For decades, mathematicians used a whole arsenal of functions—logarithms, powers, trigonometry—without realizing that they could all be reduced to a single, universal operator. Polish astrophysicist and mathematician Dr. Andrzej Odrzywołek (Jagiellonian University, Kraków) showed how to do this, and his work sparked widespread attention. In an interview for fast-takeoff.com, we discuss mathematics that looks like science fiction, whether analog computers can make a comeback, and how artificial intelligence is changing the way science is done.

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Your latest paper, “All Elementary Functions from a Single Operator”1 has caused quite a stir online. But if exponential functions, logarithms, and their connections have been known since Euler, why has such a simple, universal operator eluded mathematicians for centuries?

Dr. Andrzej Odrzywołek: Because the scientific procedure may not be clear to the broader community, let’s clarify: this is not a [scientific] paper but a preprint. The results have not yet been formally peer-reviewed and finalized.

It is common knowledge among experts that only a few operations are sufficient to provide basic “elementary” functions, which I define as those known from scientific calculators. For example, Mathematica, used for calculations by theoretical physicists, requires only four operations: addition, multiplication, exponentiation, and the base logarithm. Converting multiplication to addition and vice versa is also, and may have been obvious before calculators appeared. Every schoolchild, student, or engineer had to master the use of tables or a slide rule. The exponential function exp(x), its inverse (the natural logarithm ln(x)), and the four operations (+, −, ×, ÷). That’s enough. Further steps from the recommendation (notice of centuries to exaggeration) are simply not followed and are not implemented. The key formula for the logarithm in EML form requires 7 tools and three applications, which is borderline “natural.” EML itself requires 3 operations on a 36-key calculator, a number that can be checked with a naive algorithm up to about 6 million (2⁷ × 36³). Too many, due to the concatenation at the end, but few enough that someone had to do the math in the first place.

Some people admire the elegance of EML, others ask, “OK, but why?” How do you respond to the accusation that it’s ultimately just mathematical gymnastics, a kind of “art for art’s sake”?

Three questions in one :-)

The concept of elegance is subjective. It’s more accurate to talk about complexity or simplicity. One binary operation and the number 1 are less than a set composed of several functions of a single variable, binary operations, and constants. Formulas compiled into EML can seem ugly, like the pi formula circulating online. I’d call it rather lengthy. However, the EML operator itself is indeed elegant; its form really surprised me. I expected something worse, maybe not necessarily as strange as, say, Minkowski’s ?(x), but rather among special functions rather than school functions.

I needed a single operator for what’s called symbolic regression, i.e., searching for mathematical patterns in data sets. Recently, such techniques have been sidelined, because AI does the same thing much better. The main problem in SR is choosing a base: sine or cosine, tangent or cotangent, addition or subtraction, etc. Using too many operations complicates the search; too few, and we risk missing some valid expressions altogether. For a long time, I’ve had the idea for an automatic verifier that, before running a search, for example, in PySR, checks whether the set of constants (e.g., 2, pi, e), functions (e.g., sin, sinh, cos, cosh), and operations (e.g., +, −, ×) used will actually find any expression. While testing the verifier (VerifyBaseSet), I fed it various things and observed what happened. Then I realized that with just one operator, the problem disappears altogether.

Without a doubt, EML is a kind of intellectual gymnastics. But as Stanisław Ulam wrote:

In your publication, you also write about a prototype EML compiler and the potential for analog computer design. Does this mean that in the future, our processors could operate on a completely different, simplified architecture? Where exactly might EML find the quickest physical application?

Analog computers were long ago replaced by digital ones, but that doesn’t mean they’ve ceased to exist. Their use is limited by technical problems, one of which is the inability to generate arbitrary elementary functions. EML could fill this gap. Generating an exponent is simple: it’s the law of capacitor discharge; subtraction is performed by an operational amplifier. But along the way, we need complex numbers. The path from theory to practice won’t be straightforward. The brain undoubtedly operates in analog mode, and we can’t even replicate the brain of a fruit fly.

Just three years ago, the world of mathematics was captivated by the “einstein” tile: a single prototile that by itself forms an aperiodic set of prototiles. In a sense, you yourself have a similar invention to your name: a single operand that can express any real-world function. Do you think other fields are still hiding their “universal building blocks” from us?

