In 1896, F. Enriques discovered “Enriques surface” which is a non rational algebraic surface. In 1907 he showed that a general Enriques surface has infinitely many automorphisms, and asked whether an Enriques surface with finite automorphism group exists or not. In 1911, G. Fano presented an example of such Enriques surfaces. Later, in 1984–1986, V. Nikulin and S. Kondo classified such Enriques surfaces into 7 types. Recently this topic comes up again.

In this talk, I would like to discuss the finiteness of the automorphism groups of Enriques surfaces which is related to a geometry of a real hyperbolic space (Lobachevsky space), and a recent progress.

I will sketch the broad context and motivation of Igusa's conjecture on exponential sums, in elementary terms. The conjecture predicts bounds for finite exponential sums with index sets consisting of integers modulo a positive power of a prime number, and where uniformity is key. I will state the recent solution with Mircea Mustata and Kien Nguyen of this conjecture by Igusa. Igusa intended to use his conjecture on exponential sums to obtain new local-global principles (similar to Hasse's principle for quadratic polynomials), but this program is far from being achieved.

Let a group G act freely on a "space" X (freely means that g⋅x ≠ x if g ≠ 1). In this case, we say that X is a principal G-bundle over the quotient space X/G. In fact, this definition has to be made more precise. It turns out that principal bundles are ubiquitous in geometry: they govern vector bundles, orthogonal bundles, projective bundles etc.

After giving the precise definitions and examples, I will discuss some basic and old conjectures pertaining to local triviality of principal bundles. No knowledge of algebraic geometry or theory of algebraic groups will be assumed.

In this talk, we will review how each finite group can be disassembled into a unique set of "simple" groups, similar to the unique prime factorization of natural numbers. This raises various questions; for example: which simple pieces are there? And: in which ways can one glue these pieces together to form new finite groups? The main focus of this talk will be on this last question, and we will discuss it in the context of classifying (almost) all groups of order at most 20,000.

In the talk we offer a possible answer to this question by defining a random network model that is constructed from simple, basic principles and by showing that typical features of large networks like small-world effect, robustness and clustering emerge from this construction.

Multiphase flows are highly relevant in many biological, chemical, physical and technical processes, as the latter often involve e.g. phases of different materials, phases of different aggregate states or separation of constituents, which leads to membranes to form containers, etc. In the present talk we focus on so-called sharp-interface models, in which the phase-separating interface is described by an infinitely thin layer, i.e. by means of a hypersurface, which is then assumed to be sufficiently smooth. The resulting mathematical models turn out to be systems of partial differential equations, both in the bulk phases and on the interfaces, coupled via transmission conditions. The interfaces themselves are part of the unknowns of the problem, making it intrinsically nonlinear. We explain the derivation of this type of models, discuss a selection of our recent results concerning their mathematical analysis, and close with some open problems.