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)

November 27:

December 4:

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