C5

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.

Pancakes.

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.

Let $F \in \mathbb{Z}[x_1,\ldots,x_n]$. Suppose $F(\mathbf{x})=0$ has infinitely many integer solutions $\mathbf{x} \in \mathbb{Z}^n$. Roughly how common should be expect the solutions to be? I will tell you what your naive first guess ought to be, give a one-line reason why, and discuss the reasons why this first guess might be wrong.

I then will apply these ideas to explain the intriguing parallels between the handling of the Brauer-Manin obstruction by Heath-Brown/Skorobogotov [doi:10.1007/BF02392841] on the one hand and Wei/Xu [arXiv:1211.2286] on the other, despite the very different methods involved in each case.

Gromov and Piatetski-Shapiro proved the existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about $v^v$ such manifolds of volume at most $v$, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Moreover, by restricting attention to non-compact manifolds, our result implies the same growth type for the number of quasi- isometry classes of lattices in $SO(n,1)$. Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms.

A joint work with Arie Levit.

An invariant random subgroup in a (finitely generated) group is a

probability measure on the space of subgroups of the group invariant under

the inner automorphisms of the group. It is a natural generalization of the

the notion of normal subgroup. I’ll give an introduction into this actively

developing subject and then discuss in more detail examples of invariant

random subgrous in groups of intermediate growth. The last part of the talk

will be based on a recent joint work with Mustafa Benli and Rostislav

Grigorchuk.