Past

Stochastic processes subject to weak noise often show a metastable
behaviour, meaning that they converge to equilibrium extremely slowly;
typically, the convergence time is exponentially large in the inverse
of the variance of the noise (Arrhenius law).
  
In the case of finite-dimensional Ito stochastic differential
equations, the large-deviation theory developed in the 1970s by
Freidlin and Wentzell allows to prove such Arrhenius laws and compute
their exponent. Sharper asymptotics for relaxation times, including the
prefactor of the exponential term (Eyring–Kramers laws) are known, for
instance, if the stochastic differential equation involves a gradient
drift term and homogeneous noise. One approach that has been very
successful in proving Eyring–Kramers laws, developed by Bovier,
Eckhoff, Gayrard and Klein around 2005, relies on potential theory.
  
I will describe Eyring–Kramers laws for some parabolic stochastic PDEs
such as the Allen–Cahn equation on the torus. In dimension 1, an
Arrhenius law was obtained in the 1980s by Faris and Jona-Lasinio,
using a large-deviation principle. The potential-theoretic approach
allows us to compute the prefactor, which turns out to involve a
Fredholm determinant. In dimensions 2 and 3, the equation needs to be
renormalized, which turns the Fredholm determinant into a
Carleman–Fredholm determinant.
  
Based on joint work with Barbara Gentz (Bielefeld), and with Ajay
Chandra (Imperial College), Giacomo Di Gesù (Vienna) and Hendrik Weber
(Warwick). 

References: 
https://dx.doi.org/10.1214/EJP.v18-1802
https://dx.doi.org/10.1214/17-EJP60

It is well-known that the Teichmüller space of a compact surface can be identified with a connected component of the moduli space of representations of the fundamental group of the surface in PSL(2,R). Higher Teichmüller components are generalizations of this that exist for the moduli space of representations of the fundamental group into certain real simple Lie groups of higher rank. As for the usual Teichmüller space, these components consist entirely  of discrete and faithful representations. Several cases have been identified over the years. First, the Hitchin components for split groups, then the maximal Toledo invariant components for Hermitian groups, and more recently certain components for SO(p,q). In this talk, I will describe a general construction of (still somewhat conjecturally) all possible Teichmüller components, and a parametrization of them using Higgs bundles.

String sigma-models relevant in the AdS/CFT correspondence are highly non-trivial two-dimensional field theories for which predictions at finite coupling exist, assuming integrability and/or the duality itself.  I will discuss general features of the perturbative approach to these models, and present progress on how to go extract finite coupling information in the most possibly general way, namely via the use of lattice field theory techniques. I will also present new results on certain ``defect-CFT’' correlators  at strong coupling. 

The talk will introduce two general models of random simplicial complexes which extend the highly studied Erdös-Rényi model for random graphs. These models include the well known probabilistic models of random simplicial complexes from Costa-Farber, Kahle, and Linial-Meshulam as special cases. These models turn out to have a satisfying Alexander duality relation between them prompting the hope that information can be transferred for free between them. This turns out to not quite be the case with vanishing probability parameters, but when all parameters are uniformly bounded the duality relation works a treat. Time permitting I may talk about the Rado simplicial complex, the unique (with probability one) infinite random simplicial complex.
This talk is based on various bits of joint work with Michael Farber, Tahl Nowik, and Lewin Strauss.

Mathematicians need to talk and write about their mathematics.  This includes undergraduates and MSc students, who may be writing a dissertation or project report, preparing a presentation on a summer research project, or preparing for a job interview.  We think that it can be helpful to think of this as a form of storytelling, as this can lead to more effective communication.  For a story to be engaging you also need to know your audience.  In this session, we'll discuss what we mean by telling a mathematical story, give you some top tips from our experience, and give you a chance to think about how you might put this into practice.

