Last week we organized an intensive two-day summit Math Research in the Age of AI. It was held at the Harvard Center for Mathematical Sciences and Applications (CMSA). Thirty-two mathematicians from academia, government, and industry came together to help navigate this period of rapid change in mathematics and, most importantly, to offer ideas for building the future. There were five working groups organized around the following themes:
- Evaluation, production, and dissemination of research
- Coordinated actions for the AI era
- Open-source tools
- Formalization in mathematics
- Collaboration between academia and AI labs
Our report can be found on the summit website. There you’ll also find a link to provide feedback.
Portions of the recommendations, particularly those in the section on evaluation, production, and dissemination of research, offer immediately actionable suggestions for review and hiring in mathematics departments. We encourage departments to act on those they can implement at this time and to begin longer-term conversations to rethink evaluation practices. We recognize that many of these changes may temporarily place greater burdens on faculty. These efforts are necessary to facilitate transitions to new norms for the field and are especially important in light of the need to support our students and young colleagues. Our recommendations go beyond academia; they impact government and industry as well.
Received 30 October 2026.
- Dan Freed (Harvard), Larry Guth (MIT), Bryna Kra (Northwestern), Emily Riehl (Johns Hopkins) and Mohammed Abouzaid (Stanford)
︎
builds fine intellectual dexterity. But they also show young children that there are fundamental, unchanging facts about the world. Invariants. And starting from this, we can build up confidence that allows kids to grow into students who are grounded and questioning because they can both see and understand why there are invariants [9].
just barely fails to be integrable near the diagonal 
. (This is in contrast to, say, fractional integral operators such as 
, whose kernels go to infinity at the diagonal but remain locally integrable.) These operators arise naturally in complex analysis, Fourier analysis, and also in PDE (in particular, they include the 
is well-defined even for nice functions 
. But if 
and compactly supported, then we can restrict the integral to a compact interval ![{t \in [-R,R]}](https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+%5B-R%2CR%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
for some large 
(depending on 
and 
is uniformly bounded on the interval 
, and so the limit actually exists from the dominated convergence theorem. A variant of this argument shows that 
is also well-defined for 
at infinity. In particular, we already see that 
is not bounded on 
. However, we do see that 
.
-normalized dilations 
be a continuous linear operator which maps Schwartz functions to bounded continuous functions, and which commutes with translations and dilations as in 
is a linear multiple of the Hilbert transform and the identity. (Hint: first establish the existence of a distribution 
such that 
for all 
, and such that 
is homogeneous of degree 
. Restrict the action of 
or 
and show that 
on these functions.) (The claim is also true if we relax 
to 
, the space of distributions, but this is a little trickier to prove.) 
are compactly supported, show that 
obey a qualitative decay bound 
(say – these conditions are needed just to make 
. By taking complex conjugates we may assume that the 
sign is 
. Our task is then to show that 
and making the change of variables 
, we reduce to showing 
and obeys the decay bound 
in this region. Then Cauchy’s theorem gives 
in this case. In particular, comparing real and imaginary parts we conclude that 
and 
. Thus for reasonably decaying holomorphic functions on the upper half-plane, the real and imaginary parts of the boundary value are connected via the Hilbert transform. In particular this shows that such functions are uniquely determined by just the real part of the boundary value.
. This can be made more manifest by the following Fourier representation of the Hilbert transform.
. (Recall that 
and so its Fourier transform is defined via density by Plancherel’s theorem.) 
. It then suffices to show that 
has vanishing Fourier transform in this half-line. Define the Cauchy integral operator 
as 
; since 
also have vanishing Fourier transform on the half-line.
(we could rescale 
to be, say, 
, but we will not have need of this normalization). We can truncate and define 
converges to 
in 
and so it will suffice to show that the Fourier coefficients of 
we obtain the claim. 
. There is still the question of whether the original definition is valid for 
); in particular, 
. Then 
actually converges back to 
in some sense; informally, 
for positive frequency functions and 
for negative frequency functions. Now multiplication by 
. This is indeed the case for 
, and we now turn to the theory which will generate such bounds.
or 
. 
