What's On at UWA

Speaker: Hendrik Van Maldeghem (Universiteit Gent/Ghent University) Title: Pappian unitals in Pappian projective planes Time and place: 16:00 Friday 15/09/2017 in Weatherburn LT

Abstract: We define a class of objects in pappian projective planes by a simple algebraic formula and parametrized by quadratic field extensions. These objects turn out to be ordinary hermitian curves if the extension is separable, and projections of certain quadrics otherwise. Endowed with the secant lines, we call the resulting point-block incidence structures "Pappian unitals". These have some remarkable properties such as the lack of O'Nan configurations, the admittance of translations and a nontrivial group of projectivities, and a characterization via a geometric construction using the André representation of the projective plane relative to the quadratic extension. We classify the embeddings of all Pappian unitals in arbitrary pappian projective planes, recovering and extending a recent result by Korchmáros, Cossidente and Szönyi for finite hermitian unitals.

Speaker: Jeroen Schillewaert (University of Auckland) Title: Small maximal independent sets Time and place: 16:00 Friday 22/09/2017 in Engineering LT2 G04 (NOTE unusual location)

Abstract: We study random constructions in incidence structures using a general theorem on set systems. Our main result applies to a wide variety of well-studied problems in finite geometry to give almost tight bounds on the sizes of various substructures. This is joint work with Jacques Verstraete (UCSD).

Past and future seminars may be found at http://www.maths.uwa.edu.au/~glasby/S17.html

Speaker: Luke Morgan (University of Western Australia) Title: Cayley digraphs and Cayley index Time and place: 16:00 Friday 06/10/2017 in Weatherburn LT

Abstract: The Cayley index of a Cayley digraph on a finite group G is the index of (the regular representation of) G in the automorphism group of that digraph. The minimum Cayley index of one is attained by digraphs called DRRs (Digraphical Regular Representations). These are classified by Babai who shows that, apart from five groups, every finite group admits a DRR. That is, apart from five exceptions, every finite group G has a Cayley digraph such that G is the full automorphism group of that digraph. In this talk, we'll consider the question of what might make the Cayley index "large" relative to the number of vertices (= the order of the group). A result of Morris shows that any Cayley digraph on a cyclic $p$-group (with $p$ an odd prime) has Cayley index super-exponential in $p$, if there exists another distinct regular subgroup. This result was later generalised to $p=2$. So these results say that if a digraph is Cayley for two distinct $p$-groups, one of which is cyclic, the Cayley index is "large". In joint work with Morris and Verret, we considered if cyclic $p$-groups are exceptional in this respect. We found that, in contrast to the previous results, every non-cyclic abelian $p$-group ($p$ odd) of order at least $p^3$ admits a Cayley digraph of Cayley index $p$ that admits two distinct regular subgroups. I'll show how this result works, and give some further questions on this topic.

Past and future seminars may be found at http://www.maths.uwa.edu.au/~glasby/S17.html

There will be refreshments in Maths & Stats tea room at 15:40, and we will go for drinks at the UWA Tavern or the UniClub after the seminar. All are most welcome.

Speaker: John Bamberg (University of Western Australia) Title: q-analogues of designs and the 2-Fano plane Time and place: 16:00 Friday 13/10/2017 in Weatherburn LT

Abstract: A q-analogue of a t-design, called a t-(n,k, ambda)_q design, is a set of k-subspaces of F_q^n such that each t-subspace is contained in exactly ambda elements. If t = 1, the design is a q-analogue of a Steiner system, and is denoted S_q(t,k,n). Braun, Østergård, Vardy and Wassermann have shown that q-Steiner systems do exist, but existence is not known in the case of S_q[2; 3; 7], the q-analogue of a Fano plane. Kiermeir, Kurz and Wassermann have shown that if a S_2[2; 3; 7] exists, then the order of its automorphism group would be at most 2. We will present the progress of our search for S_2[2; 3; 7] (or its non-existence), in particular, we have found that no 2-Fano plane exists with automorphism group of order 2. This is join work with Ferdinand Ihringer, Jesse Lansdown, and Gordon Royle.

Past and future seminars may be found at http://www.maths.uwa.edu.au/~glasby/S17.html

There will be `cake' in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub or Student Tavern for a drink. Note the unusual location.

