February 01, 2024: TU Graz

Location: Seminarraum AE02, Steyrergasse 30

Schedule: 

13:30-14:30: Ingrid Vukusic (Universität Salzburg): "Consecutive triples of multiplicatively dependent integers"

Abstract: Let 1<a<b<c be multiplicatively dependent integers (i.e., there exist nontrivial integer exponents x, y, z, such that a^x b^y c^z = 1). Is it possible that a+1, b+1, c+1 are multiplicatively dependent as well? It turns out that this is easy to answer. We will discuss related, more difficult questions, which will lead to Diophantine equations. We will solve some of them using lower bounds for linear forms in logarithms. Joint work with Volker Ziegler, and work in progress.

14:30-15:00: Tea break

15:00-16:00: Victor Wang (ISTA): "Sums of three cubes over a function field"

Abstract: I will talk about joint work with Tim Browning and Jakob Glas on producing sums of three cubes over a function field, assuming the Ratios Conjecture for a geometric family of L-functions. If time permits, I may also discuss some recent developments in homological stability that could help to resolve this conjecture. 

16:00-16:15: Break

16:15-17:15: Pierre-Yves Bienvenu (TU Wien): "Intersectivity with respect to sparse sets of integers" 

Abstract: A set R of positive integers is called intersective if every set of integers of positive density must contain two (distinct) elements whose difference lies in R. For instance the set of squares is known to be intersective, theorem proven independently by Furstenberg and Sárközy. R is chromatically intersective when any coloring of the integers must have two elements of the same color differing by an element of R. In this talk, we investigate whether intersectivity may be transferred to the set of primes; for instance it is known that any relatively dense subset of the primes must contain two elements differing by a square; we say that the set of squares is prime-intersective. We discovered that this phenomenon is rather the exception, and that in general, intersective sets are not prime-intersective. This contradicts the intuition stimulated by the Green-Tao theorem that theorems from combinatorial number theory may be transfered from dense sets to relatively dense subsets of the primes. Joint work with John Griesmer, Anh Le and Lê Thái Hoàng.

18:00: Dinner at Pizzaiolo, Dietrichsteinplatz 7