Edward Frenkel is a mathematician working in representation theory, algebraic geometry, and mathematical physics. He is a professor of mathematics at University of California, Berkeley, member of the American Academy of Arts and Sciences, and author of the bestselling book Love and Math.
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I am a professor of mathematics at the University of California, Berkeley and author of the New York Times bestseller "Love and Math" which has now been published in 9 languages (with 8 more translations on the way). Two weeks ago, I earned a dubious honor as "the man who almost crashed Reddit" when my active AMA was shut down in mid-sentence. After that, the Reddit mods have kindly suggested that I redo my AMA, so I'm back!
Go ahead, Ask me Anything, and this time, pretty please, let's make sure we don't break anything. :)
Apart from the themes of love, math, applications of mathematics in today's world, and math education, I am passionate about human interactions with modern technology, and in particular, with artificial intelligence. In this regard, see the lecture I gave at the Aspen Ideas Festival two weeks ago:
UPDATE: Thank you all for your great questions. I had a lot of fun. Till next time... Sending lots of love ... and math. :)
Do you think it makes sense for so many students to take calculus but very few take discrete math or probability? Aren't the later more useful for most people?
I agree that we should diversify. Our math curriculum has not changed in decades, if not centuries, which is a shame. How come physics and biology classes get updated, and math classes do not?
Do you think math research is particularly underfunded (compared to other sciences)? It seems like a lot of math Phd students have a hard time finding jobs as mathematicians. The academic job market for math looks like a nightmare for anyone who isn't a superstar.
Yes, I agree on both counts. Academia needs a serious reform.
Do you still use chalk and slate blackboards? James Grime and other mathematicians won't touch anything else.
I do! This used to be a problem, because I would be completely covered in chalk at the end of the lecture. But now I use a special Japanese chalk, which leaves much less dust, so it's better. :)
How does Ed Witten know so much math?
Some people think he is an alien. :) Seriously though, I have collaborated with him on a paper, and I can attest that his abilities are extraordinary. Here's a quote from my book "Love and Math":
"This collaboration gave me the opportunity to observe Witten up
close. I was amazed by both his intellectual power and work ethics. I sensed that he gives a lot of thought to the choice of a problem to work on. I have talked about this earlier in the book: there are problems that may take 350 years to solve, so it is important to estimate the ratio of importance of a given problem to the probability of success within a reasonable period of time. I think Witten has a great intuition for this, as well as great taste. And once he chooses the problem, he is relentless in pursuing it, like Tom Cruise's character in the film "Collateral". His approach is thorough, methodical, with no stone left unturned. Like everyone else, he gets perplexed and confused from time to time. But he always finds his way. Working with him was inspiring and enriching on many levels."
Excluding millennium problems, what is your favorite unsolved problem in mathematics?
The Langlands Conjecture. Well, there are many versions... Let's say for function fields, but for arbitrary reductive groups.
What's your favourite unsolved problem in mathematics?
The Riemann Hypothesis.
Unfortunately, if it's this chalk, you'll need to stock up quickly or find a replacement.
Why are you giving reddit another chance after they treated you so rudely last time?
Reddit mods have apologized to me for the abrupt shutting down of my AMA two weeks ago, and they have been super-nice in arranging this AMA. I think the mods have legitimate grievances, which boiled to the surface on July 2. I hope that all these issues will be resolved and Reddit will continue providing valuable service for the community. I am happy to contribute to that.
Thank you so much for taking the time to do this AMA, and additionally for responding to my email a few months back!
I was wondering if you could address a few things for me. The first is, how would you change math education if you were the ruler of the universe? In your book, you express fondness for your teacher in high school who first discussed advanced mathematics with you (I do not have the book in front of me and can't remember the name). Do you have any ideas about how to address issues with mathematics education as it is currently done?
The second question is, how would you differentiate the mathematical concept from the philosophical one? To be certain, mathematical concepts have a variety of philosophical dimensions that are worth exploring, and lead to certain philosophical conclusions. For example, the question of the possibility of the existence of strong AI is at this juncture philosophical--it implies answers to certain metaphysical questions about consciousness, being, etc. However, it seems that it can be addressed in at least a quasi-mathematical way. Do you have any ideas about how to articulate the distinction between mathematical and philosophical thought?
In regards to math education: The key problem is that in our schools today, we do not convey to our students what mathematics is really about, what it's good for, but instead make students memorize procedures and calculations that appear to them devoid of any meaning. Mathematics, in their minds, then become a cold, lifeless, boring, and irrelevant subject. What is even worse is that many of us have traumatic experiences in our math classes as children, such as being shamed by a teacher in front of the class for incorrect solution. These memories stay with us, even if we are not consciously aware of them. And this creates the fear of mathematics.
Now, let's talk about the material. Do you know that most of mathematics we study at our schools today is more than 1,000 years old? For example, the formula for solutions of quadratic equations was in al-Khwarizmi's book published in 830, and Euclid laid the foundations of Euclidean geometry around 300 BC (2,300 years ago). If the same time lag were true in physics or biology, we wouldn't know about the solar system, the atom and DNA. I think this is unacceptable and really scandalous. Especially, today, when mathematics is all around us (think about computers, smartphones, GPS devices, video games, search algorithms and so on). But we do not teach our kids about all this stuff and instead keep feeding them the same old material. This makes no sense!