Speaker: Hongxue Liang (University of Western Australia)

Title: Flag-transitive point-primitive non-symmetric 2-(v,k,2) designs

Time and place: 16:00 Friday 27/10/2017 in Woolnough Lecture Theatre 2 GGGL:107

Abstract: A 2-(v,k,&#955;) design is a finite incidence structure D=(P, B) consisting of v points and b blocks such that every block is incident with k points, every point is incident with r blocks, and any two distinct points are incident with exactly &#955; blocks. D is called symmetric if v=b (or equivalently r=k), and non-trivial if 1< k< v. A flag of D is an incident point-block pair (&#945;, B) where &#945; is a point and B is a block. An automorphism of D is a permutation of the points which also permutes the blocks. The set of all automorphisms of D with the composition of maps is a group, denoted by Aut(D). A subgroup G&#8804;Aut( D) is called point-primitive if it acts primitively on P and flag-transitive if it acts transitively on the set of flags of D. In this talk, we will focus on the flag-transitive point-primitive automorphism groups of non-symmetric 2-(v,k,2) designs.

Past and future seminars may be found at http://www.maths.uwa.edu.au/~glasby/S17.html

Speaker: Yian Xu (University of Western Australia) Title: Constructing a 2-arc-transitive cover for a certain hypercube Time and place: 16:00 Friday 03/11/2017 in Weatherburn LT

Abstract: The canonical basis, which is a particular type of basis of a vector space will be introduced in this talk, and a sufficient and necessary condition is given to determine the existence of such a basis for a vector space. The structures of canonical bases are then used to study Cayley graphs of extraspecial $2$-groups of order $2^{2r+1}$ ($r eq 1$), which are further shown to be normal Cayley graphs and $2$-arc-transitive covers of $2r$-dimensional hypercubes. &#8203;

Speaker: Alexander Bors (University of Western Australia and University of Salzburg)

Title: Worst-case approximability of functions on finite groups by endomorphisms and affine maps

Time and place: 16:00 Friday 10/11/2017 in Weatherburn LT

Abstract: http://www.maths.uwa.edu.au/~glasby/S17.html

There will be refreshments in Maths & Stats tea room at 15:40, and we will go for drinks at the UWA Tavern or the UniClub after the seminar. All are most welcome.

Speaker: Stephen Glasby (University of Western Australia)

Title: Norman involutions and tensor products of unipotent Jordan blocks

Time and place: 16:00 Friday 17/11/2017 in Weatherburn LT

Abstract: Suppose R is an rxr unipotent matrix over some field F, i.e. its characteristic polynomial is (t-1)^r. The Jordan form of R is a sum of unipotent Jordan blocks, so we obtain some partition of r. If S is a unipotent sxs matrix over F, then so is R times S. To understand the partition of rs afforded by R times S it suffices to understand the partition afforded by J_r times J_s where J_r denotes a single rxr unipotent Jordan block. When char(F)=p, we denote this partition by lambda(r,s,p).

When p>0, the partitions lambda(r,s,p) are shrouded in mystery. Assume r<= s. We show that there is a larger set of 2^{r-1}-1 partitions (which is independent of p) and contains the mysterious partitions. These partitions correspond to involutions in the symmetric group S_r of degree r, and also to nonempty subsets of the set {1,2,...,r-1}. We also show that the group G(r,p)=<lambda(r,s,p) | s>= r>, is a wreath product, and we determine its structure.

One motivation for this research comes from representation theory: understanding the structure of the Green ring. This is joint work with Cheryl E. Praeger and Binzhou Xia.

Abstract:

Paul Dirac proposed the baryon symmetric universe in 1933. This proposal has become very attractive now since it seems that all pre-existing asymmetry would have been diluted if we had an inflationary stage in the early universe. However, if our universe began baryon symmetric, the tiny imbalance in numbers of baryons and anti-baryons which leads to our existence, must have been generated by some physical processes in the early universe. In my talk I will show why the small neutrino mass is a key for solving this long standing problem in understanding the universe we observe.



Bio:

Professor Tsutomu Yanagida is a world-renowned expert on theoretical high energy physics and cosmology. He is famous, in particular, for the Seesaw mechanism (proposed in 1979) and for the Leptogenesis (proposed in 1986). The Seesaw mechanism predicts very small neutrino masses; the 2015 Nobel Prize in Physics was awarded for the discovery of neutrino oscillations, which show that neutrinos have small masses. The Leptogenesis explains the baryon asymmetry observed in the Universe. Professor Tsutomu Yanagida has published more that 500 papers, which have generated 29,666 citations (as of 28 November 2017). His h-index is 80. He co-authored the book ``Physics of Neutrinos and Applications to Astrophysics’’ written jointly with M. Fukugita and published in 2003.

Professor Tsutomu Yanagida obtained his PhD in 1977 from Hiroshima University. In 1979, he joined Tohoku University in Japan, first as Assistant Professor, then Associate Professor (1987) and finally Professor (1990). In the period 1996—2010, he was Professor at Tokyo University. He is currently Professor at Kavli Institute for the Physics and Mathematics of the Universe, Tokyo where he has been since 2010.

