In this talk, we will be interested in measure-preserving actions of countable groups on standard probability spaces, and more precisely in the partitions of the space into orbits that they induce, also called measure-preserving equivalence relations. In 2000, Gaboriau obtained a characterization of the ergodic equivalence relations which come from non-free actions of the free group on $n>1$ generators: these are exactly the equivalence relations of cost less than n. A natural question is: how non-free can these actions be made, and what does the action on each orbit look like? We will obtain a satisfactory answer by showing that the action on each orbit can be made totipotent, which roughly means "as rich as possible", and furthermore that the free group can be made dense in the ambient full group of the equivalence relation. This is joint work with Alessandro Carderi and Damien Gaboriau.

My recent work has involved taking questions asked for finite groups and considering them for infinite groups. There are various natural directions with this. In finite group theory, there exist many beautiful results regarding generation properties. One such notion is that of spread, and Scott Harper and Casey Donoven have raised several intriguing questions for spread for infinite groups (in https://arxiv.org/abs/1907.05498). A group $G$ has spread $k$ if for every $g_1,\ldots,g_k$ we can find an $h$ in $G$ such that $\langle g_i, h\rangle=G$. For any group we can say that if it has a proper quotient that is non-cyclic, then it has spread $0$. In the finite world there is then the astounding result - which is the work of many authors - that this condition on proper quotients is not just a necessary condition for positive spread, but is also a sufficient one. Harper-Donoven’s first question is therefore: is this the case for infinite groups? Well, no. But that’s for the trivial reason that we have infinite simple groups that are not 2-generated (and they point out that 3-generated examples are also known). But if we restrict ourselves to 2-generated groups, what happens? In this talk we’ll see the answer to this question. The arguments will be concrete (*) and accessible to a general audience.

(*) at the risk of ruining the punchline, we will find a 2-generated group that has every proper quotient cyclic but that has spread zero.

Let $G$ be a group and $S$ a generating set. Then the group $G$ naturally acts on the Cayley graph $\mathrm{Cay}(G,S)$ by left multiplications. The group $G$ is said to be rigid if there exists an $S$ such that the only automorphisms of $\mathrm{Cay}(G,S)$ are the ones coming from the action of $G$. While the classification of finite rigid groups was achieved in 1981, few results were known about infinite groups. In a recent work, with M. de la Salle we gave a complete classification of infinite finitely generated rigid groups. As a consequence, we also obtain that every finitely generated group admits a Cayley graph with countable automorphism group.

Kaplansky made various related conjectures about group rings, especially for torsion-free groups. For example, the zero divisors conjecture predicts that if $K$ is a field and $G$ is a torsion-free group, then the group ring $K[G]$ has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other group properties such as orderability, and present some recent progress.

This talk is based on my upcoming book chapter of the same name, to appear in Springer handbook of the mathematics of the arts and sciences. I will sample the myriad of ways in which mathematical experiment advances my research. The examples are problems I solved with techniques inspired by the methodology and writings of the late Jonathan Borwein. I will emphasize the tools, strategies, and the broader arc that research follows: from low-dimensional, specific, and visible, to high-dimensional and general. Topics include dynamical systems, geometry, optimization, error bounds, random walks, special functions, and number theory. Because I am mainly interested in tools and creative thinking, rather than specific theory, this talk should be accessible to a wide audience.

It is a long standing question whether a group of type $F$ that does not contain Baumslag–Solitar subgroups is necessarily hyperbolic. One-relator groups are of type $F$ and Louder and Wilton showed that if the defining relator has imprimitivity rank greater than $2$, they do not contain Baumslag-Solitar subgroups, so they conjecture that such groups are hyperbolic. Cashen and I verified the conjecture computationally for relators of length at most $17$. In this talk I'll introduce hyperbolic groups and the imprimitivity rank of elements in a free group. I’ll also discuss how to verify hyperbolicity using versions of combinatorial curvature on van Kampen diagrams.

I will address two problems about recognising surface groups. The first one is the classical problem of classifying Poincaré duality groups in dimension two. I will present a new approach to this, joint with Peter Kropholler. The second problem is about recognising surface groups among one-relator groups. Here I will present a new partial result, joint with Giles Gardam and Alan Logan.

