Mathematical Institute Oxford

Representation theory of $p$-adic groups in characteristic $p$ is something which is still not understood too well. One issue, which we already see for modular representations of finite groups, is that things are not semisimple anymore, and so understanding irreducible representations is not quite enough to have full information. During the talk I will present some of my attempts at creating homological methods to understand extensions of representations of $\operatorname{SL}_2(\mathbb{Q}_p)$, and what those results imply about injective smooth representations (that are my main focus of research).

The Springer correspondence parameterises the irreducible representations of the Weyl group of a complex semisimple Lie algebra by nilpotent orbits. A key ingredient in the construction is the convolution operation, which appears in various forms throughout geometric representation theory. In this talk, we'll introduce the geometry of the Springer resolution, describe the convolution operation, and illustrate how it gives rise to a geometric construction of Weyl group representations.

I will talk about the problem of recognising when a manifold admits a finite cover that fibres over the circle, with emphasis on the case of hyperbolic manifolds in odd dimensions. I will survey the state-of-art, and discuss the role that group theory plays in the problem. Finally, I will discuss a recent result that sheds light on the analogous group-theoretic problem, that is, virtual algebraic fibring of Poincaré-duality groups. The final theorem is joint with Sam Fisher and Giovanni Italiano.

In this talk I will discuss how computational algebraic geometry and topology can be useful for studying questions arising in systems biology. In particular I will focus on the problem of comparing models and data through the lens of computational algebraic geometry and statistics. I will provide concrete examples of biological signalling systems that are better understood with the developed methods.

Please note that this will be held at Tsuzuki Lecture Theatre, St Annes College, Oxford.

Please note that you will need to register for this event via https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2018-ticke…

In this talk, I will review a number of interdisciplinary collaborations in which I have been involved over the years that have coupled mathematical
modelling with experimental studies to try to advance our understanding of processes in biology and medicine. Examples will include somatic evolution in
tumours, collective cell movement in epithelial sheets, cell invasion in neural crest, and pattern formation in slime mold. These are examples where
verbal reasoning models are misleading and insufficient, while mathematical models can enhance our intuition.

Please note that this will be held at Tsuzuki Lecture Theatre, St Annes College, Oxford.

Please note that you will need to register for this event via https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2018-ticke…