People sometimes say that we need to study the old and "boring" stuff because it is necessary to understand the new and exciting ideas. But I can tell you as a professional mathematician: this is simply not true. You do not need to know Euclidean geometry, the geometry of lines on a plane -- which is flat -- to understand the geometry of a sphere, the geometry of parallels and meridians on a globe -- which is curved, not flat. Students can grasp this non-Euclidean geometry even faster, and it's a lot more fun! And in fact this is closer to reality because the Earth is round, and its surface is spherical. It's not flat! Unfortunately, in our math classes today the world is still flat.
What we should do instead is present mathematics not as a set of calculations and procedures that need to be memorized for the exam, but as what it truly is: a parallel universe of beauty and elegance -- just like art, literature, and music. And we must show the connections between mathematics and our daily lives, to get students motivated to study.
Take a look at my Op-Ed in LA Times in which I gave 3 examples of topics that kids love, but which are not taught in schools today:
In regards to your second question, on topics such as AI, math can give us indications, but ultimately, we have to decide for ourselves. So you are right, there are philosophical, metaphysical aspects here, and each of us has to face them. There is no escape, and no hiding behind any "theories". And that's exactly what it means to be human. That's the point I made in my Aspen Lecture:
What is your best advice for someone who is horrible at math and also can never understand it ? Who needs it to pass a test
I think the root of all our problems lies in our fears. I believe that anyone can understand math, but we are traumatized as children in the hands of bad teachers and this creates fears that block us from doing math. So, we have to shed our fears and believe in ourselves. Easier said than done, I know. But I think it's important to know the "diagnosis" -- then it becomes clear what to do. Good luck on your test! I am sure you will do well.
What areas of math are particularly "hot" in terms of young people going into it and progress being made?
My area of research is called the Langlands Program, and I think it is very "hot" right now. It's not just one area, but a set of ideas on how to connect different areas of math, so there are many things to do, for all tastes.
Another area which I consider very important is the so-called "Univalent Foundations", a program of restructuring the foundations of math (moving from sets to categories) initiated by Vladimir Voevodsky.
Hey, I have a question specific to Langlands Program. I have rough familiarity with it in the arithmetic context, but not so much in the geometric context. From what I've seen, it looks like a lot of the arithmetic stuff would work out, if only we had a more geometric interpretation of these objects (hence arithmetic geometry). So it seems like the geometric side of Langlands has better, more complete tools to deal with things. My question is, what are some things that Geometric Langlands has that Arthrimetic Langlands wishes it did, and what questions can we ask only on the geometric side using these tools?
It's hard to explain in two words, but the basic difference is that in the geometric version of the Langlands Program we deal with sheaves rather than functions (see Chapter 14 of my book, for example, for more details on what this means). Therefore, we are considering categories rather than sets, and categories have a much richer structure: we have not just objects, but also relations between them; for example, symmetries of a given object (sheaf).
This is you Russian translator :)
Do you have any plans to come to Russia to present your book? I'd like to have it signed :)
Hello! You did a great job translating the book into Russian, thank you! I am now discussing with the Polytechnical Museum in Moscow the possibility of coming to visit Moscow in November. And of course I would be happy to the sign the book for you. :)
Hello Dr. Frenkel, and thank you for doing this AMA. I am entering college this fall with the intention of pursuing degrees in both chemical engineering and applied mathematics. I see all of the research being done in high level fields of mathematics and it seems very interesting, but often what I cannot see is how these new discoveries can be applied. That is why I am pursuing engineering as well. So my question is how are the newest discoveries in mathematics being applied to the real world? Thank you again.
There is no algorithm for doing this. In fact, I think that today, because of our general ignorance of math, there are many mathematical ideas which are still unknown to scientists and engineers. Physicists have been pretty good in "vacuuming" ideas of modern math -- just look at all the progress made in quantum physics with the help of high level mathematics! But in other areas, much less so. And this creates an opportunity for someone like you. My advice: study as much math as you can, following your intuition and aesthetic sense, and then I am sure you will find some areas to use it.
Thank you for the reply. I will keep this in mind through the next few years. Maybe I'll be the one doing the AMA next time!
I hope so. Good luck! :)
Isn't Langlands in New Jersey?
Correct. He is a professor at the Institute for Advanced Study in Princeton, and he and I collaborated on a project for several years:
Has there been a conjecture proposed in the last few decades that you think will be famous like the Riemann hypothesis or Poincare conjecture?
It's hard to beat Riemann. It still is the most important outstanding math problem, IMO.
I'm a graduate math student specializing in differential geometry (more specifically, 4-manifold theory). I think that modern geometry is really complicated, as in, if you take a look at John Lee's Introduction to Smooth Manifolds, you'll find that it has a lot of little lemmas and theorems that say very little, with somewhat elaborate proofs, but whose content seem trivial, given some thought. I feel this is due to the inadequacy of basing the foundations of mathematics on set theory, leading to a very cumbersome framework for geometry. Do you sometimes feel that there should be a way to base off mathematics from topological/geometrical considerations in a way to make geometry (and possibly other fields) less cumbersome?
Yes, I think so too. I think categories (or groupoids, or objects of that nature) might eventually replace sets. A proposal of this sort already exists (Voevodsky's Univalent Foundations).
Hey Ed: Fellow Math Major Alumni here, thanks for doing this AMA, I have a few questions
1) How often do you catch yourself making trivial mathmatical references?
2) How often do you catch yourself breaking down the basic logic of ideas in proof in stupid situations?
3)Why has Terrance Tao made the accomplishments of hundreds of years look like a breeze?
4) For myself, what are good arguments to present when looking for jobs with our major. It is a little weird that Mathematics is a very broad subject, so "Advanced Math Skills" is a skill that employers look for, but its practically impossible to find a job that does just that but isn't a specific niche.
5)What proof do you consider a fantastic example of "Exceptionally Beautiful Mathematics", since its all beautiful.
Thanks again :)
Being a mathematician, we are trained to use the rules of logic. So for example, when I listen to certain politicians, I can spot inconsistencies right away. :)
There are many beautiful proofs. But I think Cantor's proof that there are infinities of "different sizes" (the so-called "diagonal argument") should be on anyone's Top 10 list. It is so simple, and yet, so profound.
Who is your favourite mathematician?
I am not Prof. Frenkel, but perhaps I can give you some helpful advice anyways. (I've seen enough prodigies go through college to know that what I'm about to say is generally valuable and usually ignored.)
Don't rush into college with your eyes fixed upon a single remote and fashionable topic. This will cause you to neglect the foundations of your chosen subject, and it will cause you to neglect opportunities in other subjects. Take the time to learn the basics well, and take the time to look around.
That said, don't waste your time either. If you have learned the basics of calculus well, you're going to learn more taking a class in ODE or physics or programming. This may cause some difficulties with the bureaucracy, but you can generally find ways to work around that if you've made yourself known to the faculty as a) talented, and b) not annoying.
Do you think current Computer Science programs don't put enough emphasis on math? We have trouble hiring at our company, which does very specialized GPGPU and VHDL/Verilog programming for audio applications, because people just don't have the math skills.
Yes, I think there are many areas of math that could be used in Computer Science. An example (which I discuss in my book) is the theory of categories, which is already being used in functional programming languages such as Haskell. But there is more.
What is your favorite math solving software? Specifically for solving multiple differential equations.
programs like matlab, polymath, mathmatica..
There were two of three occasions when I used math software. Most recently, when I was writing my paper with Edward Witten, and we wanted to make sure that two different equations defined the same elliptic curve. I think we used Macaulay for that. It worked very nicely. And prior to that I used Mathematica and Maple. I used Mathematica for creating some images for my book. I recommend all three programs.
"Master and Margarita" is one of my favs. Also, books by the Russian writer Victor Pelevin -- especially, "SNUFF" and "The Sacred Book of the Werewolf". I think they have been translated into English.
Reading books about math (see below in this thread), watching videos on the Numberphile YouTube channel. They are fun!
All Kubrick's films, for sure, such as "2001: A Space Odyssey" which I have watched again recently. From more recent ones, "Interstellar", "Mr. Nobody", "Mullholland Drive"...
And yes, I am thinking of making another movie. :)
Hello Dr. Frenkel. I am a mathematics student who really like mathematical logic. However, I also am really interested in the study (the theoretical side) of artificial intelligence and consciousness, and would very much like to contribute to this field. Is there a way that mathematical logic contributes to it? If not, what math is used the most in AI research, because I would very much like to do research in such a math field.
Yes, of course mathematical logic is a very essential component of this research. Unfortunately, many AI "prophets" (like Ray Kurzweil) seem totally oblivious of the landmark results in logic of the 20th century, such as Gödel's incompleteness theorem. I talked about this in my recent talk on the AI: https://www.youtube.com/watch?v=lbLI9aX5eVg
I think mathematical logic teaches us a lot about consciousness, and this knowledge is very essential in the development of AI. So you are on the right track!
Has the perception that math is only young man's game gone away? It seems like older mathematicians have been making many contributions even at an advanced age.
It has always been sort of an urban myth, and now, with more and more older (or shall I say "more mature" :) mathematicians proving remarkable theorems it has been fully discredited.
Great to have you on here Dr. Frenkel. I just finished reading Love & Math and it is now one of my favorite books on the subject (your platonist approach exactly jives with my own philosophy as well).
I know you've done extensive work on the Langlands Program, but I wanted to ask about something that might even delve into a more abstract realm: foundations of mathematics and set/category/topos theory. I have always been interested in seeing whether we could develop an all encompassing framework despite decades of seeing a proliferation of different set theories with all sorts of independence results. I know Hugh Woodin has been working on this with his "Ultimate L" research program, while others say pluralism is here to stay (like Joel David Hamkins, who is also a mathematical platonist).
What are your thoughts on this subject? Do you think mathematicians like Hugh Woodin and Peter Koellner may find success in developing an ultimate, canonical model of set theory? Or would you more closely align yourself with the multiverse view of set theory that Hamkins advocates for?
I am fascinated by Hugh Woodin's work, which I have discussed it with him on many occasions. I think Woodin and his co-authors have been able to push Gödel's vision of logic quite far, and there is probably a lot more to come. I think it's a shame that mathematical logic is not a "hot" topic in math today. It should be! It tells us so much about the way we think.
I have a bachelor's in mathematics and my favorite areas are symbolic logic and number theory. I now work in web development use the way of thinking I learned while studying math a lot more than I engage with those fields directly.
How can I become more actively engaged in those fields again outside of academia?
Good question! If you have a university nearby, then get in touch with the faculty and students specializing in these areas, attend seminars, etc. If not, then find the schools that do have such specialists and introduce yourself via email, ask for advice, etc. These days, information is usually freely available online (articles, lectures, etc.), so you can be involved in high level research even outside of academia. Just think of Yitang Zhang. :)
Your book mentions that the analog to automorphic representations in the geometric context is automorphic sheaves. In the arithmetic context, Galois Representations have nice functorial properties under change of field, but it is very hard to find a corresponding result for automorphic forms. But when we do find such a correspondence, we can get some nice results relatively easily, eg Base Change for GL(2) gives Langlands-Tunnell.
I am not familiar with Automorphic Sheaves (yet...), but from your comment it seems that they come with these kinds of internal relations, that are difficult to prove when looking at Automorphic Forms, already built in. Is this impression in the right direction or do they have other symmetries that we can exploit? Thank you for your response.
Yes, exactly. Roughly speaking, a function does not have any symmetries (being an element of the set of functions), but a sheaf does (being an object in the category if sheaves). This gives us more tools.
If you could give one restaurant a Michelin star, which one would it be?
Chez Panisse in Berkeley. But perhaps it already has one (or two?). :)
What's the biggest brainfart you ever had while you were lecturing?
I once referred to LHC as a "large hardon collider" :)
What should we be teaching YOUNG children to prepare them for a lifelong love of mathematics?
Take a look at my Op-Ed in LA Times in which I gave 3 examples of topics that kids would love to know, but which are not taught in schools today:
Considering how there's a stigma on pure mathematics in today's society (possibly caused by mass 'traumatic' experience with early math education), how do we get people who are in disciplines not traditionally associated to math to become interested enough to make connections between math and their field?
I feel like my job is a little like the job of a therapist. I need to explain to people that there is nothing to fear in math. Which is difficult precisely because, as you said, people get traumatized in math classes in school (much more than in any other classes, IMO). Many of us are not even aware of this, as we kind of push this trauma into the unconscious... But we can overcome our traumas. The first step is to acknowledge that this is what is causing us to "hate math" -- and then we can start on the "path to recovery" by reading books, watching Numberphile videos on YouTube, etc. Professional mathematicians must do more to convey the beauty of our subject.
well im totes for that-- Dr. Carroll has a great approach-- do u read him?
as a poet-mathematician you would be a great candidate for the first wave of educators-- my dream would be to make America a nation of citizen scientists and citizen educators-- an open university for the world
instead of trying to bomb ppl into submission-- educate them! :)
Making America "a nation of citizen scientists and citizen educators" sounds good to me.
What did you think of Interstellar? it is to me the very best model for gifting real science and mathematics to the public. Did you read Kip Thorne's book on the Science of Interstellar?
Yes, I read this book. It is excellent. In "Interstellar" I especially liked the last half an hour, the part when Copper entered the black hole. I thought it was beautiful. :)
What would you say is(are) your favorite part(s) of what you do?
The favorite part is discovering something that no one else knows and then sharing it with the world. But to discover one thing you first have to fail a 100 times.
How did you choose what to specialize in? Was it a teacher? Your talents? Your interests?
Is there an additional field you would work on if you had more time?
I describe in detail how it happened in my book. At first, I was interested in quantum physics, then I realized that it'
s based on math -- so I fell in love with math. And then I was guided by the desire to connect different parts of mathematics together. That's why I started to work on the Langlands Program. Which eventually brought me back to quantum physics, my "first love".
Hello Prof Frenkel,
I am a very big fan of your writings especially your book Love and Math. I do believe it should be a necessary reading for all high school students.
My question is regarding the interaction of mathematics and artificial intelligence. What is your opinion of computer-assisted proofs and automatic theorem provers? Do you think they contribute positively to the development of mathematics?
The mathematician Timothy Gowers wrote in his GAFA visions paper "I expect computers to be better than humans at proving
theorems in 2099...In the end, the work of the math-
ematician would be simply to learn how to use theorem-proving machines
eectively and to find interesting applications for them."
Do you believe that Gower's prediction can turn out to be true?
Thanks for your kind words. I think computer-assisted proofs and automatic theorem provers are important and are becoming necessary as math is becoming more and more technical. But I strongly disagree with the quote of Timothy Gowers which you cited.
Here's my opinion:
"Computers are good—and generally much better than humans—at what they are designed to do. And that is computation. But we must realize that mathematical research is not about numbers or calculations. It is about finding connections between things that are seemingly disconnected, it’s about seeing the invisible. That can not be achieved by computation alone. As a professional mathematician, I can tell you that mathematical discoveries come to us as insights. At the actual moment of discovery, you actually stop thinking. Like an artist, you connect to something that lies far beyond logic, thinking, and computation. And that’s the beauty of math."
What topic of maths is hardest for you or you have the least amount of knowldege in?
There is so much math I don't know but would like to know more... It's a long list. LOL. Perhaps, the general area I know least is analysis.
Prof. Frenkel, I have a couple of questions:
Have you read Cédric Villani's book "Birth of a Theorem: A Mathematical Adventure"? If so, what do you think of it?
How many languages do you speak? What is the last (natural) language that you have learned and at what age did you learn it?
Thanks for doing this AMA. By the way, I really enjoyed your contributions to the 1+2+3+... = -1/12 debate/controversy.
It's a good book. People have complained that he has not even tried to explain any math. But Villani's approach -- just to show glimpses of life of a mathematician -- is a legitimate one, and he has done a great job doing it.
I speak Russian, English, and now French. Russian is my native language, I was learning English at school, but did not speak fluently until I was 21 (when I came to the US). Now it is my primary language (for example, I wrote "Love and Math" in English). And I worked hard on my French recently, because I had to give interviews to the French media after my book was published in France. :)
How long does it usually take you to write a journal paper? Do you spend more time working on developing the ideas or polishing the exposition?
