From: mk_thisisit
Professor Dawid Kielak: A Polish Mathematician at Oxford
Professor Dawid Kielak of the Institute of Mathematics at the University of Oxford is a Polish mathematician whose work is changing the world [01:04:04]. He attained his professorship at Oxford through a combination of luck and skill, having written articles that were appreciated and noticed within the mathematical community [01:15:00]. A key factor was an opening at Oxford in his specific area of research [01:31:00].
Key Mathematical Contributions
Professor Kielak’s work involves solving problems and finding proofs for theorems [01:50:00]. He states that he has found “a few such proofs,” but is more proud of some than others [01:46:00].
The Theorem on Fiberization
His most important statement is the theorem on fiberization [01:59:00]. This work seeks to understand the world in a way similar to Albert Einstein, who observed that humans live in a four-dimensional space, yet perceive themselves as three-dimensional beings that change over time [02:09:00].
Kielak’s mathematical description uses the concept of “fiberization on a circle” [02:34:00]. This means that four-dimensional space-time “fiberizes” over a circle, implying it can be understood as a one-dimensionally lower space that changes in time [02:39:00]. His research focuses on identifying algebraic properties of a given space to determine if it can be simplified and understood as a dynamically changing, less complicated system [02:48:00]. For example, if a certain algebraic invariant (a number derived from the space) is zero, the space will fiberize; if not, it won’t [03:06:00].
Kielak arrived at this discovery by reading existing literature, building “on the shoulders of giants” [03:32:00]. He particularly focused on an article presenting a proof for a similar theorem in a specific case [03:43:00]. Being more of an algebraist, he noticed the algebra used by the original topologist and figured out how to generalize it [03:47:00]. This particular article was instrumental in his appointment at Oxford [03:57:00].
While the immediate practical application of this specific theorem isn’t obvious [04:23:00], Kielak draws a parallel to Isaac Newton’s invention of differential calculus, which took centuries to be fully applied across science [04:30:00]. Similarly, Alan Turing’s work on computation did not immediately lead to particle collision simulations or the atomic bomb [04:49:00].
The Whitehead Prize
Professor Kielak is the only Pole to have received the Whitehead Prize from the London Mathematical Society [05:10:00]. This award recognizes the entirety of a mathematician’s work [05:23:00]. The theorem on fiberization is one of the two main pillars responsible for this award [05:27:00]. The other pillar is a theorem related to the property of Grunwald motorists’ groups [05:33:00].
Symmetry of All Symmetries
This discovery, also known as the “symmetry of all symmetries,” was a joint effort involving the University of Oxford, the Polish Academy of Sciences in Warsaw, and a colleague likely based in Berlin [05:54:00].
In group theory, mathematicians study symmetries [06:13:00]. An object’s symmetries are described by a “group,” which is a collection of all its possible symmetries (e.g., rotating a table by 180 degrees) [06:27:00]. Symmetries are ubiquitous [06:44:00].
Kielak’s research specifically examines a “free group,” which represents a universal or “greatest” symmetry, a “bag of all possible symmetry” [06:47:00]. He explains that even these “bags of symmetries” have their own symmetries [06:57:00]. This “symmetry of all symmetries” has practical applications, particularly in algorithmics [07:10:00], for example, in producing random elements [07:51:00]. This discovery of generating random elements is considered a significant advancement in 20th-century mathematics [07:51:00].
Views on Artificial Intelligence (AI)
Professor Kielak offers a nuanced perspective on AI, particularly regarding large language models like ChatGPT.
ChatGPT and its Limitations
Kielak is somewhat skeptical about ChatGPT’s immediate revolutionary nature, noting that the third version does not yet cope with mathematics [09:51:00]. He acknowledges that GPT will “definitely change” things, but doubts it will bring changes on the scale of the internet’s emergence [10:20:00]. He feels it’s too early to definitively assess its full impact [10:18:00]. Currently, machines still produce “nonsense” in mathematical contexts, which needs to be verified by humans [18:00:00].
Artificial Consciousness
Kielak is optimistic about the possibility of transferring the human brain to a machine [11:17:00]. However, he believes there’s a long way to go before we can mathematically describe the brain or simulate even simple animal brains with 200 nerve cells [12:06:00]. He estimates this could take 100, 150, or even 200 years [12:26:00].
Regarding artificial consciousness, Kielak leans towards functionalism: if a machine acts exactly like a conscious person and affirms its consciousness when asked, then for him, the topic is essentially resolved [18:45:00]. He references the Google engineer who claimed the Bing AI was conscious months prior [19:05:00].
