Tag - Topology

Susana Huelga: Thermal Markovian processes: from resource theories to molecular switches

23rd May, 2023

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.

Dariusz Chruscinski: Overview talk – Quantum processes: divisibility, Markovianity and classicality

23rd May, 2023

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.

Satvik Singh: Diagonal unitary covariant quantum channels

22nd May, 2023

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.

Corey Jones: K-theoretic classification of fusion category actions on locally semisimple algebras

14th May, 2023

An action of a tensor category C on an associative algebra A is a linear monoidal functor from C to the monoidal category of A-A bimodules. We consider the problem of classifying (unitary) actions of (unitary) fusion categories on inductive limits of semisimple associative algebras (called locally semisimple algebras). A theorem of Elliot classifies locally semisimple algebras by their ordered K0 groups. We extend this theorem to a K-theoretic classification of fusion category actions on locally semisimple algebras which have an inductive limit decomposition.

Oishee Banerjee: Cohomology and arithmetic of some mapping spaces

10th April, 2023

How do we describe the topology of the space of all nonconstant holomorphic (respectively, algebraic) maps F: X → Y  from one complex manifold (respectively, variety) to another? What is, for example, its cohomology? Such problems are old but difficult, and are nontrivial even when the domain and range are Riemann spheres. In this talk I will explain how these problems relate to other parts of mathematics such as spaces of polynomials, arithmetic (e.g., the geometric Batyerv-Manin type conjectures), algebraic geometry (e.g., moduli spaces of elliptic fibrations, of smooth sections of a line bundle, etc) and if time permits, homotopy theory (e.g., derived indecomposables of modules over monoids). I will show how one can fruitfully attack such problems by incorporating techniques from topology to the holomorphic/algebraic world (e.g., by constructing a new spectral sequence).

Rigoberto Zelada: Polynomial Ergodic Theorems for Strongly Mixing Commuting Transformations

4th April, 2023

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.

Adi Glücksam: Unbounded Fast Escaping Wandering Domains

21st March, 2023

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.

Henrik Kreidler: Geometric Representation of Structured Extensions in Ergodic Theory

14th March, 2023

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.

Terence Tao: Almost all Collatz orbits attain almost bounded values

13th March, 2023

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.