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1 Plenary Talks 1 Plenary talks 1 Monday 9:00, Zoom 1 A proof of the Erdos-Faber-Lov˝ asz´ conjecture Deryk Osthus University of Birmingham (This talk is based on joint work with Dong-Yeap Kang, Tom Kelly, Daniela K¨uhnand Abhishek Methuku.) MSC2000: 05D40,05C65 In 1972, Erd˝os,Faber, and Lov´aszconjectured the following equivalent statements. Let n N. ∈ (i) If A1,...,An are sets of size n such that every pair of them shares at most one n element, then the elements of i=1 Ai can be coloured by n colours so that all colours appear in each Ai. S (ii) If G is a graph that is the union of n cliques, each having at most n vertices, such that every pair of cliques shares at most one vertex, then the chromatic number of G is at most n. (iii) If is a linear hypergraph with n vertices, then the chromatic index of is at mostH n. H Here the chromatic index χ0( ) of a hypergraph is the smallest number of colours needed to colour the edges of H so that any two edgesH that share a vertex have different colours and a hypergraph is linearH if two hyperedges share at most one vertex. Erd˝oscon- sidered this to be ‘one of his three most favorite combinatorial problems’. The simplicity and elegance of its formulation initially led the authors to believe it to be easily solved. However, as the difficulty became apparent Erd˝osoffered successively increasing rewards for a proof of the conjecture, which eventually reached $500. We prove the Erd˝os-Faber-Lov´aszconjecture for every large n: Theorem 1. [1] For every sufficiently large n, every linear hypergraph on n vertices has chromatic index at most n. H In my talk, I will survey some background, related results and open problems. I will also discuss some of the ideas involved in the proof. [1] Dong-Yeap Kang, Tom Kelly, Daniela K¨uhn,Abhishek Methuku and Deryk Osthus, A proof of the Erd˝os-Faber-Lov´aszconjecture, arxiv:2101.04698. 2 Monday 13:30, Zoom 1 Decomposing the edges of a graph into simpler structures Marthe Bonamy Universit´ede Bordeaux MSC2000: 05C15 We will review various ways to decompose the edges of a graph into few simple substruc- tures. We will mainly focus on variants of edge colouring, and discuss specifically the discharging method and re-colouring techniques. 3 Tuesday 9:00, Zoom 1 Codes and designs in Johnson graphs Cheryl E. Praeger The University of Western Australia (This talk is based on joint work with R. A. Liebler, M. Neunhoeffer, and more recently J. Bamberg, A. C. Devillers and M. Ioppolo.) MSC2000: 05C25, 20B25, 94B60 The Johnson graph J(v, k) has, as vertices, all k-subsets of a v-set , with two k-subsets adjacent if and only if they share k 1 common elements of .V Subsets of vertices of J(v, k) can be interpreted as the block-set− of an incidence structure,V or as the set of codewords of a code, and automorphisms of J(v, k) leaving the subset invariant are then automorphisms of the corresponding incidence structure or code. This approach leads to interesting new designs and codes. For example, numerous actions of the Mathieu sporadic simple groups give rise to examples of Delandtsheer designs (which are both flag-transitive and anti-flag transitive), and codes with large minimum distance (and hence strong error-correcting properties). In my talk I will explore links between designs and codes in Johnson graphs which have a high degree of symmetry, and I will mention several open questions. 4 Tuesday 14:00, Zoom 1 The partition complex: an invitation to combinatorial commutative algebra Karim Adiprasito Hebrew University of Jerusalem (This talk is based on joint work with Geva Yashfe.) MSC2000: 05E40 We provide a new foundation for combinatorial commutative algebra and Stanley-Reisner theory using the partition complex introduced in [1]. One of the main advantages is that it is entirely self-contained, using only a minimal knowledge of algebra and topology. On the other hand, we also develop new techniques and results using this approach. In particular, we provide 1. A novel, self-contained method of establishing Reisner’s theorem and Schenzel’s for- mula for Buchsbaum complexes. 2. A simple new way to establish Poincar´eduality for face rings of manifolds, in much greater generality and precision than previous treatments. 3. A “master-theorem” to generalize several previous results concerning the Lefschetz theorem on subdivisions. 4. Proof for a conjecture of K¨uhnelconcerning triangulated manifolds with boundary. [1] Karim Adiprasito, Combinatorial Lefschetz theorems beyond positivity, 2018, preprint, arXiv:1812.10454. 