It depends... My last paper:
took almost 3 years to write. Usually, it's less. But polishing exposition takes a long time... In my book, I quote my teacher Boris Feigin, who said that mathematicians have to endure the tedium of writing up their results as a punishment for the pleasure of making new discoveries. :)
Can you recommend any undergrad level books? Also, hi from Kolomna. :)
From Kolomna? Great! :) I gave a list of some of my favorite math books (accessible to undergrads) in this thread.
As a mathematician working in physics, do you ever feel frustrated that you're doing an empirical science (physics) so far away from experiments? One of the complaints I hear is that theoretical physics has gotten so abstract, some people don't even consider it natural science anymore.
Well, it would be frustrating if I were a physicist. But since I am a mathematician, I tell myself that even if the physical theories I am studying have no bearing on the physical world around us, they are interesting mathematically. And perhaps they are not that far from the realistic theories, so whatever we find may be useful eventually for understanding physical reality. This is the general principle of "mathematical physics". For example, I might study 2D models even if our world is 4D. The models in 2D are simpler to study, but they can (and often do!) lead to insights that are valuable for understanding 4D models.
Tim Gowers has written about the two cultures of mathematics. One includes the problem solvers and the other the theory builders. He says there is overlap, but many people belong to one of the two cultures more than the other.
Do you think of yourself more a problem solver or a theory builder?
Here is the essay from Gowers:
While these kinds of essays are useful in order to point out different styles of doing mathematics, any divisions like this (or like pure vs. applied math) are ultimately counter-productive, IMO. For example, I don'e see myself as either a "problem solver" or a "theory builder". On the one hand, I like to work on problems which are included in a big theory (such as the Langlands Program), but on the other hand, I like solutions that are concrete. I like beautiful formulas. So... it's kind of a little bit of both. :)
I'm glad that you're doing this AMA since I'm also interested in various aspects of the Langlands program and certain topics in mathematical physics. I graduated with a non-STEM degree but right now I'm taking some post-baccalaureate courses in mathematics so that I can apply to graduate school soon. Sometimes I feel helpless because being a non-traditional student who can't necessarily take all the advanced courses I want puts me at a disadvantage compared to other applicants to grad school. Other times I feel my efforts are futile because now that I'm getting glimpses of how huge math really is, I fear that even if I do go on to do research someday it will only be in a tiny sub-sub-field of some subfield and that only five or six other people in the world will care about what I publish.
I guess my question is: what advice would you have to give to a student in my position, or someone who is worried that they can only get into a second-tier graduate school or that their research will ultimately be too obscure?
Well, I was failed at my entrance exams to Moscow University because of the policy of anti-Semitism and I had to go to a technical school. I could not even enter Moscow University through the front door, because I did not have an ID, and the entrance was guarded by the police. So in order to attend classes there, I had to sneak in by climbing over a fence on the side of the building... When we really want to achieve something, nothing can stop us.
If you had to guess, does P = NP?
If I had to bet, I would say "no".
What is your biggest complaint about current math education? For example, just last semester my calculus professor had to waste a whole class day going over state mandated materials he had to turn into the state.
Also, I am currently an engineering major considering a double major in mathematics. I love math, but I don't know of any practical reasons to justify the extra expense to my parents. Would you suggest a double major in math is useful? Why?
You can tell your parents that mathematics is penetrating our daily lives more and more (think of computers, smartphones, and all the technology we are using on a daily basis), so as an engineer you will have a big advantage if you know more math.
Hi Dr. Frenkel !
I read you book and it was awesome !
I am reading Is God a Mathematician ? at the moment, one of Mario Livio's books. It is pretty good but more focused on the history of mathematics and with its philosophical implications.
Do you have some others great books about maths to discover differents fields of maths ? :)
Here are a few:
Simon Singh, Fermat's Enigma
Mario Livio, The Equation that Couldn't be Solved
Steven Strogatz, The Joy of x
Jordan Ellenberg, How Not to Be Wrong
John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
Ian Stewart, In Pursuit of the Unknown: 17 Equations That Changed the World
Roger Penrose, The Road to Reality
Hawking was talking about transhumanism, and upgrading the legacy code. But in theory, the idea of making supersoldiers in silicon or in carbon is basically the same principle, right?
We have been trying to build better weapons and "better soldiers" for the past two millennia. And look at all the mess we have created. Isn't it time to stop focusing on killing and instead focus on what unites us. For example, mathematical formulas unite us. How about we spend more resources on math education than on weapons? I know it sounds crazy, but maybe it's time to try that. :)
Hi Professor! I was wondering, is there any subject you haven't taught a class in but would like to? Or teach a subject in such a different way from the way it's normally taught that it may well be another class?
In Berkeley, I am planning to teach a course next year together with a history professor. It will be about proof -- different perspectives: historical, legal, mathematical. I am def looking forward to that!
Hi there. Do you have any thoughts about the relationship between math and computer science?
Also, why is it that after a certain level math stops being about numbers and starts being about symbols? I love numbers, but the sight of a Greek letter outside of the Iliad gives me the shudders. This has been a significant roadblock to me in the area of advanced math and theoretical compsci. Do you have any advice about how to make this less scary?
Also, what is your favorite number? I like 49.
I would say we are on the verge of a "golden age" in the interaction between math and computer science. What gives me this confidence is how the ideas of category theory are becoming more and more wide-spread in CS. And I am sure there is more.
You maybe right that sometimes just the way we present math makes it look overly abstract. So, yeah, why not use less frightening symbols?
My favorite number? -1/12, which as you might know is the regularized sum 1+2+3+4+5+... of all positive whole numbers. :)
What do you feel are the greatest shortcomings in the US' STEM program? In particular, I feel that the 'STE' (science, technology, engineering) portions are increasingly well represented to our youth, whereas the 'M' (math) portion is still eschewed by our young students. How do we better engage the 'math' portion of STEM?
