Research Group — Zero-Dimensional Symmetry
Symmetry in Newcastle This is a series of meetings with the aim of bringing together mathematicians working on Symmetry - broadly understood - that are based around Newcastle - also broadly understood. Topics of interest include all aspects of group theory and connections to computer science, dynamics, graph theory, logic, number theory, operator algebras, topology. Getting there: The venue (search for room) on the Callaghan campus of The University of Newcastle can be reached in at least three ways: By bus, going to "Mathematics Building, Ring Rd"; by train, going to "Warabrook Station" and walking about 15-20 minutes across the campus; or by car and parking, e.g., in carpark "P2". |
Upcoming Events |
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Date | Time | Room | Speaker | Title | |
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19.04.21 | 16.30-17.30 | Zoom | Zoé Chatzidakis | A new invariant for difference fields | |
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) | |||||
18.00-19.00 | Zoom | Laura Ciobanu | Free group homomorphisms and the Post Correspondence Problem | ||
The 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. | |||||
Past Events |
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08.03.21 | 18.30-19.30 | Zoom | Charlotte Hoffmann | Short words of high imprimitivity rank yield hyperbolic one-relator groups | |
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. | |||||
19.30-21.00 | Zoom | Dawid Kielak | Recognising surface groups | ||
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. | |||||
22.02.21 | 18.30-19.30 | Zoom | Paul-Henry Leemann | Cayley graphs with few automorphisms | |
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. | |||||
20.00-21.00 | Zoom | Giles Gardam | Kaplansky's conjectures (Slides) | ||
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. | |||||
25.01.21 | 18.30-19.30 | Zoom | François Le Maître | Dense totipotent free subgroups of full groups | |
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. | |||||
20.00-21.00 | Zoom | Charles Cox | Spread and infinite groups | ||
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. |
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23.11.20 | 18.30-19.30 | Zoom | William Hautekiet | Automorphism groups of transcendental field extensions | |
It is well-known that the Galois group of an (infinite) algebraic field extension is a profinite group. When the extension is transcendental, the automorphism group is no longer compact, but has a totally disconnected locally compact structure (TDLC for short). The study of TDLC groups was initiated by van Dantzig in 1936 and then restarted by Willis in 1994. In this talk some of Willis' concepts, such as tidy subgroups, the scale function, flat subgroups and directions are introduced and applied to examples of automorphism groups of transcendental field extensions. It remains unknown whether there exist conditions that a TDLC group must satisfy to be a Galois group. A suggestion of such a condition is made. | |||||
20.00-21.00 | Zoom | Florian Breuer | Realising general linear groups as Galois groups | ||
I will show how to construct field extensions with Galois groups isomorphic to general linear groups (with entries in various rings and fields) from the torsion of elliptic curves and Drinfeld modules. No prior knowledge of these structures is assumed. | |||||
09.11.20 | 20.00-21.00 | Zoom | Henry Bradford | Quantitative LEF and topological full groups | |
Topological full groups of minimal subshifts are an important source of exotic examples in geometric group theory, as well as being powerful invariants of symbolic dynamical systems. In 2011, Grigorchuk and Medynets proved that TFGs are LEF, that is, every finite subset of the multiplication table occurs in the multiplication table of some finite group. In this talk we explore some ways in which asymptotic properties of the finite groups which occur reflect asymptotic properties of the associated subshift. Joint work with Daniele Dona. | |||||
16.10.20 | 10.00-11.00 | Zoom | Rachel Skipper | Maximal Subgroups of Thompson's group V | |
There has been a long interest in embedding and non-embedding results for groups in the Thompson family. One way to get at results of this form is to classify maximal subgroups. In this talk, we will define certain labelings of binary trees and use them to produce a large family of new maximal subgroups of Thompson's group V. We also relate them to a conjecture about Thompson's group T. This is joint, ongoing work with Jim Belk, Collin Bleak, and Martyn Quick at the University of Saint Andrews. |
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11.30-12.30 | Zoom | Lawrence Reeves | Irrational-slope versions of Thompson’s groups T and V | ||
We consider irrational slope versions of T and V. We give infinite presentations for these groups and show how they can be represented by tree-pair diagrams. We also show that they have index-2 normal subgroups that are simple. This is joint work with Brita Nucinkis and Pep Burillo. |
