Featured Lendület Researcher: Gergely Harcos

This is the second time that Harcos has been awarded a grant under the Momentum programme, and this second grant is a continuation of the first. The research is basically in the field of analytic number theory, and within that, the study of automorphic forms.

Gergely Harcos

“I work in theoretical mathematics, and in a very old field within that, number theory. Number theory is about the properties of integers. Anything that can be used to study integers falls within the scope of number theory. So number theory combines a very wide range of mathematical fields,” he said.

One of the most central concepts in mathematics

The study of automorphic forms is one of the areas of modern analytic number theory. Although automorphic forms themselves were discovered as early as the 19th century, it was not until the 1930s that their fundamental importance became clear. From the 1950s and 1960s onwards, it became absolutely clear that automorphic forms were one of the most central concepts not only in number theory but in all of mathematics. But what are automorphic forms? They are generalisations of the classical sine and cosine functions. The sine and cosine functions are periodic functions, but they play a special role among the other periodic functions, since they are solutions to special differential equations.

“All periodic functions can be mixed out of sine and cosine. So if I take any periodically recurring set of quantities (that is, functions), it can be decomposed into the sum of (usually an infinite number of) sine and cosine functions, and there is always only one such decomposition,” the mathematician continued. “The same is true in higher dimensional, more complex spaces, so there, too, all periodic functions can be mixed in a single way out of special periodic functions that are already wave-like. In other words, these periodic waves play a special role amongst the periodic functions, and we call them automorphic forms,” explained Harcos.

Automorphic forms are closely related to number theory, since some of the isometry groups that characterise them can be described by integers, and so, by implication, the behaviour of these automorphic forms is about integers. “It’s quite incredible that many deep questions about integers, which have often been asked since antiquity, can be best answered by automorphic forms,” said the team leader. “For example, in the 19th century, a much-studied question was which integers can be obtained as the sum of two, three, four or more squares, or by more complex quadratic expressions. How many ways are these numbers produced, and if many productions are possible, how are they distributed? Many questions of this type have been answered using automorphic forms, but often only in recent decades.”

According to the mathematician, it therefore seems that the deepest properties of integers are intimately linked to automorphic forms. But “this is only a very small part of the story, because we can talk about the combined effect or reflection of three areas. Automorphic forms are associated with certain spaces and certain isometry groups of spaces. In other words, automorphic forms are not only related to number theory but also to geometry,” he argued. “And isometry groups are algebraic structures. Furthermore, automorphic forms are wave functions, which typically fall within the scope of analysis or, if I go a little further, mathematical physics. So automorphic forms connect number theory, geometry, algebra, analysis and mathematical physics.”

The research team is mainly focusing on how automorphic forms can be used to understand the deep properties of integers, with an emphasis on the hyperbolic circle problem. The classical Euclidean circle problem is about how many grid points of integer coordinates fall inside a large circle, for example, around the origin. This question was already studied by Gauss more than 200 years ago. The area bounded by the circle is a good approximation of the number of grid points, but the problem of how accurate this approximation is is still unsolved. Automorphic forms and their wave function properties also help with this, but the problem becomes much more difficult when formulated on the hyperbolic plane.

The hyperbolic circle problem, prime numbers, Riemann’s hypothesis

According to Harcos, the hyperbolic circle problem is also very relevant for the history of Hungarian mathematics, since János Bolyai was one of the first to develop hyperbolic geometry. “One of the priority topics for our project is the hyperbolic circle problem, where automorphic forms are a great help,” said the mathematician. “Another priority topic is the study of the distribution of closed geodesics on hyperbolic arithmetic surfaces. The terms hyperbolic and arithmetic refer to the fact that the surface in question can be constructed from the hyperbolic plane using a number-theoretic isometry group. The physics or mathematical physics relevance of this question is also significant.”

Automorphic forms can also help in understanding prime numbers. “Prime numbers are mysterious objects. Everybody knows that prime numbers do not decompose into products, but any positive integer can be clearly decomposed into products of prime numbers. No matter how I start to decompose a positive integer, I end up with the same prime numbers, except for the order,” says Harcos. “The distribution of prime numbers shows certain well-understood statistical regularities. However, determining their exact distribution is very difficult, but it is certainly related to the automorphic forms. Just as in the case of probably one of the most famous unproven hypotheses in mathematics, the Riemann hypothesis, the application of automorphic forms is also necessary. If we want to understand the Riemann zeta function, and why the Riemann hypothesis is true (that is, why our estimate of the distribution of prime numbers is so good), a good strategy seems to be to understand automorphic L-functions. Automorphic L-functions are derived from automorphic forms and generalise the Riemann zeta function. The Riemann hypothesis itself can be formulated for all automorphic L-functions. Therefore, the study of automorphic L-functions is emphasised in the Momentum project.”