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By Dave DeFusco

Katz School researchers—Ruslan Gokhman, a student in the Ph.D. in Mathematics and 2024 graduate of the M.S. in Artificial Intelligence, and Dr. Marian Gidea, a professor of mathematical sciences—presented a new way to detect financial bubbles before they pop at the 2025 SIAM Conference on Financial Mathematics and Engineering in July in Miami.

Their secret weapon? A combination of wavelets and a cutting-edge mathematical tool with a name straight out of science fiction: Topological Data Analysis, or TDA. If you’ve ever watched the stock market soar, only to see it come crashing down days or weeks later, you’ve witnessed a financial bubble in action. These bubbles can be “positive,” where prices surge unsustainably before collapsing, or “negative,” where they plummet sharply and then snap back.

Economists have long struggled to find reliable early warning signals—patterns that hint at a looming crash or sudden rebound. “Most traditional tools either miss these turning points or pick up too much noise,” said Gokhman. “We wanted something that could see deeper patterns—not just in price, but in how the market is behaving over time.”

That’s where TDA comes in. At its heart, TDA is about shape, specifically the shape of data. Rather than looking at stock prices as a flat line moving up and down, TDA treats data like a 3D landscape, mapping out the geometry of how values change over time.

One of its key tools, persistent homology, examines the number and “visibility” of holes or loops in that shape—features that can reflect repeating patterns, sudden shifts or structural changes in the market. These geometric features show up in something called a persistence diagram, and they can signal that the market is heading for turbulence.

To get this geometric insight, researchers first need to turn a one-dimensional time series, like a stock price chart, into a cloud of points in a higher-dimensional space. That’s done using delay-coordinate embedding, where you take windows of past prices and look at them like frames in a movie. Here’s where it gets tricky: the size of that window matters—a lot.

“If the window is too small, you pick up a lot of noise,” said Gokhman. “Too large, and you lose the important features because everything gets smoothed out. You need a Goldilocks window—just right.”

To find that “just right” window size, Gokhman and Gidea turned to another powerful tool: the wavelet transform. Think of wavelets as tiny waves you can stretch and shift to match different patterns in your data—kind of like tuning a radio to hear signals buried in static.

Wavelet analysis lets researchers zoom in on dominant frequencies in the market’s behavior and see how those patterns change over time. Using a specific type of wavelet—the Morlet wavelet with fine-tuned settings—they created something called a scaleogram, a heatmap showing where and when key frequencies appear in the market data.

By combining this wavelet decomposition with TDA, they created a method to automatically pick the optimal sliding window for the analysis, meaning they could consistently identify the clearest, most meaningful geometric features.

To test their approach, the researchers ran it on historical data from the S&P 500 from late 2013 to late 2018 and Bitcoin from spring 2021, right before a crash. As each market approached a bubble and began to wobble, the TDA method produced sharp spikes in what’s called the “persistence landscape norm,” a numerical measure of the data’s geometric complexity. These spikes reliably appeared before the actual crash or rebound, offering a kind of mathematical early warning system.

“This approach allows us to detect changes that are invisible to conventional time-series tools,” said Gidea. “It reveals deep structural shifts in the data—signals that a critical transition is coming.”

Financial markets are noisy, chaotic systems influenced by everything from corporate earnings to global politics. But beneath that chaos, there may be patterns—hidden structures that can be revealed through geometry.

What makes Gokhman and Gidea’s work so significant is its ability to measure these hidden structures and tie them to real-world events like market crashes. Their method doesn’t rely on assumptions about how markets “should” behave; it finds signs that something big is about to change. Perhaps, most important, it helps answer a key question: not just whether a crash is coming, but when.

“The real challenge in applying TDA to time-series data has always been the choice of window size,” said Gokhman. “Our contribution was to solve that problem by using wavelets, which gave us a dynamic, adaptive way to analyze markets that change over time.”