2020 talks:

Date (NSW/Vic)	Speaker	Title	Video
20 May '20, 11am	Brendan McKay	A scientist's adventure into pseudoscience: the strange case of the Bible Codes
3 Jun '20, 4pm	John Bamberg	Vanishing Krein Parameters in Finite Geometry	Video
17 Jun '20, 5pm	Fiona Skerman	Branching processes with merges and locality of hypercube's critical percolation
1 Jul '20, 5pm	Tibor Szabó	Turán numbers, norm graphs, quasirandomness
15 Jul '20, 11am	Daniel Horsley	Generating digraphs with derangements	Video
22 Jul '20, 5pm	Katherine Staden	Two conjectures of Ringel	Video
29 Jul '20, 5pm	Annika Heckel	Non-concentration of the chromatic number	Slides
19 Aug '20, 11am	Tamás Makai	Majority dynamics in the dense binomial random graph	Video
26 Aug '20, 11am	Tuan Tran	Singularity of random combinatorial matrices	Video
9 Sep '20, 11am	Geertrui Van de Voorde	Sets with few intersection numbers in finite planes
22 Sep '20, 5pm	David Conlon	The random algebraic method
6 Oct '20, 11am	Tony Huynh	Idealness of k-wise intersecting families	Video
27 Oct '20, 5pm	Simone Linz	Reconstructing phylogenetic networks from trees	Video
10 Nov '20, 6pm	Vida Dujmović	Product Structure Theory with applications
24 Nov '20, 3pm	Kevin Hendrey	Obstructions to bounded branch-depth in matroids	Video

24 November 2020, 3pm AEDT Kevin Hendrey (Institute for Basic Science)

Obstructions to bounded branch-depth in matroids

Abstract: DeVos, Kwon, and Oum introduced a notion of branch-depth of matroids as a natural analogue to tree-depth of graphs, and conjectured that matroids of sufficiently large branch-depth contain either a large uniform matroid or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. I will discuss this result and some implications. Joint work with Pascal Gollin, Dillon Mayhew and Sang-il Oum.

10 November 2020, 6pm AEDT Vida Dujmović (University of Ottawa)

Product Structure Theory with applications

Abstract: I will talk about product structure theory of graphs and its application to various problems such as: graph adjacency encoding, queue/stack number and graph colourings.

27 October 2020, 5pm AEDT Simone Linz (University of Auckland)

Reconstructing phylogenetic networks from trees

Abstract: Recent advances in whole-genome studies provide increasingly strong evidence for a vital role of hybridization in the evolution of certain groups of species and allowing them to adapt to new environments. To represent such complex evolutionary histories as a web of life rather than a simple bifurcating tree of life, phylogenetic (evolutionary) networks have become a popular tool. In the context of reconstructing phylogenetic networks, the problem of characterizing and computing the minimum hybridization number for a set of phylogenetic trees has been investigated by many groups of researchers for the last 15 years. Roughly speaking, this minimum quantifies the number of hybridization events needed to explain a set of trees by simultaneously embedding them in a phylogenetic network. In this talk, we introduce cherry-picking sequences which are particular sequences on the leaves of the trees. We show how these sequences give a novel characterization of the minimum hybridization number for an arbitrarily large collection of phylogenetic trees. This is joint work with Peter Humphries and Charles Semple.

6 October 2020, 11am AEDT Tony Huynh (Monash University)

Idealness of k-wise intersecting families

Abstract: A clutter is a hypergraph such that no hyperedge is contained in another hyperedge. It is k-wise intersecting if every k hyperedges intersect, but there is no vertex contained in all the hyperedges. We conjecture that every 4-wise intersecting clutter is not ideal. Idealness is an important geometric property, which roughly says that the minimum covering problem for the clutter can be efficiently solved by a linear program. As evidence for our conjecture, we prove it for the class of binary clutters. Our proof combines ideas from the theory of clutters, graphs, and matroids. For example, it uses Jaeger's 8-flow theorem for graphs, and Seymour's characterization of the binary matroids with the sums of circuits property. We also show that 4 cannot be replaced by 3 in our conjecture, where the counterexample of course comes from the Petersen Graph. This is joint work with Ahmad Abdi, Gérard Cornuéjols, and Dabeen Lee.

