Compact flat Riemannian manifolds (i.e., quotients of real tori by the action of a finite group) exhibit a certain rigidity: they are uniquely determined by their fundamental group. In contrast, most compact complex manifolds "come in families", meaning that they allow for non-trivial (small) deformations. In my talk, I will introduce the concept of rigidity of compact complex manifolds and explain the existence of rigid hyperelliptic manifolds (i.e., quotients of complex tori by the action of a finite group). Additionally, I will outline a classification algorithm for these manifolds. This is joint work with Christian Gleissner.
A few years ago, Dimitrov proved the Schinzel-Zassenhaus Conjecture. Harry Schmidt and I showed how his general strategy can be adapted to cover some dynamical variants of this conjecture. One common tool in both results is Dubinin's Theorem on the transfinite diameter of hedgehogs. Motivated by Mahler's work on root separation, I gave an elementary proof of Dubinin's Theorem, albeit with a worse numerical constant. In this talk I will report on joint work in progress with Harry Schmidt. We find new upper bounds for the transfinite diameter of some finite topological trees. These trees are more attuned to the dynamical setting then hedgehogs and cover some post-critically finite polynomials.
Solving Diophantine equations has always been one of the most fascinating problems in number theory, since even if it can be easily formulated, it almost always requires advanced techniques. In this talk, I will focus on equations that relate sums of powers to perfect powers. After showing some examples that are in the literature, I will discuss the (non-)existence of perfect powers that are sums of cubes of a nine-term arithmetic progression. The proof involves a battery of techniques and both theoretical and computational tools. This is joint work with Mar Curcó-Iranzo, Maleeha Khawaja, Vandita Patel, Özge Ülkem.
You are cordially invited to attend the colloquium talk by Roman Shvydkoy (University of Illinois at Chicago) afterwards.