A jazz lesson

In the 1990s, when I was a student at the Rochester Institute of Technology, one of my professors was a computer scientist and working jazz trumpeter named Al Biles. He had built GenJam, short for Genetic Jammer, a genetic algorithm that learned to improvise jazz as Al rated its ideas good or bad, and in time the two of them traded fours on stage. The machine brought tireless invention and Al brought the ear. That partnership is what inspired me: a glimpse of what people and machines could do together. I am grateful to Al to this day. He let me build my graduation work around independent research on neural networks for market time series, encouraged me to take on more of it than the usual load, and did what it took to make that possible. The question I kept coming back to was a musical one: is the market playing different tunes? Three decades on, that question sits at the center of what we build.

The drum and the strike

In 1966 the mathematician Mark Kac asked a famous question: “Can One Hear the Shape of a Drum?” The shape of a drumhead fixes the tones it can make. Kac asked whether the tones, in turn, fix the shape: whether a list of numbers can give back a geometry.

The image holds two ideas. The first is shape: much of what we call understanding, in brains and in machines, turns out to be the discovery of a geometry, which things lie close together and where a space nearly splits in two. The second is the strike: a drum says nothing until something hits it, and the world reaches us the same way, as events arriving at irregular moments. Put together, they make the thesis of this essay: true intelligence lives in geometric structure and in the resolution of event-driven dynamics.

That word, resolution, carries three meanings here: how finely you can see events in time; how cleanly you can find the directions along which a complicated motion comes apart into simple pieces, which mathematicians call spectral resolution; and the moment uncertainty settles into an answer. These are one process seen from three sides.

Understanding is geometry

In 1872 Felix Klein proposed defining a geometry by its symmetries: what stays the same when you slide a figure, rotate it or change your perspective. The idea doubles as a definition of generalization. To know which changes leave something unchanged is to know which differences do not matter, and that is most of what it takes to recognize the same thing twice. A transposed melody is still the same tune, which is why GenJam could develop its ideas with such moves. Much of modern deep learning is now organized around the same principle.

Brains got there first. In 2022 a team recording hundreds of grid cells in the rat brain at once found that their joint activity lives on the surface of a doughnut, a shape it keeps even during sleep. Machines find shapes too. Language models place the days of the week on circles, and in 2025 researchers at Anthropic showed a model counting the characters in a line along a curved path and sensing the end of the line by rotating one curve against another. What gets learned is a shape, and what gets computed is motion on it. A system with the right geometry can answer questions it was never trained on, because the answers are places it can reach.

Hearing shape

How do you find a geometry that nobody drew for you? One powerful answer comes from spectral geometry, which reads shape from a list of numbers. For a drum, those numbers give away its area, the length of its rim and even how many holes it has. The strict answer to Kac’s question turned out to be no: in 1992 three mathematicians built two differently shaped drums that sound exactly alike. But such twins are rare, and in practice the spectrum gives back most of the shape.

The same idea works on any network, from friendships to the states a system passes through, and there each number comes with a direction. The first direction that is not trivially constant, known as the Fiedler vector, gives every state a coordinate, and sorting the states by that coordinate splits the network where it most nearly comes apart; a narrow bottleneck always shows up this way. The next few directions add coordinates of their own, a map in which the network’s hidden shape becomes visible. The figure in the next section shows the idea on a small network.

Markets have this geometry too. The daily moves of a few hundred stocks form a cloud of points in a space with hundreds of dimensions, and the spectrum measures the main directions of that cloud. Estimated from a limited history, even pure noise stretches that cloud in many directions, by amounts that fall inside a band mathematicians worked out in 1967. In 1999 two groups of physicists found that most of the market’s directions are stretched no more than noise would stretch them. Only a few stand out: one for the market as a whole and a handful for groups of related companies. Hearing the shape of a market starts with knowing which directions are real.

Is the market playing different tunes?

Markets move, and this is where the question from my student days comes back. Geometry extends to motion. In 1931 Bernard Koopman showed that any dynamical system, however tangled, becomes linear if you change what you watch: follow measurements made on the system rather than the system itself, and in the right coordinates the motion comes apart into independent pieces, each simply rotating or fading at its own rate. Nearly eighty years later Peter Schmid’s dynamic mode decomposition made it possible to find those coordinates directly from measured data, in systems as different as turbulent flows and power grids.