These are related. I myself invented and hold a patent for a three-dimensional polyhedron whose shape allows you to build an igloo. You can find a description in the journal Eureka (”How to Build the Perfect Igloo”2 2014), published by mathematics students at Cambridge, and its model is held by the NKF UJ. I believe that redirecting the computing power of hundreds of thousands of GPUs currently used to train AI to search the solution space of various problems could reveal many surprises.

Apparently, EML is just the beginning, and you’re already exploring its ternary variant, which doesn’t need the constant “1” at all. Why is eliminating this one extra digit so important to you, and what would it take to find a fully self-sufficient operator?

The need to consider the possibility of a specific constant at every level complicates calculations and increases the number of parameters in the EML tree. It also acts as a kind of discrete switch that removes the logarithm from EML. A model without any specific constant, containing only variables x, y, z... would be fully continuous. Such a binary operator would be even more similar to NAND. I haven’t been able to find it. Perhaps it doesn’t exist or is not an elementary function.

At the end of the publication, you openly admit that modern language models have aided you in your work, including translating code into Rust. What’s your day-to-day collaboration with AI like?

Language models have accelerated many things. Firstly, effective translations and multilingual search. Now, on X, I immediately see, for example, posts originally in Japanese that would probably be completely incomprehensible to me without them. From a circuit diagram alone, I wouldn’t have guessed that someone was presenting a proposal for a hardware EML implementation. Secondly, a literature review, including historical texts in Latin or French (Cotes, Liouville). This is very helpful, especially for interdisciplinary research in an era of narrow specialization. In a sense, LLMs fulfill the promise that Google once gave us and then took away: the ability to find any information we need.

Another issue is language proofreading. However, this mainly works with standard texts and requires extreme caution, as the problem of hallucinations still persists. Code translation is the latest AI achievement; a breakthrough occurred in December 2025. Currently, the problem isn’t writing the program itself, but rather clearly defining what it should do and how. It turns out that the best way to explain how a program should work is to have another program do the same thing. An AI agent like GPT Codex or Claude Code can independently run and test programs, allowing for efficient code translation, for example, from Mathematica to Rust. Having two versions allows for easy verification of whether they function identically. The difference is speed: the original VerifyBaseSet requires 40 minutes, but after rewriting it in Rust, it takes just seconds. AI also facilitates working with different operating systems; the repository included with the preprint works on Windows, Linux, and MacOS.

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Dr. Andrzej Odrzywołek – Polish astrophysicist and mathematician, member of the Department of General Relativity and Astrophysics at the Institute of Theoretical Physics of the Jagiellonian University in Kraków. Author of the paper “All Elementary Functions from a Single Operator,” which caused a stir in the international scientific community. Holder of a patent for a three-dimensional polyhedron for igloo construction (the model is in the Jagiellonian University collection).

Links:

1. UJ employee page

2. LinkedIn

3. X/Twitter

In an era where "AI" has become a buzzword often stripped of its technical weight, GHOST Day: Applied Machine Learning Conference stands as a beacon for those who value engineering depth. What started as a "horribly optimistic" initiative by students at Poznan University of Technology has evolved into a premier European event, attracting researchers from ETH Zürich, Meta, and Mistral AI. This is not just another tech talk—it is a space where scientific rigor meets real-world implementation. We sat down with the organizers to discuss the evolution of the "GHOST" spirit and why 2026 is the year to move past the hype and focus on what truly works.

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What exactly is GHOST DAY, and what is the story behind your intriguing name?

Piotr Wyrwiński (GHOST’s Supervisor): GHOST stands for Group of Horribly Optimistic STatisticians, and the name actually tells our origin story pretty well. It started when Mateusz Lango Ph.D. was teaching Statistics at Poznan University of Technology. A group of ambitious students wanted to go beyond the curriculum, and since Mateusz was doing his PhD in machine learning, he pointed them toward Andrew Ng’s famous ML course on Coursera. That naturally turned into regular meetings, and those meetings grew into what became the GHOST research group. The name stuck because it captures that initial spirit of students who were, let’s say, horribly optimistic about how far a statistics background and sheer enthusiasm could take them in ML. Turns out, pretty far.

