# HLFF Spotlight: 9th HLF

# BLOG: Heidelberg Laureate Forum

# The Pleasure of Abstract Structures – Paul Siewert

*The Heidelberg Laureate Forum has a single purpose: To provide some of the brightest minds in mathematics and computer science with the space and time to make connections and find inspiration. The HLFF Spotlight series shines a light on some of the brilliant young researchers attending the event, their background and research, as well as their expectations for the HLF.*

*Paul Siewert is currently working towards a Bachelor’s degree in the fields of mathematics and linguistics from the University of Göttingen. Already during his final high school years, he was involved in multiple mathematical research projects and received the first prize in the youth research competition “Jugend forscht” for challenging computational limits of FRACTRAN. In this interview, he gives us an introduction into his research, talks about his motivation for studying and explains why he is looking forward to the 9th HLF.*

Paul Siewert is attending the 9th HLF in September 2022. Image Credits: private

**What motivates you to study Mathematics and Linguistics?**Both mathematics and linguistics are great pleasures to me, being equally about rather abstract structures. Linguistics deals with structures we can directly observe around us: Language, connecting and constituting many aspects of culture and people. This fabric of complex, important and often elusive structures is something I appreciate a lot. Studying linguistics allows me to deepen this appreciation, for looking closer unveils more beauty, which is right at my fingertips when browsing through etymologies or investigating grammar. This main point holds even more for mathematics. Building theories around abstract concepts gives them substance; manipulations of strings take on meaning just by being interpreted in some abstract way; formal progress and abstract beauty are almost in congruence. For some, such abstraction may seem pointless, but to me, it justifies itself. Mathematics deals with forms of pure thought – a most wonderful art.

**Why do you want to participate in the Heidelberg Laureate Forum?**The HLF conceptually has three main ingredients: A

*forum*for

*laureates*and young

*researchers*. All of these will hopefully allow for an amazing experience. The talks about Mathematics will surely be very inspirational as a basis for thought and as insight into the state and diversity of current research. One cannot think of many better ways to learn than from leading and accomplished researchers, both young and experienced. On a more personal level, it is of course like being a VIP, but so much better: The HLF allows insight into the careers and personalities of successful people from all over the world, many of which I would otherwise probably never meet. Especially since I am still at the very beginning of my career, this inspiration could go a long way. I am really excited to meet so many very smart and extremely interesting people, to discuss mathematics, do mathematics, to find insight, inspiration, motivation, and of course friends.

**Do you have a mentor or a person who inspires you?**Many people around me and within my circle of friends inspire me in one way or the other. I am very happy to have serendipitously met two very experienced mathematicians who became my friends and, one could say, mentors. One of the two was one of the organizers of the International Tournament of Young Mathematicians in 2020, where I participated. I was lucky enough to have an inspiring chat with him at the tournament, so we stayed in touch afterwards and even became friends. We often discuss mathematical problems and have been working together on different projects since. For example, we proposed two successful projects for the German youth competition “Jugend forscht” together, one of them being awarded first prize.

**What was your prize-winning project about?**Our aim was to improve the computational limits of FRACTRAN. FRACTRAN, as introduced by John Conway, could simply be described as a programming language for abstract register machines – however, this would spoil the magic: Consider a list of fractions and take an integer. Then multiply it by the first fraction making the product an integer again. Repeat this until no such fraction is found. Perhaps surprisingly, this model is Turing-complete. In fact, one can construct a “universal list of fractions” by writing an interpreter for FRACTRAN in FRACTRAN. Two simple tricks and reworking the logic allowed us to reduce the number of fractions needed from 23 to 13 fractions. We also showed that only 2 fractions alone cannot do a lot. This gives a wonderful example for what Conway meant when he said “complexity lurks just around the corner”: Somewhere between 2 and 13 fractions is a jump at which “everything” becomes possible – although even 13 fractions would not be expected to maximize complexity in computational behavior.

Paul Siewert (right) with his colleague Caspar Kiehn (left) presenting their winning project of the “Jugend forscht” student research competition. Image Credits: Jugend forscht e.V.

**What was your second research project about?**On a computational level, a FRACTRAN program iterates a very simple arithmetic function. In the second project, we studied a phenomenon similar to FRACTRAN noted by Friedman et al. in 2019: Take the function which rounds a number down to the nearest integer and multiply it by one more than the fractional part (2.6 maps to 3.2 maps to 3.6). Starting from a specific constant described by Friedman et al., the sequence for real numbers produced by iterating this function can be rounded down – yielding precisely the sequence of all primes. Surprised by this, we tried to understand the underlying mechanism. We hypothesized that this phenomenon holds true for not only one function. Indeed, we were able to generalize the concept to a large class of functions and plan to share our insights in a publication.

**What do you do when taking a break from solving mathematical problems?**Studying linguistics also takes up a good share of my time: On the one hand, I am trying to get more into the literature and learn about the current state of the field, which I find to be much more difficult than for mathematics. On the other hand, I spend time actually learning new languages. The last two semesters, I started to study Sanskrit, which plays a vital role in Indo-European linguistics.

Apart from that, engaging with contemporary art, discourse, and culture defines much of my time not spent with mathematics. While I do not regularly make art myself, I enjoy playing the viola, which I have been doing for close to three years now.

**What part of the HLF are you looking forward to the most?**Definitely dinner. This is where people will come together, meet each other for the first time or again. And it is where the most interesting discussions lasting into the night will start, I am sure.

*More inspirational stories are to come in the HLFF Spotlight series, so stay tuned.*

Lena Sophie Schwenker wrote (08. Sep 2022):

>

[…] Paul Siewert [said in an interview] » […] In fact, one can construct a “universal list of fractions” by writing an interpreter for FRACTRAN in FRACTRAN. Two simple tricks and reworking the logic allowed us to reduce the number of fractions needed from 23 to 13 fractions. «Nice!

To operate your FRACTRAN interpreter (at least in principle) in a specific case, I gather that you’d have to select a specific input value

`n`

(in reference to this notation of the FRACTRAN algorithm)such that this input value

`n`

uniquely encodes the specific FRACTRAN program, say`P_p`

, whose operation is to be interpreted as it starts up on the specific (non-negative integer) input value`q`

.So, apparently your interpreter for FRACTRAN programs comes (at least implied) with a specific and more or less explicit list of FRACTRAN programs, matching each distinct FRACTRAN program uniquely to a corresponding index value

`p`

,and with a specific and rather explicit encoding function

`n := n[ p, q ].`

Does the specific encoding function implied with your interpreter for FRACTRAN programs strictly satisfy

`n[ p, q ] > q`

for all programs (i.e. each of the applicable index values

`p`

) and all (non-negative integer) input values`q`

?p.s.

Please discuss possible implications for trying to prove that there is no FRACTRAN program which computes the halting function

`h`

(of all the FRACTRAN programs, with any non-negative integer as input value), when given the correspondingly encoded integer`n := n[ p, q ]`

as input value, with the specific encoding as implied by your interpreter.