6 crossings, and the bottom two pictures have 5. The top two pictures emphasize the symmetry of the link, and the bottom two pictures have the smallest possible number of crossings for the Whitehead link. So the crossing number of the Whitehead link is 5.
• Can you now explain how to calculate the linking number? One way to calculate the linking number is to choose directions for the blue and yellow loops and select the crossings where the blue is on top. After that, following the chosen direction of the blue loop, at each crossing with the yellow loop, check the latter’s direction. If the direction is from right to left, count it with a plus and, otherwise, with a minus. The total is the linking number. In the case of the Whitehead link, it is zero.
• Is there a more elegant explanation for why the Whitehead link has the linking number zero? Yes. The linking number only looks at the crossings of two different strands and ignores self-crossings. If you look at the two bottom pictures, there is one self-crossing of the blue loop. Now imagine you change the crossing by moving the top blue strand underneath. After such a transformation, the crossing number doesn’t change, but the loops become unlinked. Thus, the linking number of the Whitehead link must be zero.

As you might know, my team started a project, Yulia’s Dream, in honor of Yulia Zdanovska, a young Ukrainian math talent killed in the war.

In this program, we will do what we are great at — help gifted youngsters pursue advanced math. To help the program, I started crocheting hyperbolic surfaces in the colors of the Ukrainian flag. These crochets are designed as gifts to encourage individual donors.

Fun trivia about these hyperbolic surfaces.

• Why are these surfaces so famous? These surfaces prove that Euclid’s fifth axiom is independent of the other four axioms. The fifth axiom (also known as the parallel postulate) says that if there is a line L and a point P outside of L, then there is exactly one line through P parallel to L. On these hyperbolic surfaces, the first four of Euclid’s axioms hold, while the fifth one doesn’t: if there is a line L and a point P outside of L on such a surface, then there are infinitely many lines through P parallel to L.
• But what is a line on a hyperbolic surface? A line segment connecting two points is defined as the shortest path between these points, known as a geodesic.
• How can such a surface be crocheted? I crocheted a tiny circle and continued in a spiral, making 6 stitches in each new row for every 5 stitches in the previous row. This means that each small piece of the crocheted surface is the same throughout the thingy, making these thingies hyperbolic surfaces of constant curvature.
• What is the constant curvature good for? Constant curvature makes it easy to find lines. You can just fold the thingy, and the resulting crease is a line.
• Is this thingy a hyperbolic plane? No. A cool theorem states that a hyperbolic plane can’t fit into a 3D space, so whatever someone crochets has to be finite. On second thought, anything someone crochets has to be finite anyway. But I digress. This shape can be viewed as a disc with a hole.
• The Ukrainian flag is half blue and half yellow, so why do the colors here seem so unevenly distributed? My goal was to use the same amount of blue and yellow yarn per thingy. I leave it as an exercise for the reader to calculate that regardless of how many rows of one color I crochet, to use the same amount of yarn in the second color, I need more than 3 and less than 4 rows of that color. Since I wanted the thingy to be symmetrical, sometimes I had 3, and other times, I had 4 rows of the second color. I also made 4 surfaces where I switched colors every row.

Overall, I crocheted 10 hyperbolic surfaces. If you are interested in donating to help Ukrainian students receive coaching from our program at MIT, the details will be announced on our website shortly.

Putin became the 21st century Hitler. I call him Putler.

I am an American. However, I was born in Moscow and lived my youth in the Soviet Union. I speak Russian, and I have friends in both Russia and Ukraine. The war in Ukraine is the biggest tragedy of my life. When Putler invaded Ukraine, I didn’t know what to do. I wanted to pick up a rifle and go to Ukraine to fight, but then I remembered my CPAP machine and the distilled water it needs, and I didn’t go. Instead, I ended up watching the news non-stop. Then I started sending money to different organizations supporting Ukraine.

However, I am a mathematician, so I tried to figure out whether I could help Ukraine by doing math. At first, I posted math problems from Ukraine Olympiads. Then I started discussing what we could do with my PRIMES colleges. The result was a new program, Yulia’s Dream, in honor of Yulia Zdanovska, a 21-year-old brilliant young Ukrainian mathematician killed by a Russian-fired missile. Yulia’s Dream is a free enrichment program for high-school students from Ukraine who love math.

All these activities didn’t help me with the pain. So I started crocheting. I bought yarn in the colors of the Ukrainian flag and crocheted a hyperbolic surface of constant curvature. The first picture shows the thingy from above. The second one is there for you to estimate its size: this is the biggest crocheting project I have ever finished.

For a free Ukraine! Let democracies win over dictatorships!

Oskar van Deventer is a designer of beautiful mechanical puzzles. For the recent mini-MOVES gathering at the MoMath, he asked people in his Zoom breakout room to calculate the average age in the room without revealing their actual ages. I know the following solution to this puzzle.

People agree on a large number N that is guaranteed to be greater than the sum of all the ages. The first person, say Alice, thinks of a uniformly random integer R between 0 and N. Alice adds her age to R modulo N and passes the result to the second person, say Bob. Bob adds his age modulo N and passes the result to the third person, and so on. When the result comes back to Alice, she subtracts R modulo N and announces the sum total of all the ages.

