From: mk_thisisit
A Polish mathematician at the University of Cambridge has solved a mathematical problem that remained open for 120 years [00:00:02], [00:00:50]. This breakthrough is considered to have great significance for the understanding of black holes [00:00:08]. Professor Maciej Dunajski, the mathematician in question, revealed that the key idea for the solution came to him in Austria in 2007 or 2008, specifically while skiing [00:02:04], [00:02:07]. He later completed the work with colleagues Mikee from Australia and Robert Br from Berkeley [00:02:17].
The Metrizable Problem
The problem, known as the “metrizable problem,” was posed by the French mathematician Roger Liouville at the end of the 19th century [00:01:01], [00:01:06]. It asks whether it’s possible to recreate a concept of distance from a given family of curves such that these curves represent the shortest paths [00:01:15].
An everyday analogy is straight lines on a plane [00:01:26]. Here, the concept of distance is derived from the Pythagorean theorem, and straight lines are known to be the shortest paths between two points [00:01:31]. The problem reverses this: given a family of curves (like straight lines), can one deduce a corresponding concept of distance? [00:01:44]
Connection to Einstein’s Theory of Relativity and Black Holes
This mathematical problem has profound implications for physics, especially Einstein’s theory of gravity and the theory of black holes [00:02:42], [00:02:46].
In Einstein’s theory, what we perceive as movement in a gravitational field (e.g., of galaxies, planets, comets) is actually movement along “geodesic lines” [00:02:53]. These are the shortest lines within a specific concept of space-time distance, known as the “metric” [00:03:06]. Einstein’s equations typically involve solving for this metric [00:03:15].
The problem, in the context of relativity, asks: if we know the trajectories of all celestial bodies, can we recreate the entire space-time metric from these paths alone? [00:03:40] This is particularly relevant for black holes, as the geodesics (paths) of objects falling into them are incomplete from our perspective; we only know them until they fall past the event horizon into a singularity [00:04:01], [00:04:14]. The solution to the metrizable problem indicates that usually, even from such incomplete geodesics, the universe’s metric can be recreated [00:04:26], [00:04:32].
The metric determines how distance is measured in space-time, modifying the Pythagorean theorem to account for both spatial and temporal directions and the curvature of space-time [00:05:25], [00:05:33], [00:05:46]. Nature, for unknown reasons, chooses paths where this distance is either minimal or maximal, opting for the most “economical” way of movement [00:06:11].
Professor Dunajski’s work flips the traditional approach: instead of using a known metric to calculate trajectories, it uses observed trajectories to deduce the unknown metric [00:06:35], [00:07:03].
“The problem that interested me is that we do not have this metric, I do not know what the concept of distance is, but we have all possible paths to look at. Can we open the universe from this?” [00:07:03]
The Nature of Black Holes and Unanswered Questions
Black holes are regions of space-time where gravity is so strong that nothing, not even light, can escape [00:13:00], [00:13:28]. Our understanding of physics breaks down at the singularity within black holes [00:09:09], [00:09:16]. Understanding them would represent a “quantum leap” in theoretical science [00:09:30], [00:09:35].
While Einstein’s theory of relativity predicts the existence of black holes and their singularities (points where space-time curvature becomes infinitely large, or geodesics have a beginning but no end), it does not definitively prove that these singularities must be hidden within an event horizon [00:11:24], [00:12:44]. The “cosmic censor hypothesis” suggests that all singularities are indeed hidden behind such membranes, preventing us from observing their true nature [00:13:48], [00:14:16]. However, whether this is always the case remains an open and “exceptionally difficult” problem in nonlinear differential equations [00:14:42], [00:15:06].
Another key unsolved problem related to black holes is the information paradox: what happens to information (energy, matter) that falls into a black hole when it evaporates? [00:16:11] While classical relativity suggests nothing can escape, Stephen Hawking’s work in the 1970s, applying quantum mechanics to black holes, introduced the concept of Hawking radiation [00:16:51], [00:17:58]. This suggests that black holes might slowly evaporate due to particle-antiparticle pairs being created near the event horizon, with one falling in and the other escaping [00:17:22], [00:18:18]. This implies that information might eventually be recoverable, but this problem remains a topic of intense debate [00:19:25], [00:19:35].
Despite the inability to send objects into black holes, our understanding comes from theoretical predictions aligning with astronomical observations [00:19:45], [00:20:07]. Telescopes like James Webb have provided direct and indirect evidence of black holes by observing the movement of matter and galaxies near them [00:20:34].
Mathematics as a Tool and its Independent Existence
The discussion extends to the fundamental nature of mathematics itself. While physics describes nature, mathematics is seen less as a descriptive science and more as a field that exists independently of human thought [00:28:06], [00:29:15].
“Mathematics differs from all other natural sciences in that it exists in its own world of abstract Platonic ideas… mathematical statements would be right or wrong all the time.” [00:29:15]
Mathematicians are not seen as creators but rather as discoverers of existing mathematical truths, which remain true regardless of human civilization [00:30:10], [00:31:44]. Mathematical theories are built upon axioms (postulates that are not proven), and all theorems are deduced from these [00:30:46]. This makes mathematics less controversial than physics, as a mathematical theorem, once proven, cannot be disputed unless an error in the proof is found [00:43:24].
