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February 20, 2020

What’s the Most Exciting New Mathematics?

Posted by John Baez

I’m considering writing a column on the most exciting new developments in math (for some magazine). This makes me want to ask:

What do you think are the most exciting new developments in math?

It would help a lot if you explain why you think they’re exciting. And I especially want to hear answers from outside category theory–since I roughly know what’s going on there. But there are probably new things there, too, that I haven’t heard about yet. So don’t hold back: let me know what you think!

Posted at February 20, 2020 6:24 PM UTC

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Re: What’s the Most Exciting New Mathematics?

How new is “new” for your purpose?

Posted by: Mark Meckes on February 20, 2020 7:37 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

New enough that you feel it’s new.

If you just learned a cool theorem by Apollonius, it could be new to you, but you probably wouldn’t call it a “new development in mathematics”.

Posted by: John Baez on February 20, 2020 8:00 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Here’s something I was excited to learn about, which is new by any reasonable standard. I guess this is my suggestion for “the most exciting new random matrix theory”.

Pick an n×nn \times n matrix M nM_n at random by letting each of the entries be ±1\pm 1 independently with probability 1/21/2 each. What is the probability that M nM_n will be singular?

If the matrix entries were instead picked with, say, a Gaussian distribution, it’s easy to show that the answer is 00. So all experience with the behavior of large numbers of random variables suggests that it must at least be true that lim n[M nissingular]=0. \lim_{n \to \infty} \mathbb{P}[M_n \; is \; singular] = 0. This is true, but not easy to prove. It was first proven by Komlós in the 1960s, who showed (if I remember correctly) that [M nissingular]Cn. \mathbb{P}[M_n \; is \; singular] \le \frac{C}{\sqrt{n}}. About thirty years later, this was improved by Kahn, Komlós, and Szemerédi to [M nissingular](0.999) n, \mathbb{P}[M_n \; is \; singular] \le (0.999)^n, which has the nice consequence (via the Borell–Cantelli theorem) that with probability 1, M nM_n is nonsingular for all sufficiently large nn.

What about lower bounds? Well, the most obvious way for a matrix with ±1\pm 1 entries to be singular is for two of its rows to be equal. The probability that, say, the second row is equal to the first is 2 n2^{-n}, so (12) n[M nissingular](0.999) n. \left(\frac{1}{2}\right)^n \le \mathbb{P}[M_n \; is \; singular] \le (0.999)^n. You can improve that lower bound by thinking about more than just the first two rows, or the possibility of one row being 1-1 times another, but you’ll still just get [M nissingular](12+o(1)) n \mathbb{P}[M_n \; is \; singular] \ge \left(\frac{1}{2} + o(1)\right)^n out of that kind of reasoning. A long-standing conjecture held that this lower bound was tight; that is, that an upper bound of the same form holds.

Recently, Konstantin Tikhomirov indeed proved that [M nissingular](12+o(1)) n. \mathbb{P}[M_n \; is\; singular] \ge \left(\frac{1}{2} + o(1)\right)^n. (In fact, he proved a stronger statement about the smallest singular value of random matrices with Bernoulli entries.) This was a culmination of intense work on the smallest singular values of random matrices over the past fifteen years (although still sharper versions of the conjecture remain open), by authors whose names I’ll let you read in the bibliography of the paper (to spare me the embarrassment of failing to mention someone).

As for why I find this exciting, beyond the fact that I’ve been following this story closely for a long time, I like the fact that, by the standards of the field, the statement of the result is extremely accessible. It’s no Fermat’s Last Theorem, but you only need some very basic knowledge of probability and linear algebra to understand not only the statement, but also why it was conjectured. (On that latter point, I think it has FLT beat.) The proof, on the other hand, consists of many layers of technical arguments related to additive combinatorics, geometric functional analysis, numerical linear algebra, and other areas.

Posted by: Mark Meckes on February 20, 2020 11:05 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

That should of course have said that Tikhomirov proved that [M nissingular](12+o(1)) n, \mathbb{P}[M_n\; is \; singular] \le \left(\frac{1}{2} + o(1)\right)^n, and therefore [M nissingular]=(12+o(1)) n, \mathbb{P}[M_n \; is \; singular] = \left(\frac{1}{2} + o(1)\right)^n, Otherwise there’s nothing exciting there at all.