Today GHOST is a student organization focused on studying and conducting research in machine learning. Every semester we run study groups and project teams covering different areas of ML. This semester, for example, we have sections on topics ranging from TinyML and quantum computing for ML, through LLMs and robotics, to a team building a flood prediction system using deep learning and satellite imagery. You might have also come across us at hackathons, Kaggle competitions, or other conferences.

GHOST Day: Applied Machine Learning Conference is our flagship event. The idea is to create a friendly, energetic space where ML enthusiasts can exchange experiences and stay current in a field that moves incredibly fast. Our speakers include researchers publishing at top venues like NeurIPS or ICML and experts from companies building real ML-powered products. The conference isn’t tied to any specific programming language or tech stack; it’s about techniques, methods, and algorithms. We also see it as a gateway for talented engineers who want to enter the world of AI, through intermediate-level talks, a poster session where attendees can present their own work, and direct contact with companies in the industry.

The 'Applied' subtitle seems to be a core promise. Is this a clear signal that you focus on real-world implementations and problem-solving rather than just abstract theory?

Szymon Czajkowski (Project Leader): Definitely yes - the name Applied Machine Learning Conference is, for us, a commitment not to remain solely in the realm of theory. While strong academic foundations are extremely important to us (after all, we come from Poznan University of Technology), one of our key goals is to demonstrate how these advanced concepts actually “work” in the real world.

That “Applied” aspect is the core of our community. We want participants to leave GHOST Day with the feeling that AI is not just fascinating theory found in research papers, but above all a practical tool they can use in their own projects. We combine scientific rigor with practical know-how, because we believe that only this combination enables the creation of effective solutions that truly make an impact.

Who is your primary audience? Is the program strictly designed for Computer Science students, or is there room for anyone passionate about the future of tech?

Jakub Bilski (Marketing Team): When we think of “the future of tech” in 2026, our attention most likely turns to the term “artificial intelligence.” The spark that ignited all this momentum was undoubtedly the release of ChatGPT in 2022. This has clear advantages - the AI/ML industry has accelerated at a pace that previous generations of engineers could only dream of for decades. Today, AI permeates almost every field and has become one of the primary directions of technological development.

At the same time, it’s important to keep in mind that with popularity comes a certain degree of superficiality. AI is often reduced to simple, flashy applications that perform well in the media, but do not necessarily reflect the true complexity of the field. As a result, discussions can easily become oversimplified, leading to misconceptions about what artificial intelligence really is - with Machine Learning being a core component of it.

GHOST Day existed long before AI entered the mainstream, and this largely defines the character of our conference. GHOST Day is an engineering- and research-oriented event, aimed primarily at people who already have a certain foundation and want to take the next step. After all, conferences are not about testing your knowledge, but about expanding it.

That’s why we don’t close the door to anyone, and computer science is not the only valid background. Analysts, statisticians, and engineers from other disciplines are equally welcome. However, we do expect a certain level of technological maturity. To avoid any misunderstandings, we are always transparent: GHOST Day is not a course on using commercial AI tools. We will not be teaching how to generate funny videos or how to chat with bots.

This year’s agenda is incredibly diverse—covering everything from LLM error detection and precision oncology to AI in boxing and autonomous waste collection. How did you manage to bridge such different fields into one cohesive program?

Szymon Czajkowski (Project Leader): Our philosophy is simple: we are building an event designed to be a space for every technology enthusiast, regardless of what they are currently working on. We want to unite diverse interests under the banner of AI and ML, creating one shared community.

That is why, alongside medicine or sports, the program includes topics as varied as Self-Supervised Learning—which involves teaching models to draw conclusions without labeled datasets—and neuromorphic systems, which draw inspiration directly from the biological structure of the brain. We discuss model reliability, but also very practical implementations, such as the automated analysis of millions of documents using vision-language models.

It is precisely this diversity that ensures GHOST Day is not just another closed-off conference for a narrow group of specialists. We believe that despite various technical challenges, it is this exchange of experience and shared passion that builds our unique community.

Your lineup includes experts from giants like Meta and Mistral AI, alongside researchers from top-tier institutions like ETH Zürich, University of Edinburgh, and Charles University. How did you manage to bring speakers who regularly publish at NeurIPS and ICML to Poznań?

