Past

How do you make the most of graduate supervisions?  Whether you are a first year graduate wanting to learn about how to manage meetings with your supervisor, or a later year DPhil student, postdoc or faculty member willing to share their experiences and give advice, please come along to this informal discussion led by DPhil students for the first Fridays@4 session of the term.  You can also continue the conversation and learn more about graduate student life at Oxford at Happy Hour afterwards.

Various constructions relevant to practical problems such as clustering and graph classification give rise to multiparameter persistence modules (MPPM), that is, linear representations of non-totally ordered sets. Much of the mathematical interest in multiparameter persistence comes from the fact that there exists no tractable classification of MPPM up to isomorphism, meaning that there is a lot of room for devising invariants of MPPM that strike a good balance between discriminating power and complexity of their computation. However, there is no consensus on what type of information we want these invariants to provide us with, and, in particular, there seems to be no good notion of “global” or “high persistence” features of MPPM.

With the goal of substantiating these claims, as well as making them more precise, I will start with an overview of some of the known invariants of MPPM, including joint works with Bauer and Oudot. I will then describe recent work of Bjerkevik, which contains relevant open questions and which will help us make sense of the notion of global feature in multiparameter persistence.

Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, cells in a developing tissue, or locusts in a swarm. Here I will focus on two examples of such emergent behavior in biology, specifically cell interactions during pattern formation in zebrafish skin and gametophyte development in ferns. Different modeling approaches provide complementary insights into these systems and face different challenges. For example, vertex-based models describe cell shape, while more efficient agent-based models treat cells as particles. Continuum models, which track the evolution of cell densities, are more amenable to analysis, but it is often difficult to relate their few parameters to specific cell interactions. In this talk, I will overview our models of cell behavior in biological patterns and discuss our ongoing work on quantitatively relating different types of models using topological data analysis and data-driven techniques.

We will kick off the start of the academic year and the Junior Algebra and Representation Theory seminar (JART) with a fun social event in the common room. Come catch up with your fellow students about what happened over the summer, meet the new students and play some board games. We'll go for lunch together afterwards.

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L-functions over number fields. There is an extremely similar function field analogue, worked out by Andrade-Keating. I will explain that one can relate this problem to understanding the homology of the braid group with symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove an improved range for homological stability with these coefficients. Together, these results imply the conjectured asymptotics for all moments in the function field case, for all sufficiently large (but fixed) q.

A Path Shadowing Monte-Carlo method provides prediction of future paths given any generative model.

At a given date, it averages future quantities over generated price paths whose past history matches, or “shadows”, the actual (observed) history.

We test our approach using paths generated from a maximum entropy model of financial prices,

based on the recently introduced “Scattering Spectra” which are multi-scale analogues of the standard skewness and kurtosis.

This model promotes diversity of generated paths while reproducing the main statistical properties of financial prices, including stylized facts on volatility roughness.

Our method yields state-of-the-art predictions for future realized volatility. It also allows one to determine conditional option smiles for the S&P500.

These smiles depend only on the distribution of the price process, and are shown to outperform both the current version of the Path Dependent Volatility model and the option market itself.

This work considers weighted and preconditioned GMRES. The objective is to provide a way of choosing the preconditioner and the inner product, also called weight, that ensure fast convergence. The main focus of the article is on Hermitian preconditioning (even for non-Hermitian problems).

It is indeed proposed to choose a Hermitian preconditioner H, and to apply GMRES in the inner product induced by H. If moreover, the problem matrix A is positive definite, then a new convergence bound is proved that depends only on how well H preconditions the Hermitian part of A, and on a measure of how non-Hermitian A is. In particular, if a scalable preconditioner is known for the Hermitian part of A, then the proposed method is also scalable. I will also illustrate this result numerically.

Over the years, I've often taught a first course in asymptotics and perturbation methods, even though I don't know much about the subject. In this talk, I'll discuss a textbook example of a singularly perturbed nonlinear boundary-value problem that has revealed delightful new surprises, every time I teach it. These include a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the solutions' initial conditions that can be calculated by elementary means.

If some structure, mathematical or otherwise, is giving you grief, then often the first thing to do is to attempt to break the offending object down into (finitely many) simpler pieces.

In group theory, when we speak of questions of *accessibility* we are referring to the ability to achieve precisely this. The idea of an 'accessible group' was first coined by Terry Wall in the 70s, and since then has left quite a mark on our field (and others). In this talk I will introduce the toolbox required to study accessibility, and walk you and your groups through some reasons to be accessible.

I will discuss various definitions of quantum or noncommutative graphs that have appeared in the literature, along with motivating examples.  One definition is due to Weaver, where examples arise from quantum channels and the study of quantum zero-error communication.  This definition works for any von Neumann algebra, and is "spatial": an operator system satisfying a certain operator bimodule condition.  Another definition, first due to Musto, Reutter, and Verdon, involves a generalisation of the concept of an adjacency matrix, coming from the study of (simple, undirected) graphs.  Here we study finite-dimensional C*-algebras with a given faithful state; examples are perhaps less obvious.  I will discuss generalisations of the latter framework when the state is not tracial, and discuss various notions of a "morphism" of the resulting objects

In a recent work with Sam Mellick and Amanda Wilkens, we proved that higher rank semisimple Lie groups satisfy a generalization of Gaboriau fixed price property (originally defined for countable groups) to the setting of locally compact second countable groups. As one of the corollaries, under mild conditions, we can prove that the rank (minimal number of generators) or the first mod-p Betti number of a higher rank lattice grow sublinearly in the covolume.  The proof relies on surprising geometric properties of Poisson-Voronoi tessellations in higher-rank symmetric spaces, which could be of independent interest. 

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.

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.
 

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).