This year’s Trinity Mathematical Society Symposium is running from 10:00 to 18:00 on Sunday 26nd February. We have talks by fellows and PhD students, ranging across all areas of mathematical research. The event is free and open to all; no particular specialist knowledge is assumed. There is no need to stay for the whole day – just drop in on talks you find interesting.

**The program is:**

The Trinity Clock in Great Court is quite a prominent feature of the college, at least to look at. And it has a rather curious way of announcing the hours, once for Trinity and a second time for St John’s. It is always within a second or two of the correct time and yet it hardly ever requires adjustment. Does this mean that the mechanism is unaffected by the elements? What about temperature, pressure, humidity? And does the gravitational pull of the moon make any difference? The pendulum on the Trinity Clock has been instrumented to measure period and amplitude to great accuracy. The time is compared with UTC obtained from a GPS receiver. All of this data is streamed continuously to the web at http://www.trin.cam.ac.uk/clock/. The clock has become quite a sensitive pigeon detector!

You don’t have to go further than Paul McCartney to find that it is, in fact, a changing world in which we live in. Due to recent efforts, particularly at Cambridge and Lancaster, understanding with respect to the changepoint problem is at a much more tolerable state from the perspective of the researcher. However, several key questions remain. How does one efficiently deal with multivariate data? Can new methodologies keep pace with the ever growing demands of industry? And, perhaps most importantly, why, no matter how hard we might shut our eyes to it, is everything a changepoint problem?

In 1963, Sydney Brenner, who spent 20 years of his career in Cambridge, established Caenorhabditis elegans as a model organism in biology. It is in C. elegans that many apoptotic genes were discovered. Later on, homologous genes in humans were found to also be involved in apoptosis. The work of Nobel Prize winners Andrew Fire and Craig Mello on RNA interference has also been conducted in this animal. Most of the work in C. elegans has been conducted in a strain isolated in Bristol many decades ago. Until recently, little was known about the natural habitat of this nematode and its genetic and phenotypic diversity. In this talk, I will present how we can use the natural genetic and phenotypic diversity of this animal to study the relationship between its phenotype and its genotype.

No abstract available

**12:15 – 13:15 BRUNCH**

No abstract available

Applied mathematics tries to explain the world around us with mathematical models, but what do you do if the answer from your model just doesn’t make any sense? I’ll describe two paradoxes that plagued fluid dynamics for hundreds of years, explain what we learnt from their resolution, and show you how this new understanding is being applied to current research problems.

If the integers are partitioned into finitely many pieces, what can be said about the arithmetic structure of the parts? Indeed, it is perhaps remarkable that anything interesting can be said about such general partitions. One of the first results along these lines was proved over 100 years ago by I. Schur, who proved that if the positive integers are partitioned into finitely many parts, then one of the parts must contain two integers x,y and their sum x+y. In this talk I’ll discuss a host of new results about exponential patterns that can be found in arbitrary finite partitions of the integers.

If H is a collection of graphs, we say that a graph G is H-free if it contains no copy of a graph in H as a subgraph. Many problems in extremal graph theory involve investigating the behaviour of various graph parameters, or determining the structure of H-free graphs for specific families H. It turns out that we can say quite a few fascinating things about the structure of graphs which forbid some finite collection of subgraphs and, in addition, have large minimum degree. In this talk I shall describe several results of this type, both old and new, and hopefully convey why they are interesting.

**15:30 – 16:00 BREAK**

Let $T$ be a finite subset of $\mathbb{Z}^2$. We can view $T$ as a tile consisting of $|T|$ unit squares. Must it be possible to cover the plane by non-overlapping copies of $T$? (We allow translations, rotations and reflections of $T$.) The answer is, of course, no: just take a tile with a hole. However, if we replace every unit square of $T$ by a unit cube, then we get a $3$-dimensional analogue of $T$. We now ask: does $T$ tile $\mathbb{Z}^3$? Or, does $T$ tile $\mathbb{Z}^n$ for some, possibly very large, $n$? We prove that the answer to the latter question is yes, confirming a conjecture of Chalcraft. This talk is based on joint work with Leader and Tan.

No abstract available

In a way, this academic year has been a Geometric Group Theory year, with a large scale research programme held from August to December 2016 at the Mathematical Sciences Research Institute in Berkeley, California, followed by a programme from January to June 2017 at the Newton Institute in Cambridge. These events have been particularly timely, as the area recorded a series of recent breakthrough results, ranging from the work of Ian Agol (who answers the remaining four of Thurston’s conjectures on the structure of 3-manifolds), to the work of Cheeger, Kleiner, Naor and Young (solving the Sparsest Cut Problem, and the related Goemans-Linial conjecture, in Theoretical computer science).So, what is Geometric Group Theory ? In my talk I will try to overview this area, and convey its general philosophy of reformulating in geometric terms problems from diverse areas of mathematics and then solving them with a variety of tools.

**19:45 – Annual Dinner**

(Made possible by the kind support of the Heilbronn Fund)