In 1966 Mark Kac asked a question that sounds like a riddle: can one hear the shape of a drum? A drum’s membrane can only vibrate at certain frequencies, and those frequencies are fixed by its shape. Hear all of them, every overtone, perfectly, and you hold a complete list of numbers that the shape produced. Is the list enough to give the shape back?
For twenty-six years nobody knew. Then in 1992 Carolyn Gordon, David Webb and Scott Wolpert showed that the answer is no. The two drums above are theirs. Each is assembled from seven identical triangles, arranged so that every vibration of one can be cut apart and reassembled, triangle by triangle, into a vibration of the other with exactly the same frequency. They have the same infinite list of tones. They are not the same drum.
Nothing here is quoted from a table. The frequencies come from a finite-element computation on a mesh of 29,057 points per drum, and the two lists of eigenvalues agree to fifteen digits. What you hear is synthesised from the first twenty-four of them, every tone weighted in the same way for both drums. Open the curtain and step through the modes: the numbers match exactly; the pictures never do.
I have spent my working life in two rooms that rarely talk to each other. In one I run a systematic investing business, which is a polite way of saying that I spend my days deciding which numbers to believe. In the other I do mathematics. I trained as a mathematician at Cornell, on the foundations of superstring theory, and I never stopped, moving between geometry, topology and physics. People ask how the two rooms connect. For me it is the same impulse in both: an idea takes hold, and I want to build something with it, whether a theory, a proof, or an investment process that has to survive contact with markets. Sometimes what I build is useful, sometimes it is beautiful, and once in a while it is both.
Build long enough in two places and you begin to notice things you never set out to find. Looking back across my papers, I keep finding the same drum. Somewhere there is a number everyone trusts: a sound, a track record, an entropy curve, a count. Two different things produce it. And each time there is a missing piece of information that no amount of listening to the number will recover, because it is not stored in any one place. You only see it when you look at the whole object at once.
What follows are five of those drums. At the end of each I write down what the sound left out.
I Two portfolios with the same past
A track record is the sound a strategy makes. It is the one thing an investor can hear clearly: a daily list of returns, compounded into a line that goes up or doesn’t. The temptation, always, is to read the strategy off the line.
Below are two portfolios of the same two hundred stocks. For a hundred days their returns agree to the last digit a computer can represent. Not approximately: exactly. Now look at what they hold.
Holdings of the first portfolio, stock by stock
Holdings of the second portfolio
Simulated returns from a three-factor model with stock-specific noise. The second portfolio is the first plus a long-short position chosen to be invisible to all hundred past days; each draw picks a different invisible position.
The trick is old linear algebra. A hundred days of returns on two hundred stocks are a hundred equations in two hundred unknowns, so whole directions of holdings are invisible to the past: you can add any of them to a portfolio without changing a single historical day. The record pins down a hundred numbers and leaves a hundred free. Run the future and it begins asking about exactly those hundred.
Real markets are subtler. There are more days than stocks, and the blind directions are blurred by noise rather than exactly invisible. But the lesson survives the subtlety, and much of my work in asset pricing is a precise version of it. A model that earns is not necessarily a model that explains. Performance and identification are governed by the same covariance spectrum, and they pull in different directions: a high-Sharpe stochastic discount factor need not identify which risks are priced, and the part of the cross-section that pays is so often the part nobody can name.
The sound: the track record. What it leaves out: the mechanism, meaning which risks are actually being paid.
For the mathematician
Let R ∈ ℝT×N be the in-sample return matrix with T = 100 < N = 200. For any v ∈ ker R we have R(w + v) = Rw, so every day of the history is unchanged. Here v is a random vector projected onto ker R ∩ 1⊥ (modified Gram–Schmidt with reorthogonalisation, in double precision), so both portfolios stay fully invested.
When T > N the kernel is trivial, but the small singular values of R play the same role statistically. The papers make the trade-off exact: performance and identification are complementary functionals of one spectrum, and the alpha tends to sit in the statistically unnameable bulk, at a floor set by statistical power rather than by anyone’s skill.
II Ten thousand brilliant ideas
Now let ideas be cheap. A language model will propose ten thousand trading strategies before lunch, and a backtester will score all of them by dinner. Each idea makes a sound, its backtest, and we keep the one that sounds best.
Every strategy below is pure noise: daily returns whose true average is exactly zero. Move the slider and test more of them.
Each strategy: 504 trading days of independent daily returns with mean zero and 1% volatility, then 252 fresh days. Faint lines are a sample of the strategies tested; the dark line is the winner. After the backtest ends, the faint red lines are the next year of the twenty best, and the dark line is their average. Any single winner can get lucky again; the average of the top twenty cannot.
The best backtest improves as more ideas are tried, growing roughly like the square root of the logarithm of their number, because that is how the maximum of many coin-flip records grows. It looks like discovery. It is a measurement of effort. Then the following year arrives, and the winners do no better than any other coin.
