Upcoming talks
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28th October 2024 - Debmalya Bandyopadhyay (University of Birmingham)
Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph on $n$ vertices. A tight cycle is a $k$-uniform graph with its vertices cyclically ordered so that every $k$ consecutive vertices form an edge, and any two consecutive edges share exactly $k-1$ vertices. A result by Bustamante, Corsten, Frankl, Pokrovskiy and Skokan shows that all $r$-edge coloured $K_{n}^{(k)}$ can be partition into $c_{r,k}$ vertex disjoint monochromatic tight cycles. However, the constant $c_{r,k}$ is of tower-type. In this work, we show that $c_{r,k}$ is a polynomial in $r$.
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11th November 2024 - Stelios Stylianou (University of Bristol)
Suppose we fix a finite collection $S$ of points in $\mathbb R^d$, which we regard as the locations of voters. Two players (candidates), Alice and Bob, are playing the following game. Alice goes first and chooses any point $A$ in $\mathbb R^d$. Bob can then choose any point $B$ in $\mathbb R^d$, knowing what $A$ is. A voter $V$ positioned at $x$ will vote for Alice (resp. Bob) if $d(x,A)<d(x,B)$ (resp. $d(x,B)<d(x,A)$), where $d$ denotes Euclidean distance. If $x$ is equidistant from $A$ and $B$, $V$ will not vote for anyone. The candidate with the greatest number of votes wins. If they both get the same number of votes, then (by convention) Alice is declared the winner.
When $d=1$, it is easy to see that Alice can always win this game by choosing $A$ to be the median of $S$. This observation was first made by Hotelling in the 1920s; he referred to this model (in one dimension) as the Median Voter Model. When $d > 1$ however, the game is sometimes an Alice win and sometimes a Bob win, depending on the structure of $S$. We completely characterise the sets $S$ for which Alice wins in dimensions $> 1$, showing that the game is usually won by Bob, unless $S$ has a specific structure.
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18th November 2024 - Marius Tiba (King’s College London)
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25th November 2024 - Jun Yan (University of Warwick)
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4th December 2024 - Jared León (University of Warwick)