Many mathematical objects are naturally equipped with both an algebraic and a topological structure. For example, the automorphism group of any first-order structure is, of course, a group, and in fact a topological group when equipped with the topology of pointwise convergence.
While in some cases, e.g. the additive group of the reals, the algebraic structure of the object alone carries strictly less information than together with the topological structure, in other cases its algebraic structure is so rich that it actually determines the topology (under some requirements for the topology): by a result of Kechris and Solecki, the pointwise convergence topology is the only compatible separable topology on the full symmetric group on a countable set. Which topologies are compatible with a given algebraic object has intrigued mathematicians for decades: for example, Ulam asked whether there exists a compatible locally compact Polish topology on the full symmetric group on a countable set (by the above, the answer is negative).
In the case of automorphism groups of first-order structures, the question of the relationship between the algebraic and the topological structure has been pursued actively over the past 40 years, and numerous results have been obtained: many of the most popular automorphism groups, including that of the order of the rationals and of the random graph, do have unique Polish topologies.
The endomorphism monoid of a first-order structure is algebraically not as rich as its automorphism group, and often allows many different compatible topologies. We show, however, that there is a unique compatible Polish topology on the endomorphism monoids of the random graph, the weak linear order of the rational numbers, the random poset, and many more.
This is joint work with L. Elliott, J. Jonušas, J. D. Mitchell, Y. Péresse, and C. Schindler.
Uniqueness of Polish topologies on endomorphism monoids of countable structures
Tue, Oct. 26 3pm (MATH 350)
Ezzeddine El Sai (CU Boulder)
Machine learning methods aimed at learning the underlying geometry of data predominantly assume that the underlying geometry is a smooth embedded manifold. This assumption fails for many physical models where the underlying geometry contains singularities. In this talk, we discuss learning the underlying geometry of data within the scope of algebraic geometry and we use this to detect singularities in data.