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Laura DeMarco: Intersection Theory and the Mandelbrot Set

5th October, 2023

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.

Tselil Schramm: Higher-dimensional Expansion of Random Geometric Complexes

25th July, 2023

A graph is said to be a (1-dimensional) expander if the second eigenvalue of its adjacency matrix is bounded away from 1, or almost-equivalently, if it has no sparse vertex cuts. There are several natural ways to generalize the notion of expansion to hypergraphs/simplicial complexes, but one such way is 2-dimensional spectral expansion, in which the local expansion of vertex links/neighborhoods (remarkably) witnesses global expansion. While 1-dimensional expansion is known to be achieved by, e.g., random regular graphs, very few examples of sparse 2-dimensional expanders are known, and at present all are algebraic. It is an open question whether sparse 2-dimensional expanders are natural and "abundant" or "rare." In this talk, we'll give some evidence towards abundance: we show that the set of triangles in a random geometric graph on a high-dimensional sphere yields an expanding simplicial complex of arbitrarily small polynomial degree.

Alina Vdovina: Higher structures in Algebra, Geometry and C*-algebras

23rd June, 2023

We present buildings as universal covers of certain infinite families of CW-complexes of arbitrary dimension. We will show several different constructions of new families of k-rank graphs and C*-algebras based on these complexes, for arbitrary k. The underlying building structure allows explicit computation of the K-theory as well as the explicit spectra computation for the k-graphs.

Jeremy Brent Hume: The K-theory of a rational function

22nd June, 2023

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.

Stanisław Kasjan: On Dynamics of ℬ-free Systems Generated by Behrend Sets

6th June, 2023

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.

Reinhard Werner: Inductive limits of quantum systems, equilibrium states and dynamics

26th May, 2023

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.

B.V. Rajarama Bhat: Peripheral Poisson boundary

26th May, 2023

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.

Marius Junge: Complexity and dynamics in finite and infinite dimension

26th May, 2023

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.

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.

Borys Kuca: Degree Lowering Along Arithmetic Progressions

6th March, 2023

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.