This past Thursday, the Mathematics Department hosted a presentation by Professor Emilio Elizalde of the Autonomous University of Barcelona. The title of his talk was Mathematics and Cosmology: on the Universe acceleration and the zeta function as a regularization tool. Elizalde is the father of Sergi Elizalde, Dartmouth professor and specialist in combinatorics. The younger Elizalde was on hand Thursday to deliver the opening remarks.
Unlike his son, Emilio Elizalde is a physicist by training. And his talk focused specifically on the applications of mathematics towards cosmology. He began by reviewing, at some length, the history of the universe and its study. From Galileo to Newton to Einstein, physicists and astronomers have sought to explain nature through the language of mathematics. The senior Elizalde’s own cause is to explain the expansion of the universe in terms of mathematics.
When Einstein developed his equations for general relativity, he added a constant term to compensate for what he believed was a “static” universe. Remove the constant, Einstein noticed, and the force of gravity would result in a collapsing universe. To contemporary observers, the universe was in a static equilibrium, and this constant offered a mathematical solution for their world. But after Hubble empirically proved that the universe was expanding, and the notion of the “Big Bang” was formed, Einstein threw out his constant entirely. The universe was growing, and there was no need for this check to maintain static equilibrium.
In 1998, however, a new discovery was made. Not only was the universe expanding, it was expanding at an accelerating rate. The most commonly cited reason, of course, is the existence of dark energy. This conclusion has sparked renewed interest in Einstein’s cosmological constant.
Professor Elizalde’s work examines the cosmological constant and related problems through the lens of analytical mathematics. His focus is on the Riemman zeta-function, which is also the subject of the Riemann hypothesis, arguably the greatest unsolved problem in mathematics. By extrapolating the concept of the zeta function to the domain of so-called pseudodifferential operators, it can be used as a regularization tool in quantum field theory. This method yields, for example, the vacuum energy corresponding to a quantum physical system, which could contribute to the cosmic force leading to the present acceleration of the universe.
Although Professor Elizalde was constrained by time and forced to bypass much of the theory behind his work, he was insistent on one very basic principle: the interconnections between pure mathematics and physical uses are becoming more and more profound with each passing decade. Increasingly, physicists will turn to the foundations of mathematical analysis for new inspiration.