### Upcoming Talks

#### Zahlentheoretisches Kolloquium

**Title:**On some unlikely intersections for values and orbits of rational functions

**Speaker:**Dr. Alina Ostafe (UNSW Sydney)

**Date:**Dienstag, 23. 7. 2019, 11:00 Uhr

**Room:**Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II

**Abstract:**

Abstract: For given rational functions $f_1,\ldots,f_s$ defined over a number field $\K$, Bombieri, Masser and Zannier (1999) proved that the algebraic numbers $\alpha$ for which the values $f_1(\alpha),\ldots,f_s(\alpha)$ are multiplicatively dependent are of bounded height (unless this is false for an obvious reason).

Motivated by this, we present recent finiteness results on multiplicative relations of values of rational functions at arguments restricted to the maximal abelian extension of $\K$. We go even further and discuss our work in progress on the presence of multiplicative relations modulo finitely generated groups, posing some open questions. If time allows, we will present some finiteness results regarding the presence of powers of S-integers in orbits of polynomial dynamical systems.

#### Zahlentheoretisches Kolloquium

**Title:**Finiteness results on a certain class of modular forms and applications

**Speaker:**Soumya Bhattacharya (TU Graz)

**Date:**Friday, 19.7.2019, 14:30

**Room:**Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG

**Abstract:**

Holomorphic eta quotients are certain explicit classical modular forms on suitable Hecke subgroups of the full modular group. We call a holomorphic eta quotient $f$ 'reducible' if for some holomorphic eta quotient $g$ (other than 1 and $f$), the eta quotient $f/g$ is holomorphic. An eta quotient or a modular form in general has two parameters: Weight and level.

We shall show that for any positive integer $N$, there are only finitely many irreducible holomorphic eta quotients of level $N$. In particular, the weights of such eta quotients are bounded above by a function of $N$. We shall provide such an explicit upper bound. This is an analog of a conjecture of Zagier which says that for any positive integer $k$, there are only finitely many irreducible holomorphic eta quotients of weight $k/2$ which are not integral rescalings of some other eta quotients.

This conjecture was established in 1991 by Mersmann. We shall sketch a short proof of Mersmann's theorem and we shall show that these results have their applications in factorizing holomorphic eta quotient. In particular, due to Zagier and Mersmann's work, holomorphic eta quotients of weight $1/2$ have been completely classified. We shall see some applications of this classification and we shall discuss a few seemingly accessible yet longstanding open problems about eta quotients.

This talk will be suitable also for non-experts: We shall define all the relevant terms and we shall clearly state the classical results which we use.

Remark: Soumya Bhattachary is a new member of the Institute of Analysis and Number Theory. He will stay in Graz for one year as a PostDoc researcher.

#### Zahlentheoretisches Kolloquium

**Title:**Higher-rank Bohr sets and multiplicative Diophantine approximation

**Speaker:**Niclas Technau (Tel Aviv University)

**Date:**Friday, 19.7.2019, 13:30

**Room:**Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG

**Abstract:**

Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. This talk is about joint work with Sam Chow where we provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of Diophantine approximation.