During this process, no one gets any information about other people’s ages. But two people can collude to figure out the sum of the ages of the people “sitting” between them.

I gave this problem to my grandson, and he suggested the following procedure. First, people choose two trusted handlers: Alice and Bob. Then, each person splits their age into the sum of two numbers (the splitting should be random and allow one of the numbers to be negative). They then give one number to Alice and another to Bob. Alice and Bob announce the sums of the numbers they receive. After that, the sum total of everyone’s ages is the sum of the two numbers that Alice and Bob announce.

The advantage of this method is that no two people, except Alice and Bob, can collude to get more information. The disadvantage is that if Alice and Bob collude, they would know everyone’s age. Which method would you prefer?

Many years ago, I wrote a blog post about an amusing fact: John Conway put Moscow, the former capital of the USSR, as a coauthor: A Math Paper by Moscow, USSR. Thus, Moscow got an Erdős number 2, thanks to Conway’s Erdős number 1. At that time, my Erdős number was 4, so I wondered if I should try coauthoring a paper with Moscow, my former city of birth, to improve my Erdős number.

This weird idea didn’t materialize. Meanwhile, my Erdős number became 2 after coauthoring a paper with Richard Guy, Conway’s Subprime Fibonacci Sequences. I relaxed and forgot all about my Erdős status. I couldn’t do better anyway, or could I?

I recently heard about a cheater who applied to grad schools. In addition to a bunch of fabricated grades, the cheater submitted an arXiv link to a phony paper. What is fascinating to me is that the cheater put real professors’ names from the university the cheater supposedly attended as coauthors. The professors hadn’t heard of this student and had no clue about the paper. So the cheater added fake coauthors to add legitimacy to their application and boost the perceived value of the cheater’s “research”. As a consequence, the cheater got a fake Erdős number.

I hope that the arXiv withdrew the paper. Cheating is the wrong way to improve one’s Erdős number.

But here is another story. Robert Wayne Thomason named as coauthor his dead friend, Thomas Trobaugh. The paper in question is Higher Algebraic K-Theory of Schemes and of Derived Categories and can be found at https://www.gwern.net/docs/math/1990-thomason.pdf. This paragraph in the paper’s introduction explains the situation.

The first author [Robert Wayne Thomason] must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom’s simulacrum remarked, “The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf.” Awaking with a start, I knew this idea had to be wrong, since some perfect complexes have a non-vanishing K₀ obstruction to extension. I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom’s simulacrum had been so insistent, I knew he wouldn’t let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper.

This precedent gives anyone hope that they might achieve an Erdős number 1. You just need to wait for Paul Erdős to come to you in your dreams with a genius idea.

Tim Gowers discussed the age of consent on his blog, which I can no longer find. I will talk about his post here based on my old notes and my memory. The age of consent is a legal term to protect young people from being manipulated into agreeing to sex. Having consensual sex with people under the age of consent may be considered statutory rape or child sexual abuse.

Gowers starts with several assumptions.

• Non-triviality: There should exist an age at which a person is qualified to consent to sex and, consequently, have it.
• Simplicity: Whether or not two people are allowed to have sex with each other should depend only on their age in years.
• Monotonicity: If two people are allowed to have sex with each other today, they should be allowed to have sex with each other at all times in the future.

From these assumptions, the following theorem can be deduced.

Theorem. The only possible rule satisfying these assumptions would allow any two people to have sex with each other as long as they both reached some fixed age k.

There is a problem with this type of rule. Suppose k is 18. If two people who are slightly younger than 18 have consensual sex, they can’t both be predators. These are two children with raging hormones. There is no reason to punish anyone. Now imagine that one of the partners turns 18. Society would still consider this a Romeo-and-Juliet case and would tend not to punish such a partner. Now imagine a child younger than 18 having sex with a partner over 40. The older partner has no raging hormones, knows what they are doing, and probably knows how to manipulate little children into having sex. So, it might be desirable to have a rule that differentiates between these two cases. The rule would take into account the difference in ages while forgiving younger offenders and still punishing predators.

Consider the most common type of law to resolve this issue: Anyone older than 18 can have sex, and, in addition, a person who is not older than 20 can have sex with someone between the ages of 16 and 18. This law doesn’t satisfy monotonicity. It could be that one day the older partner is not yet 20, and the next day, oops, they have a birthday. So, as a birthday gift, they are not allowed to have sex with each other anymore.

Here is a simple idea to resolve the issue by having the law focus on the age gap instead of the age of the older partner. We can have an adjusted law: Anyone older than 18 can have sex, and, in addition, a person can have sex with someone between the ages of 16 and 18 as long as the age gap is not more than four years. This rule doesn’t satisfy the simplicity assumption above, but it is simple enough. It is close in spirit to the previous rule and satisfies monotonicity. The problem with this rule is continuity.

• Continuity: If the age gap between couple A is only slightly larger than the age gap between couple B, then couple A should not have to wait significantly longer to be allowed to have sex.

According to the adjusted rule, the couple with the age gap of four years and one day might have to wait two years longer to have sex than the couple with the age gap of four years. This seems unfair.

Tim Gowers suggests dropping the simplicity rule. We can use days rather than years. For example, the rule might be that if one person in a couple is under 18, but at least 16, and has age x, then the other partner has to be not more than age y, where for example, y − x = 4 + (x − 16)/2. So when one partner turns 16, their partner has to be not older than 20. When one partner is 16 and two months, the other cannot be older than 20 and three months. With the younger partner getting older, the allowable age gap is increasing slowly. By the time the younger partner is a day from turning 18, their partner can be almost five years older.

It might be complicated for two people to calculate if they are allowed to have sex according to this formula. But Gowers’ big idea was that apps and websites could do this easily: two people plug in their birthdays and know whether they are allowed to have sex.

Puzzle. Two girls were born to the same mother, at the same time, on the same day, in the same month, in the same year, and yet somehow they’re not twins. Why not?

I won’t tell you the expected answer, but my students are inventive. They suggested all sorts of scenarios.

Scenario 1. There are two different fathers. I had to google this and discovered that, indeed, it is possible. This phenomenon is called heteropaternal superfecundation. It happens when two of a woman’s eggs are fertilized by sperm from two different men. Unfortunately for my students, such babies would still be called twins.

Scenario 2. The girls are born on the same date, but not on the same day. This could happen when transitioning from the Julian to Gregorian calendar. The difference in birth times could be up to two weeks. I had to google this and discovered that twins can be born months apart. The record holders have a condition called uterus didelphys, which means that the mother has two wombs. Unfortunately for my students, such babies would still be called twins.

Scenario 3. The second girl is a clone. This scenario can potentially happen in the future. Fortunately for that student, I suspect that such babies would be called clones, not twins.

I decided to invent my own scenario outside of the actual answer, and I did.

Scenario 4. Two girls are from the same surrogate mother, but they are not twins. I had to google this and discovered that this actually happened: Surrogate mother of ‘twins’ finds one is hers.

Sometimes life is more interesting than math puzzles.

I used to hate crocheting. Now it’s been growing on me.

I’ve been crocheting hyperbolic surfaces of constant curvature. The process is time-consuming, so while I am crocheting, I wonder about the mathematics of crocheting.

Hilbert’s theorem says that I can’t embed a hyperbolic plane in 3-dimensional space. The proof is rather involved. But here, I have an explanation from the point of view of a crochet hook. My hook starts with a tiny cycle of four stitches. Then for every x stitches the hook makes y stitches in the next row, where y is greater than x. The extra stitches should be evenly distributed to guarantee that locally every small area is approximately isomorphic to other areas, meaning that the surface has a constant curvature.

The ratio of stitches in the next row to the current row is r = y/x. Thus, the number of stitches in each row increases exponentially. But each row is a fixed height h. That means after k rows, my thingy has to fit inside a ball of radius kh. But the length of the last row is 4r^k-1. It becomes huge very fast. As the last row is a physical curve made out of stitches, there is a limit of how much of it I can fit into a given volume, creating a contradiction.

That means, if I start crocheting, something should happen that won’t allow me to continue. I decided to experiment and see what actually would happen. Being lazy, I preferred the disaster to happen sooner rather than later. So I chose the ratio of three: for each stitch on my perimeter, I added three new stitches. Shortly after I started to work, the process became more and more difficult. The ball was too tight. It was challenging to hold that thing in the place where I needed to insert the hook. And the loops were getting tighter, making it more exhausting to insert the hook into the proper hole. So each new stitch was taking more and more time to complete.

To my disappointment, the thing didn’t explode, as I was secretly hoping: I just couldn’t work on it anymore.

As you may know, I run PRIMES STEP, a local program where we do mathematical research with students in grades 6-9. Last academic year, we looked at the stable marriage problem and discovered its connection to Sudoku. Our paper The Stable Matching Problem and Sudoku (written jointly with Matvey Borodin, Eric Chen, Aidan Duncan, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, Michael Voigt) is now available at the arxiv.

Consider 3 men and 3 women who want to be married to each other in heterosexual couples. They rank each other without ties. The resulting 6 permutations of numbers 1, 2, and 3 that describe the six rankings are called the preference profile of this group of people. A matching is unstable if two people would be happier to run away together than to marry into the assigned couples. The two potential runaways are called a rogue couple. A matching is called stable if no rogue couple exists. The Gale-Shapley algorithm, invented by Gale and Shapley, finds a stable matching for any preference profile, implying that stable matching is always possible.

We discovered that preference profiles form a natural bijection with ways to place one digit into a Sudoku grid. So we wrote a paper describing the stable marriage, rogue couples, the Gale-Shapley algorithm, soulmates, and such in terms of Sudoku.

Oops, I forgot to explain who the soulmates are. We invented this term to describe two people who rank each other first. Though it is possible to have several stable matchings for the same preference profile if the soulmates exist, they must always be matched together.

We also invented a new Sudoku type, which I will explain next time.