Posted by: Mark Meckes on February 21, 2020 9:13 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Thanks for explaining this in enough detail so I can see why it’s exciting! You’re actually getting me interested to see some of the “many layers of technical arguments” required to prove these facts — from a safe distance, of course.

Btw, this is a bit ugly

[M nissingular](12+o(1)) n, \mathbb{P}[M_n is singular] \le \left(\frac{1}{2} + o(1)\right)^n,

so I’ve been taking your comments and adding \; to them, obtaining

[M nissingular](12+o(1)) n, \mathbb{P}[M_n \; is \; singular] \le \left(\frac{1}{2} + o(1)\right)^n,

This is one of the many free services offered by your nn-Café hosts.

Posted by: John Baez on February 21, 2020 10:04 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Btw, this is a bit ugly

I was surprised you said so, because on the browser I was using the first version looked fine and the second has strangely large spaces. But then I tried a different browser and I agree, the first version came out hard to read. I’ll try to remember that for the future.

If you’d like to get an idea of the technical arguments from a relatively safe distance, I suggest starting with Rudelson and Vershynin’s 2010 ICM talk. That will get you through the major ideas in about the first two thirds of Tikhomirov’s summary of his approach.

Posted by: Mark Meckes on February 21, 2020 10:27 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Here is some specificity on how different browsers handle the sample text: In Chrome, both of them look fine, and basically indistinguishable; the difference in spacing is not noticed unless you are looking for it. In Firefox, for some reason the space between “is” and “singular” disappears. so the words run together.

Posted by: Keith Harbaugh on February 26, 2020 12:41 AM | Permalink | Reply to this

Andreas

I’m using Chrome and the spacing is definitely different. The first version looks fine but the second has strangely large spaces.

Posted by: Andreas Blass on February 27, 2020 6:41 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Ludovic Patey and Keita Yokoyama’s result on Ramsey’s theorem for pairs and two colors (2016) - arguably the most important advance in logic of the last twenty years.

Posted by: Byrana on February 20, 2020 8:49 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Neat! Can you say something about why this is so important that goes beyond what’s in this article?

If I learned about it and wrote about it, I’d feel the need to write something that’s different (and better) than this article in some way. One way would be to dig deeper.

Posted by: John Baez on February 21, 2020 12:04 AM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

MIP* = RE is a very exciting result from the bleeding edge complexity theory.

Posted by: Riley on February 20, 2020 9:03 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Yes! Anyone curious about that can get some gossip here:

  • Scott Aaronson, MIP* = RE, Shtetl-Optimized, January 16, 2020.

By the way: the paper on MIP* = RE cites someone who visits the nnCafé, namely Tobias Fritz, for having helped establish the connection to Connes’ embedding conjecture.

I don’t really understand this stuff, but I could learn it….

Posted by: John Baez on February 21, 2020 1:49 AM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

I’d be excited for you to understand MIP*=RE at a blogging-about-it level. I think there’s good potential for getting a lot of idea compression from the categorical point of view. Their proof leverages a family of functors from certain categories of 2-player nonlocal games to certain categories of c* algebras. It’s not clear that anyone has written down the definition of the source category. I think if one did, then one could interpret large parts of the machinery in the MIP=RE paper as limits of appropriate diagrams. Then pass the constructions through the functor and have it look purely c-algebraic.

One branch of this research program was proving “parallel self testing” results, which should be seen as constructing products in the category of games. The “introspection game” from Anand and John’s first breakthrough NEEXP \subset MIP* is I think an exponential object in the same category.

Posted by: Jalex Stark on March 2, 2020 8:42 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Topological Data Analysis. It is quite recent and it is an amazing example of connection beetwen pure and very applied math, from bacteria classification to remote sensing.

Posted by: Roberto on February 20, 2020 11:48 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

I know a bit about topological data analysis and persistent homology. Is there a theorem in this subject, or an application of this method, that you find particularly surprising?