First, I will give a brief introduction to the Riemann zeta-function ζ(s) and its connection with prime numbers. In particular, I will mention the famous “explicit formula” that gives an explicit connection between Chebyshev’s prime-counting function ψ(x) and an infinite sum that involves the zeros of ζ(s). Using the explicit formula, many questions about prime numbers can be reduced to questions about these zeros or sums over the zeros.

Motivated by such results, in the second half of the talk I will consider sums of the form ∑φ(γ), where φ is a function satisfying mild smoothness and monotonicity conditions, and γ ranges over the ordinates of nontrivial zeros ρ = β + iγ of ζ(s), with γ restricted to be in a given interval. I will show how the numerical estimation of such sums can be accelerated, and give some numerical examples.

The new results are joint work with Dave Platt (Bristol) and Tim Trudgian (UNSW). For preprints, see arXiv:2009.05251 and arXiv:2009.13791.

You can watch a video version at https://youtu.be/0rE-EopdSyQ instead, or in addition!

A full paper describing this talk can be found at https://arxiv.org/abs/2008.01812. Mathieu functions of period π or 2π, also called elliptic cylinder functions, were introduced in 1868 by Émile Mathieu together with so-called modified Mathieu functions, in order to help understand the vibrations of an elastic membrane set in a fixed elliptical hoop. These functions still occur frequently in applications today: our interest, for instance, was stimulated by a problem of pulsatile blood flow in a blood vessel compressed into an elliptical cross-section. This talk surveys and recapitulates some of the historical development of the theory and methods of computation for Mathieu functions and modified Mathieu functions and identifies some gaps in current software capability, particularly to do with double eigenvalues of the Mathieu equation. We demonstrate how to compute Puiseux expansions of the Mathieu eigenvalues about such double eigenvalues, and give methods to compute the generalized eigenfunctions that arise there. In examining Mathieu's original contribution, we bring out that his use of anti-secularity predates that of Lindstedt. For interest, we also provide short biographies of some of the major mathematical researchers involved in the history of the Mathieu functions: Émile Mathieu, Sir Edmund Whittaker, Edward Ince, and Gertrude Blanch.

he Post Correspondence Problem (PCP) is a classical problem in computer science that can be stated as: is it decidable whether given two morphisms $g$ and $h$ between two free semigroups $A$ and $B$, there is any nontrivial $x$ in $A$ such that $g(x)=h(x)$? This question can be phrased in terms of equalisers, asked in the context of free groups, and expanded: if the `equaliser' of $g$ and $h$ is defined to be the subgroup consisting of all $x$ where $g(x)=h(x)$, it is natural to wonder not only whether the equaliser is trivial, but what its rank or basis might be. While the PCP for semigroups is famously insoluble and acts as a source of undecidability in many areas of computer science, the PCP for free groups is open, as are the related questions about rank, basis, or further generalisations. However, in this talk we will show that there are links and surprising equivalences between these problems in free groups, and classes of maps for which we can give complete answers. This is joint work with Alan Logan.

Plane partitions are a two-dimensional analogue of integer partitions introduced by MacMahon in the 1890s. Various generating functions for plane partitions admit beautiful product forms, displaying an unexpected connection to the representation theory of classical groups and Lie algebras. Cylindric partitions, defined by Gessel and Krattenthaler in the 1990s, are an affine analogue of plane partitions.

In this talk I will explain what cylindric partitions are, discuss their connection with the representation theory of infinite dimensional Lie algebras, and describe some recent results on Rogers--Ramanujan-type identities arising from the study of cylindric partitions. No knowledge of representation theory will be assumed in this talk.

Fixed and moving boundary problems for the one-dimensional heat equation are considered. A unified approach to solving such problems is proposed by embedding a given initial boundary value problem into an appropriate initial value problem on the real line with arbitrary but given functions, whose solution is known. These arbitrary functions are determined by imposing that the solution of the initial value problem satisfies the given boundary conditions. Exact analytical solutions of some moving boundary problems that have not been previously obtained are provided. Moreover, examples of fixed boundary problems over semi-infinite and bounded intervals are given, thus providing an alternative approach to the usual methods of solution.