Unification Theory and Quantum Mechanics
Professor Dunajski and Roger Penrose are working on problems in quantum theory, specifically the “reduction of the wave function” [00:34:38], [00:34:53]. Quantum theory, developed in the 1920s and 30s, introduced non-determinism to physics: elementary particles can exist in a superposition of all possible states until a measurement is performed, causing the wave function to “collapse” [00:35:08], [00:36:01].
The challenge is that this superposition is observed for microscopic objects, but not for macroscopic ones (e.g., armchairs aren’t in multiple places at once) [00:36:37]. Their theory suggests that the effects of gravity must be taken into account [00:37:05]. They believe quantum mechanics is incomplete and should be modified by gravitational theory, making wave function reduction a physical process influenced by mass [00:37:13], [00:37:42]. This is a controversial viewpoint among physicists [00:37:18], [00:42:48].
Their work contributes to the pursuit of a “grand unification theory,” aiming to connect the four fundamental forces of nature [00:39:39], [00:41:41]. They argue that to achieve this, one should re-examine the foundations of quantum mechanics rather than building upon its standard form [00:42:01].
Challenges and Discoveries in Mathematics
Mathematics, especially pure mathematics, can be difficult for the general public to appreciate due to its high level of technical specialization and hermeticism [00:44:50], [00:49:46]. Even mathematicians struggle to communicate across different subfields [00:45:07].
Infinity
One of the fascinating mysteries in mathematics is the concept of infinity [00:24:41]. While mathematics can define and work with various types of infinity, physics struggles with it, describing everything in terms of finite numbers [00:22:52]. A key question is whether all infinities are equal. For example, the infinity of natural numbers (1, 2, 3…) is “countable,” meaning each number can be assigned a unique position in a sequence [00:26:24]. However, the infinity of real numbers (including irrational numbers like pi or square root of 2) is “uncountable” and has a “greater power” in a mathematical sense [00:27:00], [00:27:11]. Whether other types of infinity exist between these two remains an open problem [00:27:25].
Prime Numbers
Another example of a persistent mathematical mystery involves prime numbers. Despite their fundamental role as “building blocks” for all other numbers, mathematicians know very little about them [00:23:25], [00:23:37]. We know there are infinitely many of them (proven by Euclid) [00:23:42]. However, there is no formula to derive the n-th prime number; they must be found empirically [00:23:53], [00:24:17].
One unsolved problem is the “twin prime conjecture,” which asks if there are infinitely many pairs of prime numbers that differ by only two (e.g., 11 and 13, 17 and 19) [00:47:03], [00:47:34]. While computer simulations suggest this is true, a formal proof is lacking [00:47:31], [00:47:43]. A significant breakthrough came in 2013 from Chinese mathematician Yitang Zhang, who showed that there are infinitely many prime pairs differing by a large number (initially 70 million, later reduced) [00:48:12], [00:48:32]. This work, however, increasingly relies on algorithmic proofs and computer verification, leading to concerns that mathematicians understand “what is happening” but not “why it is happening” [00:49:10], [00:49:43].
“It’s not enough for a mathematician to know, it’s not enough to show that something results from a long calculation. A mathematician tries to find some deeper principle of analogy.” [00:49:57]
Fermat’s Last Theorem
A historical success story is Fermat’s Last Theorem, formulated by Pierre de Fermat over 300 years ago and famously proven by Andrew Wiles at Cambridge in 1994 [00:52:25], [00:52:30], [00:52:38]. The theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than 2 [00:54:02], [00:54:42]. (For n=2, this is the Pythagorean theorem, with infinite solutions like 32 + 42 = 52) [00:53:07].
Wiles’s proof, achieved after seven years of work, utilized advanced mathematical techniques unknown in Fermat’s time, including algebraic number theory and modular forms [00:56:45]. Initially, the proof contained a subtle error, which Wiles, with the help of colleagues like Richard Taylor, spent another year patching up before the final, undisputed version was released in 1996 [00:57:27], [00:58:01].
Professor Dunajski’s Journey and Role in Africa
Professor Dunajski’s academic journey began in Łódź, Poland, where he developed a passion for physics and mathematics after attending lectures by Professor Przanowski on quantum theory [01:04:07]. He received a scholarship to Oxford in the mid-90s, where he met Roger Penrose and solved a problem Penrose suggested [01:04:56]. He later moved to Cambridge in 2002, maintaining strong ties with Oxford [01:05:50].
He identifies as a “mathematical physicist,” distinct from a theoretical physicist [01:00:10], [01:00:25]. While theoretical physicists aim to discover how the universe works, mathematical physicists use physical theories as inspiration to prove theorems in pure mathematics [01:00:30]. This synergy between physics and mathematics is a key interest for him, emphasizing that physics relies heavily on mathematics, though mathematics can exist independently [01:01:06], [01:03:07].
Professor Dunajski is also deeply involved in promoting scientific research in Africa, particularly in mathematics [01:07:24]. Recognizing the lack of scientific research despite the presence of universities, he helped establish institutes across Africa, starting in Cape Town nearly 20 years ago [01:07:50], [01:08:05]. These institutes accept top Master’s students for a year, providing lectures and guidance from visiting scientists [01:08:25]. He discovered an “exceptionally large amount of mathematical talent” in Africa [01:08:40]. His role includes suggesting research problems and connecting African scientists with international collaborators, enabling them to focus on research [01:09:14], [01:10:24]. He emphasizes keeping these initiatives within Africa to build sustainable local systems, rather than bringing all talent to Europe [01:09:54]. This effort has led to significant results, including proofs of theorems by young African mathematicians [01:10:26], [01:11:03].