Keynotes

The keynote addresses at TMT2021 will be delivered by

Saturday: Dr Julia Wolf, Counting monochromatic structures in finite abelian groups

Sunday: Professor Sir Martin Hairer, KBE FRS Title tbc

Abstracts

Dr Julia Wolf

Counting monochromatic structures in finite abelian groups

In 1928, Frank Plumpton Ramsey proved what is now a foundational theorem in combinatorics: given any integer r greater than 1, any 2-colouring of the edges of a sufficiently large complete graph will always contain a monochromatic copy of a complete subgraph of size r. As we shall see at the start of this talk, it is not too difficult to see that it will contain not only one monochromatic copy, but in fact many.

It is well known—and a result of Goodman from the 1950s—that a random 2-colouring of the edges of a large complete graph K_n contains (asymptotically, amongst all possible colourings) the minimum number of monochromatic triangles (K_3). Erdős conjectured that this was also true of monochromatic copies of K_4, but his conjecture was disproved by Thomason in the late 1980s. The question of determining for which small graphs Goodman's result holds true remains wide open, and we will survey what is known.

In the second half of this talk we explore an arithmetic analogue of this question: what can be said about the number of monochromatic additive configurations in 2-colourings of finite abelian groups?

Professor Martin Hairer

A mathematical journey through scales

You can watch this talk here.

The tiny world of particles and atoms and the gigantic world of the entire universe are separated by about forty orders of magnitude. As we move from one to the other, the laws of nature can behave in drastically different ways, sometimes obeying quantum physics, general relativity, or Newton’s classical mechanics, not to mention other intermediate theories. Understanding the transformations that take place from one scale to another is one of the great classical questions in mathematics and theoretical physics, one that still hasn't been fully resolved. In this lecture, we will explore how these questions still inform and motivate interesting problems in probability theory and why so-called toy models, despite their superficially playful character, can sometimes lead to certain quantitative predictions.

Biographies

Professor Sir Martin Hairer KBE FRS works in the field of stochastic analysis, in particular stochastic partial differential equations. He is Professor of Pure Mathematics at Imperial College London, having previously held appointments at the University of Warwick and the Courant Institute of New York University. In 2014 he was awarded the Fields Medal, and he won the 2021 Breakthrough Prize in Mathematics. He also created and maintains the Amadeus sound editing software under the name HairerSoft.

Dr Julia Wolf is Associate Professor in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge, and a Fellow of Clare College Cambridge. Her research interests are mostly discrete in nature, and broadly lie at the intersection of combinatorics, number theory and harmonic analysis. Some of her work has close connections with model theory and ergodic theory, and applications to theoretical computer science. In 2016 she was awarded the London Mathematical Society's Anne Bennett Prize "in recognition of her outstanding contributions to additive number theory, combinatorics and harmonic analysis and to the mathematical community".

For further information or to contact the organisers tmt@gre.ac.uk.

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Abstract:

Dr Julia Wolf

Counting monochromatic structures in finite abelian groups

In 1928, Frank Plumpton Ramsey proved what is now a foundational theorem in combinatorics: given any integer r greater than 1, any 2-colouring of the edges of a sufficiently large complete graph will always contain a monochromatic copy of a complete subgraph of size r. As we shall see at the start of this talk, it is not too difficult to see that it will contain not only one monochromatic copy, but in fact many.

It is well known—and a result of Goodman from the 1950s—that a random 2-colouring of the edges of a large complete graph K_n contains (asymptotically, amongst all possible colourings) the minimum number of monochromatic triangles (K_3). Erdős conjectured that this was also true of monochromatic copies of K_4, but his conjecture was disproved by Thomason in the late 1980s. The question of determining for which small graphs Goodman's result holds true remains wide open, and we will survey what is known.

In the second half of this talk we explore an arithmetic analogue of this question: what can be said about the number of monochromatic additive configurations in 2-colourings of finite abelian groups?