Discrete Geometry and Combinatorics Seminar Spring 2018

All seminars (unless otherwise stated) will take place on Tuesdays at 5.00pm in Room 707 (25 Gordon Street). See the map for further details. There will be tea afterwards in Mathematics Room 606 (25 Gordon Street). If you require any more information on the Applied seminars please contact Dr John Talbot (e-mail: j.talbot AT ucl.ac.uk or tel: 020-7679-4102).

23 January 2018

Speaker: David Ellis (QMUL)

Title: Symmetric Intersecting Families of Sets

Let X be a finite set. We say a family F of subsets of X is 'three-wise intersecting' if any three sets in F have nonempty intersection; we say it is 'symmetric' if it has transitive automorphism group. Frankl conjectured in 1981 that if F is a symmetric, three-wise intersecting family of subsets of an n-element set, then F has size o(2^n). We will discuss a recent (and surprisingly short) proof of this conjecture. Joint work with Bhargav Narayanan (Rutgers).

30 January 2018

Speaker: Samuel Porritt (UCL)

Title: Irreducible polynomials over F_q with restricted coefficients

In 2010 Mauduit and Rivat proved that, asymptotically, there are the same number of primes with an even as with an odd number of digits when written in binary (and similar results for other bases). In 2016 Maynard proved an asymptotic formula for the number of primes less than X which can be written in a given base with certain digits not used. We will discuss analogues of these two results for polynomials over a finite field. As in the integer setting, the proofs are based on the circle method but are much simpler thanks to strong estimates for exponential sums over irreducible polynomials and some technical simplifications due to the fact that we don't "carry digits" when adding polynomials over a finite field.

27 February 2018

Speaker: Prof William Jackson (QMUL)

Title: Unique low rank completability of partially filled matrices

I will consider the matrix completion problem - we are given a partially filled matrix and want to add entries in such a way that the resulting matrix has low rank. More precisely, I will assume that such a completion exists and ask whether it is unique. I will also consider the variants when the completed matrix should have the additional properties that it is a gram matrix or is skew-symmetric. I will describe how techniques from rigidity theory can be applied to help analyse these problems.

06 March 2018

Speaker: Peter Komjath, Eotvos University (Budapest)

Title: Geometric constructions in the Euclidean spaces requiring the Axiom of Choice

We present some results on Euclidean spaces whose proof require the Axiom of choice. An example is: the plane is the union of countably many parts, none containing two distinct points which span rational distance. One can see that some of the parts must be Lebesgue nonmeasurable, so no easily definable decomposition work.
We will mention results of R. O. Davies, P. Erdos, and J. Schmerl.