Below you can find information about past seminars in this series from 2016. The seminars are listed in reverse chronological order, most recent first.
Friday 14 October - Aaron Lin (LSE)
Sets with few ordinary circles
Let P be a set of n points in the plane. An ordinary circle is a circle which goes through exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least n^2/4 - O(n) ordinary circles. We also determine the exact minimum number of ordinary circles for all sufficiently large n, and describe the extremal and near-extremal examples.
In order to show this, we prove that if P spans at most Kn^2 ordinary circles, then all but O(K) many points of P lie on an algebraic curve of degree at most four.
(Joint work with Mehdi Makhul, Hossein Nassajian Mojarrad, Josef Schicho, Konrad Swanepoel, and Frank de Zeeuw.
Friday 7 October - Nóra Frankl (LSE)
Coverings by homothets of a convex body
Let K be a convex body in R^d and F be a family of its positive, smaller homothets. We define f(K,K) as the infimum of those t for which the following holds: If the total volume of F is at least t*vol(K), then F permits a translative covering of K. We improve the former upper bounds on f(K,K) and discuss further results.
Joint work with János Nagy and Márton Naszódi.
Friday 18 March
Paul Balister (Memphis)
The sharp threshold for making squares
Many of the fastest known algorithms for factoring large integers rely on finding subsequences of randomly generated sequences of integers whose product is a perfect square. Motivated by this, in 1994 Pomerance posed the problem of determining the threshold of the event that a random sequence of $N$ integers, each chosen uniformly from the set $\{1,\dots,x\}$, contains a subsequence, the product of whose elements is a perfect square. In 1996, Pomerance gave good bounds on this threshold and also conjectured that it is {\em sharp}.
In a paper published in Annals of Mathematics in 2012, Croot, Granville, Pemantle and Tetali significantly improved these bounds, and stated a conjecture as to the location of this sharp threshold. In recent work, we have confirmed this conjecture. In my talk, I shall give a brief overview of some of the ideas used in the proof, which relies on techniques from number theory, combinatorics and stochastic processes. Joint work with B\'ela Bollob\'as and Robert Morris.
Friday 26 February - Jan Hladky (Prague)
Cliques in dense inhomogeneous random graphs
The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. For a large class of graphons, we establish a formula which gives the almost sure clique number of these random graphs.
Joint work with Martin Dolezal and Andras Mathe.
Friday 19 February - Daniel Quiroz, LSE
Generalised colouring numbers of planar graphs
The generalised colouring numbers col_r(G) and wcol_r(G) were introduced by Kierstead and Yang as a generalisation of the colouring number, and have found important theoretical and algorithmic applications.
We will discuss an improvement on the known upper bounds to these numbers for graphs excluding a complete graph as a minor, from the exponential bounds of Grohe et al. to a linear bound on col_r(G) and an polynomial bound on wcol_r(G). We will look at some of the main techniques used to obtain these new bounds, with an emphasis on the results for planar graphs.
This is joint work with Jan van den Heuvel, Patrice Ossona de Mendez, Roman Rabinovich and Sebastian Siebertz.
Friday 12 February - Ahmad Abu-Khazneh, LSE
Extremal problems related to covering and matching in multipartite hypergraphs
No abstract available
Friday 22 January - Pongphat Taptagaporn (LSE)
Competitive greedy portfolio
Classical portfolio problems look at maximizing utility functions such as wealth or its risk-adjusted version. However, our studies are around regret: how far is our wealth from an optimal model (defined as having access to some future information). We study a new greedy model and analyze various statistics and their asymptotic. These have flavours from online learning, information theory, game theory.