Roger Penrose was born on August 8, 1931. For me, he is one of the figures who turned general relativity from merely a physical theory into something I could approach through geometry, topology, and symmetry.

Penrose's mathematical background is especially visible in this work. He studied mathematics at UCL and completed his PhD on algebraic geometry at Cambridge ^[1]. In his 1965 paper Gravitational Collapse and Space-Time Singularities, he showed—without relying on a particular black-hole solution—that under certain conditions gravitational collapse leads to geodesic incompleteness through causality and global geometry ^[2]. This work earned Penrose half of the 2020 Nobel Prize in Physics ^[3].

Penrose Diagrams

When I first saw a Penrose diagram, I thought of complex analysis and the Riemann sphere before I thought of general relativity. On the Riemann sphere, we add a single point at infinity and bring it into the geometry; in Penrose's conformal compactification, we can similarly study the infinity of spacetime on a finite diagram:

\[\tilde{g}_{ab}=\Omega^2g_{ab}\]

However, because spacetime has a causal structure, infinity is not a single point; it is divided into different parts such as \(\mathscr{I}^\pm\), \(i^\pm\), and \(i^0\) ^[4], ^[5]. This idea transformed the concept of “infinity” for me from something abstract into an object that could be studied geometrically.

Trapped Surfaces

What I like most about Penrose's singularity theorem is that it does not require a special symmetry. In a sufficiently strong gravitational collapse, both families of null rays emerging from a surface can have negative expansion:

\[\theta_k < 0,\qquad\theta_l < 0.\]

This idea of a trapped surface, together with the Raychaudhuri equation, leads to null geodesic incompleteness under appropriate conditions ^[2], ^[6]. This is also where I began to think of singularities not merely as “curvature becoming infinite,” but as a consequence of the global geometry of spacetime.

From Penrose to Linearization Instability

This curiosity about global geometry and consistency later led me to the work of Prof. Dr. Bayram Tekin. While reading Deser and Tekin's work on energy and conserved charges in gravitational theories ^[7], ^[8], I encountered the concept of linearization instability.

The idea is that when we linearize around a background solution,

\[g=g_0+h\]

not every perturbation necessarily corresponds to a genuine family of solutions in the full nonlinear theory. In the work of Prof. Dr. Bayram Tekin and Emel Altaş, I found it particularly interesting that Taub charges appear at this point as an integrability condition ^[9], ^[10].

What draws me along this line from Penrose diagrams to linearization instability is ultimately the same idea: equations that look locally correct are not always sufficient; global geometry and consistency at higher orders can impose additional conditions on us.

This is precisely one of the reasons I love general relativity so much.

Over the course of his 95-year life, Penrose has brought together ideas that changed the direction of modern mathematical physics, from singularity theorems and conformal geometry to twistor theory and black-hole physics. For me, however, he occupies a much more personal place: he introduced me to the field I have longed to work in throughout my life and gave me a different perspective.

Sir Roger Penrose, happy birthday.

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References

1. University of Cambridge (2020). Roger Penrose wins 2020 Nobel Prize in Physics for discovery about black holes.
2. Penrose, R. (1965). Gravitational Collapse and Space-Time Singularities. Physical Review Letters, 14, 57–59.
3. The Nobel Prize in Physics 2020. NobelPrize.org.
4. Penrose, R. (1964). Conformal treatment of infinity.
5. Hawking, S. W., & Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time.
6. Senovilla, J. M. M., & Garfinkle, D. (2015). The 1965 Penrose singularity theorem.
7. Deser, S., & Tekin, B. (2002). Gravitational Energy in Quadratic Curvature Gravities.
8. Deser, S., & Tekin, B. (2003). Energy in Generic Higher Curvature Gravity Theories.
9. Altaş, E., & Tekin, B. (2018). Linearization instability for generic gravity in AdS spacetime.
10. Altaş, E., & Tekin, B. (2019). Second order perturbation theory in general relativity: Taub charges as integral constraints.

I wanted to take a closer look at the major news that recently surfaced across mathematical communities and social media. The Jacobian Conjecture, open since 1939 in algebraic geometry and famous for its history of incorrect proofs, experienced a significant turning point on July 20, 2026. Mathematician and Anthropic researcher Levent Alpöge shared a concrete counterexample disproving the conjecture in 3 dimensions ($n=3$), announcing it via a short post on X (Twitter) rather than a formal paper.

Background of the Discovery

What makes this breakthrough particularly interesting is that Alpöge utilized Anthropic's next-generation large language model, Claude Fable 5, in the process. AI tools have been increasingly applied to unsolved mathematical problems, yielding notable results. For the Jacobian Conjecture, a 3-dimensional polynomial system—where manual verification of the degree and term count would be intractable—was identified, featuring a constant Jacobian determinant without a polynomial inverse.

What Comes Next?

This counterexample for $n=3$ opens up new directions for research:

• The 2-Dimensional ($n=2$) Case Remains Open: In the literature, the 2-variable case and the general $n$-variable case have often been treated separately. While the $n=3$ case has been disproved, $n=2$ remains unsolved and continues to be a central topic of study.
• Dixmier Conjecture: This topic is closely connected to algebraic geometry as well as the Dixmier Conjecture in quantum mechanics and algebraic structures. Disproving $n=3$ prompts a re-examination of these equivalent formulations.
• Formal Peer Review Pending: It is worth noting that this result has not yet been published in a peer-reviewed academic journal. The symbolic calculations were independently reproduced, and the counterexample was verified in the Lean proof assistant within hours; however, traditional peer review and a formal manuscript are still forthcoming.

Connecting to the Leiden Declaration

This announcement also intersects with broader discussions on AI in mathematics. Shortly before this result, on June 2, 2026, sixteen researchers from fifteen universities published the "Leiden Declaration on Artificial Intelligence and Mathematics", endorsed by the International Mathematical Union (IMU). Originating from a Lorentz Center workshop at Leiden University in September 2025, the declaration addresses verifiability, citation transparency, research autonomy, funding, and equitable access.

Alpöge's announcement relates to the declaration's core themes in several ways:

• Transparency in AI Usage: Alpöge explicitly credited Claude Fable 5 as a co-discoverer and provided Wolfram Alpha verification links, aligning with the declaration's call for clear disclosure of AI tools.
• Dissemination Channels: The result was announced on social media by a researcher at an AI company (Anthropic) prior to traditional peer review, highlighting ongoing shifts in how mathematical discoveries are shared and evaluated.
• Verification Speed: Rapid verification in proof assistants demonstrates new capabilities while raising questions about how traditional reviewing systems adapt to AI-generated results.

This milestone represents both a notable mathematical achievement and an example of evolving methodologies in research.

Below you can find links to Levent Alpöge's personal website, the original text of the Leiden Declaration, and its Turkish translation: