Past

The last Fridays@2 of the year will be the Prelims Preparation Lecture aimed at first-year undergraduates. Richard Earl and Vicky Neale will highlight some key points to be aware of as you prepare for exams, thinking both about exam technique and revision strategy, and a student will offer some tips from their personal experience.  This will complement the Friday@2 event in Week 2, on Managing exam anxiety.  As part of the Prelims Preparation session, we'll look through two past exam questions, giving tips on how to structure a good answer.  You'll find that most helpful if you've worked through the questions yourself beforehand, so this is advance notice so that you can slot the questions into your timetable for the next few days.  They are both from 2013, one is Q5 from Maths I (on the Groups and Group Actions course), and the other is Q3 from Maths IV (on the Dynamics course).  You can access these, and a large collection of other past Prelims exam questions, via the archive.

In this talk, we will consider how two very different atmospheric phenomena, the terrestrial tropical cyclone and the martian polar vortex, can be described within a single simplified dynamical framework based on the forced shallow water equations. Dynamical forcings include angular momentum transport by secondary (transverse) circulations and local heating due to latent heat release. The forcings act in very different ways in the two systems but in both cases lead to distinct annular distributions of potential vorticity, with a local vorticity maximum at a finite radius surrounding a central minimum.  In both systems, the resulting vorticity distributions are subject to shear instability and the degree of eddy growth versus annular persistence can be examined explicitly under different forcing scenarios.

PLEASE NOTE THAT THIS SEMINAR IS CANCELLED DUE TO UNFORESEEN CIRCUMSTANCES.

In 1983, Erdos and Szemerédi conjectured that for any finite subset of the integers, either the sumset or the product set has nearly quadratic growth. Applications include incidence geometry, exponential sums, compressed image sensing, computer science, and elsewhere. We discuss recent progress towards the main conjecture and related questions. 

The world of quantum invariants began in 1983 with the discovery of the Jones polynomial. Later on, Reshetikhin and Turaev developed an algebraic machinery that provides knot invariants. This algebraic construction leads to a sequence of quantum generalisations of this invariant, called coloured Jones polynomials. The original Jones polynomial can be defined by so called skein relations. However, unlike other classical invariants for knots like the Alexander polynomial, its relation to the topology of the complement is still a mysterious and deep question. On the topological side, R. Lawrence defined a sequence of braid group representations on the homology of coverings of configuration spaces. Then, based on her work, Bigelow gave a topological model for the Jones polynomial, as a graded intersection pairing between certain homology classes. We aim to create a bridge between these theories, which interplays between representation theory and low dimensional topology. We describe the Bigelow-Lawrence model, emphasising the construction of the homology classes. Then, we show that the sequence of coloured Jones polynomials can be seen through the same formalism, as topological intersection pairings of homology classes in coverings of the configuration space in the punctured disc.

Chemical reaction network (CRN) theory focusses on making claims about dynamical behaviours of reaction networks which are, as far as possible, dependent on the network structure but independent of model details such as functions chosen and parameter values. The claims are generally about the existence, nature and stability of limit sets, and the possibility of bifurcations, in models of CRNs with particular structural features. The methodologies developed can often be applied to large classes of models occurring in biology and engineering, including models whose origins are not chemical in nature. Many results have a natural algorithmic formulation. Apart from the potential for application, the results are often pleasing mathematically for their power and generality. 

This talk will concern some recent themes in CRN theory, particularly focussed on how the presence or absence of particular subnetworks ("motifs") influences allowed dynamical behaviours in ODE models of a CRN. A number of recent results take the form: "a CRN containing no subnetworks satisfying condition X cannot display behaviour of type Y"; but also, in the opposite direction, "if a CRN contains a subnetwork satisfying condition X, then some model of this CRN from class C admits behaviour of type Y". The proofs of such results draw on a variety of techniques from analysis, algebra, combinatorics, and convex geometry. I'll describe some of these results, outline their proofs, and sketch some current challenges in this area. 
 

In this talk, I am going to give an introduction to operator preconditioning as a general and robust strategy to precondition linear systems arising from Galerkin discretization of PDEs or Boundary Integral Equations. Then, in order to illustrate the applicability of this preconditioning technique, I will discuss the simple case of weakly singular and hypersingular integral equations, arising from exterior Dirichlet and Neumann BVPs for the Laplacian in 3D. Finally, I will show how we can also tackle operators with a more difficult structure, like the electric field integral equation (EFIE) on screens, which models the scattering of time-harmonic electromagnetic waves at perfectly conducting bounded infinitely thin objects, like patch antennas in 3D.