For a system that hops between states, the same logic applies to its matrix of transition probabilities. The top value is always one, the long-run balance. Values just below one belong to slow directions of motion, and a gap after them says the system has that many regimes, regions of its state space where it lingers before switching. In musical terms, each regime is a tune, and the spectrum tells you how many tunes there are and how long each tends to play. The right-hand chart in the figure below sets a walker loose on that network: two tunes, and one slow mode for the switching between them.

Two neighborhoods, two tunes. Left: each state is colored by its coordinate on the network’s slowest mode, which separates the two neighborhoods, blue and red, without being told they exist; links inside a neighborhood take its color, and the only two links between them, the narrow bridge, are drawn in black. Right: the spectrum of a random walk on the same network. After the first value, exactly one (black), comes 0.998 (orange), the slow switching between neighborhoods, with a timescale of about 590 steps; the fast modes (gray) settle within 12 steps or fewer. The gap announces two regimes.

This is spectral resolution: finding the coordinates in which a dynamics comes apart, each piece with its own clock. A fast mode fades within a few steps, while a slow one dominates every forecast beyond a certain range. Knowing which modes exist, and which horizon each one governs, is knowing the shape of the dynamics.

The world arrives as events

The third meaning of resolution is temporal, and it is where many systems quietly throw information away. Neurons do not report on a schedule; they fire when something changes. In 1996 Simon Thorpe and his colleagues showed that people can tell whether a photograph flashed for twenty milliseconds contains an animal, with brain responses diverging about 150 milliseconds later, so fast that each neuron along the way has time to fire only once or twice. Event cameras copy the trick: each pixel speaks only when its brightness changes. Control engineers showed that measuring a system each time it moves by a set amount can beat measuring it on a fixed timer. The figure below shows why on a single price path.

Clock versus events. Twenty looks taken every 3 seconds (circles) put only 5 in the burst between the dashed lines; twenty looks taken each time the price moves 3 ticks (diamonds) put 19 there, where the information is.

Markets were among the first places this was understood. In the 1960s and 1970s Benoit Mandelbrot, Howard Taylor and Peter Clark proposed that prices move in trading time rather than clock time, because a busy hour holds far more market than a quiet one, and later work showed that counting trades instead of minutes brings returns close to the bell curve.

Events need geometry

Counting events is not the end of the story. When Laszlo Gillemot, Doyne Farmer and Fabrizio Lillo studied the New York and London stock exchanges, they found that heavy tails persist even on a trade clock. Farmer and colleagues had already pointed to the reason: the largest price changes come less from unusually large orders than from gaps in the order book next to the best price. How far an event moves the price depends on the shape of the book it strikes, so the clock itself has to know the geometry.

People think in events too, and in how events relate. We carve experience into episodes, starting a new one when our predictions fail, and in 1983 the computer scientist James Allen showed that two intervals can relate in exactly thirteen ways, from “before” and “meets” to “overlaps” and “during.” That is how traders talk: “The bids pulled just before the big order hit.” Whether the bids pulled before the order arrived, or because of it, is the difference between anticipation and reaction, and a one-second bar records both as the same numbers. Now that market data arrives with nanosecond timestamps, the sequence is the information.

What this means for how we read markets

At tradeInsightAI_, this way of thinking sits at the core of AMIE, our Agentic Market Inference Engine, and of Shannon, the panel people read on top of it. AMIE is built as layers of inference that rise toward a single synthesis, and it reads far more than prices: the structure of the market itself, from the order book to the way instruments move together, and unstructured sources such as news, social media chatter and SEC filings. The same two ideas carry over. A headline, a post or a filing arrives as an event, just like an order, and text has a geometry of its own, since language models place related stories close together. AMIE reads every source at the pace it actually moves rather than on a fixed clock, and it looks for the shape of the market’s state, its regimes and the boundaries between them, rather than stacking indicators. Shannon, named after the founder of information theory, separates what matters from noise and tells the story the way traders think about it, as episodes and how they relate. Above all, Shannon describes; it does not instruct. A map tells you where you are. What you do with that is your call.

What cannot be heard

The thesis has limits. Some shapes cannot be heard: twin drums exist, and the same noise band that lets a listener discount randomness also hides any real structure that falls inside it. And markets add a problem drums never have: traders learn. When enough of them discover a geometry, the geometry changes, so any map of a market is a hypothesis about tomorrow, to be redrawn and checked honestly against what happens. Geometry is not the whole of intelligence either. Goals and judgment matter, and a map never chooses a destination.