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02.10.20 | 16.00-17.00 | Zoom | Alejandra Garrido | When is a piecewise (a.k.a. topological) full group locally compact? | |
Answer: Only when it's an ample group in the sense of Krieger (in particular, discrete, countable and locally finite) and has a Bratteli diagram satisfying certain conditions. Complaint: Wait, isn't Neretin's group a non-discrete, locally compact, topological full group? Retort: It is, but you need to use the correct topology! A fleshed-out version of the above conversation will be given in the talk. Based on joint work with Colin Reid. |
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17.30-18.30 | Zoom | Feyisayo Olukoya | The group of automorphisms of the shift dynamical system and the Higman-Thompson groups | ||
We give a survey of recent results exploring connections between the Higman-Thompson groups and their automorphism groups and the group of autmorphisms of the shift dynamical system. Our survey takes us from dynamical systems to group theory via groups of homeomorphisms with a segue through combinatorics, in particular, de Bruijn graphs. Joint work with Collin Bleak and Peter Cameron. |
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18.09.2020 | 15.00-16.00 | Zoom | Gabriel Verret | Local actions in vertex-transitive graphs | |
A graph is vertex-transitive if its group of automorphism acts transitively on its vertices. A very important concept in the study of these graphs is that of local action, that is, the permutation group induced by a vertex-stabiliser on the corresponding neighbourhood. I will explain some of its importance and discuss some attempts to generalise it to the case of directed graphs. | |||||
16.30-17.30 | Zoom | Michael Giudici | The synchronisation hierarchy for permutation groups | ||
The concept of a synchronising permutation group was introduced nearly 15 years ago as a possible way of approaching The Černý Conjecture. Such groups must be primitive. In an attempt to understand synchronising groups, a whole hierarchy of properties for a permutation group has been developed, namely, 2-transitive groups, $\mathbb{Q}$I-groups, spreading, separating, synchronsing, almost synchronising and primitive. Many surprising connections with other areas of mathematics such as finite geometry, graph theory, and design theory have arisen in the study of these properties. In this survey talk I will give an overview of the hierarchy and discuss what is known about which groups lie where. | |||||
04.09.2020 | 15.00-16.00 | Zoom | Murray Elder | Rewriting systems and geodetic graphs (Slides) | |
I will describe a new proof, joint with Adam Piggott (UQ), that groups presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 are exactly the plain groups (free products of finite and infinite cyclic groups). Our proof relies on a new result about properties of embedded circuits in geodetic graphs, which may be of independent interest in graph theory. | |||||
16.30-17.30 | Zoom | Ana Khukhro | A new characterisation of virtually free groups (Notes) | ||
A finite graph that can be obtained from a given graph by contracting edges and removing vertices and edges is said to be a minor of this graph. Minors have played an important role in graph theory, ever since the well-known result of Kuratowski that characterised planar graphs as those that do not admit the complete graph on 5 vertices nor the complete bipartite graph on (3,3) vertices as minors. In this talk, we will explore how this concept interacts with some notions from geometric group theory, and describe a new characterisation of virtually free groups in terms of minors of their Cayley graphs. | |||||
21.08.2020 | 15.00-16.00 | Zoom | Kasia Jankiewicz | Residual finiteness of certain 2-dimensional Artin groups | |
We show that many 2-dimensional Artin groups are residually finite. This includes Artin groups on three generators with labels at least 3, where either at least one label is even, or at most one label is equal 3. The result relies on decomposition of these Artin groups as graphs of finite rank free groups. | |||||
16.30-17.30 | Zoom | Simon Smith | Infinite primitive permutation groups, cartesian decompositions, and topologically simple locally compact groups (Slides) | ||
A non-compact, compactly generated, locally compact group whose proper quotients are all compact is called just-non-compact. Discrete just-non-compact groups are John Wilson’s famous just infinite groups. In this talk, I’ll describe an ongoing project to use permutation groups to better understand the class of just-non-compact groups that are totally disconnected. An important step for this project has recently been completed: there is now a structure theorem for non-compact tdlc groups G that have a compact open subgroup that is maximal. Using this structure theorem, together with Cheryl Praeger and Csaba Schneider’s recent work on homogeneous cartesian decompositions, one can deduce a neat test for whether the monolith of such a group G is a one-ended group in the class S of nondiscrete, topologically simple, compactly generated, tdlc groups. This class S plays a fundamental role in the structure theory of compactly generated tdlc groups, and few types of groups in S are known. | |||||
