Royal Holloway -
Pure Mathematics Seminar

**Autumn term 2019:**

The PM-seminar takes
place on Wednesday at 2pm in Munro Fox Lecture Theatre (unless stated
otherwise).

Here is a campus plan.

October 2: **Andrew Treglown** (Birmingham)

**Sum-free sets in the
integers**

Abstract: A sum-free set is simply a set of integers
which does not contain any solution to x+y=z. In this
talk we discuss a range of recent developments concerning sum-free sets. We
give a sharp count on the number of maximal sum-free subsets in [n], thereby
answering a question of Cameron and Erdős. We also
consider similar questions in the setting of solution-free sets (that is now we
forbid solutions to some other fixed equation). Related complexity results will
also be discussed. Underlying all this work is a connection to independent sets
in graphs and hypergraphs.

The talk includes joint work with Jozsi
Balogh, Hong Liu and Maryam Sharifzadeh; Robert
Hancock; Kitty Meeks.

October 9: **Natasha
Morrison** (Cambridge)

**The typical structure of sets with small sumset**

October 16: **Peter
Allen** (LSE)

**Packing sparse graphs**

October 23: **Jonathan
Chapman** (Manchester)

**Partition regularity and
multiplicatively syndetic sets**

Abstract: The study of partition regularity
investigates properties of sets which are preserved under finite partitions.
For example, if we are given an integer polynomial and a finite colouring of
the positive integers, can we always find a monochromatic root? In this talk I
will introduce multiplicatively syndetic sets, which are sets of integers with
'bounded multiplicative gaps', and explore their
connections with partition regularity. In particular, I will show that a homogeneous
integer polynomial has a monochromatic root under any finite colouring if and
only if the polynomial has a root over any multiplicatively syndetic set. I
will then show how this fact can be used to obtain quantitative bounds for Brauer's generalisation of van der Waerden's
theorem. I will also mention some open problems concerning minimal
multiplicatively syndetic sets.

October 30: **Ehud Meir** (Aberdeen)

**Rings of invariant and the symmetric groups**

November 6: **Kevin Buzzard** (Imperial
College)

**The future of mathematics?**

Abstract: Over the last few years, something (possibly
a mid-life crisis) has made me become concerned about the reliability of modern
mathematics, and about how the methods we mathematicians have traditionally
used to prove theorems are scaling with the advent of the internet /ArXiv, and pressure on academics to get big results out
there. I have started experimenting with a formal computer proof verification
system called Lean, integrating it into my undergraduate teaching at Imperial
and pushing it to see if it can handle modern mathematical definitions such
as perfectoid spaces and the other ideas
which got Peter Scholze a Fields Medal in
2018. I personally believe that Lean is part of what will become a paradigm
shift in the way humans do mathematics, and that people who do not switch will
ultimately be left behind. Am I right? Only time will tell. This talk will be be suitable for a general scientific audience --
mathematics undergraduates, computer scientists and philosophers will all find
it comprehensible.

November
13: **Cong Ling** (Imperial College)

**Post-Quantum Cryptography Based on Division Algebras**

The Learning with Errors (LWE) problem is the fundamental backbone of
modern lattice based cryptography. However, schemes
based on LWE are often impractical, so Ring LWE was introduced as a form of
‘structured’ LWE. Another popular variant, Module LWE, generalizes this
exchange by implementing a module structure over a Ring LWE instance. In this
work, we introduce a novel variant of LWE over cyclic algebras (CLWE). The
proposed construction is both more efficient than Module LWE and conjecturally
more secure than Ring LWE, the best of both worlds.

November 20: **Kevin
Grace** (Bristol)

**Templates for
Representable Matroids**

Abstract: The
matroid structure theory of Geelen, Gerards, and Whittle has led to an announced result that a
highly connected member of a minor-closed class of matroids representable over
a finite field is a mild modification (known as a perturbation) of a frame
matroid, the dual of a frame matroid, or a matroid representable over a proper
subfield. They introduced the notion of a template to describe these
perturbations in more detail. In this talk, we will define templates and
discuss how templates are related to each other. We define a preorder on the set of frame templates over a finite field,
and we determine the minimal nontrivial templates with respect to this preorder.

We use templates
to obtain results about representability, extremal functions, and excluded
minors for various minor-closed classes of matroids, subject to the announced
result of Geelen, Gerards,
and Whittle. These classes include the class of 1-flowing matroids and three
closely related classes of quaternary matroids -- the golden-mean matroids, the
matroids representable over all fields of size at least 4, and the quaternary
matroids representable over fields of all characteristics. This leads to a
determination of the extremal functions for these classes, verifying a
conjecture of Archer for matroids of sufficiently large rank.

This talk
will include a brief introduction to matroid theory and is partially based on
joint work with Stefan van Zwam.

November 27:

December 4:

December 11: **Joni ****Teräväinen** (Oxford)

**Chowla's conjecture at almost all scales**

An unsolved conjecture of Chowla states that the Möbius function should
not correlate with its own shifts. This can be viewed as a conjecture about the
randomness of the Möbius function.

In the last few years, there has been a lot of progress on Chowla's
conjecture, which I will survey in the talk. Nearly all of the previously
obtained results have concerned correlation sums that are weighted
logarithmically, so one wonders whether it is possible to get rid of these
weights. We show that one can indeed remove logarithmic weights from previously
known results on Chowla's conjecture, provided that one restricts to almost all
scales in a suitable sense.

This is joint
work with Terry Tao.

**Previous Pure
Mathematics Seminars **