I tried to answer this here:
Hello Dr. Frenkel,
I'm in neuroscience and machine learning, but I have a math degree (undergrad).
There is a limited selection of mathematical topics that is seen in Neuro and ML (mostly statistics, linear algebra, and optimization), whereas physicists dig much deeper into more advanced math to great success.
This surprises me, because I think 'advanced' math has so much to say for Neuro, ML, and AI. The hippocampus seems to map out space more topologically than geometrically, yet few neuroscientists can tell you the difference between the two. Brains learn to infer global features from just one or two examples (you see your first giraffe and have no problem identifying subsequent ones), which seems to suggest something local-to-global, maybe sheaf-theoretic going on. I imagine it's only a matter of time until the use of these tools become necessary for these fields (Neuro/ML) to move forward.
1) What topics or recent advances in mathematics do you think have the highest likelihood in moving from 'pure' to 'applied' in the near future? In particular for Neuro, ML and AI.
2) What equation would you get as a tattoo?
Thank you. I really enjoyed your book.
Thank you! I am sure that there are huge chunks of math that have not yet been used in these areas, which could perhaps lead to some new insights. Perhaps, you can find those?
As for the equation, I haven't decided yet. :)
Do you discriminate between Strong AI, Friendly AI and General AI?
or is it all just bad interspecies competition?
What if we could build Friendly AI into Humanist AI?
i know your argument is that we cant build an AI because of Gödel, but Hawkings argument (which he may reject today) is that someone is going to do it.
Shouldn't we be trying to make it as safe as possible?
When people say "AI", they usually mean "Strong AI" or "AGI" (there is also ANI, artificial narrow intelligence, which is basically algorithms for fulfilling specific tasks, such as translation -- that's a different "animal"). Speaking of "strong AI" -- what is meant by that is essentially a computer at the same level of intelligence as a human. Unfortunately, many scientists today believe to some extent that this kind of AI is possible. In my opinion, this flies in the face of what modern science tells us, see my Aspen Lecture:
But this belief is very TELLING: it seems to me that those who espouse such ideas believe that they are nothing but machines. There are people like Ray Kurzweil and Dmitry Itskov who seem so eager to "upload" themselves onto robots. And this is not their "personal matter". Kurzweil is the Director of Engineering at Google -- the world's most powerful information technology company.
Now, some people say: let's create "Friendly AI" -- meaning codifying sort of "rules of good behavior" for robots. Even if this were possible (of which I am not sure at all), any program can be hacked. Just look in the news in the last few days: almost every day we hear about this being hacked, that being hacked. Now it has gotten to the point of airlines flight plans and German missile systems (!)
So, I think it's naive to think we can "restrain" the AI -- unless we remember our humanity and let go of the misguided ideas of "transhumanism" ad "singularity".
Do you think there is a different way to teach Math kids at school and make it interesting than the way it is now ?
I liked doing math at school but my friends really disliked it. For them doing numbers was way out of their comfort zone.
Yes, for sure. See my other answers to similar questions in this thread.
What do you make of the current enthusiasm for Homotopy Type Theory, compared to something more orthodox, like set theory? Because to me, set theory is more convenient for automated reasoning with it's embedding in first order/higher order logic with it's mature proof calculus (particularly resolution with clauses which have nice properties that are easy for machines to manipulate) and well developed model theory that allows one to do some tests for satisfiability or the (non)existence of finite models. Is HOTT really the wave of the future?
HTT (also known as Voevodsky's program of Univalent Foundations) is very important, IMO. I see it as a new approach to mathematics in which one starts not with sets, but with categories (or groupoids) -- in other words, we put the emphasis on relations rather than objects.
The US is far behind many other countries in Math and Science. What do you think is the primary cause(s) of this, and what can be done about it?
(Personally I think most kids leave school thinking there's just no use for calculus an algebra, so why bother when they don't see themselves going beyond minimum wage jobs. If they were taught the reasons why it mattered, the jobs they could get, how it matters for college, they might be more motivated).
The reason is the way we teach math in our schools. If you were teaching an art class, and you were only teaching students how to paint fences and walls, and you never even showed them the paintings of great artists, then you should not be surprised that your students would think that they "hate art". In reality, they would be hating "painting fences and walls" but how would your students ever find out that art is much more than that?
And so it is with math.
In this Op-Ed I made some concrete suggestions:
In your talk at Aspen Ideas, which I found very interesting, you seemed to use Gödels Incompleteness theorem to argue against the possibility of AI residing in a purely physical machine. Regardless of any other problems to do with AI, I wonder if you could elaborate a bit more? As far as I understand it, science strongly indicates that humans, including our minds and thoughts, are purely "physical" in nature, I mean that as in based in the natural world. Do you believe otherwise?
Yes, humans are physical, but, number one, we are not "bags of elementary particles" walking around, and number two, we are not isolated from the rest of the world. Quantum mechanics tells us this very clearly, biology tells us this very clearly, and it's just common sense. Unfortunately, in today's world we seem to believe more and more that each of us is separate and can be deconstructed like a little lego toy. But there is NO sense whatsoever in which this is true. This is what science believed in the 17th century. If we believe in science, let's also believe what 20th century science tells us. And if we are not bags of elementary particles assembled in a certain way, then how can you be serious about uploading a human onto a robot? This idea is insane, I am sorry to say. (And I have not even touched upon Gödel's incompleteness, which gives another strong indication of the impossibility of formalizing our consciousness.)