Tensors are higher dimensional analogues of matrices, used to record data with multiple changing variables. Interpreting tensor data requires finding multi-linear stucture that depends on the application or context. I will describe a tensor-based clustering method for multi-dimensional data. The multi-linear structure is encoded as algebraic constraints in a linear program. I apply the method to a collection of experiments measuring the response of genetically diverse breast cancer cell lines to an array of ligands. In the second part of the talk, I will discuss low-rank decompositions of tensors that arise in statistics, focusing on two graphical models with hidden variables. I describe how the implicit semi-algebraic description of the statistical models can be used to obtain a closed form expression for the maximum likelihood estimate.

The famous Birch & Swinnerton-Dyer conjecture predicts that the (algebraic) rank of an elliptic curve is equal to the so-called analytic rank, which is the order of vanishing of the associated L-functions at the central point. In this talk, we shall discuss the analytic rank of automorphic L-functions in an "alternate universe". This is joint work with Kyle Pratt and Alexandru Zaharescu.

Your brain has its own waterscape: whether you are reading or sleeping, fluid flows around or through the brain tissue and clears waste in the process. These physiological processes are crucial for the well-being of the brain. In spite of their importance we understand them but little. Mathematics and numerics could play a crucial role in gaining new insight. Indeed, medical doctors express an urgent need for modeling of water transport through the brain, to overcome limitations in traditional techniques. Surprisingly little attention has been paid to the numerics of the brain’s waterscape however, and fundamental knowledge is missing. In this talk, I will discuss mathematical models and numerical methods for the brain's waterscape across scales - from viewing the brain as a poroelastic medium at the macroscale and zooming in to studying electrical, chemical and mechanical interactions between brain cells at the microscale.
 

This talk aims to provide a simple introduction on how to probe the
explicit geometry of certain moduli schemes arising in enumerative
geometry. Stable pairs, introduced by Pandharipande and Thomas in 2009, offer a curve-counting theory which is tamer than the Hilbert scheme of
curves used in Donaldson-Thomas theory. In particular, they exclude
curves with zero-dimensional or embedded components.

Ribbons are non-reduced schemes of dimension one, whose non-reduced
structure has multiplicity two in a precise sense. Following Ferrand, Banica, and Forster, there are several results on how to construct
ribbons (and higher non-reduced structures) from the data of line
bundles on a reduced scheme. With this approach, we can consider stable
pairs whose underlying curve is a ribbon: the remaining data is
determined by allowing devenerations of the line bundle defining the
double structure.

In this talk I will present a Perron-Frobenius type result for nonlinear eigenvector problems which allows us to compute the global maximum of a class of constrained nonconvex optimization problems involving multihomogeneous functions.

I will structure the talk into three main parts:

First, I will motivate the optimization of homogeneous functions from a graph partitioning point of view, showing an intriguing generalization of the famous Cheeger inequality.

Second, I will define the concept of multihomogeneous function and I will state our main Perron-Frobenious theorem. This theorem exploits the connection between optimization of multihomogeneous functions and nonlinear eigenvectors to provide an optimization scheme that has global convergence guarantees.

Third, I will discuss a few example applications in network science and machine learning that require the optimization of multihomogeneous functions and that can be solved using nonlinear Perron eigenvectors.

Computers can now beat humans at chess and at go. Surely one day they will beat us at proving theorems. But when will it happen, how will it happen, and what should humans be doing in order to make it happen? Furthermore -- do we actually want it to happen? Will they generate incomprehensible proofs, which teach us nothing? Will they find mistakes in the human literature?

I will talk about how I am training undergraduates at Imperial College London to do their problem sheets in a formal proof verification system, and how this gamifies mathematics. I will talk about mistakes in the modern pure mathematics literature, and ask what the point of modern pure mathematics is.

In this talk I will discuss some aspects of the potential theory, fine properties and boundary behaviour of the solutions to the Total Variation Flow. Instead of the classical Euclidean setting, we intend to work mostly in the general setting of metric measure spaces. During the past two decades, a theory of Sobolev functions and BV functions has been developed in this abstract setting.  A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as weighted Euclidean spaces, Riemannian manifolds, Heisenberg groups, graphs, etc.