07.08.2020 | 15.00-16.00 | Zoom | Tony Guttmann | On the amenability of Thompson's Group $F$ | |
In 1967 Richard Thompson introduced the group $F$, hoping that it was non-amenable, since then it would disprove the von Neumann conjecture. Though the conjecture has subsequently been disproved, the question of the amenability of Thompson's group F has still not been rigorously settled. In this talk I will present the most comprehensive numerical attack on this problem that has yet been mounted. I will first give a history of the problem, including mention of the many incorrect "proofs" of amenability or non-amenability. Then I will give details of a new, efficient algorithm for obtaining terms of the co-growth sequence. Finally I will describe a number of numerical methods to analyse the co-growth sequences of a number of infinite, finitely-generated groups, and show how these methods provide compelling evidence (though of course not a proof) that Thompson's group F is not amenable. I will also describe an alternative route to a rigorous proof. (This is joint work with Andrew Elvey Price). | |||||
16.30-17.30 | Zoom | Collin Bleak | On the complexity of elementary amenable subgroups of R. Thompson's group $F$ | ||
The theory of EG, the class of elementary amenable groups, has developed steadily since the class was introduced constructively by Day in 1957. At that time, it was unclear whether or not EG was equal to the class AG of all amenable groups. Highlights of this development certainly include Chou's article in 1980 which develops much of the basic structure theory of the class EG, and Grigorchuk's 1985 result showing that the first Grigorchuk group $\Gamma$ is amenable but not elementary amenable. In this talk we report on work where we demonstrate the existence of a family of finitely generated subgroups of Richard Thompson’s group $F$ which is strictly well-ordered by the embeddability relation in type $\varepsilon_{0}+1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In this way, for each $\alpha<\varepsilon_{0}$, we obtain a ﬁnitely generated elementary amenable subgroup of F whose EA-class is $\alpha+2$. The talk will be pitched for an algebraically inclined audience, but little background knowledge will be assumed. Joint work with Matthew Brin and Justin Moore. | |||||
17.07.2020 | 15.00-16.00 | Zoom | Harry Hyungryul Baik | Normal generators for mapping class groups are abundant in the fibered cone (Notes) | |
We show that for almost all primitive integral cohomology classes in the fibered cone of a closed fibered hyperbolic 3-manifold, the monodromy normally generates the mapping class group of the fiber. The key idea of the proof is to use Fried’s theory of suspension flow and dynamic blow-up of Mosher. If the time permits, we also discuss the non-existence of the analog of Fried’s continuous extension of the normalized entropy over the fibered face in the case of asymptotic translation lengths on the curve complex. This talk is based on joint work with Eiko Kin, Hyunshik Shin and Chenxi Wu. | |||||
16.30-17.30 | Zoom | Federico Vigolo | Asymptotic expander graphs (Slides) | ||
A sequence of expanders is a family of finite graphs that are sparse yet highly connected. Such families of graphs are fundamental object that found a wealth of applications throughout mathematics and computer science. This talk is centred around an "asymptotic" weakening of the notion of expansion. The original motivation for this asymptotic notion comes from the study of operator algebras associated with metric spaces. Further motivation comes from some recent works which established a connection between asymptotic expansion and strongly ergodic actions. I will give a non-technical introduction to this topic, highlighting the relations with usual expanders and group actions. | |||||
26.06.2020 | 14.00-15.00 | Zoom | Tianyi Zheng | Neretin groups admit no non-trivial invariant random subgroups (Slides) | |
We explain the proof that Neretin groups have no nontrivial ergodic invariant random subgroups (IRS). Equivalently, any non-trivial ergodic p.m.p. action of Neretin’s group is essentially free. This property can be thought of as simplicity in the sense of measurable dynamics; while Neretin groups were known to be abstractly simple by a result of Kapoudjian. The heart of the proof is a “double commutator” lemma for IRSs of elliptic subgroups. | |||||
16.00-17.00 | Zoom | Hiroki Matui | Various examples of topological full groups (Slides) | ||