What advice would you give a young, aspiring high school student who is interested in math and thinks about pursuing it further?
My advice is follow your passion and do not be discouraged if you hit any "bumps" on the road. People often get frustrated with math because they feel they "don't understand." But I say: good! Life would be so boring if we could understand everything easily. :) If anything, I should be worried about what I think I understand. 90% of the time, when I do mathematics, I do not understand, and this is what drives my passion for learning and discovery.
Do you think there should different forms of math classes at university's? Like should there should be math calculus, engineering calculus, ect?
There are certain things which are universal. But we need to "diversify" our math classes and present new material (see my other answers on this topic in this thread).
Do you have any tips for people who struggle with math ?
I talked about this already in this thread. It is important to realize the source of this struggle. I believe it is rooted (as are many other problems) in traumatic experiences while studying math at school (such as being called to the blackboard and not being able to solve an equation in front of the entire class). We push these experiences down into the unconscious, but they come back in the form of our fear and "hatred" of math. As the result, we block ourselves. So, once we realize this, we need to work on overcoming this fear. For example, reading some popular math books, watching videos on YouTube (such as those on the Numberphile channel). Not trying to understand everything right way, just getting the "impression", the "big picture". And little by little, this fear will go away.
Same as the world without artists.
Your position on the matter seems to make total sense, more sense than strong A.I advocates. My question is, what kind of implications would this have for the evolution of intelligence/consciousness in organisms? I would imagine that the most basic of cells had commutable functions that, with enough effort, we could model. At what point would this stop happening. Does evolution have the ability to form consciousness from something that was once potentially computable that is now no longer computable?
Modeling is great. We should model live organisms. This is useful for curing diseases for example. I am all for it. But we go astray when we forget about the difference between the organism and the model. As Alan Watts used to say, there is a difference between a meal and a menu. :)
Dr. Frenkel, what would you recommend students looking for education in more advanced topics do when the resources are not provided? I am an incoming senior in high school and I find math fascinating. I have done some independent studying outside of my math class, however when I tried to ask my teacher about some of the trickier stuff to clear up the finer details it became all to evident that she had an education degree and not a math degree. Fortunately I have found some other resources to help with my understanding (I just finished a summer program where I was working with a maths prof.) but those are temporary solutions. What can I do as a student to learn these things?
Ps: thanks for taking the time to do this ama
I suggest reading books (see the list I gave in this thread, for example) and watching Numberphile videos on YouTube. In college, there are more resources for independent study.
Hahaha. Nice. Except that I don't smoke. :)
I'll go ahead and ask a silly question: what's your favorite function?
Mine is sin(pi/x) * e^(-1/x^2) -- can you guess why? But I also like e^x.
I love them all, the way parents love their children. :) But I like yours too.
Is there any area of mathematics that's, to this day, still especially underdeveloped?
One area, which is not exactly underdeveloped, but which does not attract enough attention, is mathematical logic. And yet, it is incredibly important, in my opinion.
Do you try to have a few problems that you're actively working on and make progress here and there, or do you try to focus on one problem until you solve it?
I usually have several problems I am working on.
Disclaimer: I know a fair bit of graduate level mathematics but I am a physics novice. And this question is vague and poorly worded because I am deeply confused about it myself:
What kind of a mathematical object is the universe ("spacetime"?) of General Relativity?
Presumably it's a 4 dimensional manifold (perhaps with some singularities) but it's also seen as an object, or a 3-manifold, that's evolving in time according to some flow equations. What I want is an intrinsic description of the universe. In particular, this description would not treat time any differently than the three spatial dimensions.
I started thinking about this question because I could not find a satisfactory (intrinsic) definition of Schwarzschild radius.
The spacetime of General Relativity is a 4D manifold with a Lorentzian metric (or signature -+++).
In this clip, Maxim Kontsevich gives an explanation for why he believes that we are living in a computer simulation. He basically points out that nature should be a manifold and not a bunch of vector spaces, because vector spaces are very limited objects. Do you agree with his opinion on this? If not, why not?
I have to watch it to answer this question. Here's my take on the whole "computer simulation" discussion:
Ultimately, though, I think the world is MUCH more complicated than anything we can imagine. As Goethe wrote: "All theory is gray, my friend, but evergreen is the tree of life."
Hi! I'm part of a non-profit organization that organizes math summercamps for high school kids that are interested in math, so I've seen the gap between "high school math" and "interesting, fun math" very often (we try to focus on the latter) and I agree with your views on the problems with math education not reflecting the true nature of what makes the field so interesting. Math education primarily seems to be aimed at the application of math to other fields: physics, economy, etc... This is, however, very important as one can't do physics without a decent background in calculus for example.
So my question is: How would you reshape math education to make it more interesting for the students, while still keeping the basics necessary for other fields? Split the subject in two (math applications & pure math)? Put the prerequisite math knowledge into the curriculum of the course that requires it (e.g. teach calculus in physics class)? Simply spend more time on math courses (that's a good idea anyway, I'd say)?
It's great that you do this work. We need more of this!
In regards to your question, I think we should update our curriculum: in addition to the "perennial" stuff (like Euclidean geometry) include some easily explainable and fun topics of modern math, as well as a discussion of applications of math in today's world (of which there are plenty!). I give more details in this article:
I'm an upper-division mathematics undergrad who has a mathematical result* that I think is an interesting enough curiosity to write up for the arXiv. But every time I sit down to do so, I'm plagued by relentless self-doubt and fear that it's wrong or stupid or trivial.
