Let λ be the Liouville function and P(x) any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence λ(P(n)) changes sign infinitely often. We present a solution to this problem for new classes of polynomials P, including any product of linear factors or any product of quadratic factors of a certain type. The proofs also establish some nontrivial cancellation in Chowla and Elliott type correlation averages.
Riehl and Verity introduced ∞-cosmoi - certain simplicially enriched categories - as a framework in which to give a model-independent approach to ∞-categories. For instance, there is an infinity cosmos of ∞-categories with finite limits or colimits, or of cartesian fibrations. In this talk, I will introduce the notion
of an accessible ∞-cosmos and explain that most, if not all, ∞-cosmoi arising in practice are accessible. Applying results of earlier work, it follows that accessible ∞-cosmoi have homotopy weighted colimits and admit a broadly applicable homotopical adjoint functor theorem.
An LMS online lecture course in model categories.
In this talk we discuss various model categories of locally cartesian closed (lcc) categories and their relevance to coherence problems, in particular the coherence problem of categorical semantics of dependent type theory. We begin with Lcc, the model category of locally cartesian closed (lcc) sketches. Its fibrant objects are precisely the lcc categories, though without assigned choices of universal objects. We then obtain a Quillen equivalent model category sLcc of strict lcc categories as the category of algebraically fibrant objects of Lcc. Strict lcc categories are categories with assigned choice of lcc structure, and their morphisms preserve these choices on the nose. Conjecturally, sLcc is precisely Lack’s model category of algebras for a 2-monad T , where T is instantiated with the free lcc category functor on Cat. We then discuss the category of algebraically cofibrant objects of sLcc and show how it can serve as a "gros" model of dependent type theory.
The homotopy hypothesis is a well-known bridge between topology and category theory. Its most general formulation, due to Grothendieck, asserts that topological spaces should be "the same" as infinity-groupoids. In the stable version of the homotopy hypothesis, topological spaces are replaced with spectra.
In this talk we will review the classical homotopy hypothesis, and then focus on the stable version. After discussing what the stable homotopy hypothesis should look like on the categorical side, we will use the Tamsamani model of higher categories to provide a proof.
An LMS online lecture course in number theory and dynamics.
The main goal of this mini-course is to illustrate a proof of Furstenberg's ×2,×3 theorem: The ×2,×3 orbit of any irrational number on the unit interval is dense. Key results that will be needed for the proof are topological properties of irrational rotation on the unit interval. We will discuss those results and provide detailed backgrounds as well as proofs. At the end of the course, I will introduce various results and problems on digit expansions of integers. The following topics will be covered:
1. Irrational rotations on torus;
2. Diophantine approximation: Dirichlet theorem, Roth's theorem, Baker's theory of linear forms of logarithms;
3. Furstenberg's ×2,×3 theorem;
4. Results and problems on digit expansions of integers;
5. Furstenberg's theorem on 2-dimensional torus (if time permits).
Note: For 2., I will mostly state the results without giving proofs as they are out of the scope of this mini-course.
We introduce the notion of M-locally generated category for a factorization system (E,M) and study its properties. We offer a Gabriel-Ulmer duality for these categories, introducing the notion of nest. We develop this theory also from an enriched point of view. We apply this technology to Banach spaces showing that it is equivalent to the category of models of the nest of finite-dimensional Banach spaces.
Among Banach spaces approximate injectivity is more important than injectivity. We will treat it from the point of view of enriched category theory - as enriched injectivity over complete metric spaces.
An LMS online lecture course in holomorphic dynamics.
The series will consist of 6 one-hour lectures which will focus on the iteration of entire functions. We explore, among other things, some famous fractal Julia sets and the well-known Mandelbrot set. In particular, we will cover the following topics:
1. Equicontinuity, normal families, Montel's theorem, Riemann mapping theorem, the Riemann sphere.
2. Iteration of polynomials. Definition of the Fatou set and the Julia set for a polynomial. Examples.
3. The filled Julia set. Fixed and periodic points.
4. An introduction to the properties of the Fatou set and the Julia set.
5. The Mandelbrot set: its definition and properties.
6. Introduction to the iteration of transcendental entire functions.
7. Similarities and differences between polynomials and transcendental entire functions.
8. The escaping set: definition, properties, and its important role.
9. Examples of the Fatou, Julia and escaping sets for transcendental entire functions.
The lecture series is addressed to PhD students from diverse mathematical backgrounds. We shall assume a basic knowledge of complex analysis and a little topology. Some more advanced background in complex analysis will be covered in the first lecture. No knowledge of dynamics will be assumed.
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