The total variation flow can be understood as a process diminishing the total variation using the gradient descent method.  This idea can be reformulated using parabolic minimizers, and it gives rise to a definition of variational solutions.  The advantages of the approach using a minimization formulation include much better convergence and stability properties.  This is a very essential advantage as the solutions naturally lie only in the space of BV functions. Our main goal is to give a necessary and sufficient condition for continuity at a given point for proper solutions to the total variation flow in metric spaces. This is joint work with Vito Buffa and Juha Kinnunen.

Limit groups are a powerful tool in the study of free and hyperbolic groups (and even broader classes of groups). I will define limit groups in various ways: algebraic, logical and topological, and draw connections between the different definitions. We will also see how one can equip a limit group with an action on a real tree, and analyze this action using the Rips machine, a generalization of Bass-Serre theory to real trees. As a conclusion, we will obtain that hyperbolic groups whose outer automorphism group is infinite, split non-trivially as graphs of groups.

Let $X$ be a countable discrete metric space, and think of operators on $\ell^2(X)$ in terms of their $X$-by-$X$ matrix. Band operators are ones whose matrix is supported on a "band" along the main diagonal; all norm-limits of these form a C*-algebra, called uniform Roe algebra of $X$. This algebra "encodes" the large-scale (a.k.a. coarse) structure of $X$. Quasi-locality, coined by John Roe in '88, is a property of an operator on $\ell^2(X)$, designed as a condition to check whether the operator belongs to the uniform Roe algebra (without producing band operators nearby). The talk is about our attempt to make this work, and an expander-ish condition on graphs that came out of trying to find a counterexample. (Joint with: A. Tikuisis, J. Zhang, K. Li and P. Nowak.)
 

This presentation introduces a rigorous framework for the study of commonly used machine learning techniques (kernel methods, random feature maps, etc.) in the regime of large dimensional and numerous data. Exploiting the fact that very realistic data can be modeled by generative models (such as GANs), which are theoretically concentrated random vectors, we introduce a joint random matrix and concentration of measure theory for data processing. Specifically, we present fundamental random matrix results for concentrated random vectors, which we apply to the performance estimation of spectral clustering on real image datasets.

We introduce a new and generic approximation to Schur complements arising from inf-sup stable mixed finite element discretizations of self-adjoint multi-physics problems. The approximation exploits the discretization mesh by forming local, or element, Schur complements and projecting them back to the global degrees of freedom. The resulting Schur complement approximation is sparse, has low construction cost (with the same order of operations as a general finite element matrix), and can be solved using off-the-shelf techniques, such as multigrid. Using the Ladyshenskaja-Babu\v{s}ka-Brezzi condition, we show that this approximation has favorable eigenvalue distributions with respect to the global Schur complement. We present several numerical results to demonstrate the viability of this approach on a range of applications. Interestingly, numerical results show that the method gives an effective approximation to the non-symmetric Schur complement from the steady state Navier-Stokes equations.
 

Let us fix a prime $p$. The Erdős-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero.

In this talk, we discuss a new upper bound for this problem for any fixed prime $p\geq 5$ and large $n$. Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method, as well as some new combinatorial ideas.

In STFC's Computational Mathematics Group, we work alongside scientists at large-scale National Facilities, such as ISIS Neutron and Muon source, Diamond Light Source, and the Central Laser Facility. For each of these groups, non-linear least squares fitting is a vital part of their scientific workflow. In this talk I will describe FitBenchmarking, a software tool we have developed to benchmark the performance of different non-linear least squares solvers on real-world data. It is designed to be easily extended, so that new data formats and new minimizers can be added. FitBenchmarking will allow (i) scientists to determine objectively which fitting engine is optimal for solving their problem on their computing architecture, (ii) scientific software developers to quickly test state-of-the-art algorithms in their data analysis software, and (iii) mathematicians and numerical software developers to test novel algorithms against realistic datasets, and to highlight characteristics of problems where the current best algorithms are not sufficient.
 

