The aim of this talk is to introduce the area of homogeneous dynamics to a general audience and to survey some key instances in which number theory interacts with it. I hope to touch upon the following:

(1) Margulis' resolution of Oppenheim's conjecture regarding values of indefinite irrational quadratic forms at integral points.

(2) Einsiedler-Lindenstrauss-Katok work towards Littlewood's conjecture in Diophantine approximation.

(3) Linnik/Duke theorem that says (loosely) that integral points on large spheres are spread randomly on these spheres.

(4) I will also try to mention a few results that I have been involved in proving.

The set of all increasing functions on the rational numbers carries an algebraic structure (the standard composition operation of functions, yielding a semigroup) and a topological structure (the topology of pointwise convergence in which a sequence of functions converges if and only if it is eventually constant at every argument). These structures are compatible in the sense that the operation is continuous with respect to the topology. Whenever one considers such a combined algebraic-topological structure, one can ask the following general question: Is the topology predetermined by the compatibility with the operations, i.e., does the algebra have a unique (Polish?) topology? In other words: Can the (Polish) topology be "reconstructed" from the algebra? I will present several examples of Polish groups and semigroups that do (not) have this property and introduce the known proof techniques. Afterwards, I will discuss why these successful techniques fail for the space of all increasing functions on the rational numbers and outline how this shortcoming can be mitigated to show that, indeed, the increasing functions on the rational numbers carry a unique Polish topology.

This is joint work with Michael Pinsker.

Unique Polish semigroup topology: novel techniques to crack the semigroup of increasing functions on the rational numbers

Nov. 28, 2023 3:30pm (MATH 3…

Topology

Howy Jordan

X

Given a space, consider a decomposition into a pair of complementary open and closed subspaces. Artin gluing tells us which open sets of the subspaces can glue to an open set of the whole space and what data is needed to glue sheaves on the subspaces to obtain a sheaf on the whole space. This can be readily generalized to decompositions into locally closed subspaces.

Artin gluing describes how to glue sheaves, hence etale bundles. In this talk we will consider how to also apply Artin gluing to fiber bundles. If the fibers are isomorphic we can obtain another fiber bundle, but in general we obtain stratified fiber bundles. In the case of tangent bundles, properties of the Artin gluing data are related to the Whitney regularity conditions.

Artin Gluing of Fiber Bundles and the Whitney Conditions