Posted by: John Baez on February 20, 2020 11:59 PM | Permalink | Reply to this

If you are looking for a general direction of research, I’d say the attempts to use algebraic geometry for computational complexity theory (falls under the umbrella named ‘geometric complexity theory’) are all very interesting.

Posted by: Some guy on February 21, 2020 2:01 PM | Permalink | Reply to this

Re:

Thanks. Is there some particularly exciting result in this area that I could focus on?

Posted by: John Baez on February 21, 2020 11:38 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

I have two general suggestions, and a bunch of suggested results in set theory.

The general suggestions:

  1. In order to avoid bias, it might be helpful to divide the question into subareas (i.e., instead of asking for our favorite exciting mathematical results, ask for our favorite exciting results in number theory, set theory, functional analysis, etc.

  2. Avoid relying on Quanta magazine. That website is basically following a few popular math blogs (e.g., Tao’s, Kalai’s, etc) and the updates coming from departments with strong PR. This results in an echo chamber that gives the reader a quite biased and twisted view of the recent advances in mathematics (see for example the above comment, describing a result of Patey and Yokoyama as “the most important advance in logic of the last twenty years”. I can’t think of many logicians who would agree with this claim).

Some favorite exciting recent results in set theory, in no particular order:

  1. MM^++ implies Woodin’s (*) axiom (Aspero and Schindler).
  2. Viale’s extension of Woodin’s absoluteness theorem for the Chang model.
  3. Gitik’s solution of the PCF conjecture.
  4. Recent advances in the study of cardinal characteristics of the continuum, leading to a model with a maximal number of pairwise distinct cardinal invariants (due to Goldstern, Kellner and Shelah).
  5. Neeman’s work on forcing with side conditions.
  6. Recent breakthroughs in the descriptive set theoretic study of sets of reals with maximality properties (such as maximal almost disjoint families).
  7. Malliaris and Shelah proving the equality between the invariants p and t.

I’ll hopefully find some time during the weekend to add some references and elaborate on at least some of the above results.

Posted by: Anonymous Set Theorist on February 21, 2020 3:09 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Anonymous Set Theorist:

In order to avoid bias, it might be helpful to divide the question into subareas (i.e., instead of asking for our favorite exciting mathematical results, ask for our favorite exciting results in number theory, set theory, functional analysis, etc.

I’m just looking for a few exciting topics to write about. Ultimately the choices will be completely biased by what I happen to find exciting, and can understand well enough to explain. I don’t have the energy to attempt an even-handed survey of different areas.

Avoid relying on Quanta magazine.

On the contrary, if they write about something I’m less likely to write about it, because that would be “old news”. There are lots of things that have not yet been popularized, I’ll give those get higher priority. (Of course I might be tempted to write about something if I feel they did a really bad job.)

Thanks for listing those set-theoretic results. I’d like to hear more about them.

Posted by: John Baez on February 21, 2020 8:39 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Two areas which seem quite exciting to me (other than my own, which is obviously the most exciting) are : 1. Exact WKB, and relations with the geometry of the Hitchin system. 2. 3d mirror symmetry and symplectic duality.

Posted by: Francis on February 21, 2020 6:38 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Thanks! Could you say a bit more about what these are and (just as important) why they are exciting?

I know what a WKB approximation is, and I can imagine that situations where the WKB approximation is exact would be exciting. I don’t know what the Hitchin system is. I have a vague general idea of mirror symmetry in string theory. I don’t know what 3d mirror symmetry is (something about some sort of 3-dimensional field theory?). I don’t know what symplectic duality is.

Posted by: John Baez on February 21, 2020 8:29 PM | Permalink | Reply to this

Re: Whats the Most Exciting New Mathematics?

I wanted to leap in with a beautiful explanation of the Hitchin system, but I doubt I could give a better one than Hitchin’s 1987 paper.

Although I wouldn’t be very surprised if you had already heard of this but just forgotten the name…

Posted by: Will Sawin on February 22, 2020 8:38 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Duminil-Copin and his many coworkers are making lots of progress on phase transitions in a wide class of models. I’m no expert, but the pace of new results seems quite high, with far reaching new ones announced a few days ago. Some of these have been covered by e.g. the Bourbaki seminar (which I could follow at the time and found indeed impressive), Princeton,…

Posted by: bonody101 on February 21, 2020 7:57 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Picking up on some recent themes not already mentioned in the thread…

  • The contributions of amateur mathematicians in topics like permutations, universal cover geometries, graph coloring, etc.

