### CMSA Day (Workshop and AGM)

Register here to receive meeting details. |

Instructions on accessing the talks have been emailed to registered participants on Friday 11 December at 13:05 AEDT.

The Council of the Combinatorial Mathematics Society of Australasia (CMSA) will hold a one-day workshop on Tuesday 15 December 2020, as a partial replacement for the postponed conference 43ACC. The CMSA Day schedule follows (times AEDT), whereas talk abstracts can be found below.

10:30-11:30 AEDT | Noga Alon, Princeton University and Tel Aviv University |
Equipartitions |

11:30-12:30 | Lisa Sauermann, Institute for Advanced Study |
On polynomials that vanish to high order on most of the hypercube |

13:00-13:45 | CMSA Annual General Meeting | |

14:00-15:00 | Nick Cavenagh, University of Waikato |
Mutually orthogonal frequency squares (MOFS) |

15:00-16:00 | Alice Devillers, University of Western Australia |
Some recent results on flag-transitive and block-transitive 2-designs |

CMSA Day is a free event but provides a great opportunity to renew your CMSA membership or become a new member of the CMSA! When registering, you will have the option of a free ticket (without CMSA membership) or a paid ticket of approximately $26 AUD which comes with CMSA membership for 2021. If you are already a CMSA *life member* then you should select the free ticket option. If you have access to research funding then we encourage you to purchase a paid ticket.

The talk abstracts follow.

Noga Alon (Princeton University and Tel Aviv University) |

Equipartitions |

Abstract:
A substantial number of problems deal with the existence of
a set of prescribed type containing a fair share of an object according
to several different measures, or the existence of a partition
of the object in which this is the case for every part. Examples include the
Ham-Sandwich Theorem in Measure Theory, the Hobby-Rice Theorem in
Approximation Theory, the Necklace Theorem and
the Ryser Conjecture in Discrete Mathematics, and more. The techniques
in the study of the subject combine combinatorial, topological, geometric
probabilistic and algebraic tools. I will describe the topic, focusing on several recent existence results and their algorithmic aspects. |

Lisa Sauermann (Institute for Advanced Study) |

On polynomials that vanish to high order on most of the hypercube |

Abstract:
Motivated by higher vanishing multiplicity generalizations of Alon's Combinatorial Nullstellensatz and its applications, we study the following problem: for fixed k and n large with respect to k, what is the minimum possible degree of a polynomial P in R[x_{1},...,x_{n}] such that P(0,…,0) is non-zero and such that P has zeroes of multiplicity at least k at all points in {0,1}^{n} except the origin? For k=1, a classical theorem of Alon and Füredi states that the minimum possible degree of such a polynomial equals n. We solve the problem for all k>1, proving that the answer is n+2k-3. Joint work with Yuval Wigderson. |

Alice Devillers (University of Western Australia) |

Some recent results on flag-transitive and block-transitive 2-designs |

Abstract:
t-designs are useful combinatorial objects, born out of statistical experiment needs. They consist in points and subsets of points called blocks, such that each block has the same size and every subset of t points is in the same number of blocks. I will explain several transitivity properties that the automorphism group of a t-design can have. Then I will concentrate on 2-designs admitting a point-imprimitive group of automorphisms which is either flag-transitive or block-transitive. I will survey some known results, and explain some recent results from 3 papers on this topic. Joint work with Cheryl Praeger, Carmen Amarra, Hongxue Liang, and Binzhou Xia. |

Nick Cavenagh (University of Waikato) |

Mutually orthogonal frequency squares (MOFS) |

Abstract:
A frequency square of type (n; λ) is a n*n array such that each symbol from a set of size n/λ occurs λ times in each row and λ times each in column. Thus a frequency square of type (n; 1) is a Latin square of order n. Two frequency squares of type (n; λ) are said to be orthogonal if each possible ordered pair occurs λ^{2} times when the squares are overlapped. Sets of mutually orthogonal frequency squares (MOFS) occur more easily than sets of mutually orthogonal Latin squares (MOLS); for example there is a set of nineteen MOFS of type (6; 3)); but, as Euler knew, there are no pairs of MOLS of order 6. We give new results on the existence and non-existence of MOFS and maximal sets of MOFS. These results feature Hadamard matrices, integral polytopes and row-column factorial designs.
Joint work with Ian Wanless, Adam Mammoliti and Thomas Britz. |