I. Schur's 1921 Problem for a polyomial f with integer coefficients was to describe all such $f$ that mapped one-one on the integers mod $p$ for almost all primes $p$. Example: $w\to {w}^{5}$ does if $p$-1 is prime to 5. Schur's problem was solved in 1969 over all number fields $F$.
I will explain why the analogue for rational functions had more major consequences, by showing aspects of the solution for $f$ of two types:
Of prime degree $\ell$; and of degree $\ell}^{2$.
The principal point will be its relation with Serre's Open Image Theorem, and in particular, with the theory of Complex multiplication.
In both cases, the problem is to describe which rational functions have a particular relation between the Galois groups of the splitting fields of $f(w)-z$ over $F(z)$ and over ${F}^{c\phantom{\rule{0}{0ex}}l}(z)$, $F}^{c\phantom{\rule{0}{0ex}}l$ an algebraic closure of $F$.
This problem raises a question on realizing dihedral groups as Galois groups; in turn an unsolved mystery about hyperelliptic jacobians.
Then, I will point out that the same question has a formulation to every finite group $G$ with its order divisible by a prime $\ell$ for which it has no Z/$\ell$ quotient. That relates generalizing Serre's Open Image Theorem to the Regular Version of the Inverse Galois Problem, the topic of a book I am writing on while I am here.
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The Ran space Ran(X) is the space of finite subsets of X, topologized so that points can collide. Ran spaces have been studied in diverse works from Borsuk–Ulam and Bott, to Beilinson–Drinfeld, Gaitsgory–Lurie and others. The alpha form of factorization homology takes as input a manifold or variety X together with a suitable algebraic coefficient system A, and it outputs the sheaf homology of Ran(X) with coefficients defined by A. Factorization homology simultaneously generalizes singular homology, Hochschild homology, and conformal blocks or observables in conformal field theory. I'll discuss applications of this alpha form of factorization homology in the study of mapping spaces in algebraic topology, bundles on algebraic curves, and perturbative quantum field theory.
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We give a description of subdirect products of several factors in congruence permutable varieties using Fleischer's Lemma and higher commutators. As an application we will obtain a sufficient condition for such a subdirect product to be finitely generated. This is joint work N. Ruskuc.
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Since its introduction, supercharacter theory has been used to study a wide variety of problems. However, the structure of supercharacter theories themselves remains mysterious. In this talk, we will discuss supercharacter theories that are able to detect nilpotence, in some sense. By defining analogs of the center and commutator subgroup for a given supercharacter theory $S$ of $G$, one may use these to define a coarser version of nilpotence, which we call $S$- nilpotence. The supercharacter theories $S$ of a nilpotent group $G$ for which $G$ is $S$-nilpotent will be classified, with particular emphasis on $p$-groups. Then some potential applications and further generalizations will be discussed, as time permits.
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I'll also describe a beta form of factorization homology, where one replaces the Ran space with a moduli space of stratifications, designed to overcome certain strict limitations of the alpha form. The key notion is that of a constructible bundle: a K-point of this moduli space classifies a constructible bundle over K. One application is to proving the Cobordism Hypothesis, after Baez–Dolan, Costello, Hopkins–Lurie, and Lurie. This is joint work with David Ayala.
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Intersection homology was developed by Goresky and MacPherson to extend Poincare duality and related invariants such as signatures from manifold theory to "singular spaces" such as non-smooth algebraic varieties. We'll provide an introduction to these topics, including an overview of some early applications and more recent contributions by the speaker and collaborators.