Past

Many real-world networks contain small recurring connectivity patterns also known as network motifs. Although network motifs are widely considered to be important structural features of networks that are closely connected to their function methods for characterizing and modelling the local connectivity structure of complex networks remain underdeveloped. In this talk, we will present a non-parametric approach that is based on generative models in which networks are generated by adding not only single edges but also but also copies of larger subgraphs such as triangles to the graph. We show that such models can be formulated in terms of latent states that correspond to subgraph decompositions of the network and derive analytic expressions for the likelihood of such models. Following a Bayesian approach, we present a nonparametric prior for model parameters. Solving the resulting inference problem results in a principled approach for identifying atomic connectivity patterns of networks that do not only identify statistically significant connectivity patterns but also produces a decomposition of the network into such atomic substructures. We tested the presented approach on simulated data for which the algorithm recovers the latent state to a high degree of accuracy. In the case of empirical networks, the method identifies concise sets atomic subgraphs from within thousands of candidates that are plausible and include known atomic substructures.

Ginzburg–Landau type functionals provide a relaxation scheme to construct harmonic maps in the presence of topological obstructions. They arise in superconductivity models, in liquid crystal models (Landau–de Gennes functional) and in the generation of cross-fields in meshing. For a general compact manifold target space we describe the asymptotic number, type and location of singularities that arise in minimizers. We cover in particular the case where the fundamental group of the vacuum manifold in nonabelian and hence the singularities cannot be characterized univocally as elements of the fundamental group. The results unify the existing theory and cover new situations and problems.

This is a joint work with Antonin Monteil and Rémy Rodiac (UCLouvain, Louvain- la-Neuve, Belgium)

We recall the impact of stabilizing a 4-manifold with $S^2 \times S^2$. The corresponding local situation concerns knots in the 3-sphere which bound (nullhomotopic) discs in a stabilized 4-ball. We explain how these discs arise, and discuss bounds on the minimal number of stabilizations needed. Then we compare this minimal number to the 4-genus.
This is joint work with A. Conway.

In this talk I present work in progress with Wolfgang König and Nicolas Perkowski on the parabolic Anderson model (PAM) with white noise potential in 2d. We show the behavior of the total mass as the time tends to infinity. By using partial Girsanov transform and singular heat kernel estimates we can obtain the mass-asymptotics by using the eigenvalue asymptotics that have been showed in another work in progress with Khalil Chouk. 

We prove an a priori bound for solutions of the dynamic $\Phi^4_3$ equation.

This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions.

We treat the  large and small scale behaviour of solutions with completely different arguments.For small scales we use bounds akin to those presented in Hairer's theory of regularity structures. We stress immediately that our proof is fully self-contained, but we give a detailed explanation of how our arguments relate to Hairer's. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure.

The fact that our bounds do not depend on space-time boundary conditions makes them useful for the analysis of large scale properties of solutions. They can for example be used in a compactness argument to construct solutions on the full space and their invariant measures

We consider the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. We construct smooth solutions with concentrated vorticities around $k$ points which evolve according to the Hamiltonian system for the Kirkhoff-Routh energy,  using an outer-inner solution gluing approach. The asymptotically singular profile  around each point resembles a scaled finite mass solution of Liouville's equation.
We also discuss the {\em vortex filament conjecture} for the three-dimensional case. This is joint work with Juan D\'avila, Monica Musso and Juncheng Wei.

In this talk I will present a large class of non-supersymmetric AdS7 solutions of IIA supergravity, and their (in)stabilities. I will start by reviewing supersymmetric AdS7 solutions of 10D supergravity dual to 6D (1,0) SCFTs. I will then focus on their non-supersymmetric counterpart, discussing how they are related. The connection between supersymmetric and non-supersymmetric solutions leads to a hint for the SUSY breaking mechanism, which potentially allows to evade some of the assumptions of the Ooguri-Vafa Conjecture about the AdS landscape. A big subset of these solutions shows a curious pattern of perturbative instabilities whenever many open-string modes are considered. On the other hand an infinite class remains apparently stable.

Would you employ you? What are employers looking for in Mathematical graduates? What kind of work can use your skills? This workshop will get your minds thinking about the possibilities after you have finished studying and will cover:

·         General careers’ information starting from a mathematical sciences degree

·         Things to think about at CV and interview stage

·         How membership of a professional body (the IMA) supports your applications and career development.

·         Information about the Mathematics Teacher Training Scholarships

Here Z is Zermelo’s set theory of 1908, as later formulated: full separation, but no replacement or collection among its axioms. PROVI was presented in lectures in Cambridge in 2010 and later published with improvements by Nathan Bowler, and is, I claim, the weakest subsystem of ZF to support a recognisable theory of set forcing: PROV is PROVI shorn of its axiom of infinity. The provident sets are the transitive non-empty models of PROV. The talk will begin with a presentation of PROV, and then discuss more recent applications and problems: in particular an answer in the system Z + PROV to a question posed by Eugene Wesley in 1972 will be sketched, and two proofs (fallacious, I hope) of 0 = 1 will be given, one using my slim models of Z and the other applying the Spector–Gandy theorem to certain models of PROVI. These “proofs”, when re-interpreted, supply some arguments of Reverse Mathematics.