Real networks are highly clustered (large number of short cycles) in contrast with their random counterparts. The Erdős–Rényi model and the Configuration model will generate networks with a tree like structure, a feature rarely observed in real networks. This means that traditional random networks are a poor choice as null models for real networks. Can we do better than that? Maximum entropy random graph ensembles are the natural choice to generate such networks. By introducing a bias with respect to the number of short cycles in a degree constrained graph, we aim to get a random graph model with a tuneable number of short cycles [1,2]. Nevertheless, the story is not so simple. In the same way random unclustered graphs present undesired topology, highly clustered ones will do as well if one is not careful with the scaling of the control parameters relative to the system size. Additionally the techniques to generate and sample numerically from general biased degree constrained graph ensembles will also be discussed. The topological transition has an important impact on the computational cost to sample graphs from these ensembles. To take it one step further, a general approach using the eigenvalues of the adjacency matrix rather than just the number of short cycles will also be presented, [2].

[1] López, Fabián Aguirre, et al. "Exactly solvable random graph ensemble with extensively many short cycles." Journal of Physics A: Mathematical and Theoretical 51.8 (2018): 085101.
[2] López, Fabián Aguirre, and Anthony CC Coolen. "Imaginary replica analysis of loopy regular random graphs." Journal of Physics A: Mathematical and Theoretical 53.6 (2020): 065002.

The goal of this talk is to introduce a way to use the philosophy of Random Matrix Theory to understand, pose, and maybe even solve problems about p-adic matrices.

I will talk about my work arxiv:1911.05038. We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \omega) \in H^s\times H^s\times H^{s'}$, $2<s'<s$. The result extends the sharp result of Smith-Tataru and Wang, established in the irrotational case, i.e $ \omega=0$, which is known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e. with a constant density, the result is known to hold for $ \omega\in H^s$, $s>3/2$ and fails for $s\le 3/2$, see the work of Bourgain-Li. It is thus natural to conjecture that the optimal result should be $(v,\varrho, \omega) \in H^s\times H^s\times H^{s'}$, $s>2, \, s'>\frac{3}{2}$. We view our work here as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustic spacetime.
 

Given two metric spaces $X$ and $Y$, it is natural to ask how faithfully, from the point of view of the metric, one can embed $X$ into $Y$. One way of making this precise is asking whether there exists a coarse embedding of $X$ into $Y$. Positive results are plentiful and diverse, from Assouad's embedding theorem for doubling metric spaces to the elementary fact that any finitely generated subgroup of a finitely generated group is coarsely embedded with respect to word metrics. Moreover, the consequences of admitting a coarse embedding into a sufficiently nice space can be very strong. By contrast, there are few invariants which provide obstructions to coarse embeddings, leaving many seemingly elementary geometric questions open.
I will present new families of invariants which resolve some of these questions. Highlights of the talk include a new algebraic dichotomy for connected unimodular Lie groups, and a method of calculating a lower bound on the conformal dimension of a compact Ahlfors-regular metric space.
 

The optimal matching problem is about the rate of convergence
in Wasserstein distance of the empirical measure of iid uniform points
to the Lebesgue measure. We will start by reviewing the macroscopic
behaviour of the matching problem and will then report on recent results
on the mesoscopic behaviour in the thermodynamic regime. These results
rely on a quantitative large-scale linearization of the Monge-Ampere
equation through the Poisson equation. This is based on joint work with
Michael Goldman and Felix Otto.
 

We "review" how one can expand a filtration by continuously adding a stochastic process. The new results (obtained with Léo Neufcourt) relate to the seimartingale decompositions after the expansion. We give some possible applications. 