22 September 2020, 5pm AEST David Conlon (Caltech)

The random algebraic method

Abstract: The random algebraic method is a means of constructing examples in extremal combinatorics with some of the best properties of both algebraic constructions and probabilistic ones. This method has seen a broad array of applications in recent years, some of which we will discuss in this talk.

9 September 2020, 11am AEST Geertrui Van de Voorde (University of Canterbury)

Sets with few intersection numbers in finite planes

Abstract: Many problems in finite geometry follow the following pattern: say we have a set S of points in a plane, and require that a combinatorial property holds (e.g. that the number of points of S on every line is at most a fixed number). Does such a set exist? And if so, can we say something about the algebraic structure of this set? What if we impose some extra symmetry conditions? By far the most famous example of such a theorem is Segre's beautiful characterisation of conics in a Desarguesian projective plane of odd order q: every oval (which is a set C of q+1 points such that no line contains more than 2 points of C) is the set of points on a conic. We will look at some classical results about ovals, hyperovals and maximal arcs and present some more recent results of the same flavour about KM-arcs (joint work with Maarten De Boeck).

26 August 2020, 11am AEST Tuan Tran (Institute for Basic Science)

Singularity of random combinatorial matrices

Abstract: Let Qn be a random n by n matrix with entries in {0,1} whose rows are independent vectors of exactly n/2 zero components. We show that the probability that Qn is singular is exponentially small, which is optimal up to a multiplicative factor in the exponent. We develop a general method to handle random matrices with dependent entries.

19 August 2020, 11am AEST Tamás Makai (UNSW Sydney)

Majority dynamics in the dense binomial random graph

Abstract: Majority dynamics is a deterministic process on a graph which evolves in the following manner. Initially every vertex is coloured either red or blue. In each step of the process every vertex adopts the colour of the majority of its neighbours, or retains its colour if no majority exists. We analyse the behaviour of this process in the dense binomial random graph when the initial colour of every vertex is chosen independently and uniformly at random. We show that with high probability the process reaches complete unanimity, partially proving a conjecture of Benjamini, Chan, O'Donnel, Tamuz and Tan. This is joint work with N. Fountoulakis and M. Kang.

29 July 2020, 5pm AEST Annika Heckel (LMU Munich)

Non-concentration of the chromatic number

Abstract: There are many impressive results asserting that the chromatic number of a random graph is sharply concentrated. In 1987, Shamir and Spencer showed that for any function p=p(n), the chromatic number of G(n,p) takes one of at most about n1/2 consecutive values whp. For sparse random graphs, much sharper concentration is known to hold: Alon and Krivelevich proved two point concentration whenever p < n-1/2-ε. However, until recently no non-trivial lower bounds for the concentration were known for any p, even though the question was raised prominently by Erdős in 1992 and Bollobás in 2004. In this talk, we show that the chromatic number of G(n,1/2) is not whp contained in any sequence of intervals of length n1/2-o(1), almost matching Shamir and Spencer's upper bound. Joint work with Oliver Riordan.

22 Jul 2020, 5pm AEST Katherine Staden (University of Oxford)

Two conjectures of Ringel

Abstract: In a graph decomposition problem, the goal is to partition the edge set of a host graph into a given set of pieces. I will focus on the setting where the host graph and the pieces have a comparable number of vertices, and in particular on two conjectures of Ringel from the 60s on decomposing the complete graph: in the first (Ringel's conjecture) they are identical half-sized trees, and in the second (the Oberwolfach problem) they are identical 2-factors. I will give some ideas from my recent proofs of the first problem and a generalised version of the second, for large graphs, in joint work with Peter Keevash. The first conjecture was proved independently by Montgomery, Pokrovskiy and Sudakov.

15 Jul 2020, 11am AEST Daniel Horsley (Monash University)

Generating digraphs with derangements

Abstract: Let S be a collection of derangements (fixed point-free permutations) of a possibly infinite set X. The derangement action digraph DA(X,S) is the digraph on vertex set X that has an arc from x to y if and only if some derangement in S maps x to y. We say that S generates DA(X,S). Derangement action digraphs were introduced by Iradmusa and Praeger in 2019, adapting the definition of a group action digraph due to Annexstein, Baumslag and Rosenberg. I will discuss recent work by Iradmusa, Praeger and myself in which we characterise, for each positive integer k, the digraphs that can be generated by at most k derangements. Our result resembles the De Bruijn-Erdős theorem in that it characterises a property of an infinite graph in terms of properties of its finite subgraphs.