We investigate monotone solutions of the moral hazard problems without the monotone likelihood ratio property. The solutions are explicitly characterised by a concave envelope relaxation approach for a two-action model in which the principal is risk neutral or exhibits constant absolute risk aversion.  

The classical wave equation is derived from the system of three equations: The equation of motion of a (one-dimensional) deformable body, the Hook law as a constitutive equation, and the  strain measure, and describes wave propagation in elastic media. 
Fractional wave equations describe wave phenomena when viscoelasticity of a material or non-local effects of a material comes into an account. For waves in viscoelastic media, instead of Hook's law, a constitutive equation for viscoelastic body,  for example, Fractional Zener model or distributed order model of viscoelastic body, is used. To consider non-local effects of a media, one may replace classical strain measure by non-local strain measure. There are other constitutive equations and other ways to describe non-local effects which will be discussed within the talk.  
The system of three equations subject to initial conditions, initial displacement and initial velocity, is equivalent to one single equation, called fractional wave equation. Using different models for constitutive equations, and non-local measures, different fractional wave equations are obtained. After derivation of such equations, existence and uniqueness of their solution in the spaces of distributions is proved by the use of Laplace and Fourier transforms as main tool. Plots of solutions are presented. For some of derived equations microlocal analysis of the solution is conducted. 

I will give an introduction to the theory of definable parameterization of definable sets in the o-minimal context and its application to diophantine problems. I will then go on to discuss uniformity issues with particular reference to the subanalytic case. This is joint work with Jonathan Pila and Raf Cluckers

The last decade or so has seen substantial progress in the theory of (outer) automorphism groups of right-angled Artin groups (RAAGs), spearheaded by work of Charney and Vogtmann. Many of the techniques used for RAAGs also apply to a wider class of groups, graph products of finitely generated abelian groups, which includes right-angled Coxeter groups (RACGs). In this talk, I will give an introduction to automorphism groups of such graph products, and describe recent developments surrounding the outer automorphism groups of RACGs, explaining the links to what we know in the RAAG case.

Let $X$ be a K3 surface and let $Z_X(q)$ be the generating series of the topological Euler characteristics of the Hilbert scheme of points on $X$. It is known that $q/Z_X(q)$ equals the discriminant form $\Delta(\tau)$ after the change of variables $q=e^{2 \pi i \tau}$. In this talk we consider the equivariant generalization of this result, when a finite group $G$ acts on $X$ symplectically. Mukai and Xiao has shown that there are exactly 81 possibilities for such an action in terms of types of the fixed points. The analogue of $q/Z_X(q)$ in each of the 81 cases turns out to be a cusp form (after the same change of variables). Knowledge of modular forms is not assumed in the talk; I will introduce all necessary concepts. Joint work with Jim Bryan.

Neural Network algorithms have achieved unprecedented performance in image recognition over the past decade. However, their application in real world use-cases, such as self driving cars, raises the question of whether it is safe to rely on them.

We generally associate the robustness of these algorithms with how easy it is to generate an adversarial example: a tiny perturbation of an image which leads it to be misclassified by the Neural Net (which classifies the original image correctly). Neural Nets are strongly susceptible to such adversarial examples, but when the architecture of the target neural net is unknown to the attacker it becomes more difficult to generate these examples efficiently.

In this Black-Box setting, we frame the generation of an adversarial example as an optimisation problem solvable via derivative free optimisation methods. Thus, we introduce an algorithm based on the BOBYQA model-based method and compare this to the current state of the art algorithm.

Unitary highest weight modules were classified in the 1980s by Enright-Howe-Wallach and independently by Jakobsen. The classification is based on a version of the Dirac inequality, but the proofs also require a number of other techniques and are quite involved. We present a much simpler proof based on a different version of the Dirac inequality. This is joint work with Vladimir Soucek and Vit Tucek.
 