Is this experience a familiar feeling to you or any of the other mathematicians you work with? I don't feel it's the case for any of the professors I've talked to.
* I think I have a categorical definition of the derivative that only uses the complete partial order property of the continuum.
Well, we all get "plagued by self-doubt" from time to time. There is not magic formula for this. I'd say try to find the source. Where do you think it's coming from? Was there a particular experience in the past that is influencing you? If so, you can go to the source and resolve it, and then you will be liberated.
Dear Professor Frenkel, Your numberphile video "Why Do People Hate Mathematics?" is one of my favorites on youtube. I showed it to son who hates maths - it almost convinced him... Still working on it, but thanks all the same.
My questions are what is the most beautiful line of reasoning in all of mathematics? And what is a good area for a young research mathematicians to get into?
Thank you! I think Georg Cantor's "diagonal argument" (showing that the cardinality of the set of all real numbers between 0 and 1 is greater than the cardinality of all natural numbers) should be on any Top 10 list of the most beautiful mathematical proofs.
Добрый день, Эдуард Владимирович!
Вопрос по поводу разрыва между актуальными исследованиями и университетской программой. Этот разрыв уже огромен и он увеличивается. Скоро ли наступит момент когда студентам будет почти невозможно войти в научную среду? Какие варианты решения этой проблемы?
This question is about the gap between modern math and the math curriculum. It is growing, and the question is how to deal with it?
I agree completely that this problem will soon become untenable. There is only one way to deal with it, and that is to modify the school curriculum bringing it more up to date. There are several obvious "entry points" for modern math which can be easily explained to students, such as non-Euclidean geometry, symmetry groups, and clock arithmetic. More details are in this article:
and this video:
Love your book. What should every mathematician know about mirror symmetry? I'm a grad student in topology and know a bit about it but don't know any big picture. What other stringy dualities do you think mathematicians should know about?
Thanks! I would start with the basic example, which is the toric case (I try to explain the gyst of it at the end of my book). It's simple and beautiful. Kind of like Fourier transform. And then perhaps SYZ mechanism... Another duality important for mathematicians is electromagnetic duality (also known as S-duality).
What advice would you give to a soon to be math teacher?
Your job is incredibly important. And it's tough, because there are so many contradictory requirements and pressures that are put on you. But in a way, that's what makes it so exciting, doesn't it? I'd say first and foremost, you have to love the subject. And then second, try to educate yourself about some interesting topics (by reading popular books or watching Numberphile videos on YouTube: https://www.youtube.com/user/numberphile). Then you will be able to give your students something beyond the standard curriculum, and also convey your passion. Then, the students will love you, and they will love math.
How does math work hand and hand with love? How can it work for someone like me who's been hurt too many times and had given up on love?
You have identified the basic mechanism: we get hurt and then we don't want to be hurt anymore, so we give up or develop some defense mechanisms. This applies to love, and this applies to math as well. So here's one example of how math and love go hand in hand. :) How to overcome this? We have to keep going, because there is no other way. Getting hurt is part of being human. But if we don't know sadness, we will never know joy. (Watch the movie "Inside Out" -- it's about THAT.)
For people working on 'new' math, what is the process like? What is being written down? What is being thought? Is it "I think it works this way, but how can I prove it.."
There is no formula or algorithm for this. I tried to explain this in my book this way (Chapter 7):
"The ability to see patterns and connections that no one had seen before does not come easily. It is usually the product of months, if not years, of hard work. Little by little, the inkling of a new phenomenon or a theory emerges, and at first you don't believe it yourself. But then you say: "what if it's true?'" You try to test the idea by doing sample calculations. Sometimes these calculations are hard, and you have to navigate through mountains of heavy formulas. The probability of making a mistake is very high, and if it does not work at first, you try to redo it, over and over again.
More often than not, at the end of the day (or a month, or a year), you realize that your initial idea was wrong, and you have to try something else. These are the moments of frustration and despair. You feel that you have wasted an enormous amount of time, with nothing to show for it. This is hard to stomach. But you can never give up. You go back to the drawing board, you analyze more data, you learn from your previous mistakes, you try to come up with a better idea. And every once in a while, suddenly, your idea starts to work. It's as if you had spent a fruitless day surfing, when you finally catch a wave: you try to hold on to it and ride it for as long as possible. At moments like this, you have to free your imagination and let the wave take you as far as it can. Even if the idea sounds totally crazy at first."
I just want to say that i absolutely loved your book. It was very inspiring. I bought it to try to develop a love for math, but i ended up being much more interested in your personal struggles to study math in Russia. Would you say that those early days really developed your drive to excel? If it was me, probably half my drive would be to spite those that said i couldn't. Or was it purely for your love of math?
Thank you for your kind words. I think it was both: my love for math and the drive to show that I can do it despite the obstacles that were placed in front of me. Such experiences shape us and make us who we are.
Hey Prof. Frenkel. I took your Math 53 course at Berkeley like 5 years ago. I learned a lot in that course. So here's a belated thanks!
Anyways, what is your favorite restaurant to eat in Berkeley? Also, how do you stay motivated to learn more about math?
Glad to hear you learned a lot from that course! My favorite restaurant is Cesar's, but I also like Chez Panisse very much (Alice Waters is a good friend :) How do I stay motivated? Curiosity, I guess. :)
What is you favourite equation and why?
There are many beautiful equations. For example, I like the one which we used as a "formula of love" in the film "Rites of Love and Math":