The TMS Symposium is a day of lectures, held in the Winstanley Lecture Theatre, about a wide array of mathematical subjects.

This year the Symposium will be split over 2 days on the weekend of the 23^{rd}-24^{th} February. The first day will be dedicated to the history of the TMS, with 10 distinguished speakers, representing or talking about each decade of the TMS. The second day will be for current members of the society to talk about their research and contribution to mathematics.

## Schedule 23^{rd} February

Morning Session | ||
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09:00-09:45 | Piers Bursill-Hall | Industrial Strength Total Failure |

What was the state of mathematics – and particularly the foundations of mathematics – when Russell came up to Cambridge, and what would have driven him to the project of the Principia Mathematica … one of the most extraordinary and important failures of 20th century mathematics. | ||

09:45-10:30 | Miles Reid | Trinity Around 1970 - student life and math activities |

The talk will consist mainly of self-indulgent recollections of student life in Trinity 1966-1972, including a few anecdotes about Peter Swinnerton-Dyer and some of the many other great characters who were around, and with a few brief sections on math studies. | ||

10:30-10:50 | Break | |

10:50-11:35 | Christopher Tout | Milne and the Dawn of the Theory of Stellar Structure |

Between 1921 and 1937 Edward Arthur Milne published around 90 papers on stars. This was a period of very rapid change in our understanding of these fascinating objects that make up the visible Universe and Milne was a key player in these changes. In 1925 Cecilia Payne found the first indications that hydrogen is the major constituent of the Sun. Though Eddington was already convinced that nuclear fusion is the source of a star’s luminosity it was not until 1928 that Gamow applied quantum mechanical tunnelling to alpha particle emission and it became feasible that fusion could operate at the temperatures expected at the centres of stars. Milne’s early contributions concerned stellar atmospheres, including the formation of spectral lines and limb darkening. Later he delved into many aspects of stellar structure, including rotating stars, pulsating stars, binary stars and collapsed degenerate stars. It was Milne’s work on collapsed stars that caught the attention of Chandrasekhar and led him to determine the maximum mass for a white dwarf in 1931. I shall endeavour to portray these exciting times and how the ideas have developed into our present-day theory of stellar structure and evolution. | ||

11:35-12:20 | Richard Pinch | From CATAM to Quantum |

I plan to explore how mathematics, both pure and applied, has interacted with computation from the 1970s to the present day. I’ll illustrate with some of my own experiences from student days through to advising government on post-quantum cryptography, and how my research related to (but did not solve) three of the notorious Clay Millennium Problems. | ||

Afternoon Session | ||

13:45-14:45 | Richard Chapling | G.H. Hardy: The leading mathematician in England |

Hilbert is reported to have said that Hardy was “the best mathematician, not only in Trinity but in England”. But there’s more to Hardy than just brilliant research. I will discuss the larger role that Hardy played in mathematics, from Tripos reform to international relations. | ||

14:45-15:30 | Sir David Cox | 14:45 - Frank Anscombe and Cambridge in the 1940's and 50's |

The talk will be in two parts, one about undergraduate life in Cambridge in the early 1940’s and the second about Frank Anscombe and the Statistical Laboratory in the early 1950’s. | ||

15:30-15:50 | Break | |

15:50-16:35 | Julia Gog | Mathematical biology: the trickiest branch of mathematics? |

Mathematical Biology certainly would not have made sense to the members of TMS as a research field 100 years ago. Now, most decent maths degree include a course on it, and Math Bio continues to grow as a research field. Aside from a bit of wild speculation and generalising, I will focus on the bits of Math Bio that I know: the dynamics of infectious disease. I will introduce some recent work, and some thoughts on the potential for this area in future. | ||

16:35-17:20 | Colm-Cille Caulfield | How the Titanic Tragedy transformed Trinity: Turbulence Theory and Taylor in the Teens |

