Past

Complex networks have become the main paradigm for modeling the dynamics of interacting systems. However, networks are intrinsically limited to describing pairwise interactions, whereas real-world systems are often characterized by interactions involving groups of three or more units. In this talk, I will consider social systems as a natural testing ground for higher-order network approaches (hypergraphs and simplicial complexes). I will briefly introduce models of social contagion and norm evolution on hypergraphs to show how the inclusion of higher-order mechanisms can lead to the emergence of novel phenomena such as discontinuous transitions and critical mass effects. I will then present some recent results on the role that structural features play on the emergent dynamics, and introduce a measure of hyper-coreness to characterize the centrality of nodes and inform seeding strategies. Finally, I will delve into the microscopic dynamics of empirical higher-order structures. I will study the mechanisms governing their temporal dynamics both at the node and group level, characterizing how individuals navigate groups and how groups form and dismantle. I will conclude by proposing a dynamical hypergraph model that closely reproduces the empirical observations.
 

Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. In this talk, we study the growth rate of these functions adapted to finite characteristic quotients. One potential application of this result is towards linearity of the mapping class group.

We introduce a sparse and very high order hp-finite element method for the weak form of the Helmholtz equation.  The domain may be a disk, an annulus, or a cylinder. The cells of the mesh are an innermost disk (omitted if the domain is an annulus) and concentric annuli.

We demonstrate the effectiveness of this method on PDEs with radial direction discontinuities in the coefficients and data. The discretization matrix is always symmetric and positive-definite in the positive-definite Helmholtz regime. Moreover, the Fourier modes decouple, reducing a two-dimensional PDE solve to a series of one-dimensional solves that may be computed in parallel, scaling with linear complexity. In the positive-definite case, we utilize the ADI method of Fortunato and Townsend to apply the method to a 3D cylinder with a quasi-optimal complexity solve.

Let $G$ be a $d$-regular graph of growing degree on $n$ vertices, and form a random subgraph $G_p$ of $G$ by retaining edge of $G$ independently with probability $p=p(d)$. Which conditions on $G$ suffice to observe a phase transition at $p=1/d$, similar to that in the binomial random graph $G(n,p)$, or, say, in a random subgraph of the binary hypercube $Q^d$?

We argue that in the supercritical regime $p=(1+\epsilon)/d$, $\epsilon>0$ being a small constant, postulating that every vertex subset $S$ of $G$ of at most $n/2$ vertices has its edge boundary at least $C|S|$, for some large enough constant $C=C(\epsilon)>0$, suffices to guarantee the likely appearance of the giant component in $G_p$. Moreover, its asymptotic order is equal to that in the random graph $G(n,(1+\epsilon)/n)$, and all other components are typically much smaller.

We further give examples demonstrating the tightness of this result in several key senses.

A joint work with Sahar Diskin, Joshua Erde and Mihyun Kang.

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time. 

Students presenting are:

Adrian Martini, supervisor Alison Ethridge

Fang Rui Lim, supervisor Rama Cont

Thomas Groves, supervisor Dmitry Beylaev

Sarah-Jean Meyer, supervisor Massimiliano Gubinelli

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time.

We establish exponentially-fast mixing for passive scalars driven by two well-known examples of random divergence-free vector fields. The first one is the alternating shear flow model proposed by Pierrehumbert, in which case we set up a dynamics-based framework to construct such space-time smooth universal exponential mixers. The second example is the statistically stationary, homogeneous, isotropic Kraichnan model of fluid turbulence. In this case, the proof follows a new explicit identity for the evolution of negative Sobolev norms of the scalar. This is based on joint works with Alex Blumenthal (Georgia Tech) and Michele Coti Zelati (ICL), and Michele Coti Zelati and Theodore Drivas (Stony Brook), respectively.

The twin prime conjecture asserts that there are infinitely many primes p for which p+2 is also prime. This conjecture appears far out of reach of current mathematical techniques. However, in 2013 Zhang achieved a breakthrough, showing that there exists some positive integer h for which p and p+h are both prime infinitely often. Equidistribution estimates for primes in arithmetic progressions to smooth moduli were a key ingredient of his work. In this talk, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinement of the q-van der Corput method can be used to improve on equidistribution estimates of the Polymath project for primes in APs to smooth moduli.

We describe the compact scaling limits of uniformly random quadrangulations with boundaries on a surface of arbitrary fixed genus. These limits, called Brownian surfaces, are homeomorphic to the surface of the given genus with or without boundaries depending on the scaling regime of the boundary perimeters of the quadrangulation. They are constructed by appropriate gluings of pieces derived from Brownian geometrical objects (the Brownian plane and half-plane). In this talk, I will review their definition and discuss possible alternative constructions. This is based on joint work with Jérémie Bettinelli.

In a classical work, Bowen and Margulis proved the equidistribution of
closed geodesics in any hyperbolic manifold. Together with Jeremy Kahn
and Vladimir Marković, we asked ourselves what happens in a
three-manifold if we replace curves by surfaces. The natural analog of a
closed geodesic is then a minimal surface, as totally geodesic surfaces
exist only very rarely. Nevertheless, it still makes sense (for various
reasons, in particular to ensure uniqueness of the minimal
representative) to restrict our attention to surfaces that are almost
totally geodesic.