I will discuss the geometric interpretation of the twisted index of 3d supersymmetric gauge theories on a closed Riemann surface. In the first part of the talk, I will show that the twisted index computes the virtual Euler characteristic of the moduli space of solutions to vortex equations on the Riemann surface, which can be understood algebraically as quasi-maps to the Higgs branch. I will explain 3d N=4 mirror symmetry in this context, which implies non-trivial relations between enumerative invariants associated to these moduli spaces. Finally, I will present a wall-crossing formula for these invariants derived from the gauge theory point of view.
 

I will describe work exploring the spectrum of two-dimensional unitary conformal field theories(CFT) with no extended chiral algebra and central charge larger than one. I will revisit a classical result from analytic number theory by Rademacher, which provides an exact formula for the Fourier coefficients of modular forms of non-positive weight. Generalizing this, I will explain how we employed Rademacher's idea to study the spectral density of two-dimensional CFT of our interest. The expression is given in terms of a Rademacher expansion, which converges for nonzero spin. The implications of our spectral density to the pure gravity in AdS3 will be discussed.

When your lecturers say that they expect you to study your notes between lectures, what do they really mean?  There is research on how mathematicians go about reading maths effectively.  We'll look at a technique (self-explanation training) that has been shown to improve students' comprehension of proofs, and in this interactive workshop we'll practise together on some examples.  Please bring a pen/pencil and paper!

The pRED Clinical Pharmacology Disease Modelling Group (CPDMG) aims to better understand the biological basis of inter-patient variability of clinical response to drugs.  Improved understanding of how our drugs drive clinical responses informs which combination dosing regimens (“right drugs”) specific patient populations (“right patients”) are most likely to benefit from. Drug evoked responses are driven by drug-molecular-target interactions that perturb target functions. These direct, "proximal effects" (typically activation and/or inhibition of protein function) propagate across the biological processes these targets participate in via “distal effects” to drive clinical responses. Clinical Systems Pharmacology approaches are used by CPDMG to predict the mechanisms by which drug combinations evoke observed clinical responses. Over the last 5 years, CPDMG has successfully applied these approaches to inform key decisions across clinical development programs. Implementation of these approaches requires: (i) integration of prior relevant biological/clinical knowledge with large clinical and “omics” datasets; (ii) application of supervised machine learning (specifically, Artificial Neural Networks (ANNs)) to transform this knowledge/data into actionable, clinically relevant, mechanistic insights.  In this presentation, key features of these approaches will be discussed by way of clinical examples.  This will provide a framework for outlining the current limitations of these approaches and how we plan to address them in the future.

We show that two rules are sufficient to automate gradient descent: 1) don't increase the stepsize too fast and 2) don't overstep the local curvature. No need for functional values, no line search, no information about the function except for the gradients. By following these rules, you get a method adaptive to the local geometry, with convergence guarantees depending only on smoothness in a neighborhood of a solution. Given that the problem is convex, our method will converge even if the global smoothness constant is infinity. As an illustration, it can minimize arbitrary continuously twice-differentiable convex function. We examine its performance on a range of convex and nonconvex problems, including matrix factorization and training of ResNet-18.

RF-engineering defines the “perfect” parabolic shape a foldable reflector antenna (e.g. the membrane) should have. In practice it is virtually impossible to design a deployable backing structure that can meet all RF-imposed requirements. Inevitably the shape of the membrane will deviate from its ideal parabolic shape when material properties and pragmatic mechanical design are considered. There is therefore a challenge to model such membranes in order to find the form they take and then use the model as a design tool and perhaps in an optimisation objective function, if tractable. 

The variables we deal with are:
Elasticity of the membrane (anisotropic or orthotropic typ)
Boundary forces (by virtue of the interaction between the membrane and it’s attachment)
Elasticity of the backing structure (e.g. the elasticity properties of the attachment)
Number, location and elasticity of the membrane fixing points

There are also in-orbit environmental effects on such structures for which modelling could also be of value. For example, the structure can undergo thermal shocks and oscillations can occur that are un-dampened by the usual atmospheric interactions at ground level etc. There are many other such points to be considered and allowed for.

