I will discuss a result with Bonatti and Crovisier from 2009 showing that the C1 generic diffeomorphism f of a closed manifold has trivial centralizer; i.e., fg = gf implies that g is a power of f. I'll discuss features of the C1 topology that enable our proof (the analogous statement is open in general in the Cr topology, for r > 1). I'll also discuss some features of the proof and some recent work, joint with Danijela Damjanovic and Disheng Xu that attempts to tackle the non-generic case.
One of the most famous – and still not fully understood – objects in mathematics is the Mandelbrot set. It is defined as the set of complex numbers c for which the polynomial fc(z)=z2+c has a connected Julia set. But the Mandelbrot set turns out to be related to many different areas of mathematics. Inspired by recent results in arithmetic geometry, I will describe how the tools of arithmetic intersection theory can be applied in the setting of these complex dynamical systems to give new information about the Mandelbrot set.
The dynamics of iterating a rational function exhibits complicated and interesting behaviour when restricted to points in its Julia set. Kajiwara and Watatani constructed a C*-algebra from a rational function restricted to its Julia set in order to study its dynamics from an operator algebra perspective. They showed the C*-algebras are Kirchberg algebras that satisfy the UCT, and are therefore classified by K-theory. The K-theory groups of these algebras have been computed in some special cases, for instance by Nekrashevych in the case of a hyperbolic and post-critically finite rational function. We compute the K-theory groups for a general rational function using methods different to those used before. In this talk, we discuss our methods and results.
Given a set ℬ of natural numbers, not containing 1, we denote by ℱℬ the set of ℬ-free numbers, that is, ℱℬ = ℤ \ ⋃b∈ℬ bℤ. Let Xη be the ℬ-free subshifts, that is the subshift induced by η, where η denotes the characteristic function of ℱℬ . That means, Xη is the closure of the set of all shifts of η in the space {0, 1}ℤ equipped with the product topology. We are interested in the case when B is a Behrend set, that is, when the set of ℬ-free numbers has zero density. It turns out that this is the case precisely when Xη is proximal and has zero entropy. We prove that the complexity of Xη, with ℬ being a Behrend set, can achieve any subexponential growth. Together with the ℬ-free shift we investigate the ℬ-admissible shift Xℬ and we show that it is transitive if and only if the set B is pairwise coprime, which allows one to characterize dynamically the subshifts generated by the Erdős sets (infinite, coprime and not Behrend). We also estimate the complexity for some classical subshifts (the subshift of primes or semi-primes). The lower estimates are obtained conditionally on Hardy-Littlewood Conjecture or Dickson’s Conjecture. We remark on a recent result of Tao and Ziegler (not assuming the conjectures) that the shift of primes is uncountable.
Several limits of quantum theory can be understood from a simple framework in which some of the basic features of these limits can be established by abstract general results. Often the limit may have a very different structure from its approximants, so one cannot simply let some parameter go to a limiting value. We focus on the unified definition of the limit theory with its states and observables, its dynamics and its equilibrium states. Examples that will be mentioned are (1) the mean field with or without tagged particles, (2) The classical limit (ħ → 0), (3) the limit of infinite lattice systems, (4) Some continuum limits/renormalization theories.
It is shown that the operator space generated by peripheral eigenvectors of a unital completely positive map on a von Neumann algebra has a C*-algebra structure. This extends the notion of non-commutative Poisson boundary by including the point spectrum of the map contained in the unit circle. The main ingredient is dilation theory. This theory provides a simple formula for the new product. The notion has implications to our understanding of quantum dynamics. For instance, it is shown that the peripheral Poisson boundary remains invariant in discrete quantum dynamics.
In recent joint work with Yidong Chen, we discovered spectral gap estimates and concentration inequalities for for dynamics with few generators. Some of these estimates are dimension free and then can be used to feed in the recent theory of complexity initiated by Lloyd and Jaffe, and adapted more recently for specific resources. The goal is to find a viable theory of complexity which holds in type II1 and III1 von Neumann algebras, both of which come naturally in quantum field theory and Witten's take on black holes.
Quantum resource theory formulations of thermodynamics offer a versatile tool for the study of fundamental limitations to the efficiency of physical processes, independently of the microscopic details governing their dynamics. Despite the ubiquitous presence of non-Markovian dynamics in open quantum systems at the nanoscale, rigorous proofs of their beneficial effects on the efficiency of quantum dynamical processes are scarce. Here we combine the resource theory of athermality with concepts from the theory of divisibility classes for quantum channels, to prove that memory effects can increase the efficiency of photoisomerization to levels that are not achievable under a purely thermal Markovian (i.e. memoryless) evolution. This provides rigorous evidence that memory effects can provide a resource in ultrafast biological quantum dynamics, and, more generally, quantum thermodynamics at the nanoscale.
In my talk I introduce fundamental concepts concerning the divisibility of quantum and classical dynamical maps. I discuss the notion of quantum Markovianity in terms of dynamical maps (divisibility) and explore the multitime statistics of a process using the quantum regression formula. Additionally, I delve into the concept of classicality, providing illustrations and discussions specifically focusing on amplitude damping and dephasing processes.
