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.

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

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

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, Řstergĺrd, 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

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,λ) 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 λ 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 (α, B) where α 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≤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

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. ​

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

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.

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.