), or conversely map a tall narrow function down to a broad shallow function of comparable 
). It turns out that these two phenomena are more or less dual to each other, and so when working with an operator which is “comparable” to its adjoint (which is for instance the case with the skew-adjoint Hilbert transform, 
. The dual approach is to show that if 
norm of 
for various sets 
, but we will not discuss those here as they do not extend well to the more general class of singular kernels considered here.)
lies outside of the support of 
(defined away from the diagonal 
and the Hölder-type regularity estimates 
. 
and 
are both singular kernels. In higher dimensions, the kernel 
is a singular kernel whenever 
is a Hölder-continuous function. 
) from the estimates 
is 
, these estimates are then consistent with 
(in which case the Hölder type conditions become Lipschitz type conditions). It is more desirable 
than for small 
version of the estimate already follows from 
is not significant; it is not hard to show that the regularity conditions are equivalent to the condition in which 
which is bounded on 
, 
is compactly supported, and 
, the physical space multiplier 
is a CZO with kernel 
. However, this is the only source of ambiguity; see Exercise 
, but we will not do so here for sake of concreteness. 
. We will discuss other examples of CZOs, such as the Riesz transforms, Hörmander–Mikhlin multipliers, and pseudodifferential operators of order zero, later on.
, and the translates 
and dilates 
of 
and 
. (What happens to the kernels 
be a ball in 
have mean zero, thus 
. Then we have the pointwise bound 
, but we shall not do so here. For 
we can use the mean zero hypothesis to obtain ![\displaystyle Tf(y) = \int_B K(x,y) f(x)\ dx = \int_B [ K(x,y) - K(x_0,y) ] f(x)\ dx,](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++Tf%28y%29+%3D+%5Cint_B+K%28x%2Cy%29+f%28x%29%5C+dx+%3D+%5Cint_B+%5B+K%28x%2Cy%29+-+K%28x_0%2Cy%29+%5D+f%28x%29%5C+dx%2C&bg=ffffff&fg=000000&s=0&c=20201002)
). 
, with an operator norm of 
. 
(the 
and 
to both be 
, where the “good” function 
obeys the bounds 
is supported on 
, has mean zero, and has the 
, we can split 
has 
of 
depending only on 
and summing in 
. But here is the great advantage of weak-type estimates: we can tolerate arbitrarily bad behavior as long as it is localized to small sets. Indeed we have 
for 
. The self-adjointness of the CZOs as a class then gives the same statement for 
, thus we have proven
. 
let 
be the Dirichlet summation operator, defined on Schwartz functions 
converges in 
for all 
. 
an 
is true by Plancherel’s theorem, but a surprising 
. The almost everywhere convergence problem remains open (and is considered very difficult) in higher dimensions. 
be two CZOs with the same kernel 
such that 
for all 
and 
. Now observe that 
is an absolutely continuous measure, and use the Radon–Nikodym theorem.) 
for some 
, then Corollary 
instead of 
. In particular this apparent expansion of the notion of a CZO does not in fact increase the class of CZOs (up to changes in the implied constants). 
. At first glance things are rather bad; not only is 
for 
directly. A naive implementation of the kernel formula 
between 
, the expression 
decays like 
as 
, and so the integral will be absolutely integrable as long as 
. At the other extreme, if 
(up to the usual measure zero ambiguities).
are equivalent modulo a constant if there exists a complex number 
for almost every 
is any ball which contains both 
, and that 
) on the common domain of definition (i.e. 
or 
).
be a function defined modulo a constant. The 
(or 
the space of all functions with finite 
ranges over constants; thus the statement 
is equivalent (up to changes in the implied constant) to the assertion that for any ball 
such that 
, thus 
on the average.
. But it also contains unbounded functions:
is in 
, on which CZOs are often ill-behaved. Consider for instance the Hilbert transform applied (formally) to the signum function 
. The resulting function is not bounded, but lies in 
modulo constants. 
and 
. By definition of the 
. We split 
. Since 
we have 
; since 
, and so 
.