Coda

GenJam made music because a machine and a musician each did what they do best. Reading a market calls for the same partnership. The machine does the listening that no person could, to every order, headline and filing as it arrives, learning the market’s shape from what moves with what and its rhythm from the events themselves. The person brings what GenJam drew from Al: judgment, and the final call.

The question I first asked as Al’s student, whether the market is playing different tunes, has a sharper form now: can one hear the shape of a market? For drums, the answer to Kac’s question is not always. For markets, the answer that matters in practice is closer to yes, for a listener who attends to the right modes, at the moments the drum is struck, and who knows when it cannot hear any more.

Further reading

Al Biles, jazz trumpeter, Professor Emeritus at the Rochester Institute of Technology, and creator of GenJam. https://genjam.org/

John A. Biles, “GenJam: A Genetic Algorithm for Generating Jazz Solos”, Proceedings of the International Computer Music Conference, 1994. https://genjam.org/wp-content/uploads/2019/07/bilesicmc94.pdf

John A. Biles, “Interactive GenJam: Integrating Real-Time Performance with a Genetic Algorithm”, Proceedings of the International Computer Music Conference, 1998. https://genjam.org/wp-content/uploads/2019/07/bilesicmc98.pdf

Mark Kac, “Can One Hear the Shape of a Drum?”, The American Mathematical Monthly, 1966. https://doi.org/10.1080/00029890.1966.11970915

Carolyn Gordon, David Webb and Scott Wolpert, “One Cannot Hear the Shape of a Drum”, Bulletin of the American Mathematical Society, 1992. https://doi.org/10.1090/S0273-0979-1992-00289-6

Michael Bronstein, Joan Bruna, Taco Cohen and Petar Veličković, “Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges”, 2021. https://arxiv.org/abs/2104.13478

Richard Gardner and colleagues, “Toroidal Topology of Population Activity in Grid Cells”, Nature, 2022. https://doi.org/10.1038/s41586-021-04268-7

Joshua Engels and colleagues, “Not All Language Model Features Are One-Dimensionally Linear”, 2025. https://arxiv.org/abs/2405.14860

Wes Gurnee and colleagues, “When Models Manipulate Manifolds: The Geometry of a Counting Task”, Anthropic, 2025. https://transformer-circuits.pub/2025/linebreaks/index.html

B. O. Koopman, “Hamiltonian Systems and Transformation in Hilbert Space”, Proceedings of the National Academy of Sciences, 1931. https://doi.org/10.1073/pnas.17.5.315

Peter Schmid, “Dynamic Mode Decomposition of Numerical and Experimental Data”, Journal of Fluid Mechanics, 2010. https://doi.org/10.1017/S0022112010001217

Laurent Laloux, Pierre Cizeau, Jean-Philippe Bouchaud and Marc Potters, “Noise Dressing of Financial Correlation Matrices”, Physical Review Letters, 1999. https://doi.org/10.1103/PhysRevLett.83.1467

Simon Thorpe, Denis Fize and Catherine Marlot, “Speed of Processing in the Human Visual System”, Nature, 1996. https://doi.org/10.1038/381520a0

Karl Johan Åström and Bo Bernhardsson, “Comparison of Riemann and Lebesgue Sampling for First Order Stochastic Systems”, 2002. https://doi.org/10.1109/CDC.2002.1184824

Peter Clark, “A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices”, Econometrica, 1973. https://doi.org/10.2307/1913889

Thierry Ané and Hélyette Geman, “Order Flow, Transaction Clock, and Normality of Asset Returns”, The Journal of Finance, 2000. https://doi.org/10.1111/0022-1082.00286

J. Doyne Farmer and colleagues, “What Really Causes Large Price Changes?”, Quantitative Finance, 2004. https://doi.org/10.1080/14697680400008627

Laszlo Gillemot, J. Doyne Farmer and Fabrizio Lillo, “There’s More to Volatility than Volume”, Quantitative Finance, 2006. https://doi.org/10.1080/14697680600835688

Jeffrey Zacks and colleagues, “Event Perception: A Mind-Brain Perspective”, Psychological Bulletin, 2007. https://doi.org/10.1037/0033-2909.133.2.273

James Allen, “Maintaining Knowledge about Temporal Intervals”, Communications of the ACM, 1983. https://doi.org/10.1145/182.358434