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.
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