I will begin with the definition of topological full groups and explain various examples of them. The topological full group arising from a minimal homeomorphism on a Cantor set gave the first example of finitely generated simple groups that are amenable and infinite. The topological full groups of one-sided shifts of finite type are viewed as generalization of the Higman-Thompson groups. Based on these two fundamental examples, I will discuss recent development of the study around topological full groups. | |||||
05.06.2020 | 15.00-16.00 | Zoom | Federico Berlai | From hyperbolicity to hierarchical hyperbolicity | |
Hierarchically hyperbolic groups (HHGs) and spaces are recently-introduced generalisations of (Gromov-) hyperbolic groups and spaces. Other examples of HHGs include mapping class groups, right-angled Artin/Coxeter groups, and many groups acting properly and cocompactly on CAT(0) cube complexes. After a substantial introduction and motivation, I will present a combination theorem for hierarchically hyperbolic groups. As a corollary, any graph product of finitely many HHGs is itself a HHG. Joint work with B. Robbio. | |||||
16.30-17.00 | Zoom | Mark Hagen | Hierarchical hyperbolicity from actions on simplicial complexes | ||
The notion of a "hierarchically hyperbolic space/group" grows out of geometric similarities between CAT(0) cubical groups and mapping class groups. Hierarchical hyperbolicity is a "coarse nonpositive curvature" property that is more restrictive than acylindrical hyperbolicity but general enough to include many of the usual suspects in geometric group theory. The class of hierarchically hyperbolic groups is also closed under various procedures for constructing new groups from old, and the theory can be used, for example, to bound the asymptotic dimension and to study quasi-isometric rigidity for various groups. One disadvantage of the theory is that the definition --- which is coarse-geometric and just an abstraction of properties of mapping class groups and cube complexes --- is complicated. We therefore present a comparatively simple sufficient condition for a group to be hierarchically hyperbolic, in terms of an action on a hyperbolic simplicial complex. I will discuss some applications of this criterion to mapping class groups and (non-right-angled) Artin groups. This is joint work with Jason Behrstock, Alexandre Martin, and Alessandro Sisto. | |||||
15.05.2020 | 15.00-16.00 | Zoom | Alex Bishop | Geodesic Growth in Virtually Abelian Groups | |
Bridson, Burillo, Elder and Šunić asked if there exists a group with intermediate geodesic growth and if there is a characterisation of groups with polynomial geodesic growth. Towards these questions, they showed that there is no nilpotent group with intermediate geodesic growth, and they provided a sufficient condition for a virtually abelian group to have polynomial geodesic growth. In this talk, we take the next step in this study and show that the geodesic growth for a finitely generated virtually abelian group is either polynomial or exponential; and that the generating function of this geodesic growth series is holonomic, and rational in the polynomial growth case. To obtain this result, we will make use of the combinatorial properties of the class of linearly constrained language as studied by Massazza. In addition, we show that the language of geodesics of a virtually abelian group is blind multicounter. | |||||
16.30-17.00 | Zoom | James East | Presentations for tensor categories | ||
Many well-known families of groups and semigroups have natural categorical analogues: e.g., full transformation categories, symmetric inverse categories, as well as categories of partitions, Brauer/Temperley-Lieb diagrams, braids and vines. This talk discusses presentations (by generators and relations) for such categories, utilising additional tensor/monoidal operations. The methods are quite general, and apply to a wide class of (strict) tensor categories with one-sided units. | |||||
01.05.2020 | 15.00-16.00 | Zoom | Yeeka Yau | Minimal automata for Coxeter groups | |
In their celebrated 1993 paper, Brink and Howlett proved that all finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognising the language of reduced words in a Coxeter group. This automaton is not minimal in general, and recently Christophe Hohlweg, Philippe Nadeau and Nathan Williams stated a conjectural criteria for the minimality. In this talk we will explain these concepts, and outline the proof of the conjecture of Hohlweg, Nadeau, and Williams. We will also describe an alternative algorithm to minimise any finite state automaton recognising the language of reduced words in a Coxeter group, which utilises the associated root system of the group. This work is joint with James Parkinson. |
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16.30-17.30 | Zoom | Adam Piggott | The automorphism groups of the easiest infinite groups still present many mysteries (Slides) | ||