The talk will present a study of (finite-dimensional) quantum channels which are covariant under the action of the diagonal unitary group. Many salient examples, such as the depolarizing channels, dephasing channels, amplitude damping channels, and mixtures thereof, lie in this class. The first part of the talk will be devoted to the study of entanglement properties of these channels. In particular, by reformulating the entanglement-breaking property of such channels in terms of the cone of pairwise completely positive matrices, I will show that the well-known PPT-squared conjecture holds for channels in this class. I will also unravel an interesting connection between the entanglement-breaking property of such channels and triangle-free graphs. The second half of the talk will deal with the ergodic properties of these channels. I will show that the ergodic behaviour of a channel in this class is essentially governed by a classical stochastic matrix, thus allowing us to exploit tools from classical ergodic theory to study quantum ergodicity of such channels.
The quantum unique ergodicity conjecture of Rudnick and Sarnak concerns the mass equidistribution in the large eigenvalue limit of Laplacian eigenfunctions on negatively curved manifolds. This conjecture has been resolved by Lindenstrauss when this manifold is the modular surface assuming these eigenfunctions are additionally Hecke eigenfunctions, namely Hecke-Maass cusp forms. I will discuss a variant of this problem in this arithmetic setting concerning the mass equidistribution of Hecke-Maass cusp forms on submanifolds of the modular surface.
I will discuss pointwise ergodic theory as it developed out of Bourgain's work in the 80s, leading up to my work with Mirek and Tao on bilinear ergodic averages.
In its dynamical formulation, the Furstenberg-Sárközy theorem states that for any invertible measure-preserving system (X,μ,T), any set A⊆X with μ(A)>0, and any integer polynomial P with P(0)=0,
c(A)=limN−M→∞ 1/(N−M) ∑n=MN−1 μ(A∩TP(n)A)>0.
The limit c(A) obtains the 'correct' value μ(A)2 when T is totally ergodic. In fact, when T is totally ergodic, one has an ergodic theorem for polynomial actions: for any integer polynomial P and any f∈L2(μ),
limN−M→∞ 1/(N−M) ∑n=MN−1 TP(n)f= ∫X f dμ,
where the limit is taken in L2(μ). We will explain that the correct notion of total ergodicity for polynomial actions of more general rings depends on the dynamical behavior of actions along finite index ideals. From this point of view, the action of a large finite field on itself is asymptotically totally ergodic, since the index of the only proper ideal {0} grows with the size of the field. Guided by ergodic-theoretic results about polynomial (multiple) recurrence in totally ergodic systems, we then obtain several new results about polynomial configurations in large subsets of finite fields.
The goal of this talk is to present new results dealing with the asymptotic joint independence properties of commuting strongly mixing transformations along polynomials. These results form natural strongly mixing counterparts to various weakly and mildly mixing polynomial ergodic theorems. A decisive role in the proofs is played by a new notion of convergence that is adequate for dealing with strong mixing and, as we will see, cannot be avoided while working with commuting polynomial actions. This talk is based on joint work with Vitaly Bergelson.
Complex dynamics explores the evolution of points under iteration of functions of complex variables. In this talk I will introduce into the context of complex dynamics, a new approximation tool allowing us to construct new examples of entire functions and show new possible dynamical behaviours. In particular, we answer a question of Rippon and Stallard from 2012 about unbounded wandering domains with unbounded orbits, and provide a collection of examples supporting a conjecture of Baker.
The Mackey-Zimmer representation theorem is a key structural result from ergodic theory: Every compact extension between ergodic measure-preserving systems can be written as a skew-product by a homogeneous space of a compact group. This is used, e.g., in Furstenberg's original ergodic theoretic proof of Szemerédi's theorem, as well as in the classical proofs of the Host-Kra-Ziegler structure theorem for characteristic factors. Inspired by earlier work of Ellis, we discuss a topological approach, first to the original theorem, and then to a generalization relaxing the ergodicity assumptions due to Austin.
Define the Collatz map Col on the natural numbers by setting Col(n) to equal 3n+1 when n is odd and n/2 when n is even. The notorious Collatz conjecture asserts that all orbits of this map eventually attain the value 1. This remains open, even if one is willing to work with almost all orbits rather than all orbits. We show that almost all orbits n, Col(n), Col2(n), ... eventually attain a value less than f(n), for any function f that goes to infinity (no matter how slowly). A key step is to obtain an approximately invariant (or more precisely, self-similar) measure for the (accelerated) Collatz dynamics.
We will discuss a version of the Green-Tao arithmetic regularity lemma and counting lemma which works in the generality of all linear forms. In this talk we will focus on the qualitative and algebraic aspects of the result.
Ever since Furstenberg proved his multiple recurrence theorem, the limiting behaviour of multiple ergodic averages along various sequences has been an important area of investigation in ergodic theory. In this talk, I will discuss averages along arithmetic progressions in which the differences are elements of a fixed integer sequence. Specifically, I will give necessary and sufficient conditions under which averages of fixed length of the aforementioned form have the same limit as averages along arithmetic progressions of the same length. The result relies on a higher-order version of the degree lowering argument, which is of independent interest. The talk is based on a joint work with Nikos Frantzikinakis.
Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner’s theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood. We consider ℤ-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of non-trivial horocycle orbit closures is known. Based on joint work with James Farre and Yair Minsky.