Wiktor Kamzela (Speakers Coordinator): Each year, we strive to build an agenda that is, above all, as engaging as possible - we look for individuals who not only conduct cutting-edge research but are also eager to share their knowledge and inspire others. The fact that our invited speakers include experts who publish at top-tier conferences is the result of tremendous effort from the entire organizing team.

The greatest recognition of our work is that many speakers are not only willing to return as participants, but also choose to collaborate with us and recommend our event to their peers, who later become speakers themselves. The conference continues to grow year by year, allowing us to invite individuals who, just a few years ago, might not have even considered attending a conference organized by students from Poznań - and who now see us as a compelling alternative to mainstream conferences.

There will be several partner booths on-site. Is this a good opportunity for attendees to bring their CVs and discuss specific job openings or internships?

Adam Dobosz (Sponsors Coordinator): Absolutely! At the sponsor booths, participants will be able to speak directly with company representatives and learn about current job and internship opportunities. This year, exhibitors include (in alphabetical order): Allegro, AMD, Datarabbit, Google, and Pearson.

GHOST Day attracts top students and AI/ML professionals, making it one of the largest concentrations of talent in this field in Poland. Companies are well aware of this and treat the event as a genuine recruitment opportunity, while participants - both those taking their first steps in the industry and experienced specialists - see it as a chance to explore career opportunities.

Conferences are more than just lectures. What are your plans for integrating the community? Will there be networking sessions or social events to talk to the experts face-to-face?

Lidia Wiśniewska (Finance and Logistics Coordinator): Absolutely. We believe that the most valuable ideas are born at the intersection of different perspectives, often in more informal settings. That’s why, after the first day of inspiring talks, we are inviting our participants to a networking event in the center of Poznań. Our goal is to create a space for casual exchange of experiences and for establishing relationships that last for years, and perhaps even evolve into collaborative projects in the future.

Additionally, the agenda for both days includes short coffee and networking breaks. These are the perfect moments to discuss the talks or simply to catch up with others in the industry.

Finally, I would like to ask you to invite the readers of fast-takeoff.com (and beyond) to take part in your event—I personally invite you to attend as well!

Jędrzej Ogrodowski (Marketing Coordinator): We invite you to GHOST Day: AMLC 2026 - a great opportunity to see what research teams from the top institutions mentioned above are working on, connect with others, and further develop your interests. Ahead of us are two days full of inspiring talks, conversations with experts, knowledge sharing, good coffee—and more. The event will take place on May 8–9 at the Lecture Center of Poznan University of Technology. See you there!

Join us: https://ghostday.pl/#tickets

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Next Frontier AI is the Europe’s most radical initiative designed to discover and define the next paradigm of artificial intelligence.

Managed by SPRIND (Germany’s Federal Agency for Breakthrough Innovation), this initiative provides the funding, freedom, and ecosystem required to establish three new Frontier AI Labs in Europe.

After successful stops in Paris and Munich, Next Frontier AI is landing in Warsaw on April 22. A €125 million funding pool is now accessible to Polish teams:

• 10 selected European teams will each receive €3 million in non-dilutive funding (zero equity taken).

• The top 3 teams can receive an additional €23.5 million over two years.

• The Ultimate Goal: A structured path to raise up to €1B to build a full-scale European Frontier AI lab.

If you are an AI researcher or deep-tech founder working on the next breakthrough—whether it’s radical new architectures, world models, or agentic systems—this is the platform to scale your hypothesis.

The event features a high-caliber lineup of researchers, operators, and investors:

• Carsten Dirks (SPRIND): Next Frontier AI Lead and former COO of Aleph Alpha.

• Prof. Tomasz Trzciński (Warsaw University of Technology): Director of ELLIS Unit Warsaw and NeurIPS 2025 Best Paper recipient.

• Prof. Krzysztof Pyrć (Foundation for Polish Science): Renowned virologist and head of the Jagiellonian University Virology Lab.

• Aleksandra Pedraszewska (vastpoint): Deep tech VC and early leader at ElevenLabs.

• Callum Hill (Creator Fund): Investor specializing in early-stage scientific founders and AI.

Registration is free of charge, and you can sign up at the link below:
https://luma.com/nfai-warsaw?tk=5IRfMv

I hope you will be there.