Free groups, and free products of finite groups, are the easiest non-abelian infinite groups to think about. Yet the automorphism groups of such groups still present significant mysteries. We discuss a program of research concerning automorphisms of easily understood infinite groups. | |||||
06.03.2020 | 10.00-11.00 | V 107 | Waltraud Lederle | Conjugacy and dynamics in the almost automorphism group of a tree | |
We define the almost automorphism group of a regular tree, also known as Neretin's group, and determine when two elements are conjugate. (joint work with Gil Goffer) | |||||
11.30-12.30 | V 107 | Mark Pengitore | Translation-like actions on nilpotent groups | ||
Whyte introduced translation-like actions of groups as a geometric generalization of subgroup containment. He then proved a geometric reformulation of the von Neumann conjecture by demonstrating that a finitely generated group is non amenable if and only it admits a translation-like action by a non-abelian free group. This provides motivation for the study of what groups can act translation-like on other groups. As a consequence of Gromov’s polynomial growth theorem, virtually nilpotent groups can act translation-like on other nilpotent groups. We demonstrate that if two nilpotent groups have the same growth, but non-isomorphic Carnot completions, then they can't act translation-like on each other. (joint work with David Cohen) | |||||
14.00-15.00 | V 107 | Jeroen Schillewaert | Fixed points for group actions on $2$-dimensional affine buildings | ||
We prove a local-to-global result for fixed points of groups acting on $2$-dimensional affine buildings (possibly non-discrete, and not of type $\tilde{G}_{2}$). In the discrete case, our theorem establishes two conjectures by Marquis. (joint work with Koen Struyve and Anne Thomas) | |||||
15.30-16.30 | V 107 | Francois Thilmany | Lattices of minimal covolume in $\mathrm{SL}_n$ | ||
A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices of $\mathrm{SL}_2(\mathbb{R})$. In general, given a semisimple Lie group $G$ over some local field $F$, one may ask which lattices in $G$ attain the smallest covolume. A complete answer to this question seems out of reach at the moment; nevertheless, many steps have been made in the last decades. Inspired by Siegel's result, Lubotzky determined that a lattice of minimal covolume in $\mathrm{SL}_2(F)$ with $F=\mathbb{F}_q((t))$ is given by the so-called characteristic $p$ modular group $\mathrm{SL}_2(\mathbb{F}_q[1/t])$. He noted that, in contrast with Siegel’s lattice, the quotient by $\mathrm{SL}_2(\mathbb{F}_q[1/t])$ was not compact, and asked what the typical situation should be: "for a semisimple Lie group over a local field, is a lattice of minimal covolume a cocompact or nonuniform lattice?". In the talk, we will review some of the known results, and then discuss the case of $\mathrm{SL}_n(\mathbb{R})$ for $n > 2$. It turns out that, up to automorphism, the unique lattice of minimal covolume in $\mathrm{SL}_n(\mathbb{R})$ ($n > 2$) is $\mathrm{SL}_n(\mathbb{Z})$. In particular, it is not uniform, giving a partial answer to Lubotzky’s question in this case. |
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01.11.2019 | 12.00-13.00 | V 109 | Anthony Dooley | Classification of non-singular systems and critical dimension | |
A non-singular measurable dynamical system is a measure space $X$ whose measure $\mu$ has the property that $\mu $ and $\mu \circ T$ are equivalent measures (in the sense that they have the same sets of measure zero). Here $T$ is a bimeasurable invertible transformation of $X$. The basic building blocks are the \emph{ergodic} measures. Von Neumann proposed a classification of non-singular ergodic dynamical systems, and this has been elaborated subsequently by Krieger, Connes and others. This work has deep connections with C*-algebras. I will describe some work of myself, collaborators and students which explore the classification of dynamical systems from the point of view of measure theory. In particular, we have recently been exploring the notion of critical dimension, a study of the rate of growth of sums of Radon-Nikodym derivatives $\Sigma_{k=1}^n \frac{d\mu \circ T^k}{d\mu}$. Recently, we have been replacing the single transformation $T$ with a group acting on the space $X$. | |||||
14.00-15.00 | V 109 | Colin Reid | Piecewise powers of a minimal homeomorphism of the Cantor set | ||
