For a pragmatic, technological species such as ours, reality constantly proves to be mathematically inconvenient. We therefore approximate, sample, digitize, and compute to analyze a universe that is not really parceled out in such easy-to-handle little bits (at least at the scales of everyday life). Such expediencies of analysis underlie everything from forecasting the weather to transmitting cell phone calls.
Just because we routinely transform reality into somewhat choppy models, that doesn’t mean we want to abandon a sense of natural or mathematical fidelity. In his work with partial differential equations (PDEs) and with harmonic analysis, Francesco Di Plinio, newly arrived Tamarkin assistant professor of mathematics, strives to provide the necessary mathematical grounding for such modeling efforts.
“One staple of my work in PDE is to provide theoretical justification to numerical models,” Di Plinio said.
For example, consider the problem of modeling the long-term changing movement, salinity, and temperature within a large patch of ocean, maybe thousands of kilometers on a side. In a rectangular or triangular patch of those dimensions, the lengths and widths would so vastly exceed the depths, that it could be expedient to model it as two-dimensional. And while it seems unnatural to consider an arbitrary “patch” of ocean, computation requires finding a way to work with this arbitrarily distinct entity, even while acknowledging that it’s really part of a larger system.
“You would like that this introduction of this artificial boundary does not change the dynamics of the patch,” Di Plinio said. “Otherwise you simulate a system that does not correspond to a real one.”
Based on his studies of the underlying mathematics of the fluid dynamics involved – some rather complicated PDEs – Di Plinio has published two papers that provide theoretical justifications for how the ocean can be modeled this way. The papers aren’t meant for weather forecasters or climate modelers – they are highly theoretical works – but they can still help ensure that the tools such folks employ have a solid basis.
Now consider a signal such as the one of interest in your telephone. The device and its network purport to transmit your natural voice, in some pragmatic approximation, to whomever you call. Your voice is a sound wave that, in theory, can be broken up into simpler waves that, if added back together, would reconstitute your original voice. In the digital age, we really just sample the voice at some interval and then we approximate the original wave further with trains of 1’s and 0’s. In other words, we take a long, smooth string and cut it to segments and then turn those segments into bits. In his research on harmonic analysis and “time-frequency analysis” Di Plinio again works to explore what theory really allows for in terms of approximating mathematical functions such as sound waves and their constituent frequencies.
Di Plinio knows a thing or two about creating numerical models for science and engineering applications because that’s what he studied in college in Milan, Italy. But he found that he loved the pure mathematics he encountered more than the applications and so pursued a doctorate in the field at Indiana University.
At Brown, he’s finding the best of both those worlds, plus one more. He said he’s proud to join such strong groups as the Department of Mathematics and the Division of Applied Mathematics. But then there’s also the Institute for Computational and Experimental Research in Mathematics (ICERM), where math and computation meet.
“It provides us with additional motivation and exposure to different kinds of mathematics,” Di Plinio said. “The presence of these three strong mathematical poles is what made me motivated.”
It’s not easy for Di Plinio and his young family to remain away from Italy, but his Tamarkin professorship is too great an opportunity.
“For me it’s a dream come true,” he said. “It’s worth it.”