Speaker: Lutz Strüngmann (Mannheim University of Applied Sciences, Germany)

Title: Transitivity Notions for Groups

Time and place: 16:00 Friday 02/02/2018 in Social Sciences LT (Note the unusual location!)

Abstract: http://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

There will be `cake' in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub or Student Tavern for a drink.

Past and future seminars may be found at http://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

Symmetry is one of the essential and most visible patterns that can be seen in nature. Starting from the left-right symmetry of the human body, all types of symmetry can be found in crystals, plants, animals and nature as a whole.

Similarly, principals of symmetry are also some of the fundamental and most useful tools in modern mathematical natural science that play a major role in theory and applications. As a consequence, it is not surprising that the desire to understand the origin of life, based on the genetic code, forces us to involve symmetry as a mathematical concept.

The genetic code can be seen as a key to biological self-organisation. All living organisms have the same molecular bases - an alphabet consisting of four letters (nitrogenous bases): adenine, cytosine, guanine, and thymine. Linearly ordered sequences of these bases contain the genetic information for synthesis of proteins in all forms of life. Thus, one of the most fascinating riddles of nature is to explain why the genetic code is as it is.

Genetic coding possesses noise immunity which is the fundamental feature that allows to pass on the genetic information from parents to their descendants. Hence, since the time of the discovery of the genetic code, scientists have tried to explain the noise immunity of the genetic information. In this talk we will discuss recent results in mathematical modelling of the genetic code with respect to noise immunity, in particular error-detection and error-correction.

Cheese and wine to follow in the Maths common room (5pm-6pm)

Speaker: John Bamberg (UWA)

Title: Abstract ovals

Time and place: 16:00 Friday 09/02/2018 in Woolnough LT (note unusual location)

Abstract: This talk will be about “abstract ovals” which were introduced in 1966 by Francis Buekenhout. In particular, we will present some new results, that have some interesting connections with the theory of Moufang sets and classical permutation group theory. This is joint work with Tim Harris and Tim Penttila.

Speaker: Scott Harper (University of Bristol)

Title: Uniform Domination for Simple Groups

Time and place: 16:00 Friday 23/02/2018 in Robert Street LT

Abstract: It is well known that every finite simple group can be generated by just two elements. In fact, by a theorem of Guralnick and Kantor, there is a conjugacy class C such that for each non-identity element x there exists an element y in C such that x and y generate the entire group. Motivated by this, we introduce a new invariant for finite groups: the uniform domination number. This is the minimal size of a subset S of conjugate elements such that for each non-identity element x there exists an element s in S such that x and s generate the group. This invariant arises naturally in the study of generating graphs.

In this talk, I will present recent joint work with Tim Burness, which establishes best possible results on the uniform domination number for finite simple groups, using a mix of probabilistic and computational methods, together with recent results on the base sizes of primitive permutation groups.

Past and future seminars may be found at http://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

There will be cake in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub or Student Tavern for a drink.

Speaker: Tim Boykett (Johannes Kepler University, Linz, Austria)

Title: Algebraic Models of Computation: Nearrings of Group Automata

Time and place: 16:00 Friday 2/3/2018 in Weatherburn LT

Abstract: Input-Output Automata are a simple but powerful model of processes. Inputs arrive and, depending upon the current state, the automaton outputs a symbol and moves to a new state. In the case that the input and output alphabets are identical, we can compose them by connecting the input of one automaton to the output of another. By equipping the alphabet with a group operation, we can compose automata in parallel, feeding each the same input and summing their output using the group operation. We obtain a collection of synchronous state automata, equivalently a set of mappings on infinite sequences.

The algebraic tool that best allows us to model, manipulate and analyse collections of such automata are nearrings, the nonlinear analogue of rings. A (right) nearring (N,+,*) is a (not necessarily abelian) group (N,+), a semigroup (N,*) with one distributive law (a+b)*c= a*c+b*c. Standard examples are the set of all mappings of a group to itself under pointwise addition and functional composition, the closed subalgebras of this, or the class of nearfields as used in describing certain projective planes including the Hall Plane. In this talk we will outline this construction in more detail, demonstrate the complexity of the nearring of automata and introduce radicals as simplifying tools. In particular we will show that the amnesiac mapping and the set of delay automata help bound the Jacobson 2-radical, defining it in certain cases. In these cases we can write down the 2-semisimple automata nearrings explicitly. Input-Output Automata are a simple but powerful model of mechanical, electronic and other processes. In the case that the input and output alphabets are identical, we can compose them by connecting the input of one automaton to the output of another. By equipping the alphabet with a group operation, we can compose automata in parallel, feeding each the same input and summing their output using the group operation.

