In this talk, as a continuation of my talk in the Members' Colloquium but with a specialized audience in mind, I will discuss in more detail some of the general geometric and dynamical structures underlying the theoretical aspects of the restricted 3-body problem, and outline new research directions.
Compact hyperbolic manifolds are very interesting geometric objects. Maybe surprisingly, they are also interesting from an algebraic point of view: They are completely determined by their fundamental groups (this is Mostow's Theorem), which is naturally a subgroup of the rational valued invertible matrices in some dimension, GLn(ℚ). When the fundamental group essentially consists of the integer points of some algebraic subgroup of GLn we say that the manifold is arithmetic. A question arises: is there a simple geometric criterion for arithmeticity of hyperbolic manifolds? Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen. In a recent work with Fisher, Miller and Stover we proved this conjecture. Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which Alex Furman and I have developed in recent years. In my talk I will survey the subject and focus on the relation between the geometric, algebraic and arithmetic concepts
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.
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.
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.
Classically, heights are defined over number fields or transcendence degree one function fields. This is so that the Northcott property, which says that sets of points with bounded height are finite, holds. Here, expanding on work of Moriwaki and Yuan-Zhang, we show how to define arithmetic intersections and heights relative to any finitely generated field extension 𝐾/𝑘, and construct canonical heights for polarizable arithmetic dynamical systems 𝑓:𝑋→𝑋. These heights have a corresponding Northcott property when 𝑘 is ℚ or 𝔽𝑞. When 𝑘 is larger, we show that Northcott for canonical heights is conditional on the non-isotriviality of 𝑓:𝑋→𝑋, generalizing work of Lang-Neron, Baker, and Chatzidakis-Hrushovski. Additionally, we prove the Hodge Index Theorem for arithmetic intersections relative to 𝐾/𝑘. Since, when Northcott holds, pre-periodic points are the same as height zero points, this has applications to dynamical systems. By the Lefschetz principle, these results can be applied over any field.
An LMS online lecture course in Poisson structures.
The Liouville function is a multiplicative function that encodes important information related to distributional properties of the prime numbers. A conjecture of Chowla states that the values of the Liouville function fluctuate between plus and minus in such a random way, that all sign patterns of a given length appear with the same frequency. The Chowla conjecture remains largely open and in this talk we will see how ergodic theory combined with some feedback from number theory allows us to establish two variants of this conjecture. Key to our approach is an in-depth study of measure preserving systems that are naturally associated with the Liouville function. The talk is based on joint work with Bernard Host.
One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. We will explore a dynamical approach to this topic. After introducing a new dynamical framework for treating questions in multiplicative number theory, we will present an ergodic theorem which contains various classical number-theoretic results, such as the Prime Number Theorem, as special cases. This naturally leads to a formulation of an extended form of Sarnak's conjecture, which deals with the disjointness of actions of (ℕ,+) and (ℕ,*). This talk is based on joint work with Vitaly Bergelson.
The classical Linnik problems are concerned with the equidistribution of adelic torus orbits on the homogeneous spaces attached to inner forms of GL2, as the discriminant of the torus gets large. When specialized, these problems admit beautiful classical interpretations, such as the equidistribution of integer points on spheres, of Heegner points or packets of closed geodesics on the modular surface, or of supersingular reductions of CM elliptic curves. In the mid 20th century, Linnik and his school established the equidistribution of many of these classical variants through his ergodic method, under a congruence condition on the discriminants modulo a fixed auxiliary prime. More recently, the Waldspurger formula and subconvex estimates on L-functions were used to remove these congruence conditions, and provide effective power-savings rates.
In their 2006 ICM address, Michel and Venkatesh proposed a variant of this problem in which one considers the product of two distinct inner forms of GL2, along with a diagonally embedded torus. One can again specialize the setting to obtain interesting classical reformulations, such as the joint equidistribution of integer points on the sphere, together with the shape of the orthogonal lattice. This hybrid context has received a great deal of attention recently in the dynamics community, where, for instance, the latter problem was solved by Aka, Einsiedler, and Shapira, under supplementary congruence conditions modulo two fixed primes, using as critical input the joinings theorem of Einsiedler and Lindenstrauss.
In joint (ongoing) work with Valentin Blomer, we remove the supplementary congruence conditions in the joint equidistribution problem, conditionally on the Riemann hypothesis, while obtaining a logarithmic rate of convergence. The proof uses Waldsurger’s theorem and estimates of fractional moments of L-functions in the family of class group twists.
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