, thus 
, otherwise the trivial bound of 
is superior. It asserts, for instance, that 
on at most 
of any given ball.
that 
on at most 
of the ball), 
on at most 
, and so forth, leading to a bound of the form 
. Then for any ball 
depending only on 
. For each 
denote the best constant for which we have 
to be chosen later, and assume that 
. Let 
. Then we have 
. Let 
are a finite collection of balls intersecting 
and 
if we choose 
large enough.
is pointwise bounded almost everywhere by 
(by the Lebesgue differentiation theorem), and so 
, and hence 
, we obtain the claim. 
and all locally integrable 
. 
and 
. Then 
and 
. Let us understand the size of the distribution function 
. From the 
, such that 
almost everywhere outside of 
, and such that 
on each cube 
also, and thus 
. From the John–Nirenberg inequality we thus see that 
norm we obtain the claim. 
. It is now easy to check that 
, the boundedness on 
; in particular, note how the Calderón–Zygmund decomposition continues to make an appearance.)
be a sequence of non-zero frequencies in 
which are lacunary in the sense that 
for all 
. Let 
be a sequence of complex numbers which obeys the 
bound 
. To avoid issues of convergence let us assume that only finitely many of the 
are non-zero. Let 
be the exponential sum 
is an interval in 
is a bump function adapted to 
. (Hints: split the 
, and the low frequencies where 
. For the high frequencies, forget about 
. For the low frequencies, choose 
and any locally integrable 
, define the dyadic BMO norm 
of 
. Show that we have the John–Nirenberg inequality 
, and in particular 
as well.) 
be the shifted dyadic meshes from the previous week’s notes, and set 
to be the associated shifted dyadic BMO norms, show that 
![{V: [0,1] \rightarrow {\bf R}}](https://s0.wp.com/latex.php?latex=%7BV%3A+%5B0%2C1%5D+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
be the function 
. Use a Bellman function approach to establish the John–Nirenberg type inequality 
, all intervals 
with 
. 
, define the function 
as 
, and let 
for at least one 
for at least one 
. Use the Hardy–Littlewood maximal inequality to show that for any sufficiently large 
, that 
. Use this to give an alternate proof of Lemma 
, but their behavior is much more elementary to analyze thanks to Hölder’s inequality; for instance, a spatial multiplier 
is bounded on 
is in 
is a tempered distribution we define the Fourier multiplier 
on Schwartz functions 
by the formula 
. 
, and that is enough to make 
a tempered distribution, and so 
is well-defined as a tempered distribution. If 
is a Fourier multiplier with symbol 
. More generally, any constant-coefficient differential operator 
, where 
is a polynomial, has symbol 
; thus one can formally view 
in 
. The translation operator 
is a Fourier multiplier with symbol 
, thus we formally have 
. A convolution operator 
, with 
; conversely, every Fourier multiplier is a convolution operator with the distribution 
. The Hilbert transform is a Fourier multiplier with symbol 
, thus 
.
. We say that 
to be the 
. Thus for instance the 
norm is the same as the 
, which has the same 
we can check that 
is an affine transformation rather than a linear one.
and 
be 
is also an 
or 
.) 
or 
, it is also not hard to show that 
does not lie in 
, although this is a little tricky. Thus for 
for some fixed bounded domain 
for all 
, thus 
. Then the 
-dimensional Fourier multiplier 
are related by the identity ![\displaystyle [m(D) f]_{x_d} = \tilde m(D) [f_{x_d}]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Bm%28D%29+f%5D_%7Bx_d%7D+%3D+%5Ctilde+m%28D%29+%5Bf_%7Bx_d%7D%5D&bg=ffffff&fg=000000&s=0&c=20201002)
and Schwartz 
is the slice 
is in 
, then 
with 
is continuous, then 
for any 
. For any 
be the linear transformation 
pointwise, where 
is the (non-invertible) projection 
are Schwartz functions, we see from Hölder’s inequality that 
), basically by exploiting the fact that there was a projection from 
) 
is bounded, define the multiplier 
by the formula 
, 
to be the 
operator norm of 
. (Hint: take functions 
, extend them periodically to 
), apply 
.) 
for which the multiplier 
is an 
is singular on a large, curved set (in this case, the sphere), which necessarily leads to some delicate and still incompletely understood geometric issues (most notably the Kakeya conjecture, which we will also not discuss here).
and 
. (In particular 
.) Then 
for all 
derivatives here, but in practice this type of control is sufficient for many applications. The conditions 
, although it essentially allows different annuli to behave independently. For instance, if 
is a bump function adapted to the annulus 
are arbitrary signs, then 
obeys the homogeneous symbol estimates of order 
. 
