Given a field K and an ordered abelian group G, we can form the field K((G)) of generalised formal power series with coefficients in K and indices in G. When is this field decidable? In certain cases, decidability reduces to that of K and G. We survey some results in the area, particularly in the case char K > 0, where much is still unknown.
Inexact hardware trades reduced numerical precision against a reduction
in computational cost. A reduction of computational cost would allow
weather and climate simulations at higher resolution. In the first part
of this talk, I will introduce the concept of inexact hardware and
provide results that show the great potential for the use of inexact
hardware in weather and climate simulations. In the second part of this
talk, I will discuss how rounding errors can be assessed if the forecast
uncertainty and the chaotic behaviour of the atmosphere is acknowledged.
In the last part, I will argue that rounding errors do not necessarily
degrade numerical models, they can actually be beneficial. This
conclusion will be based on simulations with a model of the
one-dimensional Burgers' equation.
The common convention when dealing with hyperbolic groups is that such groups are finitely
generated and equipped with the word length metric relative to a finite symmetric generating
subset. Gromov's original work on hyperbolicity already contained ideas that extend beyond the
finitely generated setting. We study the class of locally compact hyperbolic groups and elaborate
on the similarities and differences between the discrete and non-discrete setting.
Since their introduction, Shimura varieties have proven to be important landmarks sitting right at the crossroads between algebraic geometry, number theory and representation theory. In this talk, starting from the yoga of motives and Hodge theory, we will try to motivate Deligne's construction of Shimura varieties, and briefly survey some of their zoology and basic properties. I may also say something about the links to automorphic forms, or their integral canonical models.
We state a conjecture about the size of the intersection between a bounded-rank progression and a sphere, and we prove the first interesting case, a result of Chang. Hopefully the full conjecture will be obvious to somebody present.
The Balog-Szemeredi-Gowers theorem states that, given any finite subset of an abelian group with large additive energy, we can find its large subset with small doubling constant. We can ask how this constant depends on the initial additive energy. In the talk, I will give an upper bound, mention the best existing lower bound and, if time permits, present an approach that gives a hope to improve the lower bound and make it asymptotically equal to the upper bound from the beginning of the talk.
At the end of the 70s, Gross and Deligne conjectured that periods of geometric Hodge structures with multiplication by an abelian number field are always products of values of the gamma function at rational numbers, with exponents determined by the Hodge decomposition. I will explain a proof of an alternating variant of this conjecture for the cohomology groups of smooth, projective varieties over the algebraic numbers acted upon by a finite order automorphism.