Let $X$ be the Cantor set and let $g$ be a minimal homeomorphism of $X$ (that is, every orbit is dense). Then the topological full group $\tau[g]$ of $g$ consists of all homeomorphisms $h$ of $X$ that act 'piecewise' as powers of $g$, in other words, $X$ can be partitioned into finitely many clopen pieces $X_1,...,X_n$ such that for each $i$, $h$ acts on $X_i$ as a constant power of $g$. Such groups have attracted considerable interest in dynamical systems and group theory, for instance they characterize the homeomorphism up to flip conjugacy (Giordano-Putnam-Skau) and they provided the first known examples of infinite finitely generated simple amenable groups (Juschenko--Monod). My talk is motivated by the following question: given $h\in\tau[g]$ for some minimal homeomorphism $g$, what can the closures of orbits of $h$ look like? Certainly $h\in\tau[g]$ is not minimal in general, but it turns out to be quite close to being minimal, in the following sense: there is a decomposition of $X$ into finitely many clopen invariant pieces, such that on each piece $h$ acts a homeomorphism that is either minimal or of finite order. Moreover, on each of the minimal parts of $h$, then either $h$ or $h^{-1}$ has a 'positive drift' with respect to the orbits of $g$; in fact, it can be written in a canonical way as a conjugate of a product of induced transformations (aka first return maps) of $g$. No background knowledge of topological full groups is required; I will introduce all the necessary concepts in the talk. | |||||
15.30-16.30 | V 109 | Michael Barnsley | Dynamics on Fractals | ||
I will outline a new theory of fractal tilings. The approach uses graph iterated function systems (IFS) and centers on underlying symbolic shift spaces. These provide a zero dimensional representation of the intricate relationship between shift dynamics on fractals and renormalization dynamics on spaces of tilings. The ideas I will describe unify, simplify, and substantially extend key concepts in foundational papers by Solomyak, Anderson and Putnam, and others. In effect, IFS theory on the one hand, and self-similar tiling theory on the other, are unified. The work presented is largely new and has not yet been submitted for publication. It is joint work with Andrew Vince (UFL) and Louisa Barnsley. The presentation will include links to detailed notes. The figures illustrate 2d fractal tilings. By way of recommended background reading I mention the following awardwinning paper: M. F. Barnsley, A. Vince, Self-similar polygonal tilings, Amer. Math. Monthly 124 (1017) 905-921. |
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06.09.2019 | 14.30-15.30 | V 109 | Sasha Fish | Patterns in sets of positive density in trees and buildings | |
We will discuss what are the patterns that necessarily occur in sets of positive density in homogeneous trees and certain affine buildings. Based on joint work with Michael Bjorklund (Chalmers) and James Parkinson (University of Sydney). | |||||
16.00-17.00 | V 109 | Michael Coons | Integer sequences, asymptotics and diffraction | ||
For some time now, I have been trying to understand the intricacy and complexity of integer sequences from a variety of different viewpoints and at least at some level trying to reconcile these viewpoints. However vague that sounds - and it certainly is vague to me - in this talk I hope to explain this sentiment a bit. While a variety of results will be considered, I will focus closely on two examples of wider interest, the Thue-Morse sequence and the set of $k$-free integers. | |||||
02.08.2019 | 12.00-13.00 | V 109 | Brian Alspach | Honeycomb Toroidal Graphs | |
The honeycomb toroidal graphs are a family of graphs I have been looking at now and then for thirty years. I shall discuss an ongoing project dealing with hamiltonicity as well as some of their properties which have recently interested the computer architecture community. | |||||
14.00-15.00 | V 109 | John Bamberg | Symmetric finite generalised polygons | ||
Finite generalised polygons are the rank 2 irreducible spherical buildings, and include projective planes and the generalised quadrangles, hexagons, and octagons. Since the early work of Ostrom and Wagner on the automorphism groups of finite projective planes, there has been great interest in what the automorphism groups of generalised polygons can be, and in particular, whether it is possible to classify generalised polygons with a prescribed symmetry condition. For example, the finite Moufang polygons are the 'classical' examples by a theorem of Fong and Seitz (1973-1974) (and the infinite examples were classified in the work of Tits and Weiss (2002)). In this talk, we give an overview of some recent results on the study of symmetric finite generalised polygons, and in particular, on the work of the speaker with Cai Heng Li and Eric Swartz. | |||||
15.30-16.30 | V 109 | Marston Conder | Edge-transitive graphs and maps | ||
In this talk I'll describe some recent discoveries about edge-transitive graphs and edge-transitive maps. These are objects that have received relatively little attention compared with their vertex-transitive and arc-transitive siblings. First I will explain a new approach (taken in joint work with Gabriel Verret) to finding all edge-transitive graphs of small order, using single and double actions of transitive permutation groups. This has resulted in the determination of all edge-transitive graphs of order up to 47 (the best possible just now, because the transitive groups of degree 48 are not known), and bipartite edge-transitive graphs of order up to 63. It also led us to the answer to a 1967 question by Folkman about the valency-to-order ratio for regular graphs that are edge- but not vertex-transitive. Then I'll describe some recent work on edge-transitive maps, helped along by workshops at Oaxaca and Banff in 2017. I'll explain how such maps fall into 14 natural classes (two of which are the classes of regular and chiral maps), and how graphs in each class may be constructed and analysed. This will include the answers to some 18-year-old questions by Širáň, Tucker and Watkins about the existence of particular kinds of such maps on orientable and non-orientable surfaces. |