, but we run into the formal difficulty that 
be a fixed bump function which equals one on the ball 
and is supported on the ball 
, and write 
. Then we have the partition of unity 
, and hence we have the pointwise convergent sum 
. Using Plancherel and dominated convergence we conclude that 
we have 
are also bounded and compactly supported with disjoint supports. We can write 
are bounded with compact support; we omit the details. Now observe that 
is supported on the annulus 
, and multiple applications of the Leibnitz rule give us the bounds 
case of this bound and the triangle inequality we obtain the bound 
times (using the identity 
) we obtain the bound 
, and hence 
factor essentially just contributes an extra factor of 
to the analysis, and we end up with 
is convergent in 
to a kernel 
obeying the bounds 
is a singular kernel, and 
for all 
certainly obeys the symbol estimates. Another useful instance of the theorem is to conclude that the homogeneous fractional derivative operators 
, with symbol 
and 
real, are also bounded on 
(in particular, it grows at most polynomially in 
. A similar argument also applies to the inhomogeneous fractional derivative operators 
with symbol 
. (The factor of 
is of course irrelevant here and the same results hold if this factor is dropped. Indeed, in PDE it is customary to drop the 
for 
, with symbol 
. These symbols are easily verified to obey the homogeneous symbol estimates and are thus in 
and Schwartz 
is bounded on 
, Schwartz 
is the Fourier multiplier with symbol 
.
for all 
, obeys the homogeneous symbol estimates and is thus in 
, we let 
be the space of measurable functions 
such that 
in the usual manner. One can (rather laboriously) rederive most of the scalar 
with the inner product 
). For instance, one can verify the duality relationship 
. Also, the real interpolation theory continues to be valid in this setting (the basic reason being that the layer-cake decompositions, which depend only on the magnitude of the underlying function, continue to be valid).
, then one can define the vector-valued integral 
in several ways, for instance by using duality 
for balls or cubes 
denote the bounded linear operators of 
, with the usual operator norm.
be Hilbert spaces. A singular kernel from 
(defined away from the diagonal 
which is bounded, 
is compactly supported, and 
to 
with bound 
, let 
be a bump function adapted to an annulus 
. Then for any 
for 
defined by 
, we quickly see from Plancherel’s theorem that 
with operator norm 
to 
in the obvious manner. But as in the proof of the Hörmander–Mikhlin multiplier theorem we have 
. Then for any 
be bump functions adapted to a slightly wider annulus than 
is also adapted to the same annulus as 
. This leads to the identity 
is known as a Littlewood–Paley square function of 
as the component of 
. The point of the inequality is that even though each component 
, let 
be a function obeying the estimates 
, all 
, and some 
, improve this bound to 
is extremely likely to have magnitude about 
, and can be viewed as a variant of the law of large numbers:
be independent random signs, drawn from 
with the uniform distribution. Then for any 
are real. When 
has expectation 
and 
. This proves the claim for 
and the lower bound for 
. Hölder also shows us that the lower bound for 
. Now consider the expression 
) 
. 
for some 
for each 
and then take 
, that the sum 
obeys the homogeneous symbol estimates of order 
. (Hint: look at 
for a real parameter 
we have 
; assume that all but finitely many of the 
are non-zero. Let 
; in particular it will only be defined modulo a constant.) 
be an integer with 
, and let 
finite. Show that 
, show that 
. 
means “greater than”; and 
, “greater than or equal to.” Don’t translate 
and 
.
. Evolve it under 
…”).
defines a regime, not a limit. The conditions 
and 
define limits and are inequivalent to 
Then Y.”
then Y.
. Substituting into Eq. (1) yields [equation].
,” “the spin quantum number 
,” “the observable 
” “the magnetic spin quantum number 
” and “the observable 
.” Leave no ambiguity about which object you have in mind.
, 
denote the measurement outcome.
denote the Pauli-
operator of qubit 
:
, encoding directionality.
