Past

Social surveys are widely used in today's society as a method for obtaining opinions and other information from large groups of people. The questions in social surveys are usually presented in either multiple-choice or free-response formats. Despite their advantages, free-response questions are employed less commonly in large-scale surveys, because in such situations, considerable effort is needed to categorise and summarise the resulting large dataset. This is the so-called coding problem. Here we propose a survey framework in which, respondents not only write down their own opinions, but also input information characterising the similarity between their individual responses and those of other respondents. This is done in much the same way as ``likes" are input in social network services. The information input in this simple procedure constitutes relational data among opinions, which we call the opinion graph. The diversity of typical opinions can be identified as a modular structure of such a graph, and the coding problem is solved through graph clustering in a statistically principled manner. We demonstrate our approach using a poll on the 2016 US presidential election and a survey given to graduates of a particular university.

In this talk we discuss the Type I blow up and the related problems in the 3D Euler equations. We say a solution $v$ to the Euler equations satisfies Type I condition at possible blow up time $T_*$ if $\lim\sup_{t\nearrow T_*} (T_*-t) \|\nabla v(t)\|_{L^\infty} <+\infty$. The scenario of Type I blow up is a natural generalization of the self-similar(or discretely self-similar) blow up. We present some recent progresses of our study regarding this. We first localize previous result that ``small Type I blow up'' is absent. After that we show that the atomic concentration of energy is excluded under the Type I condition. This result, in particular, solves the problem of removing discretely self-similar blow up in the energy conserving scale, since one point energy concentration is necessarily accompanied with such blow up. We also localize the Beale-Kato-Majda type blow up criterion. Using similar local blow up criterion for the 2D Boussinesq equations, we can show that Type I and some of Type II blow up in a region off the axis can be excluded in the axisymmetric Euler equations. These are joint works with J. Wolf.

We prove that every rational homology cobordism class in the subgroup generated
by lens spaces contains a unique connected sum of lens spaces whose first homology embeds in
any other element in the same class. As a consequence we show that several natural maps to
the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility
condition between the determinants of certain 2-bridge knots and other knots in the same
concordance class. This is joint work with Daniele Celoria and JungHwan Park.

In this talk I will discuss some recent results that allow to approximate a real singularity given by polynomial equations of degree d (e.g. the zero set of a polynomial, or the number of its critical points of a given Morse index) with a singularity which is diffeomorphic to the original one, but it is given by polynomials of degree O(d^(1/2)log d).
The approximation procedure is constructive (in the sense that one can read the approximating polynomial from a linear projection of the given one) and quantitative (in the sense that the approximating procedure will hold for a subset of the space of polynomials with measure increasing very quickly to full measure as the degree goes toinfinity).

The talk is based on joint works with P. Breiding, D. N. Diatta and H. Keneshlou      

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators

The stack of Higgs bundles of a given rank and degree over a non-singular projective curve can be stratified in two ways: according to its Higgs Harder-Narasimhan type (its instability type) and according to the Harder-Narasimhan type of the underlying vector bundle (instability type of the underlying bundle). The semistable stratum is an open stratum of the former and admits a coarse moduli space, namely the moduli space of semistable Higgs bundles. It can be constructed using Geometric Invariant Theory (GIT) and is a widely studied moduli space due to its rich geometric structure.

In this talk I will explain how recent advances in Non-Reductive GIT can be used to refine the Higgs Harder-Narasimhan and Harder-Narasimhan stratifications in such a way that each refined stratum admits a coarse moduli space. I will explicitly describe these refined stratifications and their intersection in the case of rank 2 Higgs bundles, and discuss the topology and geometry of the corresponding moduli spaces

I will describe recent advances in the study of quantum gravity where one can explicitly show in examples that superpositions of states with fixed topology can change the topology of spacetime. These effects lead to paradoxes that are resolved in effective field theory by the introduction of code subspaces. I will also talk about more typical states and issues related on how to decide if a black hole horizon is smooth or not.

Valérie Voorsluijs
Deterministic limit of intracellular calcium spikes
Abstract: In non-excitable cells, global calcium spikes emerge from the collective dynamics of clusters of calcium channels that are coupled by diffusion. Current modeling approaches have opposed stochastic descriptions of these systems to purely deterministic models, while both paradoxically appear compatible with experimental data. Combining fully stochastic simulations and mean-field analyses, we demonstrate that these two approaches can be reconciled. Our fully stochastic model generates spike sequences that can be seen as noise-perturbed oscillations of deterministic origin while displaying statistical properties in agreement with experimental data. These underlying deterministic oscillations arise from a phenomenological spike nucleation mechanism.

Matthias Nagel
Knots in dimensions three and four
Abstract: Knot theory studies the various embeddings of a circle into three-dimensional space. I will describe an equivalence relation on knots, called "concordance", which takes the fourth dimension into account. The study of concordance is intimately related with many problems at the heart of the topology of four-manifolds, such as the difference between the smooth and the topological category, and I will discuss results that illuminate this relation.

This week’s Fridays@2 session, led by Dr Richard Earl and Dr Vicky Neale, is intended to provide advice on exam preparation and how to approach the Part A and Part B exams.

This session is aimed at second years and third years who will be sitting exams this term. Next week’s Fridays@2 will be for first years and will look at preparing for Prelims papers.
 