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03.05.2019 | 14.00-15.00 | W 238 | Heiko Dietrich | Quotient algorithms (a.k.a. how to compute with finitely presented groups) | |
In this talk, I will survey some of the famous quotient algorithms that can be used to compute efficiently with finitely presented groups. The last part of the talk will be about joint work with Alexander Hulpke (Colorado State University): we have looked at quotient algorithms for non-solvable groups, and I will report on the findings so far. | |||||
15.30-16.30 | W 238 | Youming Qiao | Isomorphism testing problems: in light of Babai's graph isomorphism breakthrough | ||
In computer science, an isomorphism testing problem asks whether two objects are in the same orbit under a group action. The most famous problem of this type has been the graph isomorphism problem. In late 2015, L. Babai announced a quasipolynomial-time algorithm for the graph isomorphism problem, which is widely regarded as a breakthrough in theoretical computer science. This leads to a natural question, that is, which isomorphism testing problems should naturally draw our attention for further exploration? | |||||
12.00-13.00 | W 238 | Nicole Sutherland | Computations of Galois groups and splitting fields | ||
The Galois group of a polynomial is the automorphism group of its splitting field. These automorphisms act by permuting the roots of the polynomial so that a Galois group will be a subgroup of a symmetric group. Using the Galois group the splitting field of a polynomial can be computed more efficiently than otherwise, using the knowledge of the symmetries of the roots. I will present an algorithm developed by Fieker and Klueners, which I have extended, for computing Galois groups of polynomials over arithmetic fields as well as approaches to computing splitting fields using the symmetries of the roots. | |||||
05.04.2019 | 12.00-13.00 | W 238 | Arnaud Brothier | Jones' actions of the Thompson's groups: applications to group theory and mathematical physics | |
Motivating in constructing conformal field theories Jones recently discovered a very general process that produces actions of the Thompson groups $F$,$T$ and $V$ such as unitary representations or actions on $C^{\ast}$-algebras. I will give a general panorama of this construction along with many examples and present various applications regarding analytical properties of groups and, if time permits, in lattice theory (e.g. quantum field theory). | |||||
14.00-15.00 | W 238 | Lawrence Reeves | An irrational-slope Thompson's group | ||
Let $t$ be the the multiplicative inverse of the golden mean. In 1995 Sean Cleary introduced the irrational-slope Thompson's group $F_t$, which is the group of piecewise-linear maps of the interval $[0,1]$ with breaks in $Z[t]$ and slopes powers of $t$. In this talk we describe this group using tree-pair diagrams, and then demonstrate a finite presentation, a normal form, and prove that its commutator subgroup is simple. This group is the first example of a group of piecewise-linear maps of the interval whose abelianisation has torsion, and it is an open problem whether this group is a subgroup of Thompson's group $F$. | |||||
15.30-16.30 | W 238 | Richard Garner | Topos-theoretic aspects of self-similarity | ||
A Jonsson-Tarski algebra is a set $X$ endowed with an isomorphism $X\to XxX$. As observed by Freyd, the category of Jonsson-Tarski algebras is a Grothendieck topos - a highly structured mathematical object which is at once a generalised topological space, and a generalised universe of sets. In particular, one can do algebra, topology and functional analysis inside the Jonsson-Tarski topos, and on doing so, the following objects simply pop out: Cantor space; Thompson's group V; the Leavitt algebra L2; the Cuntz semigroup S2; and the reduced $C^{\ast}$-algebra of S2. The first objective of this talk is to explain how this happens. The second objective is to describe other "self-similar toposes" associated to, for example, self-similar group actions, directed graphs and higher-rank graphs; and again, each such topos contains within it a familiar menagerie of algebraic-analytic objects. If time permits, I will also explain a further intriguing example which gives rise to Thompson's group F and, I suspect, the Farey AF algebra. No expertise in topos theory is required; such background as is necessary will be developed in the talk. |