-axis
, which means 
to physicists. Use 
instead, unless you’re writing for engineers (who denote 
(\ell) instead.
, –, 
, or 
. Find the symbol immediately rightward of the initial = sign. Begin the new line directly below that symbol.
in the example under (A). Put the corresponding 
means “approximately equals”; and ~, “scales as.” Approximations convey more information than scaling relations do.
For example, if 
is odd, then 
and 
fit the template.
isn’t a matrix; a matrix represents an operator in terms of a particular basis. Therefore, no equals sign should interrelate an 
of real numbers
and 
the number of primes 
is close to
is the number of primes 
the recently proved 
and 
If the Riemann hypothesis is true, any 
will do.
But in 1914, Littlewood shocked the world by sketching a proof that 
changes sign infinitely often!
He gave his student Stanley Skewes the task of figuring that out.
Their paper includes many fascinating graphs—but the printing is of very poor quality, unpleasant to look at. I think the world deserves better pictures of their amazing result, so yesterday I decided to redo their work with the help of AI.
could have a big effect. However, a smoothed version of 
for some 
What we can compute is an approximation coming from Riemann’s explicit formula, which expresses 
of the zeta function. Assume the Riemann hypothesis, so 
is real. Write 
and measure 
:![\begin{array}{ccl} D(u)\! &:= & \bigl(\pi(x) - \mathrm{li}(x)\bigr)\,\frac{\log x}{\sqrt{x}} \\ \\ &\approx & \!\!-\Bigl(1 + \frac{2}{u} + \frac{8}{u^2}\Bigr) - 2\sum_{0 <\gamma\le T} \mathrm{Re}\!\left[\frac{e^{i\gamma u}}{\rho}\Bigl(1 + \frac{1}{\rho u}\Bigr)\right] . \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bccl%7D++D%28u%29%5C%21+%26%3A%3D+%26+%5Cbigl%28%5Cpi%28x%29+-+%5Cmathrm%7Bli%7D%28x%29%5Cbigr%29%5C%2C%5Cfrac%7B%5Clog+x%7D%7B%5Csqrt%7Bx%7D%7D+%5C%5C+%5C%5C++%26%5Capprox+%26+%5C%21%5C%21-%5CBigl%281+%2B+%5Cfrac%7B2%7D%7Bu%7D+%2B+%5Cfrac%7B8%7D%7Bu%5E2%7D%5CBigr%29++-+2%5Csum_%7B0+%3C%5Cgamma%5Cle+T%7D+%5Cmathrm%7BRe%7D%5C%21%5Cleft%5B%5Cfrac%7Be%5E%7Bi%5Cgamma+u%7D%7D%7B%5Crho%7D%5CBigl%281+%2B+%5Cfrac%7B1%7D%7B%5Crho+u%7D%5CBigr%29%5Cright%5D+.++%5Cend%7Barray%7D++&bg=ffffff&fg=333333&s=0&c=20201002)
Higher powers of primes give extra corrections that are negligible when 
with frequency 
These oscillations are the key: 
means 
and the vertical axis is in units of 
which is about 10155 near x = 10316. Each graph shows two curves:
in these units, assuming the omitted terms behave like random noise.
with 
This is a Gaussian average of 
which is the quantity Lehman’s trick from 1966 controls rigorously.
operations. Instead the program uses a nonuniform fast Fourier transform from the 
and it computes each phase 
modulo 
carefully, because 
for 
usually hovers around 
but throughout this window the low zeros happen to be nearly in phase, lifting it close to zero. Even so it is a narrow escape: the blue curve rises only about 0.017 above zero. The red curve peaks at about 
which is what lets Lehman’s theorem prove that 
In this graph the blue curve rises only about 0.0018 above zero at that spike. That is no larger than the error from using only two million zeros, so this graph alone could not prove the spike is positive. Their proof used about 5 × 108 zeros. The red curve, smoothed too heavily to see so narrow a spike, stays below zero there.