In this presentation, I will give an overview of the Ubiratan D'Ambrosio's concept of ethnomathematics and Elder Albert Marshal's concept of “two-eyed seeing.” I will address some of the dynamics between these two concepts and illustrate them with several examples that will include a brief analysis of geometry evident in a traditional Haida hat currently on display at the SFU Museum of Anthropology and the work of contemporary Salish artist Dylan Thomas.

A positive cone on a group $G$ is a subsemigroup $P$, such that $G$ is the disjoint union of $P$, $P^{-1}$ and the trivial element. Positive cones codify naturally $G$-left-invariant total orders on $G$. When $G$ is a finitely generated group, we will discuss whether or not a positive cone can be described by a regular language over the generators and how the ambient geometry of $G$ influences the geometry of a positive cone. This will be based on joint works with Juan Alonso, Joaquin Brum, Cristobal Rivas and Hang Lu Su.

Being of type $FP_2$ is an algebraic shadow of being finitely presented. A long standing question was whether these two classes are equivalent. This was shown to be false in the work of Bestvina and Brady. More recently, there are many new examples of groups of type $FP_2$ coming with various interesting properties. I will begin with an introduction to the finiteness property $FP_2$. I will end by giving a construction to find groups that are of type $FP_2(F)$ for all fields $F$ but not $FP_2(Z)$.

As educators, we need to assess our students for a variety of reasons from the mundane requirement to submit ranked scores to the arcane desire to encourage and track learning. I have developed my approach to oral examinations for undergraduate students in an attempt to support collaborative analysis of my student's understanding, as well as an opportunity for growth and discovery right up to the final moments of a course. I will present my experiences using oral assessments both alone and in combination with written work in multivariable calculus with mid-level students, in partial differential equations with upper-level students, and in introductory calculus with first-year students.

If $(K,f)$ is a difference field, and $a$ is a finite tuple in some difference field extending $K$, and such that $f(a)$ in $K(a)^{\mathrm{alg}}$, then we define $dd(a/K)=lim[K(f^k(a),a):K(a)]^{1/k}$, the distant degree of $a$ over $K$. This is an invariant of the difference field extension $K(a)^{\mathrm{alg}}/K$. We show that there is some $b$ in the difference field generated by $a$ over $K$, which is equi-algebraic with $a$ over $K$, and such that $dd(a/K)=[K(f(b),b):K(b)]$, i.e.: for every $k>0$, $f(b)$ in $K(b,f^k(b))$. Viewing $Aut(K(a)^{\mathrm{alg}}/K)$ as a locally compact group, this result is connected to results of Goerge Willis on scales of automorphisms of locally compact totally disconnected groups. I will explicit the correspondence between the two sets of results. (Joint with E. Hrushovski)

How difficult is it to solve a given computational problem? In a large class of computational problems, including the fixed-template Constraint Satisfaction Problems (CSPs), this fundamental question has a simple and beautiful answer: the more symmetrical the problem is, the easier is to solve it. The tight connection between the complexity of a CSP and a certain concept that captures its symmetry has fueled much of the progress in the area in the last 20 years. I will talk about this connection and some of the many tools that have been used to analyze the symmetries. The tools involve rather diverse areas of mathematics including algebra, analysis, combinatorics, logic, probability, and topology.

This discussion will revolve around Michael Donovan’s experience as a Fulbright Visiting Scholar in 2019-2020 and the important role of being an ambassador for Australia and Fulbright’s international relationship building. This talk will feature an aspect of the Fulbright application process called the personal statement. The personal statement is not about your academic standing or research tasks but a process to allow the Fulbright review committee to see who you are. It may appear to be a small additional element, but it highlights you and how you can fit within the Fulbright values as a ‘cultural ambassador’ for yourself, your institution, your research and the Fulbright program.

I will first introduce the idea of integer relations and discuss practical concerns for their computation by numeric techniques (i.e., using floating point arithmetic). To this end I will discuss the PSLQ and LLL algorithms (and will mention, in passing, some others). I will then extend the idea of integer relations into the relations consisting of algebraic integers. I will discuss, in particular, the case of algebraic integers from quadratic extension fields. As with the first part of the talk, I will discuss practical concerns for calculation by numeric techniques of these quadratic integer relations.