When hypotheses are free, generating them is no longer the scarce input. Validating them is. My paper The Validation Bottleneck works out what that does to discovery, including a twist specific to language models: a model has read everything published before its training cutoff, so any test on data it may have seen is contaminated, and clean evidence lives only on the far side of that wall. The paper’s two experiments are pre-registered, which is the oldest remedy there is: write down in advance how many things you will try.
The sound: the best backtest. What it leaves out: how many backtests were run to find it.
For the mathematician
If S1, …, SN are annualised Sharpe ratios of independent zero-mean strategies observed for T years, each is approximately normal with variance 1/T, so the expected maximum is about √(2 ln N ⁄ T). Here T = 2. The out-of-sample year is drawn independently, so its expected Sharpe ratio is zero whatever the in-sample winner looked like.
III Same radiation, different fate
Black holes are the most unforgiving drums in physics. In 1974 Hawking showed that they glow, faintly, and slowly evaporate. The glow looks thermal, which is to say featureless, and that seemed to mean that whatever fell in, a diary or a star, is erased. If quantum mechanics is right, that cannot happen.
In 1993 Don Page described what an information-preserving black hole would have to look like from outside. Track the entanglement entropy of the radiation: it rises as the hole evaporates, peaks at the Page time, when about half is gone, and falls back to zero. In 2019 semiclassical gravity was shown to reproduce this curve, through islands and replica wormholes, and the curve became the emblem of the paradox’s resolution.
But the Page curve is a sound.
Both worlds look exactly like this from outside.
Information preserved Input erased
Schematic. Top: Page’s average radiation entropy for 40 qubits, the same in both worlds. Bottom: what an observer with all the radiation can learn about a two-qubit diary thrown in early. In the preserving world it comes back shortly after the Page time; in the erasing world it never does.
In the third of my papers on the subject I constructed exact finite examples of two processes that produce the same radiation trajectory for a given preparation. One is an isometry: nothing is lost, and after the Page time the diary can be read back out of the radiation. The other quietly erases its input. Their entropy curves are indistinguishable. The difference lives entirely in the correlations between the radiation and what fell in, and only the full channel, reconstructed by complete tomography, tells them apart.
The sound: the Page curve. What it leaves out: the correlation between the radiation and what fell in.
For the mathematician
From The Page Curve Is Not Enough: two ensembles can share every averaged spectral moment yet have different entropy fluctuations, and an isometry and an input-erasing channel can reproduce the entire radiation trajectory for one preparation while preserving different reference correlations. Full Choi-state tomography gives the positive result: it identifies the channel.
The plot uses Page’s formula S(k) ≈ k − 22k−n⁄(2 ln 2) for k ≤ n⁄2 radiated qubits out of n = 40, and a Hayden–Preskill-style recovery of the diary after the Page time in the isometric world. It illustrates the shape of the statement, not the paper’s specific examples.
IV Every corner real
Here is the picture that, for me, holds this whole page together.
Drag to turn it. Arrow keys work too.
Each corner of this triangle is honest geometry. You could build it from wood. In fact you can build the whole thing: turn it and the trick appears, three straight beams that close only from one direction. Oscar Reutersvärd drew it from cubes in 1934, and Roger Penrose rediscovered it two decades later. Then, in 1991, Penrose did something stranger. He showed that the impossibility is a cohomology class. Every piece of the figure is consistent on its own; the inconsistency lives in how the pieces fit together around the loop, and it is measured by a class that vanishes exactly when the figure could exist.
String theory asks that the six unseen dimensions of the universe be curled into a Calabi–Yau threefold, and the mathematics of those spaces was the subject of my doctorate. Many natural examples are singular: there are points where the space pinches, like the tip of a double cone. The question is whether the pinches can be smoothed away. At a single point the answer is local, and usually yes. Whether the whole space can be smoothed at once is a different question.
Each node contributes an arrow, the class of its small curve. Smoothing a node gives its arrow a nonzero weight. Smoothing all of them at once needs weights under which the arrows, placed tip to tail, return to the start.
Robert Friedman found the rule for the simplest pinches, called nodes, in 1986. Each node comes with a small curve, and each curve has a homology class, which you can think of as an arrow. Smoothing every node at once requires a relation among the arrows: weights, all nonzero, under which they close into a loop. A node whose arrow has no partner can be smoothed on its own and still never be smoothed together with the others. Every corner real; the whole impossible.
My Calabi–Yau papers carry this rule beyond nodes, to cone points over del Pezzo surfaces, where the condition becomes that the relation must avoid every local discriminant. The sharpest example is a threefold with twenty-six nodes and four cone points, every one of which can be smoothed on its own; the global relation rules out a smoothing of the whole over a formal disc. In the most recent paper the relations turn out to have a mirror image. On the mirror Calabi–Yau the small curves become collapsing three-spheres, and the spheres obey exactly the same relations: a local case of a prediction David Morrison made in 1999.
The sound: each singularity, examined one at a time. What it leaves out: the relation among all of them.
For the mathematician
Friedman (1986): let X be a Calabi–Yau threefold with nodes pi and a small resolution X̂ with exceptional curves Ci. A smoothing exists exactly when there are λi, all nonzero, with Σ λi[Ci] = 0 in H2(X̂). The arrows above live in a two-dimensional lattice for the sake of the drawing; in the theorem the lattice is H2 of the resolution.
