Abstract: The Lovasz Local Lemma is a probabilistic tool that gives a sufficient condition for when all bad events in a given set may be avoided with positive probability. In combinatorics, the Lovasz local lemma hence also gives a sufficient condition for when a combinatorial object contains a substructure avoiding given properties. In this talk, we will demonstrate how the Lovasz local lemma has been applied to various types of graph coloring problems in the past, and we will show recent applications of this lemma used to prove new results and solve some open problems. In particular, we will show that the Lovasz local lemma can be used to answer a question of Dvorak, Louis Esperet, Ross J. Kang, Kenta Ozeki about single-conflict colorings, and we will show that the Lovasz local lemma can also be applied to improve a result of Aharoni, Berger, Chudnovsky, Havet, and Jiang about cooperative colorings.

Bio : Peter is a 2nd-year PhD candidate advised by Bojan Mohar and Ladislav Stacho. His interests include graph colouring, rainbow graph structures, and games on graphs. Previously, Peter completed a Bachelor's degree at the University of Kansas and a Master's degree at Simon Fraser University.

Abstract: Promotion is an operator that acts on the set of linear extensions of a fixed partially ordered set (poset); it is one of the primary operators that has been studied in the blossoming area of dynamical algebraic combinatorics. The primary focus of this talk will be on two new variants of promotion. The first is a noninvertible extension of promotion that I have studied with Noah Kravitz. Iterating this extended promotion operator sorts arbitrary labelings of a poset into linear extensions. The second variant is a cyclic analogue of promotion called toric promotion, which acts on the labelings of a graph. It turns out that toric promotion exhibits a remarkably simple orbit structure when the graph is a forest.

Bio: Colin is a 5th-year PhD candidate advised by Noga Alon. He has broad interests within combinatorics, especially when it is enumerative, algebraic, or dynamical in nature. Colin graduated Summa Cum Laude with a B.S. degree in Mathematics from the University of Florida.

Abstract: The queens domination problem asks for the minimum number of queens one needs to place on an (n x n) chessboard to dominate all squares according to standard chess rules. However, in our rulebook squares are dominated if they lie on a straight line which passes through a pair of pawns and lines are not restricted to horizontal, vertical and diagonal lines!

As we try to find the minimum number of pawns needed to dominate the whole board, we will see counting arguments, number theory and probabilistic methods at work and encounter lots of open problems suited for research at any level of study.

Based on joint work with Oswin Aichholzer and David Eppstein.

Bio: Eva is a 2nd-year PhD candidate supervised by Michael Drmota. Her work is mainly in analytic combinatorics but she also enjoys problems with a geometric and number-theoretic flavour. Eva did her bachelor's and master's studies in discrete mathematics at TU Graz with a year at Charles University Prague as an Erasmus student. She also has a degree in stage and costume design at the Graz University for Music and Performing Arts.

Abstract: In recent years, spectacular results in Ramsey theory have been obtained by various researchers, using a combination of geometry over finite fields and the study of (pseudo)random graphs. We will give an overview of these breakthroughs and the strength of this interplay. We will see how recent results from colleagues in finite geometry and myself contribute to this story and discuss some avenues of further progress. No preliminary knowledge about finite geometry is required, as we will give a gentle introduction to the topic.

Bio: Sam is a PhD candidate working under the supervision of Philippe Cara and Jan De Beule. His research interests are finite geometry, algebraic and extremal graph theory and the areas in which they overlap. Sam obtained both a Bachelor's and Master's degree in Mathematics from Universiteit Gent.

Abstract: Combinatorics is a broad field with wide applicability to solve many applied challenges in the modern world. This talk highlights the journey of using combinatorics to solve problems, showcasing interesting applications from the relationship of percolation theory with the collaboration networks of artists on Spotify and optimal Candy Crush strategies, to the future of the internet using zero-knowledge proofs and computational democracy. This plenary will be a lighthearted exploration of how combinatorics can touch even the most obscure areas of the world and how you can play a role in solving the most pressing challenges.

Bio: Tobin is a 1st-year PhD student in the Human Dynamics group at the MIT Media Lab and Connection Science in the Institute for Data, Systems, and Society and is advised by Sandy Pentland. His research is primarily in data science with interests in topics including privacy-preserving machine learning, studying algorithms in decentralized systems, and computational social science. Tobin was the valedictorian when he completed his Bachelor's degree in Mathematical and Computer Sciences and more recently obtained a Master's of Philosophy in Applied Mathematics and Data Science, both from the University of Adelaide.

Abstract: Phylogenetics is the study of evolutionary relationships between present-day organisms. Evolution can occur via a number of mechanisms, and one such mechanism is the application of large-scale genome rearrangement events, in which sections of the genome are broken apart and reattached in a different order or orientation. By considering the possible combinations of rearrangements that could convert one genome into another, we can identify how closely related the organisms are.

I will give an overview of genome rearrangement modelling as an applied combinatorial problem, and then present our algebraic framework for modelling genomes. I will discuss some of my recent work in characterising `biologically reasonable' rearrangements and determining how many of these there are for a given number of regions.

Bio: Josh is a 3rd-year PhD candidate in applied mathematics at the University of Tasmania working under the supervision of Jeremy Sumner and Venta Terauds. His research interests are primarily in the application of algebraic ideas to problems in evolutionary biology. Josh completed a combined Bachelor of ICT and Bachelor of Science (Honours) from the University of Tasmania.