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of $GL(n, K)$ ($K$ - a field, $n \geq 3$) which is not contained in the center, contains $SL(n, K)$. A. Rosenberg gave description of normal subgroups of $GL(V)$, where $V$ is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations $g$ such that $g-id_V$ has finite dimensional range the proof is not complete. We fill this gap for countably dimensional $V$ giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field. Similar results for Lie algebras of matrices will be surveyed. The is based on results presented in https://arxiv.org/abs/1808.06873 and https://arxiv.org/abs/1806.01099. (joint work with Martyna Maciaszczyk and Sebastian Zurek.)

Highly transitive groups, i.e. groups admitting an embedding in $\mathrm{Sym}(\mathbb{N}) with dense image, form a wide class of groups. For instance, M. Hull and D. Osin proved that it contains all countable acylindrically hyperbolic groups with trivial finite radical. After an introduction to high transitiviy, I will present a theorem (from joint work with P. Fima, F. Le Maître and S. Moon) showing that many groups acting on trees are highly transitive. On the one hand, this theorem gives new examples of highly transitive groups. On the other hand, it is sharp because of results by A. Le Boudec and N. Matte Bon.

The Euler-Poincaré characteristic of a discrete group is an important (but also quite mysterious) invariant. It is usually just an integer or a rational number and reflects many quite significant properties. The realm of totally disconnected locally compact groups admits an analogue of the Euler-Poincaré characteristic which surprisingly is no longer just an integer, or a rational number, but a rational multiple of a Haar measure. Warning: in order to gain such an invariant the group has to be unimodular and satisfy some cohomological finiteness conditions. Examples of groups satisfying these additional conditions are the fundamental groups of finite trees of profinite groups. What arouses our curiosity is the fact that - in some cases - the Euler-Poincaré characteristic turns out to be miraculously related to a zeta-function. A large part of the talk will be devoted to the introduction of the just-cited objects. We aim at concluding the presentation by facing the concrete example of the group of F-points of a split semisimple simply connected algebraic group $G$ over $F$ (where $F$ denotes a non-archimedean locally compact field of residue characteristic $p$). Joint work with Gianmarco Chinello and Thomas Weigel.

The talk will review the history of parameterized complexity theory and its steady trajectory towards real-world accountability in the design and analysis of algorithms. The original motivation for the definition of the central concept of fixed parameter tractability (FPT) came from the graph minors project of Robertson and Seymour, with its key results on:

1. well-quasi ordering, and
2. FPT order tests.

The functor project explores how these can be widely generalized and used as complexity classification tools on a number of different levels, and how this approach may provide an opening for an "encountered-instance" (as contrasted with "worst–case") framework for complexity analysis and algorithm design.

We provide a new axiomatic framework, inspired by the work of Ol'shanskii, to describe explicitly certain irreducible unitary representations of second-countable non-discrete unimodular totally disconnected locally compact groups. We show that this setup applies to various families of automorphism groups of locally finite semiregular trees and right-angled buildings. The talk is based on material presented in arxiv.org/abs/2106.05730.

In the 90's, Nebbia conjectured that a group of tree automorphisms acting transitively on the tree's boundary must be of type I, that is, its unitary representations can in principal be classified. For key examples, such as Burger-Mozes groups, this conjecture is verified. Aiming for a better understanding of Nebbia's conjecture and a better understanding of representation theory of groups acting on trees, it is natural to ask whether there is a characterisation of type I groups acting on trees. In 2016, we introduced in collaboration with Cyril Houdayer a refinement of Nebbia's conjecture to a trichotomy, opposing type I groups with groups whose von Neumann algebra is non-amenable. For large classes of groups, including Burger-Mozes groups, we could verify this trichotomy. In this talk, I will motivate and introduce the conjecture trichotomy for groups acting on tress and explain how von Neumann algebraic techniques enter the picture.

In 1993 Brink and Howlett proved that finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognising the language of reduced words in the Coxeter group. This automaton is constructed in terms of the remarkable set of "elementary roots" in the associated root system. In this talk we outline the construction of Brink and Howlett. We also describe the minimal automaton recognising the language of reduced words, and prove necessary and sufficient conditions for the Brink-Howlett automaton to coincide with this minimal automaton. This resolves a conjecture of Hohlweg, Nadeau, and Williams, and is joint work with Yeeka Yau.