The statistics of these surfaces then depend very strongly on how we
order them: by genus, or by area. If we focus on surfaces whose *area*
tends to infinity, we conjecture that they do indeed equidistribute; we
proved a partial result in this direction. If, however, we focus on
surfaces whose *genus* tends to infinity, the situation is completely
opposite: we proved that they then accumulate onto the totally geodesic
surfaces of the manifold (if there are any).

Classically, topological vector bundles are classified by homotopy classes of maps into infinite Grassmannians. This allows us to study topological vector bundles using obstruction theory: we can detect whether a vector bundle has a trivial subbundle by means of cohomological invariants. In the context of algebraic geometry, one can ask whether algebraic vector bundles over smooth affine varieties can be classified in a similar way. Recent advances in motivic homotopy theory give a positive answer, at least over an algebraically closed base field. Moreover, the behaviour of vector bundles over general base fields has surprising connections with the theory of quadratic forms.

Transfer learning is an emerging and popular paradigm for utilizing existing knowledge from  previous learning tasks to improve the performance of new ones. In this talk, we will first present transfer learning in the early diagnosis of eye diseases: diabetic retinopathy and retinopathy of prematurity.  

We will discuss how this empirical  study leads to the mathematical analysis of the feasibility and transferability  issues in transfer learning. We show how a mathematical framework for the general procedure of transfer learning helps establish  the feasibility of transfer learning as well as  the analysis of the associated transfer risk, with applications to financial time series data.

Come and meet the Mathematrix committee and find out more about what the society is for and what events we are putting on this term. There will be free pizza! 

https://www.maths.ox.ac.uk/study-here/postgraduate-study/mathematrix

Interpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Key steps in using mechanistic mathematical models to interpret data include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction. Here we present a systematic, computationally efficient likelihood-based workflow that addresses all three steps in a unified way. Recently developed methods for constructing profile-wise prediction intervals enable this workflow and provide the central linkage between different workflow components. These methods propagate profile-likelihood-based confidence sets for model parameters to predictions in a way that isolates how different parameter combinations affect model predictions. We show how to extend these profile-wise prediction intervals to two-dimensional interest parameters, and then combine profile-wise prediction confidence sets to give an overall prediction confidence set that approximates the full likelihood-based prediction confidence set well.  We apply our methods to a range of synthetic data and real-world ecological data describing re-growth of coral reefs on the Great Barrier Reef after some external disturbance, such as a tropical cyclone or coral bleaching event.
 

Interacting particle systems provide flexible and powerful models that are useful in many application areas such as sociology (agents), molecular dynamics (proteins) etc. However, particle systems with large numbers of particles are very complex and difficult to handle, both analytically and computationally. Therefore, a common strategy is to derive effective equations that describe the time evolution of the empirical particle density, the so-called Dean-Kawasaki equation.

Our aim is to derive and study continuum models for the mesoscopic behavior of particle systems. In particular, we are interested in finite size effects. We will introduce nonlinear and non-Gaussian models that approximate the Dean-Kawasaki equation, in the special case of non-interacting particles. We want to study the well-posedness of these nonlinear SPDE models and to control the weak error of the SPDE approximation.  This is the joint work with H. Kremp (TU Wien) and N. Perkowski (FU Berlin).

We discuss $C^1$-regularity and developability of isometric immersions of flat domains into $\mathbb{R}^3$ enjoying a local fractional Sobolev $W^{1+s;2/s}$-regularity for $2/3 \leq s < 1$, generalising the known results on Sobolev (by Pakzad) and H\"{o}lder (by De Lellis--Pakzad) regimes. Ingredients of the proof include analysis of the weak Codazzi equations of isometric immersions, the study of $W^{1+s;2/s}$-gradient deformations with symmetric derivative and vanishing distributional Jacobian determinant, and the theory of compensated compactness. Joint work with M. Reza Pakzad and Armin Schikorra.

Floer theory, which mimics an infinite dimensional Morse approach to the study of critical points of smooth functions, appeared as an attempt to prove Arnold conjecture. The theory is more or less well understood in some compact cases.

Non-compact symplectic manifolds can sometimes be compactified as singular symplectic manifolds where the symplectic form "blows up" along a hypersurface in a controlled way (b^m-symplectic manifolds). In natural examples in Celestial mechanics such as the 3-body problem, these compactifications are given by regularization transformations à la Moser/Mc Gehee etc.

I will use the theory of b^m-symplectic/b^m-contact manifolds (introduced by Scott, Guillemin-Miranda Weitsman, and Miranda-Oms) as a guinea pig to propose ways to extend the study of Hamiltonian/Reeb Dynamics to singular symplectic/contact manifolds. This, in particular, yields new results for non-compact symplectic manifolds and for special (but, yet, meaningful) classes of Poisson manifolds.

Inspiration comes from several results extending the Weinstein conjecture to the context of b^m-contact manifolds and its connection to the study of escape orbits in Celestial mechanics and Fluid Dynamics. Those examples motivate a model for (singular) Floer homology.

I'll describe the motivating examples/results and some ideas to attack the general questions.

The Hardy Lectureship was founded in 1967 in memory of G.H. Hardy (LMS President 1926-1928 & 1939-1941). The Hardy Lectureship is a lecture tour of the UK by a mathematician with a high reputation in research.