This work is supported by the grant P29931 and SFB Project F5004 of the Austrian Science Foundation, FWF.

Speaker: Alejandra Ramos Rivera (UWA and University of Primorska, FAMNIT, Koper, Slovenia)

Title: Structural results on tetravalent half-arc-transitive graphs

Time and place: 16:00 Friday 09/03/2018 in Weatherburn LT

Abstract: In this talk we focus on tetravalent graphs admitting a half-arc-transitive subgroup of automorphisms, that is a subgroup acting transitively on its vertices and its edges but not on its arcs. One of the most fruitful approaches for the study of structural properties of such graphs is the well known paradigm of alternating cycles and their intersections which was introduced by Maruši&#269; 20 years ago.

We introduce a new parameter for such graphs, giving a further insight into their structure. Various properties of this parameter are given. The obtained results are used to establish a link between two frameworks for a possible classification of all tetravalent graphs admitting a half-arc-transitive subgroup of automorphisms, the one proposed by Maruši&#269; and Praeger in 1999, and the much more recent one proposed by Al-bar, Al-kenai, Muthana, Praeger and Spiga which is based on the normal quotients method.

We also present results on the graph of alternating cycles of a tetravalent graph admitting a half-arc-transitive subgroup of automorphisms. A considerable step towards the complete answer to the question of whether the attachment number necessarily divides the radius in tetravalent half-arc-transitive graphs is made.

Past and future seminars may be found at http://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

There will be cake in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub or Student Tavern for a drink.

There's a widespread belief that people who are good at maths can go into finance and become obscenely rich. This talk tries to put that in perspective by describing a common approach to algorithmic trading, and exploring some of the reasons why it's harder than it sounds.

Speaker: Wei Jin (Jiangxi University of Finance and Economics and University of Western Australia)

Title: Finite two-distance-transitive circulants

Time and place: 16:00 Friday 16/03/2018 in Weatherburn LT

Abstract: In this talk, we give a complete classification of the family of 2-distance-transitive circulants. We show that a 2-distance-transitive circulant is a cycle, a Paley graph of prime order, a regular complete multipartite graph, or a regular complete bipartite graph of order twice an odd integer minus a 1-factor.

This is joint work with J.Y. Chen and C.H. Li.

Speaker: Prof. Michael Giudici (University of Western Australia)

Title: Bases for permutation groups and the Saxl graph

Time and place: 16:00 Friday 23/03/2018 in Weatherburn LT

Abstract: Let G be a permutation group on a set X. A base for G, is a subset B of X such that the pointwise stabiliser of the elements of B is trivial. There has been a large amount of recent research on the size of a base of a primitive permutation group, culminating with the recent proof of Pyber’s Conjecture. At the same time there has been a large amount of work devoted to finding the primitive groups with a base of size two. For such groups we can define the Saxl graph of G to be the graph with vertex set X and two elements are joined by an edge if they are a base. I will discuss some recent work with Tim Burness that investigates some of the properties of this graph.

Please come at 15:00 this Friday to meet our BPhil students over cake (which may go quickly).

All welcome.

Speaker: David Roberson (Technical University of Denmark)

Title: Vector Colorings of the Categorical Product of Graphs

Time and place: 16:00 Friday 06/04/2018 in Weatherburn LT

Abstract:In 1966 Hedetniemi conjectured the that chromatic number of the categorical product (sometimes called the direct product) of two graphs is equal to the minimum of their individual chromatic numbers. This remains one of the major open questions in the field of graph colorings to this day. We will prove this conjecture for the vector chromatic number, a vector/semidefinite relaxation of the chromatic number. Moreover, we will provide a necessary and sufficient condition for when every optimal vector coloring of the product is induced by optimal vector colorings of the factors.

Fluid models are widely employed in many fields of science, ranging from astronomy and physics to biology and chemistry. The fundamental principle, and motivation, behind fluid models is to provide an effective macroscopic representation of the collective behaviour arising from a large number of microscopic events. Thus, the main advantage of fluid models is a reduction in complexity, while still capturing the essential characteristics of the macroscopic system. The difficulty in constructing fluid models from kinetic theory arises from the presence of the collision operator. A systematic treatment of the collisional effects in a plasma is presented to derive fluid models beyond the usual assumption of “thermal equilibrium”. Such extended models will greatly help model cold and moderate temperature plasmas, for example in the study of astrophysical phenomena or in industrial applications.