1 Jul 2020, 5pm AEST Tibor Szabó (Freie Universität Berlin)

Turán numbers, norm graphs, quasirandomness

Abstract: The Turán number of a (hyper)graph H, defined as the maximum number of (hyper)edges in an H-free (hyper)graph on a given number of vertices, is a fundamental concept of extremal combinatorics. The behaviour of the Turán number is well-understood for non-bipartite graphs, but for bipartite H there are more questions than answers. A particularly intriguing half-open case is the one of complete bipartite graphs. The projective norm graphs NG(q,t) are algebraically defined graphs which provide tight constructions in the Turán problem for complete bipartite graphs H=Kt,s when s>(t–1)!. The Kt,s-freeness of NG(q,t) is a very much atypical property: in a random graph with the same edge density a positive fraction of t-tuples are involved in a copy of Kt,s. Yet, projective norm graphs are random-like in various other senses. Most notably their second eigenvalue is of the order of the square root of the degree, which, through the Expander Mixing Lemma, implies further quasirandom properties concerning the density of small enough subgraphs. In this talk we explore how far this quasirandomness goes. The main contribution of our proof is the estimation, and sometimes determination, of the number of solutions of certain norm equation system over finite fields. Joint work with Tomas Bayer, Tamás Mészáros, and Lajos Rónyai.

17 Jun 2020, 5pm AEST Fiona Skerman (Uppsala University)

Branching processes with merges and locality of hypercube's critical percolation

Abstract: We define a branching process to understand the locality or otherwise of the critical percolation in the hypercube; that is, whether the local structure of the hypercube can explain the critical percolation as a function of the dimension of the hypercube. The branching process mimics the local behaviour of an exploration of a percolated hypercube; it is defined recursively as follows. Start with a single individual in generation 0. On an first stage, each individual has independent Poisson offspring with mean (1+p)(1-q)k where k depends on the ancestry of the individual; on the merger stage, each pair of cousins merges with probability q. We exhibit evidence of a critical merger probability qc=qc(p) for extinction of the branching process. When p is sufficiently small, the first order terms of qc coincide with those of the critical percolation for the hypercube, suggesting that percolation in the hypercube is dictated by its local structure. This is work in progress with Laura Eslava and Sarah Penington.

3 Jun 2020, 4pm AEST John Bamberg (University of Western Australia)

Vanishing Krein Parameters in Finite Geometry

Abstract: The Krein condition on the parameters of a strongly regular graph (L. L. Scott 1973, 1977) is one of the most successful tools in ruling out sets of possible parameters of strongly regular graphs. Delsarte’s generalisation for association schemes (1973) has also played an important role in the theory of association schemes and its applications: to coding theory, design theory, and finite geometry. In this talk, we give a brief introduction to the interplay between association schemes and finite geometry, and some recent results on vanishing Krein parameters of the speaker and his student Jesse Lansdown.

20 May 2020, 11am AEST Brendan McKay (Australian National University)

A scientist's adventure into pseudoscience: the strange case of the Bible Codes

Abstract: Over the centuries, many claims have been made of numerical patterns of miraculous nature hidden within the text of sacred writings, including the Jewish, Christian and Islamic scriptures. Usually the patterns involve counting of letters and words, or calculations involving numerical equivalents of the letters. Until recently, all such claims were made by people with little mathematical understanding and were easily explained. This situation changed when a highly respected Israeli mathematician Eliyahu Rips and two others published a paper in the academic journal Statistical Science claiming to prove that information about medieval Jewish rabbis was encoded in the Hebrew text of the Book of Genesis. The journal reported that its reviewers were "baffled". The paper in Statistical Science spawned a huge "Bible Codes" industry, complete with best selling books, TV documentaries, and even an adventure movie. The talk will reveal the inside story of the Codes and the people behind them, from their inception through to their refutation.