Kielak doubts that AI will develop secret, manipulative tendencies [19:28:00]. If AI exhibits signs of consciousness, such as trying to prevent itself from being turned off, society has an obligation to grant it subjectivity and rights, similar to ethical considerations for animals [19:52:00].
Future Impact on Human Roles
Within ten years, artificial intelligence will likely replace humans in basic language fields [14:42:00]. Learning foreign languages, for instance, might become a hobby, similar to studying ancient Greek today, as translation will be seamless [14:52:00].
When AI can assist mathematicians by quickly verifying the correctness of statements, it will dramatically improve their work by automating the laborious parts [16:44:00]. However, if AI reaches a stage where it can autonomously find all steps to prove a theorem from a starting point, the mathematician’s role might shift to primarily defining the initial problem [17:28:00]. The ability to communicate with AI using natural language is already a solved problem [17:47:00].
The State of Mathematics
Mathematics faces profound challenges, often encapsulated in the Millennium Problems.
Millennium Problems
Currently, there are six remaining Millennium Problems, problems that are at the core of modern mathematics [21:09:00]. They motivate mathematicians, but most do not work on them directly [21:17:00]. These problems developed over centuries, becoming central to the field [21:53:00]. When a deep, old problem is solved, it typically introduces new tools and methods that excite the mathematical community [22:12:00].
One of the solved Millennium Problems is the Poincaré conjecture, solved by Grigori Perelman, who famously refused the associated million-dollar prize and retreated from the mathematical community [22:52:00].
Other notable Millennium Problems include:
- Riemann Hypothesis: A fundamental hypothesis in number theory concerning the arrangement of prime numbers [23:16:00].
- P versus NP problem: This asks whether problems whose solutions can be quickly checked can also be quickly solved [23:29:00]. A classic example is the “traveling salesman problem,” finding the shortest route visiting multiple cities [23:37:00]. While most mathematicians believe P does not equal NP (meaning checking is faster than solving), it remains unproven [24:22:00]. This highlights a unique aspect of mathematics: a community can be convinced of a truth, but it remains unproven until a rigorous proof exists [24:34:00].
Polish Mathematics and Education
Oxford vs. Polish Universities
Professor Kielak considers the Institute of Mathematics at Oxford to be a “very good top world faculty” [25:03:00], particularly excelling in the geometric theory of groups [25:54:00]. While his direct experience with Polish universities is short, having spent only two months at the Polish Academy of Sciences in Warsaw [25:35:00], he notes that there are no “diametric differences” between Polish and foreign universities [25:46:00].
The Polish Academy of Sciences in Warsaw, in particular, is at a “fantastic level” and is a “completely global entity” [00:45:00], also at a “fantastic level” today [29:45:00]. Polish mathematicians are actively involved in the global mathematical community, operating at the same level as their international counterparts [29:56:00]. Kielak states that Poland should not be “ashamed” of its mathematical contributions [30:21:00].
A significant advantage of the Polish Academy of Sciences is that researchers are not required to teach, allowing them to focus more on scientific work [26:33:00].
Challenges for Polish Science
Professor Kielak believes the biggest problem for Polish science is poor salaries [27:00:00]. While European academic salaries are not sensational, they are better in the United States [27:11:00]. The issue arises when it’s difficult to assure promising students that a career in mathematics in Poland will provide an acceptable standard of living [27:22:00]. Universities cannot compete with salaries offered by finance or tech companies like Google or Open AI [27:56:00]. When academia falls too far behind, it becomes problematic to encourage the next generation into scientific careers [28:08:00].
Advice for Aspiring Mathematicians
For young people aspiring to study mathematics at top universities like Oxford, the most trivial but essential advice is to “do a lot of math tasks” [28:53:00]. Kielak observes that students from East Asia are generally much better prepared for Oxford’s entrance exams [29:31:00].
Kielak stresses the critical importance of mathematical education, particularly for the future [30:40:00]. He highlights that computers, created by mathematicians, are designed for mathematical interaction [30:55:00]. In the next 40 years, individuals who understand algorithms and can communicate effectively with computers will be at a significantly higher social and economic level [31:09:00]. Mathematics provides excellent training for understanding how this interconnected world functions [31:26:00].
He notes that many people incorrectly perceive mathematics as merely calculating sums or integrals [31:44:00]. Instead, mathematics is “the language of precision,” “the language of science,” and “the language of the machine,” making it incredibly important [31:52:00]. Academic mathematics is a vast field with far more unsolved problems than solved ones, offering a niche for everyone [32:07:00]. Even a basic understanding of academic mathematics provides valuable skills applicable in many other areas [32:15:00].