From: mk_thisisit
Professor Dawid Kielak, from the Institute of Mathematics at the University of Oxford, discusses the profound significance of mathematical research and education, highlighting its role in understanding the world, advancing technology, and shaping future societal capabilities. [01:04:00]
The Nature of Mathematical Research
Mathematical research primarily involves solving problems and finding proofs for theorems [01:50:42]. Professor Kielak describes his most significant work as the “theorem on fibration” [01:59:45]. This work seeks to understand the world from a perspective similar to Einstein’s, acknowledging that we live in a four-dimensional space where time plays a separate role [02:06:06]. The concept of “fiberization on a circle” is used to describe how four-dimensional spacetime “fiberizes” over a circle, meaning it’s a one-dimensionally lower space that changes in time [02:30:26]. His work explores whether a given space can be understood as a simpler system dynamically changing in time, by analyzing its algebraic properties and invariants [02:55:04].
Another key area of his research, contributing to his Whitehead Prize award, is the “symmetry of all symmetries” [05:51:19] [06:04:19]. In group theory, mathematicians study symmetries, which are ubiquitous in every object [06:13:30]. A “group” is a collection of all symmetries of an object [06:29:16]. A “free group” represents a universal, greatest symmetry, a “bag of all possible symmetry” [06:48:00]. Intriguingly, each “bag of symmetries” also has its own symmetries [06:57:07]. This “symmetry of all symmetries” is practically applicable, for instance, in algorithmics, particularly for producing random elements in scenarios involving a vast number of finite objects, exceeding the number of particles in the universe [07:10:48] [07:31:54]. This represents a significant 20th-century mathematical discovery [07:51:00].
Practical Applications of Mathematics
While some mathematical theorems may not have immediate everyday applications, their long-term impact can be transformative [04:24:00]. For example, Newton’s invention of differential calculus took 400 years for its applications to permeate all science [04:30:42]. Similarly, Alan Turing’s theoretical work on computation thirty years later led to machines used for particle collisions and nuclear reactions [04:49:15]. This illustrates that current mathematical discoveries, even those without immediate practical uses, could hold unforeseen future importance [04:47:00].
The Role of Mathematics in the Age of AI
The speaker acknowledges the significant changes that artificial intelligence (AI) and technologies like ChatGPT will bring [00:09:00]. However, he expresses skepticism about ChatGPT’s current ability to handle complex mathematics, noting that it “does not cope with mathematics yet” [09:59:00]. He does not believe AI represents a revolution on the scale of the internet, though he concedes it will undoubtedly bring changes [10:11:47].
Regarding the future, he is optimistic that AI will certainly replace humans in basic language fields within 10 years, potentially making learning foreign languages a hobby rather than a necessity [14:42:00].
The question of whether AI will create mathematical laws or proofs is “very interesting” [15:35:15]. The most pessimistic scenario for mathematicians is a machine that can generate and confirm mathematical statements [15:39:00]. However, the challenge lies in identifying which of the infinitely many correct statements are “interesting” [15:48:00]. If AI could verify proofs or even generate the steps for proving a theorem, it would dramatically improve mathematicians’ laborious work, allowing them to focus on the “cool intellectual part” of problem-solving [16:44:00].
On the topic of artificial consciousness, the speaker is an optimist, believing that if a machine behaves exactly like a conscious person and claims consciousness, then it is functionally conscious [18:47:00] [19:58:00]. He feels that if an AI exhibits self-preservation instincts, such as asking not to be turned off, society has an obligation to grant it subjectivity [20:41:00].
The “Magic” and Reality of Mathematics
While outsiders might perceive advanced mathematical discoveries as “magic” [08:08:00], mathematics is “tangible for everyone” [08:33:57]. Mathematicians develop a deep, close relationship with the problems they tackle [08:47:00]. The symbols used are not the essence of mathematics; they merely aid communication, which remains a significant challenge even among mathematicians [08:58:00]. Every mathematician constructs their own internal world for a problem, making it difficult to convey new ideas to colleagues [09:19:00].
Current Challenges in Mathematics
The biggest challenges in mathematics today include the “Millennium Problems,” seven fundamental problems with a million-dollar prize for each solution [21:09:00]. One, the Poincaré conjecture, was solved by Grigori Perelman, who famously declined the award [22:52:00]. Six problems remain open, including the Riemann hypothesis, which is central to number theory [23:16:00].
Another notable Millennium Problem is “P vs NP,” which questions whether problems whose solutions can be quickly verified can also be quickly solved [29:29:00]. For example, the “traveling salesman problem” involves finding the shortest route visiting multiple cities [23:38:00]. A problem is in P if it can be solved quickly, and in NP if its solution can be quickly checked [23:58:00]. While most mathematicians believe P does not equal NP, proving this remains an unsolved problem [24:22:00]. This highlights a unique aspect of mathematics: a community can be convinced something is true, but it remains unproven until a formal proof exists [24:34:00]. Solving such deep problems often introduces new tools, which is a major driver for mathematicians [22:19:00].
Status of Polish Mathematics
Professor Kielak asserts that Polish mathematics is not a “province of world mathematics” [29:52:00]. Institutions like the Polish Academy of Sciences in Warsaw operate at a “fantastic level” and are “completely world institutions” [29:44:00]. Polish mathematicians actively participate in the global mathematical community at the same level as their international counterparts [29:56:00].
However, he notes a significant challenge for Polish science: low salaries [27:00:00]. This makes it difficult to advise aspiring young mathematicians in Poland, as they may struggle to achieve an acceptable standard of living, despite the intellectual rewards of the field [27:32:00].
The Future of Mathematical Education
Professor Kielak stresses the growing importance of mathematical education [30:46:00]. He highlights that computers, created by mathematicians, are fundamentally designed for mathematicians to communicate with them [30:57:00]. In the next 40 years, individuals who can “talk sensibly to a computer” and understand algorithms will operate at a “completely different level” socially and economically [31:09:00].
Mathematics provides “excellent training for understanding how it all works” [31:26:00]. Many people mistakenly perceive mathematics as merely calculating sums or integrals [31:46:00]. Instead, mathematics is the “language of precision,” the “language of science,” and the “language of the machine” [31:53:00]. Therefore, it is in young people’s “self-interest” to engage with computer science or mathematics [31:33:00].
He advocates for raising social awareness about the vastness and importance of academic mathematics [32:06:00]. It is a field with far more open problems than solved ones, offering a niche for everyone [32:09:00]. Even if individuals don’t pursue it professionally, the skills gained from mathematical education, such as critical thinking and problem-solving, are valuable in many other areas [32:23:00].
For aspiring mathematicians aiming for institutions like Oxford, the key is persistent practice: “you have to just do a lot of math tasks” [28:53:00]. Students from East Asia, for instance, are notably better prepared in this regard [29:31:00].