Geospatial artificial intelligence (geoAI) is an emerging scientific discipline that combines innovations in spatial science, artificial intelligence methods in machine learning and high-performance computing to extract knowledge from spatial big data. In this talk, I will discuss potential applications for environmental epidemiology, including the ability to incorporate large amounts of big spatial and temporal data in a variety of formats; computational efficiency; flexibility in algorithms and workflows to accommodate relevant characteristics of spatial (environmental) processes including spatial nonstationary; and scalability to model other environmental exposures across different geographic areas.

The contraction subgroup for $x$ in the locally compact group, $G$, $\mathrm{con}(x) = \left\{ g\in G \mid x^ngx^{-n} \to 1\text{ as }n\to\infty \right\}$, and the Levi subgroup is $\mathrm{lev}(x) = \left\{ g\in G \mid \{x^ngx^{-n}\}_{n\in\mathbb{Z}} \text{ has compact closure}\right\}$. The following will be shown. Let $G$ be a totally disconnected, locally compact group and $x\in G$. Let $y\in{\sf lev}(x)$. Then there are $x'\in G$ and a compact subgroup, $K\leq G$ such that:
-$K$ is normalised by $x'$ and $y$,
-$\mathrm{con}(x') = \mathrm{con}(x)$ and $\mathrm{lev}(x') = \mathrm{lev}(x)$ and
-the group $\langle x',y,K\rangle$ is abelian modulo $K$, and hence flat.
If no compact open subgroup of $G$ normalised by $x$ and no compact open subgroup of $\mathrm{lev}(x)$ normalised by $y$, then the flat-rank of $\langle x',y,K\rangle$ is equal to $2$.

Space weather events are associated with strong concentrations of magnetic field on the surface of the Sun. Currently, only low-fidelity space-weather forecasts are possible. Predicting the emergence time and size of the Sun’s active regions would be a significant step forward for space weather forecasting. The physics behind how these active regions emerge from the interior to the surface of the Sun is poorly understood. Only since the advent of a space-based monitoring campaign has it been possible to capture the emergence process of the magnetic field, Doppler velocity and intensity continuum of hundreds of active regions. By measuring the average motion of the polarities, surface velocities and pattern of convection, it is clear that convection plays an important role in the emergence of magnetic field on the Sun. Using machine learning and sophisticated simulations we aim to identify the convection cell dynamics associated with the emergence process, moving towards improved space weather forecasting.

Many astrophysical and laboratory plasmas have a very high conductivity. In the limit of infinite conductivity the magnetic field is “frozen” to the plasma, and consequently the magnetic field topology (defined by the connectivity and linkage of the field lines) is preserved. A breakdown in this “ideal” behaviour permits “magnetic reconnection” to occur, which is behind explosive energy release processes such as solar flare and sawtooth crashes in tokamaks. There exist analogous processes in high Reynolds number fluids and superfluids. Here I will develop the mathematical basis for understanding these concepts. I will also provide some illustrative examples of the importance of characterising magnetic complexity in determining where and how magnetic reconnection occurs.

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This national and international two-day symposium will address the pressing challenge of how to Indigenise mathematical practice at Universities, both in education and research. The methodology is of collaboration and sharing of knowledge and worldviews from within both Indigenous cultures and the cultures of mathematics and its allied disciplines.

The symposium will be organised around a collection of interconnected themes, each chaired by a partnership of Indigenous and non-Indigenous practitioners.

The physical location of this blended face to face and online symposium is significant. The Birabahn building of the Wollotuka Institute blends indoor and outdoor spaces, inviting new perspectives, whilst also having the capabilities for an international video-linked conference.

