Number Theory

A vanishing sum of roots of unity (VSRU) is a finite list $z_1,\ldots,z_K$ of $N$-th complex roots of unity whose sum is zero. While there are many simple examples—including the famous "beautiful equation" of Euler, $e^{i \pi} + 1 = 0$—such sums become extremely complex as the parameter $N$ attains more complex prime power divisors (and we will see several classical examples illustrating this idea, as well as new examples from my work).

One fruitful line of inquiry is to seek a quantitative relationship between the prime divisors of $N$, their associated exponents, and the cardinality parameter $K$. A theorem of T.Y. Lam and K.H. Leung from the early '90's states: $K$ must always be (at least) as large as the smallest prime dividing $N$. This generalizes the well known observation that that sum of all $p$-th roots of unity (where $p$ is any prime number) must vanish; and, one notices that Euler's equation is one example of this fact.

In this talk, we will discuss two significant strengthenings of this result (one due to myself and I. Łaba, another due to myself, G. Kiss, I. Łaba and G. Somlai), which are derived from complexity measurements for polynomials with integer coefficients which have many cyclotomic polynomial divisors. As applications, we give connections in two other areas of mathematics. The first is in the study of integer tilings: additive decompositions of the integers $Z = A+B$ as a sum set, where each integer is represented uniquely. The second application is to the Favard length problem in fractal geometry, which asks for bounds upon the average length of the projections of certain dynamically-defined fractals onto lines.

This talk is based upon my individual work, as well as my joint work with I. Łaba, as well as my joint work with G. Kiss, I. Łaba and G. Somlai. All are welcome, and the first 15-20 minutes will include introductory ideas and examples for all results discussed in the latter portion of the talk.

We will give an exposition on the recent progress in the study of unimodal sequences, beginning with the work of Isaac Newton and then to the contemporary papers of June Huh. We will also relate this topic to the Riemann hypothesis. In the process, we will connect many areas of mathematics ranging from number theory, commutative algebra, algebraic geometry and combinatorics.

A set {𝑎1,𝑎2,…,𝑎𝑚}of distinct positive integers is a Diophantine 𝑚-tuple if the product of any two distinct elements in the set is one less than a square. In this talk, I will discuss some recent results related to Diophantine tuples and their generalizations. Joint work with Ernie Croot, Seoyoung Kim, and Semin Yoo.

We will talk about Drinfeld modules, and how they compare to elliptic curves for algorithms and computations.

Drinfeld modules can be seen as function field analogues of elliptic curves. They were introduced in the 1970's by Vladimir Drinfeld, to create an explicit class field theory of function fields. They were instrumental to prove the Langlands program for GL2 of a function field, or the function field analogue of the Riemann hypothesis.

Elliptic curves, to the surprise of many theoretical number theorists, became a fundamental computational tool, especially in the context of cryptography (elliptic curve Diffie-Hellman, isogeny-based post-quantum cryptography) and computer algebra (ECM method).

Despite a rather abstract definition, Drinfeld modules offer a lot of computational advantages over elliptic curves: one can benefit from function field arithmetics, and from objects called Ore polynomials and Anderson motives.

We will use two examples to highlight the practicality of Drinfeld modules computations, and mention some applications.

It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height $T$, is $\frac{1}{2} (e^2-5) \log T$ as $T \rightarrow \infty$. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer.