Scientific

Exponential sums play a role in many different problems in number theory. For instance, Gauss sums are at the heart of some early proofs of the quadratic reciprocity law, while Kloosterman sums are involved in the study of modular and automorphic forms. Another example of application of exponential sums is the circle method, an analytic approach to problems involving the enumeration of integer solutions to certain equations. In many cases, obtaining upper bounds on the modulus of these sums allow us to draw conclusions, but once the modulus has been bounded, it is natural to ask the question of the distribution of exponential sums in the region of the complex plane in which they live. After a brief overview of the motivations mentioned above, I will present some results obtained with Emmanuel Kowalski on the equidistribution of exponential sums indexed by the roots modulo p of a polynomial with integer coefficients.

A Beurling number system consists a non-decreasing unbounded sequence of reals larger than 1, which are called generalized primes, and the sequence of all possible products of these generalized primes, which are called generalized integers. With both sequences one associates counting functions. Of particular interest is the case when both counting functions are close to their classical counter parts: namely when the prime-counting function is close to Li(x), and when the integer-counting function is close to ax for some positive constant a.

A Beurling number systems is well-behaved if it admits a power saving in the error terms for both these counting functions. In this talk, I will discuss some general theory of these well-behaved systems, and present some recent work about examples of such well-behaved number systems. This talk is based on joint work with Gregory Debruyne and Szilárd Révész.

The size function $h^0$ for a number field is analogous to the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. Van der Geer and Schoof conjectured that $h^0$ attains its maximum at the trivial class of Arakelov divisors if that field is Galois over $\mathbb{Q}$ or over an imaginary quadratic field. This conjecture was proved for all number fields with the unit group of rank $0$ and $1$, and also for cyclic cubic fields which have unit group of rank two. In this talk, we will discuss the main idea to prove that the conjecture also holds for totally imaginary cyclic sextic fields, another class of number fields with unit group of rank two. This is joint work with Peng Tian and Amy Feaver.

By using an identity relating a sum to an integral, we obtain a family of identities for the averages \(M(X)=\sum_{n\leq X} \mu(n)\) and \(m(X)=\sum_{n\leq X} \mu(n)/n\). Further, by choosing some specific families, we study two summatory functions related to the Möbius function, \(\mu(n)\), namely \(\sum_{n\leq X} \mu(n)/n^s\) and \(\sum{n\leq X} \mu(n)/n^s \log(X/n)\), where \(s\) is a complex number and \(\Re s >0\). We also explore some applications and examples when s is real. (joint work with O. Ramaré)

I will report on a simple model for collective self-organization in colonies of myxobacteria. Mechanisms include only running, to the left or to the right at fixed speed, and tumbling, with a rate depending on head-on collisions. We show that variations in the tumbling rate only can lead to the observed qualitatively different behaviors: equidistribution, rippling, and formation of aggregates. In a second part, I will discuss in somewhat more detail questions pertaining to the selection of wavenumbers in the case where ripples are formed, in particular in connection with recent progress on the marginal stability conjecture for front invasion.