  • Advances in quantum computing and the applications to which they’re being put.

  • The growing use of sabermetrics and statistics-based strategies in sports and games.

  • The realm of mathematical physics. I’m specifically thinking of the recent proof of the Cartan-Hadamard conjecture for all dimensions, but I am also looking at recent work involving dynamical systems.

Good luck with the new column!

Posted by: Ironman on February 22, 2020 4:39 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

The automatic “related entries” feature is pointing to my guest posts from last year about sporadic SICs and exceptional Lie algebras, so maybe I should take that suggestion as an endorsement! :-)

The side of SIC research that may be more in line with the spirit of the question might, however, be the study of the non-sporadic examples. These relate to algebraic number theory in ways that nobody foresaw. When I went to physicist school, I learned about Hilbert spaces, but nobody told me they would have anything to do with Hilbert’s 12th problem!

  • M. Appleby, S. Flammia, G. McConnell and J. Yard, “SICs and Algebraic Number Theory,” Foundations of Physics 47 (2017), 1042–59, arXiv:1701.05200.
  • G. S. Kopp, “SIC-POVMs and the Stark conjectures,” International Mathematics Research Notices (2019), arXiv:1807.05877.
  • I. Bengtsson, “Algebraic units, anti-unitary symmetries, and a small catalogue of SICs,” arXiv:2001.08487 (2020).

Figuring this business out was recently chosen as the first item in a list of five open problems in quantum information, with a prize attached.

Posted by: Blake Stacey on February 22, 2020 6:37 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

The work of Marcus, Spielman, and Srivastava on interlacing families of polynomials has had a lot of ramifications well beyond its initial applications to solve the Kadison-Singer problem.

The work of Adiprasito/Huh and collaborators on Hodge theory and and Lefschetz theorems in combinatorial contexts (e.g. matroids) and log-concavity also seems likely to have ramifications well beyond its initial areas of application.

Posted by: Dan F. on February 26, 2020 7:39 AM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Saw Huh give a great talk a while ago about matroids (about which I know little). And recently an amazing talk by Adiprasito who explained applications in geometric topology (closer to me; he outlined an approach towards the Singer conjecture). Super exciting.

Posted by: Simon on March 8, 2020 5:42 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

In Euclidean harmonic analysis (not representation theory):

Bourgain–Demeter’s 2\ell^2 decoupling for the paraboloid (2014).

Bourgain–Demeter–Guth’s solution to the Vinogradov mean value theorems via decoupling techniques (2015).

Wooley’s contributions to this at the same time via his efficient congruencing technique should not go left out.

To give context, Vinogradov’s mean value conjectures arose in the late 1950s (or maybe 60s?) and were one of the big problems in the circle method which itself is one of the big areas of analytic number theory. In particular, this gives progress towards Waring’s problem which is circa 1770. Wooley was steadily making progress via the introduction of several methods when Bourgain–Demeter–Guth solved the problem. Bourgain–Demeter–Guth’s perspective arose out of Bourgain–Demeter’s solution to the decoupling problem which was considered a very difficult problem at the time (prior to 2014). BD’s result had numerous applications to number theory, PDEs, etc. BDG and Wooley’s results give quantitative progress towards the Riemann hypothesis. In particular, they improve the known zero-free region. Unfortunately, this is not enough to settle RH.

In CS, Tang’s solution to the Netflix problem (2018) was quite exciting as it dequantized a quantum algorithm that was supposed to give a significant improvement. This led to further results in the area.

In number theory, Koukoulopoulos–Maynard’s solution to the Duffin-Schaeffer conjecture (2019) stimulated a lot of excitement.

Posted by: K Hughes on March 1, 2020 5:30 PM | Permalink | Reply to this

Re: What’s the Most Exciting New Mathematics?

Thanks for all of these! A lot of interesting things for me to study.

Posted by: John Baez on March 1, 2020 10:17 PM | Permalink | Reply to this

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