Competition between peptides for binding and presentation by MHC class I molecules decides the immune response to foreign or tumor antigens. Many previous studies have attempted to classify the immunogenicity of a peptide using machine learning algorithms to predict the affinity, or half-life, of the peptide binding to MHC. However immunopeptidome analyses have shown a poor correlation between sequence based predictions and the abundance on the cell surface of the experimentally identified peptides. Such metrics are, for instance, only comparable when the abundance of competing peptides can be accurately quantified. We have developed a model for predicting the relative presentation of competing peptides that takes into account off-rate, source protein abundance and turnover and cofactor-assisted MHC assembly with peptides. This model is mechanism based so that it can accommodate complex biology phenomena such as inflammation, up or downregulation of peptide loading complex chaperones, appearance of a mutanome. We have used aspects of the model to drive an investigation of the precise molecular mechanism of peptide selection by MHC I and its associated intracellular cofactors.

We discuss a nonlinear variant of Roth’s theorem on the existence of three-term progressions in dense sets of integers, focusing on an effective version of such a result. This is joint work with Sarah Peluse.
 

My goal for the talk is to give a "from the ground-up" introduction to symplectic topology. We will cover the Darboux lemma, pseudo-holomorphic curves, Gromov-Witten invariants, quantum cohomology and Floer cohomology.

We present a novel approach to the solution of time-dependent PDEs via the so-called monolithic or all-at-once formulation.

This approach will be explained for simple parabolic problems and its utility in the context of PDE constrained optimization problems will be elucidated.

The underlying linear algebra includes circulant matrix approximations of Toeplitz-structured matrices and allows for effective parallel implementation. Simple computational results will be shown for the heat equation and the wave equation which indicate the potential as a parallel-in-time method.

This is joint work with Elle McDonald (CSIRO, Australia), Jennifer Pestana (Strathclyde University, UK) and Anthony Goddard (Durham University, UK)

According to the Harish-Chandra philosophy, cuspidal representations are the basic building blocks in the representation theory of finite reductive groups.  Similarly for supercuspidal representations of p-adic groups.  Self-dual representations play a special role in the study of parabolic induction.  Thus, it is of interest to know whether self-dual (super)cuspidal representations exist.  With a few exceptions involving some small fields, I will show precisely when a finite reductive group has irreducible cuspidal representations that are self-dual, of Deligne-Lusztig type, or both.  Then I will look at implications for the existence of irreducible, self-dual supercuspidal representations of p-adic groups.  This is joint work with Manish Mishra.

The Einstein equations in wave coordinates are an example of a system 
which does not obey the "null condition". This leads to many 
difficulties, most famously when attempting to prove global existence, 
otherwise known as the "nonlinear stability of Minkowski space". 
Previous approaches to overcoming these problems suffer from a lack of 
generalisability - among other things, they make the a priori assumption 
that the space is approximately scale-invariant. Given the current 
interest in studying the stability of black holes and other related 
problems, removing this assumption is of great importance.

The p-weighted energy method of Dafermos and Rodnianski promises to 
overcome this difficulty by providing a flexible and robust tool to 
prove decay. However, so far it has mainly been used to treat linear 
equations. In this talk I will explain how to modify this method so that 
it can be applied to nonlinear systems which only obey the "weak null 
condition" - a large class of systems that includes, as a special case, 
the Einstein equations. This involves combining the p-weighted energy 
method with many of the geometric methods originally used by 
Christodoulou and Klainerman. Among other things, this allows us to 
enlarge the class of wave equations which are known to admit small-data 
global solutions, it gives a new proof of the stability of Minkowski 
space, and it also yields detailed asymptotics. In particular, in some 
situations we can understand the geometric origin of the slow decay 
towards null infinity exhibited by some of these systems: it is due to 
the formation of "shocks at infinity".

In the spirit of the Ax-Kochen-Ershov principle, one could conjecture that the imaginaries in equicharacteristic zero Henselian fields can be entirely classified in terms of the Haskell-Hrushovski-Macpherson geometric imaginaries, residue field imaginaries and value group imaginaries. However, the situation is more complicated than that. My goal in this talk will be to present what we believe to be an optimal conjecture and give elements of a proof.

It is often fruitful to study an infinite discrete group via its finite quotients.  For this reason, conditions that guarantee many finite quotients can be useful.  One such notion is residual finiteness.
A group is residually finite if for any non-identity element g there is a homomorphism onto a finite group, which doesn’t map g to e. I will mention how this relates to topology, present an argument why the surface groups are residually finite and I’ll show that in this case it is enough to consider homomorphisms onto alternating groups.

This talk is a brief introduction to the analysis of Donaldson theory, a branch of gauge theory. Roughly, this is an area of differential topology that aims to extract smooth structure invariants from the geometry of the space of solutions (moduli space) to a system of partial differential equations: the Yang-Mills equations.

I will start by discussing the differential geometric background required to talk about Yang-Mills connections. This will involve introducing the concepts of principal fibre bundles, connections and curvature. In the second half of the talk I will attempt to convey the flavour of the mathematics used to address technical issues in gauge theory. I plan to do this by presenting a sketch of the proof of Uhlenbeck's compactness theorem, the main technical tool involved in the compactification of the moduli space.