Abstract : Motivated by online decision-making in time-varying combinatorial environments, we study the problem of transforming offline algorithms to their online counterparts. We focus on offline combinatorial problems that are amenable to a constant factor approximation using a greedy algorithm that is robust to local errors. For such problems, we provide a general framework that efficiently transforms offline robust greedy algorithms to online ones using Blackwell approachability. We show that the resulting online algorithms have $O(\sqrt{T})$ (approximate) regret under the full information setting. We further introduce a bandit extension of Blackwell approachability that we call Bandit Blackwell approachability. We leverage this notion to transform greedy robust offline algorithms into a $O(T^{2/3})$ (approximate) regret in the bandit setting. Demonstrating the flexibility of our framework, we apply our offline-to-online transformation to several problems at the intersection of revenue management, market design, and online optimization, including product ranking optimization in online platforms, reserve price optimization in auctions, and submodular maximization. We show that our transformation, when applied to these applications, leads to new regret bounds or improves the current known bounds.

Bio: Susan is a 4th-year PhD candidate at MIT's Operations Research Center advised by Negin Golrezaei. Her research interests include online learning, machine learning, combinatorial algorithms, approximation algorithms, and mechanism design with applications to revenue management and online markets. She also did her undergraduate studies in Mathematics and Electrical Engineering and Computer Science at MIT.

Abstract: In 1941, Kelly and Ulam conjectured that all graphs on at least 3 vertices are uniquely determined by (or reconstructible from) their vertex-deleted subgraphs. While this conjecture remains very much open, it has since blossomed into a rich area with many variants being actively studied. At the heart of these lies the basic question: given some object, what collection of subobjects suffices to uniquely determine the whole? This talk will provide a brief survey of some key results, tools, and variants of graph reconstruction, with a view to discussing recent progress on two particular variants in which we start with even less information than in the classical problem.

Bio: Jane is a DPhil (PhD) candidate at the University of Oxford working in combinatorics under the supervision of Alex Scott. Previously, she was an undergraduate at the Australian National University, where her Honours thesis in graph theory was supervised by Brendan McKay.

Abstract: This talk will be a tour of some of the ways in which statistical mechanics (also known as statistical physics) can provide a useful perspective on problems in combinatorics and computer science. Extremal questions like ``Which $d$-regular graph has the most independent sets?'' and algorithmic questions about counting and sampling can be generalized and answered using statistical mechanics tools. We'll start with an introduction to spin systems and combinatorially relevant examples such as the Ising, Potts, and hardcore models before exploring some of these questions and emerging methods to tackle them. I'll also mention my recent work in this area, joint with Charlie Carlson, Ewan Davies, Nicolas Fraiman, Alexandra Kolla, and Aditya Potukuchi.

Bio: Corrine is a 5th-year PhD candidate advised by Bhargav Narayanan. She works in discrete mathematics with a particular interest in problems that lie in the intersection of combinatorics and other areas, such as probability, topology, and statistical physics. Corrine completed a Bachelor of Arts from Sarah Lawrence College with concentrations in Mathematics and Theater.

Abstract: A Latin square is a matrix of symbols such that every symbol occurs exactly once in each row and column. A Latin square can also be considered as a set of triples of the form (row, column, symbol). A conjugate of a Latin square is obtained by uniformly permuting the elements of each triple, hence each Latin square has 6 conjugates. Every pair of rows in a Latin square induces a permutation in 2-line form. We call a Latin square $L$ row-Hamiltonian if the permutation induced by each pair of distinct rows of $L$ is a full cycle permutation. Row-Hamiltonian Latin squares are equivalent to perfect 1-factorisations of complete bipartite graphs. For a Latin square $L$, we define $\nu(L)$ to be the number of conjugates of $L$ which are row-Hamiltonian. We have constructed an infinite family of row-Hamiltonian Latin squares with $\nu(L) = 4$ for each Latin square $L$ in the family. No such infinite family was previously known. This construction allows us to solve two published open problems, one posed by Wanless and the other by Falconer.

Bio: Jack is a 1st-year PhD candidate working under the supervision of Ian Wanless and Daniel Horsley. His research is mainly on Latin squares and perfect 1-factorisations of graphs. Jack completed his undergraduate studies at Monash University with an extended major in mathematics and a minor in computational science.

Abstract: In the theory of random graphs, it is a common objective to prove that almost all graphs of a certain type contain a specified subgraph. A powerful tool for this purpose is the small subgraph conditioning method which was introduced by Robinson and Wormald in 1992. This technique been used to prove the existence of subgraphs such as perfect matchings, Hamilton cycles and spanning trees in almost all regular graphs even when the second moment method fails to do so.

We prove a new threshold result for the asymptotically almost sure existence of a $2$-factor in a random uniform regular hypergraph by using the small subgraph conditioning method. This extends results of Robalewska (1996), that almost all regular graphs contain a $2$-factor, and Cooper et al. (1996), that almost all random uniform regular hypergraphs subject to a degree threshold contain a perfect matching. We also explain the consequences of our result for a version of qualitative equivalence between random hypergraph models known as `contiguity'.

Bio: Adi is currently an Honours student in Software Engineering under the supervision of Haris Aziz. Prior to this year, he completed an Honours year in Advanced Mathematics, also at UNSW, under the supervision of Catherine Greenhill working on random graphs.