George Ingram Taylor (Trinity undergraduate 1905-1908; elected fellow 1910) was one of the most influential applied mathematicians of the 20th century, who made a huge number of contributions to fluid and solid mechanics. This talk will focus on the significance, for both Cambridge Mathematics and the world at large, of the work presented in his Adams Prize Essay of 1915 on “Turbulent Motion in Fluids”. Some of the key results presented in this essay, the partial manuscript of which is held in the Trinity Library, rely on data taken by Taylor himself on the first “Ice Patrol” cruise, triggered by the tragedy of the sinking of the Titanic. The essay was actually written when Taylor was participating in the first world war effort designing aircraft at Farnborough for the precursor of the Royal Air Force, and the talk will also consider the lasting influence on applied mathematics of Taylor’s philosophical approach to research. | ||

17:20-18:05 | Francis Woodhouse | Mechanics meets biology |

Unlike an ordinary gas, biological systems are never in equilibrium: cells constantly use chemical energy to grow and move, forming a clear arrow of time. The recent creation of artificial versions of these ‘active’ systems raises the tantalising prospect of soft robotic systems fuelled by as simple a source as oxygen. After a tour through the mathematics of elastic networks, marrying linear algebra, graph theory and dynamics and invoking plenty of tenuous Trinity connections, we will see how endowing such a network with biologically-inspired activity can create intricate self-actuating mechanisms. |

## Schedule 24^{th} February

Morning Session | ||
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10:30-11:05 | Mary Fortune | Designing Dementia Trials Embedded Within a Cohort |

Clinical trial recruitment sounds like it should be a job for the medics, not the mathematicians. But what if you already have access to decades of data about your prospective participants, and want to select those who you wish to invite to take part based on mathematical predication of how their disease will progress? In many dementia trials, the focus is upon patients who are asymptomatic or at an early stage in the disease. We would like to choose trial participants who are likely to benefit from the treatment. However, if our recruitment is too selective, we may not be able to say anything about the effect of the treatment in the whole population. Any strategy needs to balance the pay-off between these two factors | ||

11:05-11:40 | Oliver Feng | Maximum likelihood estimation of a log-concave density |

A density on R^d is said to be log-concave if its logarithm is a concave function, and the estimation of a unknown log-concave density based on i.i.d. observations represents a central problem in the area of non-parametric inference under shape constraints. In contrast to traditional smoothing techniques, the log-concave maximum likelihood estimator is a fully automatic estimator which does not require the choice of any tuning parameters and therefore has the potential to offer practitioners the best of the parametric and non-parametric worlds. I will discuss some recent theoretical results on the performance of this estimator, with a particular focus on its ability to adapt to structural features of the target density. | ||

11:40-12:15 | Carl Turner | Introduction to Topological Phases of Matter |

I will talk a little about how we traditionally classify states of matter, and then blow this out of the water by introducing the idea of a topological phase of matter. The idea is that even fairly innocuous interactions between particles can result in striking collective behaviours at low energies; this includes sensitivity to the topology of the surface that the system is built on, and emergent "anyonic" particle-like objects, exhibiting properties that are impossible for fundamental particles in our universe. These phases of matter have connections to many areas of ongoing research, from pure mathematical classification problems, through dreams of quantum computation and quantum error correction, to condensed matter experiments. I will try to give a brief overview of how all this arises through simple examples. | ||

12:15-12:50 | Kirill Kalinin | Networks of non-equilibrium condensates for global optimisation of spin Hamiltonians |

The majority of optimisation problems are computationally impractical for conventional classical computers and known as NP-hard optimisation problems. Such problems deal with scheduling, the dynamic analysis of neural networks and financial markets, the prediction of new chemical materials, and machine learning. Incredibly, it is possible to reformulate these optimisation problems into the problem of finding the ground state of a particular spin Hamiltonian. In my talk I will address various physical platforms that can simulate such spin Hamiltonians in order to solve optimisation problems orders of magnitude faster than can be achieved on a classical computer. In particular, the spin Hamiltonians can be simulated experimentally with polariton condensates. These are effectively comprised of a “mix” of the states of light and matter, and can be explicitly mapped into problems such as the travelling salesman problem. Using such mappings, one can study physical systems experimentally and effectively “read out” the solution to an optimisation problem one wishes to solve. A possible speedup opens a path to global minimisation of large-scale, real-world problems not accessible by classical simulations. | ||

