This mathematics PhD student is focusing on answering just one equation

“Saying you’re pursuing a PhD in Mathematics would usually get people scratching their heads. Why would you need to get a PhD in that subject? Thomas Caussade , a PhD candidate in Mathematics at UCL , understands this sentiment. “People usually think that doing mathematics would mean it’s just you, your laptop, and your calculations against the world,” he shares. “That’s not true. It’s pretty interactive. While those in the field have different questions on different subjects to solve, we’ll come together to talk about what we’re doing.” It’s easy to forget just how much mathematics shapes the world around us. Behind every space mission, smartphone signal, and weather prediction lies an equation waiting to be solved. History even shows us how powerful mathematical thinking can be — Alan Turing, who completed a PhD in mathematical logic, used his work to crack the Enigma code and help end World War II. For Thomas, mathematics isn’t just about abstract numbers or symbols. It’s a language — one that helps decode the invisible forces of the universe. Caussade graduated with an undergraduate and Master’s degree in Mathematical Engineering from Pontificia Universidad Católica de Chile. Source: Thomas Caussade Solving the “unsolvable” with a PhD in Mathematics Waves are often pictured as ripples across a pond, or maybe even sound waves bouncing off the walls of concert halls. However, Thomas views it as a profound mathematical mystery. His PhD work dives into the equations that describe how waves move, scatter, and interact with objects. These equations, called partial differential equations (or PDEs for short), are at the heart of how we understand the physical world — from light bending around a corner to seismic waves travelling through the Earth. Now, here’s the catch: solving these equations exactly is rarely possible. They’re just too complex. So mathematicians like Thomas turn to numerical methods — clever ways of getting computers to approximate the answer. The computer doesn’t solve the equation in one go; it chops space into tiny pieces, does a lot of math on each piece, and stitches the results together to estimate how the wave behaves. The trouble is that when waves become more energetic — what scientists call “high frequency” — this process becomes messy. This is the NSD motivation. Caussade often uses this image to explain the focus of his PhD in Mathematics. Source: Thomas Caussade Higher frequency means smaller wavelengths, and smaller wavelengths mean you need way more points to capture all the details. Suddenly, what used to take a few seconds to compute can take weeks. That’s where the PhD in Mathematics candidate’s work comes in. He’s designing smarter, faster numerical methods that can handle these high-frequency waves without crashing your computer. Think of it like teaching the computer some shortcuts — mathematical tricks that let it predict what the wave is doing without calculating every tiny detail. To achieve this, he combines theory and computation: First, he writes down the equations that describe how a wave interacts with an obstacle. Then, he develops algorithms that efficiently approximate those equations. Finally, he tests them through simulations to assess their effectiveness. The payoff could be huge. Better methods for handling high-frequency waves could benefit fields ranging from acoustics to radar, seismology, and even medical imaging. In a sense, Thomas is helping mathematicians and scientists hear the world more clearly. Caussade is originally from Chile and moved so that he could pursue his PhD in Mathematics. Source: Thomas Caussade This is why what he’s doing matters. Waves power every Wi-Fi signal, radar pulse, or echo of sound — and understanding how those waves behave is no small feat. Caussade studies the equations that describe them, working to make computer simulations faster and more accurate. His goal? To help engineers test designs virtually before building anything in the real world. “Building an antenna and realising it doesn’t work is expensive,” Caussade explains. “But if you can simulate it on your laptop, you can run thousands of tests and find the best design before you even start.” At the core of his research is the Helmholtz equation, a classic model that describes how waves propagate and scatter. It has been known for more than a century, but solving it efficiently at high frequencies remains a significant challenge. As frequency increases, so does computational cost — what once took seconds can suddenly take days. Caussade’s work aims to change that. By developing numerical methods whose cost doesn’t skyrocket with frequency, he’s helping make simulations faster, more innovative, and more scalable. The results could transform everything from antenna design to medical imaging, demonstrating that even the oldest equations can still yield cutting-edge innovation.

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