For 
it says
Proofs are in the books of Edwards and Ingham. Möbius inversion gives
and pairing each zero with its complex conjugate gives the formula for 
and higher roots, the constants, and the zero sum belonging to 
Assuming the Riemann hypothesis, they are smaller than the terms kept by factors like 
which is about 
near 
The error in expanding each 
is of order 
negligible even when summed over all zeros. This step too can be made rigorous with explicit remainder bounds.
give an error of order 
Making that smaller than 
would need 
around 
here, while the 2,001,052nd zero only has height 
which is the setting Rubinstein and Sarnak use. A Parseval identity for such functions should then give the long-run mean square of the omitted tail as about 
That is a statement about averages over very long stretches of 
It does not bound the error at any particular point, or even the average error across our window. But we can say 
So the blue curve is just a good guess, not something proved.
centred at some point 
The theorem says this average equals 
minus a sum over the zeros, with each zero damped by a factor 
plus an error term with explicit bounds that hold for every 
then 
near 
so stopping at height 
for all 
So between 1019 and about 1.397 × 10316 the question remains open!
Acta Arithmetica 11 (1966): 397–410.
complex zeros on 
Mathematics of Computation 80, no. 276 (2011). 
II. Proceedings of the London Mathematical Society, 3rd ser., 5, no. 1 (1955): 48–70. 
with 
and 
are distinct. Equivalently, all the differences 
, 
of a Sidon set in 
. Several algebraic constructions of Sidon sets (by Singer, Bose, and Ruzsa) are known with 
and special choices of 
; see 
, with the bound 
also proposed by Erdős as “perhaps too optimistic”.
denotes the reflection of 
to be chosen later, we conclude that 
with a further weight 
. This gives an inequality 
appropriately. For instance if one selects ![\displaystyle w := 1_{[1, k^{3/2}]}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++w+%3A%3D+1_%7B%5B1%2C+k%5E%7B3%2F2%7D%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle t := 1_{[-k^{3/2}, N]}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++t+%3A%3D+1_%7B%5B-k%5E%7B3%2F2%7D%2C+N%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, and the above bound becomes (after ignoring some lower order terms) 
with more complicated cutoffs, which obey some biconvex constraints which one can explore using various optimization algorithms. It turns out that one can do even better by working not with deterministic choices of 
are only true after performing an expectation. The calculations get quite messy (and highly computer-assisted), but eventually give 
-Golomb rulers, in which a given non-zero difference is permitted to appear up to 
to be a sequence 
of independent events that leads to human extinction, where 
has probability 
. (For simplicity I’m assuming that 
doesn’t depend on 
. If we now have a set of independent scenarios 
, then the probability that none of them occurs is 
. So if that is small, which roughly speaking holds if the sum of the 
is significantly larger than 1, then extinction is almost certain.
and 
, then my 
is basically 100% and unlikely to change, my 
is medium, and subject to change (probably in an upward direction unfortunately), and my 
is where the main uncertainty lies for me. Soares and Yudkowsky argue that 
is close to 1. 
must be accounted for by work 
and heat 
,
, we also change how we split that energy change into 
and 
. For each of the three choices, however, the definitions are constructed so that work and heat add up to the total change in internal energy. All three therefore satisfy the first law.
. The difference between the actual work and this ideal minimum,
where 
:
has a 10-dimensional representation, the ‘vector’ representation 
, and also two 16-dimensional representations, the ‘left and right-handed spinor’ representations 
There’s an isomorphism of representations
but we might as well work with 
Here 
means the kth exterior power. For any vector space 
we have
for larger 
In fact this isomorphism is part of a pattern! The next one involves the vector and left-handed spinor representations of 
But it’s this:
while 14 is the dimension of 
is a subgroup of 
because 
were isomorphic to the 6th exterior power of the adjoint representation of 
with
to be a perfect square!
is an integer when 
are the Fibonacci numbers 0, 1, 1, 2, 3, … and 
are the 
with 
Apparently this happens iff 
I won’t think about this now… but this is the last hard step.
case had been proved by Euler over 200 years earlier, I decided that that would be a good place to start: once I had sorted that out, I was optimistic that I would be ready to tackle the general case.
had no solution. Rather, I just worked out some successive differences and stared at them, trying to get some idea of why none of them was a perfect cube. (I should be clear that this story is a reconstruction of what I think probably happened given the few memory traces that remain half a century later rather than a completely reliable account.) At some point, I had the idea of taking the difference sequence of the difference sequence, and discovered that it formed an arithmetic progression. That felt like progress, so I investigated difference sequences a bit more and discovered, purely empirically, the rule that if you start with 
.