We adapt convergent look-ahead and backward finite difference formulas to compute future eigenvectors and eigenvalues of piecewise smooth time-varying matrix flows $A(t)$. This is based on the Zhang Neural Network model for time-varying problems and uses the associated error function

$E(t) =A(t)V(t)−V(t)D(t)$

with the Zhang design stipulation

$\dot{E}(t) =−\eta E(t)$.

Here $E(t)$ decreased exponentially over time for $\eta >0$. It leads to a discrete-time differential equation of the form $P(t_k)\dot{z}(t_k) = q(t_k)$ for the eigendata vector $z(t_k)$ of $A(t_k)$. Convergent high order look-ahead difference formulas then allow us to express $z(t_k+1)$ in terms of earlier discrete $A$ and $z$ data. Numerical tests, comparisons and open questions follow.

The time bottleneck in the manufacturing process of Besi (company involved in ESGI 149 Innsbruck) is the extraction of undamaged dies from a component wafer. The easiest way for them to speed up this process is to reduce the number of 'selections' made by the robotic arm.  Each 'selection' made by this robotic arm can be thought of as choosing a 2x2 submatix of a large binary matrix, and editing the 1's in this submatrix to be 0's.  The quesiton is: what is the fewest number of 2x2 submatrices required to cover the full matrix, and how can we find this number. This problem can be solved exactly using integer programming methods, although this approach proves to be prohibitively expensive for realistic sizes. In this talk I will describe the approach taken by my team at EGSI 149, as well as directions for further improvement.

  Our understanding of physical phenomena is intimately linked to the way we understand the relevant observables describing them. While a big deal of progress has been made for processes occurring in flat space-time, much less is known in cosmological settings. In this context, we have processes which happened in the past and which we can detect the remnants of at present time. Thus, the relevant observable is the late-time wavefunction of the universe. Questions such as "what properties they ought to satisfy in order to come from a consistent time evolution in cosmological space-times?", are still unanswered, and are compelling given that in these quantities time is effectively integrated out. In this talk I will report on some recent progress in this direction, aiming towards the idea of a formulation of cosmology "without time". Amazingly enough, a new mathematical structure, we called "cosmological polytope", which has its own first principle definition, encodes the singularity structure we ascribe to the perturbative wavefunction of the universe, and makes explicit its (surprising) relation to the flat-space S-matrix. I will stress how the cosmological polytopes allow us to: compute the wavefunction of the universe at arbitrary points and arbitrary loops (with novel representations for it); interpret the residues of its poles in terms of flat-space processes; provide a  general geometrical proof for the flat-space cutting rules; reconstruct the perturbative wavefunction from the knowledge of the flat-space S-matrix and a subset of symmetries enjoyed by the wavefunction.

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.

The quantum unipotent coordinate ring has a cluster algebra structure. On the other hand, this ring is isomorphic to the Grothendieck ring of the module category of quiver Hecke algebras (QHA). We can prove that cluster monomials of the quantum unipotent coordinate ring correspondi to real simple modules. This is a joint work with Seok-Jin Kang, Myungho Kim and Se-jin Oh.

Numerical simulations indicate that deep artificial neural networks (DNNs) seem to be able to overcome the curse of dimensionality in many computational  problems in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy and the dimension of the function which the DNN aims to approximate. However, there are only a few special situations where results in the literature can rigorously explain the success of DNNs when approximating high-dimensional functions.

In this talk it is revealed that DNNs do indeed overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients. A crucial ingredient in our proof of this result is the fact that the artificial neural network used to approximate the PDE solution really is a deep artificial neural network with a large number of hidden layers.

For $G$ connected, reductive algebraic group defined over $\mathbb{C}$ the Springer Correspondence gives a bijection between the irreducible representations of the Weyl group $W$ of $G$ and certain pairs comprising a $G$-orbit on the nilpotent cone of the Lie algebra of $G$ and an irreducible local system attached to that $G$-orbit. These irreducible representations can be concretely realised as a W-action on the top degree homology of the fibres of the Springer resolution. These Springer fibres are geometrically very rich and provide interesting Weyl group combinatorics: for instance, the irreducible components of these Springer fibres form a basis for the corresponding irreducible representation of $W$. In this talk, I'll give a general survey of the Springer Correspondence and then discuss recent joint projects with Daniele Rosso, Vinoth Nandakumar and Arik Wilbert on Kato's Exotic Springer correspondence.