Bartosz Naskręcki
@nasqret • 13,204 subscribers
Mathematician | Vice-Dean @UAM_Poznan | Researcher @ccaiwut | Owner of https://t.co/lEspgf36Pg | Mathematics, AI and programming
Videos
Repeating the same experiment but one year later. GPT 5.6 Pro cannot be used vanilla for this prompt because it immediately fires the right CAS command to compute the Galois group. So instead I used a very restrictive prompt. One year has passed since GPT 5 Pro premiere, GPT Pro 5.6 is now able to compute the result from first principles, including the resolvent construction. This is like seeing a model building a whole Galois theory course live. It's so nice to learn from this output, very detailed! Prompt: Compute the Galois of the polynomial x^8-x^7+3*x^6-3*x^5+2*x^4-2*x^3+5*x^2+5*x+1 Don't look up the result in the internet nor any specialized database. Don't use ready made commands from SymPy or any other program. Develop a full resolvent computation. Do it all the way down to the definitions, no excuses, like there is no tomorrow.
Bartosz Naskręcki70,809 次观看 • 2 天前
I have made a repository with landing page where the three approaches by levent, Andy Jiang and Aaron Lou to the Jacobian Conjecture counterexample are uniformly explained. There is a video tutorial and a Jupyter Book. Feel free to share and expand or modify. Comments welcome. The video was done with the new Manim Voiceover feature based on the actual review paper. Landing page: GitHub repo: Knowledge book:
Bartosz Naskręcki41,869 次观看 • 21 天前
Enough of the fluff. Time for some real advice. Launch your GPT 5.6 Sol Pro. Use the following prompt (feel free to swap the second and third lines for something of your choice): Prove for me that the group of rational points on the elliptic curve (y^2=x^3+17) is finitely generated. I want to tailor the general proof to this particular case to narrow down my understanding to a particular case. Write down a full mathematical proof, step by step, in Leslie Lamport style. Unpack all the definitions and notions so that it is fully self-contained. An example here: Notice how Sol is spawning all the subagents to complete the task. What is "Leslie Lamport style"? It's a technique where you split your proof into short, one-line sentences (claims) that are easy to justify. In a situation where your proof is not a 1-to-1 translation of, e.g., Python code, the Leslie Lamport-style proof can be treated as such. It tremendously improves the possibility of verifying each claim and helps the model structure the arguments in a very clear way. Leslie Lamport, a legendary mathematician and computer scientist, was a promoter of this way of structuring proofs. Partly because of that, he created an extension of Donald Knuth’s typesetting system TeX called LaTeX. Here you can read about "Leslie Lamport-style proofs":
Bartosz Naskręcki31,115 次观看 • 1 个月前
I think one of the roles we play as mathematicians in society is to help people become acquainted with the underlying secret patterns. I have been working for several years on projects in crystallography, where we study crystal structures. But few people know that such periodic patterns come with severe constraints on their symmetry. In the plane, there are 17 different symmetry types, a fact well known even to the designers of the mesmerizing patterns of the Alhambra. Here you can find and experiment with such tilings in the plane, gaining insight into the intrinsic beauty of the so-called wallpaper groups, the crystal symmetries in dimension 2. The app interactively helps you design symmetric patterns with colors and shows how changes in the structure of the unit cell propagate via symmetry. In dimension 3, if you look into International Tables for Crystallography, Vol. 1, you will find a theorem due to Schoenflies and Fedorov stating that there are 230 such symmetry types, a cornerstone of modern chemistry. Beyond that, in dimensions 4 and higher, a count can be made, but it requires a proof of the general theorem due to Frobenius and Bieberbach. This was an answer to the first part of Hilbert’s famous eighteenth problem. One of the fun consequences of such a classification is that in dimensions 2 and 3, 5-fold symmetry is forbidden in regular periodic arrangements. Intrinsically, this fact is related to the existence of matrices with a fifth-root-of-unity eigenvalue. For integral matrices, this is possible only in dimensions 4 and higher. If you generalize the square and cube tilings to dimensions 4 and 5, obtaining hypercubic tilings, the 5-fold symmetry pattern emerges. Skew projections of the 5D hypercubic tiling onto a 2-dimensional plane give rise to a quasicrystalline tiling known as the Penrose tiling. You can find such patterns in front of the Andrew Wiles Building at the Oxford Mathematical Institute. In later posts this summer, I will take a deep dive into group homology, a modern tool for studying the geometry of crystals. There are still many open questions, for example, how many symmetry types exist exactly in dimensions beyond 6. This is still largely unknown; at present, we only have asymptotic lower bounds.
Bartosz Naskręcki44,042 次观看 • 2 个月前
Introducing TikZBot. The birth of an agent that produces its own TikZ creations. Fully interactive and extensible. It can fix scenarios, expand its own work, learn from diagrams, and talk to you. For now, it’s a PoC. The manual version works like a charm. I’m now adding the self-play and chat components. Programmed in a few hours in Codex CLI using the newest 5.2-Codex model. So much fun.
Bartosz Naskręcki18,444 次观看 • 6 个月前
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