Speakers:

• Associate Professor Kathryn Butler of the Bundjalung and Worimi people, Director of the Wollotuka Institute, University of Newcastle
• Miss Tammy Small, student advancement manager, Wollotuka Institute, University of Newcastle
• Dr Henry Fowler of the Navajo people, Diné College, Tsaile, Arizona
• Associate Professor Edward Doolittle of the Mohawk people from Six Nations in southern Ontario, First Nations University of Canada
• Mr Dan Collins of the Worimi/Biripi nation, Yapug program convener, Wollotuka Institute, University of Newcastle
• Mr Nathan Towney of the Wiradjuri people from Wellington in NSW, Pro Vice Chancellor Indigenous Strategy and Leadership, University of Newcastle
• Dr Michael Donovan of the Gumbaynggir people, Macquarie University, Australia
• Mr Jade Kennedy, of the Yuin people of the Illawarra and South Coast of NSW, University of Wollongong
• Professor Rowena Ball, ANU
• Professor Mark Maclean, UBC
• Dr Veselin Jungic, Simon Fraser University
• Dr Naomi Borwein, Western Ontario University (ECR)
• Miss Jo-Ann Larkins, Federation University (ECR)
• Miss Amber Hughes, University of Newcastle (ECR)
• Dr Maureen Edwards, Wollongong University

The notion of amenable actions by discrete groups on $C^{\ast}$-algebras has been introduced by Claire Amantharaman-Delaroche more than thirty years ago, and has become a well understood theory with many applications. So it is somewhat surprising that an established theory of amenable actions by general locally compact groups has been missed for a very long time. We now present a theory which extends the discrete case and unifies several notions of approximation properties of actions which have been discussed in the literature. We also discuss the weak containment problem which asks wether an action $\alpha:G\to\mathrm{Aut}(A)$ is amenable if and only if the maximal and reduced crossed products coincide. In this lecture we report on joint work with Alcides Buss and Rufus Willett.

For a non-amenable group $G$, there can be many group $C^{\ast}$-algebras that lie naturally between the universal and the reduced $C^{\ast}$-algebra of $G$. These are called exotic group $C^{\ast}$-algebras. After a short introduction, I will explain that if $G$ is a simple Lie group or an appropriate locally compact group acting on a tree, the $L^p$-integrability properties of different spherical functions on $G$ (relative to a maximal compact subgroup) can be used to distinguish between exotic group $C^{\ast}$-algebras. This recovers results of Samei and Wiersma. Additionally, I will explain that under certain natural assumptions, the aforementioned exotic group $C^{\ast}$-algebras are the only ones coming from $G$-invariant ideals in the Fourier-Stieltjes algebra of $G$.
This is based on joint work with Dennis Heinig and Timo Siebenand.

We are all surrounded by data and the easiest way to understand what it is trying to tell us is to draw a picture that helps summarise them in some way. Often, standard visual techniques (like histograms and bar charts) are used, and used well but sometimes they aren’t. With the advent of new technologies that are now widely accessible to industry and the general public, there have been many attempts at proposing new ways of visualising data. Some of them have been very helpful while others have been less than useful. In this talk I shall present an overview of good, bad, and plain ugly attempts to visualise data in the news and social media and explain what makes them so. There will be zero maths involved, but plenty of pretty (and not so pretty) pictures . . . in colour.

We prove that for each finite index subgroup $H$ of the mapping class group of a closed hyperbolic surface, and for each real number $r>0$ there does not exist a faithful $C^{1+r}$--action of $H$ on a circle. (Joint with Thomas Koberda and Cristobal Rivas)

The famous Lehmer problem asks whether there is a gap between 1 and the Mahler measure of algebraic integers which are not roots of unity. Asked in 1933, this deep question concerning number theory has since then been connected to several other subjects. After introducing the concepts involved, we will briefly describe a few of these connections with the theory of linear groups. Then, we will discuss the equivalence of a weak form of the Lehmer conjecture and the "uniform discreteness" of cocompact lattices in semisimple Lie groups (conjectured by Margulis). Joint work with Lam Pham.

Artin groups emerged from the study of braid groups and complex hyperplane arrangements. Artin groups have very simple presentation, yet rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric aspects of these groups. No previous knowledge on Artin groups or Garside groups is required. This is joint work with D. Osajda.

Let $G$ be the fundamental group of a closed orientable surface of genus at least 2, and $\alpha$ an automorphism of $G$. In a celebrated result, Thurston showed that the mapping torus $G \rtimes_{\alpha} \mathbb{Z}$ is hyperbolic if and only if no power of $\alpha$ preserves a non-trivial conjugacy class. In this talk, I will describe joint work with François Dahmani, where we show that if $G$ is torsion-free hyperbolic, then $G \rtimes_{\alpha} \mathbb{Z}$ is relatively hyperbolic with ``optimal'' parabolic subgroups.