Paper 5 gives a necessary and sufficient criterion for nodes and exact anticanonical cone points over del Pezzo surfaces of degrees 5, 6 and 7 and over ℙ¹×ℙ¹: a homological relation must avoid every local discriminant. Paper 4 rules out a smoothing over a formal disc of the 26-node, four-cone mirror member from Paper 3. Paper 6 proves, near the maximally degenerate limit and locally in the coefficient space of the mirror family, that the exceptional curves and the mirror vanishing three-spheres have the same integral relation lattice.
Penrose’s paper is On the cohomology of impossible figures, Structural Topology 17 (1991), reprinted in Leonardo 25 (1992). The tribar defines a nonzero class in H¹ with coefficients in the positive reals: the ratios of depths on the overlaps.
Explore further in the Conifold Transitions explorer.
V One knot, two cuts
Knot theory has its own version of hearing the shape of a drum. There are two very different ways to count solutions attached to a knot. One counts instantons, solutions of the four-dimensional Yang–Mills equations, the physics of the strong force turned into geometry. The other cuts the knot in two and works in a small space called the pillowcase, a sphere with four corners shaped like a pillow, on which each half of the knot draws a curve. The crossings of the two curves are the candidates, and a Floer complex sorts out which of them survive. A conjecture of Hedden, Herald and Kirk, part of the Atiyah–Floer program, says the two counts agree.
For the torus knots T(3,n) I computed the pillowcase side for every n and found that it matches the instanton side in every grading, including the cases where the complex has a nonzero differential, which is where agreement has teeth.
Then a surprise. The knot 819 is the torus knot T(3,4). Cut it a different way and, without extra data, the numerical computation suggests that the pillowcase answer changes. Same knot, two cuts, two answers, unless the curve carries a bounding cochain: a piece of algebraic data that repairs it. In the examples I have computed, it amounts to smoothing the curve at a single self-intersection.
Schematic, not the actual curves of 819. The blue curve comes from one half of the knot, the red from the other; each crossing is a generator. In this cartoon the instanton side has three.
The sound: the crossings of two curves in the pillowcase. What it leaves out: a bounding cochain.
For the mathematician
Pillowcase Floer homology of the torus knots T(3,n): for small perturbations in an explicit range the complex has exactly the generators predicted by the instanton side; depending on n modulo 6 it has either no differential or exactly one; combined with a result of Daemi and Scaduto, the pillowcase complex is chain equivalent to the instanton complex over the field with two elements. A numerical computation suggests that, without a bounding cochain, the answer for 819 depends on the decomposition.
Maurer–Cartan elements in the pillowcase: for Smith’s decomposition of the pretzel knots P(−2,3,q), explicit Maurer–Cartan deformations, each a smoothing of the curve at a single self-intersection, give the correct answer for P(−2,3,5) and P(−2,3,7), conditionally on the conjecture of Cazassus, Herald, Kirk and Kotelskiy.
Explore further in the Pillow Case explorer.
Coda What the sound leaves out
| The sound | What it leaves out |
|---|---|
| A drum’s frequencies | its shape |
| A track record | which risks are being paid |
| The best backtest | how many were tried |
| The Page curve | the correlation with what fell in |
| Each singularity alone | the relation among all of them |
| Crossings in the pillowcase | a bounding cochain |
Read down the right-hand column and the entries begin to resemble one another. None of them is a bigger number or a finer measurement. Each is information about how pieces fit together: which holdings combine into which exposures, which trials belong to one search, which correlations bind the radiation to its source, which relation ties the curves, which cochain glues the two halves of a knot. Penrose’s triangle is the cleanest picture of the idea: data that is invisible on every piece and unavoidable on the whole. Mathematicians call such things gluing data, or cocycles. I have come to think of them as the part of the truth that is never local.
That is why the same ideas keep working in both rooms. A portfolio manager and a geometer are both handed a sound and asked for a shape. The discipline is the same in both: to know which questions the sound has already answered, and which it never will.
Not all of my work is about the sound failing. Sometimes the job is the reverse, to prove that a sound is enough. The Atiyah–Floer program compares two theories, instantons and holomorphic curves, and a theorem of Daemi, Fukaya and Lipyanskiy says that they produce the same groups. In two long manuscripts I argue that the agreement goes further: it respects every four-dimensional cobordism, first with coefficients modulo two and then over the integers, up to sign. That is a claim of enough, and claims of enough are a different kind of animal. A counterexample checks itself; anyone can compute it. A universal statement asks other people to trust a hundred and fifty pages. Those manuscripts have not yet been refereed, and I say so on the papers page.
I work fast, and I hunt. I build things, I get some of them wrong, I fix them, and I would rather pay for a mistake than for timidity. What I have learned is where that temperament is safe: in the search for the drum that sounds the same and is not. There the example is its own proof. When I claim that something is enough, I want other eyes on it.
Understanding is not hearing the sound more precisely. It is learning what else to listen for.
The comparison of cobordism maps has its own guided tour in the Atiyah–Floer Cobordism explorer.