Meeting of the LMS Joint Research Groups

Tuesday 20th June 2017, University of Bath

For our 17th meeting we have a program of three exciting talks
covering a wide range of disciplines.

**Program:**
**Optional lunch (only free for speakers but everyone
welcome):
**

12:00 - 13:00 Congregate in the 4 west level 4 social space

13:00 - 13:45 Lunch in the Claverton rooms

Tropical Intersection Theory and the Correspondence Theorem for Rational Curves

One of the major motivations to study tropical geometry as an algebraic geometer has been its power to turn intricate counting problems in algebraic geometry into problems of essentially combinatorial, discrete nature. Tropical intersection theory provides a set of tools to understand this transition from algebraic to tropical enumerative geometry. In my talk, I will – after briefly recalling the foundations of tropical intersection theory and its relation to algebraic geometry – present the main ideas of how tropical intersection theory can be used to prove the correspondence theorem for rational curves.

Online Bayesian inference of phylogenies

Whole genome sequencing has had a big impact in studying the evolutionary history of pathogens. At the core of many studies is the need to infer the clonal ancestry of a sample from sequence data. This is usually performed using a Bayesian approach, with a coalescent prior on ancestries, and using Markov chain Monte Carlo (MCMC) for inference, as implemented in software such as BEAST. This talk begins by outlining the generic problem of inference of phylogenies, then goes on to discuss work on a sequential Monte Carlo approach to inference in models with coalescent priors, with the aim of inferring clonal ancestries “online” as data arrives. The proposed approach is also applicable to Bayesian model comparison, and this will also be discussed briefly.

The Tropical Nullstellensatz

The Nullstellensatz is one of the most fundamental results in elementary algebraic geometry. A naive tropical generalization is badly false, however. In this talk I will explain how the theory of "tropical ideals" introduced with Felipe Rincon building on work of the Giansiracusas gives a setting where the weak Nullstellensatz holds tropically. I will also discuss progress towards the strong Nullstellensatz in this setting.

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