5 Wednesday 9:00, Zoom 1 Base sizes and relational complexity of finite permutation groups Colva Roney-Dougal University of St Andrews MSC2000: 20D05 This talk will start by briefly surveying what is known about the maximal subgroups of the finite simple groups. We will then see how this knowledge has been applied to bound some combinatorial invariants of finite permutation groups. A base for a subgroup G of the symmetric group S(Ω) is a subset ∆ of Ω whose pointwise stabiliser in G is trivial. The first invariant we will look at is the size b(G) of a minimal base for G. We will see that b(G) gives a coarse estimate of the size of G, and survey results both old and new which bound b(G). Next, we’ll see how large an irredundant base for G can be: this is an ordered base ∆ = (δ1, . , δk) such that the stabiliser Gα1,...,αi 1 = Gα1,...,αi 1,αi , for all i. − 6 − We’ll end by linking these ideas to model theory, via the idea of relational complexity. 6 Thursday 9:00, Zoom 1 Hasse-Weil type theorems and relevant classes of polynomial functions Daniele Bartoli Universit`adegli Studi di Perugia MSC2000: 14-02 Several types of functions over finite fields have relevant applications in applied areas of mathematics, such as cryptography and coding theory. Among them, planar func- tions, APN permutations, permutation polynomials, and scattered polynomials have been widely studied in the last few years. In order to provide both non-existence results and explicit constructions of infinite fam- ilies, sometimes algebraic varieties over finite fields turn out to be a useful tool. In a typical argument involving algebraic varieties, the key step is estimating the number of their rational points over some finite field. For this reason, Hasse-Weil type theorems (such as Lang-Weil’s and Serre’s) play a fundamental role. 7 Thursday 13:30, Zoom 1 Borel combinatorics Oleg Pikhurko University of Warwick MSC2000: 05C63, 03E05, 28A05 We give an introduction, aimed at non-experts, to Borel combinatorics (that studies definable graphs on topological spaces and looks for constructive assignments satisfying some given local combinatorial constraints). This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. The aim of this talk is to advertise this field to a wider research community. 8 Friday 9:00, Zoom 1 Generating graphs randomly Catherine Greenhill UNSW Sydney MSC2000: 05C85, 60J10, 68R05, 68W20, 68W40 Graphs are used in many disciplines to model the relationships that exist between objects in a complex discrete system. Researchers often wish to compare their particular graph to a “typical” graph from a family (or ensemble) of graphs which are similar to theirs in some way. Such a family might be the set of all graphs with a given number of vertices and edges, or the set of all graphs with a particular degree sequence. One way to do this is to take several random samples from the family, to gather infor- mation about what is “typical”. This motivates the search for an algorithm which can generate graphs uniformly (or approximately uniformly) at random from the given fam- ily. Since many random samples may be required, the algorithm should also be efficient. Rigorous analysis of such algorithms is often challenging, involving both combinatorial and probabilistic arguments. I will discuss some algorithms for sampling graphs, and the methods used to analyse them. 9 Friday 16:00, Zoom 1 Recent Advances on the Graph Isomorphism Problem Martin Grohe RWTH Aachen University MSC2000: 05C60, 68R10, 20B25 The question of whether there is a polynomial time algorithm deciding if two graphs are isomorphic has been a one of the best known open problems in theoretical computer science for almost 50 years. Indeed, the graph isomorphism problem is one of the very few natural problems in the complexity class NP that is neither known to be solvable in polynomial time nor known to be NP-complete. Five years ago, Babai gave a quasipoly- nomial time isomorphism algorithm. Despite of this breakthrough result, the question for a polynomial algorithm remains wide open. My talk will be a survey of recent progress on the isomorphism problem. I will focus on two generic algorithmic strategies that have proved to be useful and interesting in various contexts. The first is the combinatorial Weisfeiler-Leman algorithm with a wide range of applications from practical graph isomorphism testing to machine learning. The second is the group theoretic divide-and-conquer strategy, going back to Luks (1983), that is the foundation of Babai’s quasi-polynomial time isomorphism test. In subsequent developments, it led to the design of isomorphism algorithms with a quasi-polynomial parameterised running time of the form npolylog k, where k is a graph parameter such as the maximum degree. 10 2 Minisymposia talks 11 Tuesday 10:30, Zoom 1 Diagonal semilattices and their graphs Peter J.
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