Given a "random" polynomial over the integers, it is expected that, with high probability, it is irreducible and has a big Galois group over the rationals. Such results have been long known when the degree is bounded and the coefficients are chosen uniformly at random from some interval, but the case of bounded coefficients and unbounded degree remained open. Very recently, Emmanuel Breuillard and Peter Varju settled the case of bounded coefficients conditionally on the Riemann Hypothesis for certain Dedekind zeta functions. In this talk, I will present unconditional progress towards this problem, joint with Lior Bary-Soroker and Gady Kozma.

Many fluid flows in natural systems are highly complex, with an often beguilingly intricate and confusing detailed structure. Yet, as with many systems, a good deal of insight can be gained by testing the consequences of simple mathematical models that capture the essential physics.  We’ll tour two such problems.  In the summer melt seasons in Greenland, lakes form on the surface of the ice which have been observed to rapidly drain.  The propagation of the meltwater in the subsurface couples the elastic deformation of the ice and, crucially, the flow of water within the deformable subglacial till.  In this case the poroelastic deformation of the till plays a subtle, but crucial, role in routing the surface meltwater which spreads indefinitely, and has implications for how we think about large-scale motion in groundwater aquifers or geological carbon storage.  In contrast, when magma erupts onto the Earth’s surface it flows before rapidly cooling and crystallising.  Using analogies from the kitchen we construct, and experimentally test, a simple model of what sets the ultimate extent of magmatic intrusions on Earth and, as it turns out, on Venus.  The results are delicious!  In both these cases, we see how a simplified mathematical analysis provides insight into large scale phenomena.

Since differentiation generally lowers exponents, it is straightforward that the space of Laurent polynomials $\mathbb{C}[x, x^{-1}]$ is a finitely generated module over the ring of differential operators $\mathbb{C}[x, \mathrm{d}/\mathrm{d}x]$. This innocent looking fact has been vastly generalized to a statement about holonomic D-modules, using the beautiful theory of b-functions (or Bernstein—Sato polynomials). I will give an overview of the classical theory before discussing some recent developments concerning a $p$-adic analytic analogue, which is joint work with Thomas Bitoun.

When the linear system in Newton’s method is approximately solved using an iterative method we have an inexact or truncated Newton method. The outer method is Newton’s method and the inner iterations will be the iterative method. The Inexact Newton framework is now close to 30 years old and is widely used and given names like Newton-Arnoldi, Newton-CG depending on the inner iterative method. In this talk we will explore convergence properties when the outer iterative method is Gauss-Newton, the Halley method or an interior point method for linear programming problems.

In this talk, which is based on joint work with Benjamin Gess, I will describe a pathwise well-posedness theory for stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise. Such equations arise in the theory of mean field games, as an approximation to the Dean–Kawasaki equation in fluctuating hydrodynamics, to describe the fluctuating hydrodynamics of a zero range process, and as a model for the evolution of a thin film in the regime of negligible surface tension.  Our methods are motivated by the theory of stochastic viscosity solutions, which are applied after passing to the equation’s kinetic formulation, for which the noise enters linearly and can be inverted using the theory of rough paths.  I will also mention the application of these methods to nonlinear diffusion equations with linear, multiplicative noise.

When groups may be built up as graphs of 'simpler' groups, it is often 
of interest to study how good residual finiteness properties of simpler 
groups can imply residual properties of the whole. The essential case of 
this theory is the study of residual properties of finite groups. In 
this talk I will discuss the question of when a graph of finite 
$p$-groups is residually $p$-finite, for $p$ a prime. I describe the 
previous theorems in this area for one-edge and finite graphs of groups, 
and their method of proof. I will then state my recent generalisation of 
these theorems to potentially infinite graphs of groups, together with 
an alternative and more natural method of proof. Finally I will briefly 
describe a usage of these results in the study of accessibility -- 
namely the existence of a finitely generated inaccessible group which is 
residually $p$-finite.

The Helmholtz equation models waves propagating with a fixed frequency. Discretising the Helmholtz equation for high frequencies via standard finite-elements results in linear systems that are large, non-Hermitian, and indefinite. Therefore, when solving these linear systems, one uses preconditioned iterative methods. When one considers uncertainty quantification for the Helmholtz equation, one will typically need to solve many (thousands) of linear systems corresponding to different realisations of the coefficients. At face value, this will require the computation of many preconditioners, a potentially expensive task.

Therefore, we investigate how well a preconditioner for one realisation of the Helmholtz equation works as a preconditioner for another realisation. We prove that if the two realisations are 'nearby' (with a precise meaning of 'nearby'), then the preconditioner is robust (that is, preconditioned GMRES converges in a number of iterations that is independent of frequency). We also give some preliminary computational results indicating the speedup one obtains in uncertainty quantification calculations.

A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of k-colorings of a graph G on n vertices with maximum degree \Delta is rapidly mixing for k \ge \Delta+2. In 1999, Vigoda showed rapid mixing of flip dynamics with certain flip parameters on the set of proper k-colorings for k > (11/6)\Delta, implying rapid mixing for Glauber dynamics. In this paper, we obtain the first improvement beyond the (11/6)\Delta barrier for general graphs by showing rapid mixing for k > (11/6 - \eta)\Delta for some positive constant \eta. The key to our proof is combining path coupling with a new kind of metric that incorporates a count of the extremal configurations of the chain. Additionally, our results extend to list coloring, a widely studied generalization of coloring. Combined, these results answer two open questions from Frieze and Vigoda’s 2007 survey paper on Glauber dynamics for colorings. 

This is joint work with Michelle Delcourt and Luke Postle.