Afternoon Session | ||

13:30-14:05 | Irene Li | The non-equilibrium characteristics of active matter |

Active matter consists of a large number of "active" particles, each of which is capable of consuming energy locally, causing them to move with deterministic or stochastic rules that break the time reversal symmetry. Examples of active matter are commonplace in biology: bird flocks, bacteria colonies, tumour growth, self-organising bio-polymers etc. As a result of this constant injection and dissipation of energy at the microscopic level, these systems are permanently driven away from equilibrium, where many familiar principles of statistical physics don't apply. Recent progresses have enabled us to capture the collective behaviours of many complex biological systems in terms of simpler models, making it easier to trace the non-equilibrium nature of such systems. Specifically, the talk will focus on self-propelled bacteria with quorum sensing and population dynamics. | ||

14:05-14:40 | Alex Chamolly | Why sperm doesn’t have fins |

Have you never wondered? Alright, I forgive you. But, if you somehow did produce sperm with fins you’d soon realise that it's actually an existential question — literally — as with such equipment you’d never have come to be. Lacking inertia, the physics of fluids on the microscale is vastly different from our everyday experience, and so sperm, bacteria and other microswimmers had to develop unique strategies to survive. I will present some surprising theorems, highlight how nature has adapted in response to them and how they challenge current designs for artificial microrobots. Expect wet jokes and dry puns. | ||

14:40-15:15 | Michael Gomez | Elastic snap-through: from the Venus flytrap to jumping popper toys |

Snap-through buckling is a type of instability in which an elastic object rapidly jumps from one state to another. Such instabilities are familiar from everyday life: children’s popper toys rapidly ‘pop’ and jump after being turned inside-out, while snap-through is harnessed to generate fast motions in applications ranging from soft robotics to artificial heart valves. In biology, snap-through has long been exploited to convert energy stored slowly into explosive movements: both the leaf of the Venus flytrap and the beak of the hummingbird snap-through to catch prey unawares. Despite the ubiquity of snap-through in nature and engineering, its dynamics is usually only understood qualitatively, with many examples reported of delay phenomena in which snap-through occurs much more slowly than would be expected for an elastic instability. To explain this discrepancy, it is commonly assumed that some dissipation mechanism (such as material viscoelasticity) must be causing the system to lose energy and slow down. In this talk we examine how viscoelasticity influences the snap-through dynamics of a simple truss-like structure. We present a regime diagram that characterises when the timescale of snap-through is governed by viscous or elastic effects, and relate this to the creep behaviour we see in jumping popper toys. | ||

15:15-15:45 | Break | |

15:45-16:20 | Marius Leonhardt | Why is $e^{\pi \sqrt{163}}$ almost an integer? |

$e^{\pi \sqrt{163}}=262537412640768743.99999999999925$ is very close to an integer. Coincidence? Not at all. In this talk, we will see how this is related to the $j$-function, a certain holomorphic function on the upper half plane. We will interpret the $j$-function as a function on the space of elliptic curves, and then see how symmetries of elliptic curves and of this space have far-reaching consequences -- not only for $e^{\pi \sqrt{163}}$. | ||

16:20-16:55 | Oliver Janzer | A generalization of the off-diagonal Ramsey numbers |

The Ramsey number R(s,t) stands for the smallest positive integer N such that whenever the edges of the complete graph on N vertices are 2-coloured with red and blue, then there is a red clique on s vertices or a blue clique on t vertices. In this talk, I will review the known results about this function, mentioning some famous open problems. I will also talk about how random graphs can be used to prove lower bounds on R(s,t). Then I will present a generalisation of this function and state a recent result of Gowers and myself, which is based on an unusual random graph construction. | ||

16:55-17:30 | Andrew Carlotti | Uniform Bounds for Non-negativity of the Diffusion Game |

I will discuss a variant of the chip-firing game known as the diffusion game. In the diffusion game, we begin with some integer labelling of the vertices of a graph, interpreted as a number of chips on each vertex, and then for each subsequent step every vertex simultaneously fires a chip to each neighbour with fewer chips. In general, this could result in negative vertex labels. In this talk I will answer the following question: do there exist values f(n), for each n, such that whenever we have a graph on n vertices and an initial allocation with at least f(n) chips on each vertex, then the number of chips on each vertex will remain non-negative. I will also consider the possibility of a similar bound g(d) for each d, where d is the maximum degree of the graph. |