(where there are 
there exists an integer 
such that 
be two 
–isomorphism if there exists 
such that 
for every 
. By rescaling we can always take 
to be 1, in which case we have that 
for every 
, then this tells us that 
between 
from 
. (The idea of the proof is as follows: pick inside the unit ball of 
for any two 
and 
. The identity map between these two spaces is an 
Hadamard matrix, then the corresponding linear map is a ![p,q\in[1,\infty]](https://s0.wp.com/latex.php?latex=p%2Cq%5Cin%5B1%2C%5Cinfty%5D&bg=ffffff&fg=333333&s=0&c=20201002)
the Banach-Mazur distance between 
and 
is 
. It is also easy to show that 
, so 
-spaces hardly improve on the easy lower bound, and do not improve on it at all in dimensions 
,
for the set of all 
.) In words, there is a positive constant 
such that for every positive integer 
.
for the set of primes. And yet, Vinogradov’s result is unquestionably a theorem, while Gluskin’s result is unquestionably a counterexample, or at least an example.
, whereas for Gluskin’s theorem the statement to be proved is
and 
,
.
can be converted into an existentially quantifed statement 
. (For this to be an equivalence one needs the axiom of choice, but it is certainly a sufficient condition.) This is not just a piece of logical trickery, but it often reflects quite accurately how we think about some problems. For instance, it is more natural to think of Gluskin’s example as a recipe for constructing (or at least proving the existence of) a pair of suitable normed spaces for any given dimension 
to pairs of normed spaces by giving its value at each 
does not belong to the convex hull of a certain compact set 
and a 
such that 
and 
for every 
. In both these cases it feels natural to regard the result as a theorem that is proved via an existential statement, perhaps because it is the theorem that is ultimately what interests us. But using “what interests us” as a criterion to determine what counts as a counterexample seems a little vague, and is a difficult criterion to use if we want to explain convincingly why AI should be good at finding counterexamples.
, then I may well look for a property 
that follows from 
, it is often the case that the main question in our minds is less, “Why is 
is a counterexample to the universally quantified statement 
. However, we do not describe all existential statements as counterexamples. For example, if I were to say, “The 
are all separable, as is 
, but 
is not separable,” I would not describe the second part of that assertion as a counterexample to the claim that all Banach spaces are separable. Rather, I would present it as probably the most basic example of a non-separable space. The important point seems to be that there was no particular reason to think that all Banach spaces would be separable, and finding an example of a non-separable space is not very difficult.
of them, but that bound can be improved if instead you observe that a complete 5-partite graph can be written as a union of two triangle-free subgraphs, and therefore it is possible to write the complete graph as a union of 
triangle-free graphs. It is then tempting to try to do better, with triangle-free graphs that are less dense but that make up for it with unbounded chromatic number — a necessary condition if one wishes to use a sublogarithmic number of graphs, which is equivalent to showing a superexponential lower bound for 
. All this is to say that when I worked on the problem, my efforts were concentrated on what turned out to be the right direction, so for me OpenAI found an example of what I (weakly) expected, rather than a counterexample.
that is small enough for later arguments to work. But when we search for an example of an object 
), it is often not a good strategy to specify 
. If we are lucky we can describe in a nice way a very general class of objects 
and show that 
satisfies 
of a certain type. The problem is then reduced to finding 
, it can be surprisingly helpful to start by attempting to prove the statement 
. The reason this can be helpful is that using our standard methods of attempting to prove something, we may end up identifying a key lemma that would suffice: that is, we may find an intermediate property 
in a non-trivial way and thus reduce the problem 
. Turning things round again, it may well then be that finding a counterexample to 
not because we think it has a chance of working (if we did, then we would be using the first strategy), but because we hope that if 
that does not have that defect. Again, this strategy can either be iterated or combined with one or more of the other strategies.
, each of which is, individually, quite easy to satisfy. In such situations, we often “build” 
.