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In this edition, three 2017 ARC Laureate Fellows in mathematics outline their projects. Descriptions of these projects are given as abstracts below. |
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15.03.2019 | 12.00-13.00 | Purdue Room | Mathai Varghese | Advances in Index Theory | |
The project aims to develop novel techniques to investigate Geometric analysis on infinite dimensional bundles, as well as Geometric analysis of pathological spaces with Cantor set as fibre, that arise in models for the fractional quantum Hall effect and topological matter, areas recognised with the 1998 and 2016 Nobel Prizes. Building on the applicant's expertise in the area, the project will involve postgraduate and postdoctoral training in order to enhance Australia's position at the forefront of international research in Geometric Analysis. Ultimately, the project will enhance Australia's leading position in the area of Index Theory by developing novel techniques to solve challenging conjectures, and mentoring HDR students and ECRs. | |||||
14.00-15.00 | Purdue Room | Fedor Sukochev | Breakthrough methods for noncommutative calculus | ||
This project aims to solve hard, outstanding problems which have impeded our ability to progress in the area of quantum or noncommutative calculus. Calculus has provided an invaluable tool to science, enabling scientific and technological revolutions throughout the past two centuries. The project will initiate a program of collaboration among top mathematical researchers from around the world and bring together two separate mathematical areas into a powerful new set of tools. The outcomes from the project will impact research at the forefront of mathematical physics and other sciences and enhance Australia's reputation and standing. | |||||
15.30-16.30 | Purdue Room | George Willis | Zero-Dimensional Symmetry and its Ramifications | ||
This project aims to investigate algebraic objects known as 0-dimensional groups, which are a mathematical tool for analysing the symmetry of infinite networks. Group theory has been used to classify possible types of symmetry in various contexts for nearly two centuries now, and 0-dimensional groups are the current frontier of knowledge. The expected outcome of the project is that the understanding of the abstract groups will be substantially advanced, and that this understanding will shed light on structures possessing 0-dimensional symmetry. In addition to being cultural achievements in their own right, advances in group theory such as this also often have significant translational benefits. This will provide benefits such as the creation of tools relevant to information science and researchers trained in the use of these tools. | |||||
01.03.2019 | 12.00-13.00 | W 104 | Marcelo Laca | An introduction to KMS states and two suprising examples | |
The KMS condition for equilibrium states of C*-dynamical systems has been around since the 1960's. With the introduction of systems arising from number theory and from semigroup dynamics following pioneering work of Bost and Connes, their study has accelerated significantly in the last 25 years. I will give a brief introduction to C*-dynamical systems and their KMS states and discuss two constructions that exhibit fascinating connections with key open questions in mathematics such as Hilbert's 12th problem on explicit class field theory and Furstenberg's x2 x3 conjecture. | |||||
14.00-15.00 | W 104 | Zahra Afsar | KMS states of $C^*$-algebras of $*$-commuting local homeomorphisms and applications in $k$-graph algebras | ||
In this talk, I will show how to build $C^*$-algebras using a family of local homeomorphisms. Then we will compute the KMS states of the resulted algebras using Laca-Neshveyev machinery. Then I will apply this result to $C^*$-algebras of $K$-graphs and obtain interesting $C^*$-algebraic information about $k$-graph algebras. This talk is based on a joint work with Astrid an Huef and Iain Raeburn. | |||||
15.30-16.30 | W 104 | Aidan Sims | What equilibrium states KMS states for self-similar actions have to do with fixed-point theory | ||
Using a variant of the Laca-Raeburn program for calculating KMS states, Laca, Raeburn, Ramagge and Whittaker showed that, at any inverse temperature above a critical value, the KMS states arising from self-similar actions of groups (or groupoids) $G$ are parameterised by traces on C*(G). The parameterisation takes the form of a self-mapping $\chi$ of the trace space of C*(G) that is built from the structure of the stabilisers of the self-similar action. I will outline how this works, and then sketch how to see that $\chi$ has a unique fixed-point, which picks out the ``preferred" trace of C*(G) corresponding to the only KMS state that persists at the critical inverse temperature. The first part of this will be an exposition of results of Laca-Raeburn-Ramagge-Whittaker. The second part is joint work with Joan Claramunt. |