Some of the simplest expected cases of Langlands functoriality are the symmetric power liftings Sym^r from automorphic representations of GL(2) to automorphic representations of GL(r+1). I will discuss some joint work with Jack Thorne on the symmetric power lifting for holomorphic modular forms.

Nonlinear generalizations of the Schrödinger equation are of wide applicability to a range of areas including atomic and optical systems, 
plasma physics and water waves.  In this  talk we revisit some principal excitations in atomic and optical systems (such as Bose-Einstein condensates and photo-refractive crystals), namely dark solitonic fronts in single-component systems, and dark-bright waves in multi-component systems. Upon introducing them and explaining their existence and stability properties in one spatial dimension, we will extend them both in the form of stripes and in that rings in two-dimensions, presenting an alternative (adiabatic-invariant based) formulation of their stability and excitations. We will explore their filamentary dynamics, as well as the states that emerge from their transverse (snaking) instability. Then, we will consider these structures even in three dimensions, in the form of planar, as well as spherical shell wave patterns and generalize our adiabatic invariant formulation there. Finally, time permitting, we will give some glimpses of how some of these dynamical features in 1d and 2d generalize in a multi-orbital, time-dependent quantum setting.

We build on the literature on financial contagion using models of cross-holdings of equity participations and debt in different seniority classes, and extend them to include bail-ins and contingent convertible debt instruments, two mechanisms of debt-to-equity conversion. We combine these with recently proposed methods of network valuation under stochastic external assets, allowing for the pricing of debt instruments in each seniority layer and the calculation of default probabilities. We show that there exist well-defined valuations for all financial assets cross-held within the system. The full model constitutes an extension of classic asset pricing models that accounts for cross-holdings of debt securities. Our contribution is to add convertible debt to this framework.

In 2013, Bhargava-Shankar proved that (in a suitable sense) the average rank of elliptic curves over Q is bounded above by 1.5, a landmark result which earned Bhargava the Fields medal. Later Bhargava-Gross proved similar results for hyperelliptic curves, and Poonen-Stoll deduced that most hyperelliptic curves of genus g>1 have very few rational points. The goal of my talk is to explain how simple curve singularities and simple Lie algebras come into the picture, via a modified Grothendieck-Brieskorn correspondence.

Moreover, I’ll explain how this viewpoint leads to new results on the arithmetic of curves in families, specifically for certain families of non-hyperelliptic genus 3 curves.

Discussion of https://arxiv.org/abs/1812.04716

Systems of nonlinear polynomial equations arise in a variety of fields in mathematics, science, and engineering.  Many numerical techniques for solving and analyzing solution sets of polynomial equations over the complex numbers, collectively called numerical algebraic geometry, have been developed over the past several decades.  However, since real solutions are the only solutions of interest in many applications, there is a current emphasis on developing new methods for computing and analyzing real solution sets.  This talk will summarize some numerical real algebraic geometric approaches as well as recent successes of these methods for solving a variety of problems in science and engineering.

In recent years it has become abundantly clear that mathematics can do "things" in society; indeed, many more things than in the past. Deep mathematical work now underpins some of the most important aspects of the way society functions. And, as mathematically-trained people, we are constantly promoting the positive impact of mathematics. But if such work is capable of good, then is it not also capable of harm? So how do we begin to identify such potential harm, let alone address it and try and avoid, or at least reduce, it? In this session we will discuss how mathematics is a powerful double-edged sword, and why it must be wielded responsibly.

The second fundamental form of an embedded manifold must satisfy a set of constraint equations known as the Gauß-Codazzi equations. Since work of Chen-Slemrod-Wang, these equations are known to satisfy a particular div-curl structure: under suitable L^p bound on the second fundamental form, the curvatures are weakly continuous. In this talk we explore generalisations of this original result under weaker assumptions. We show how techniques from fluid dynamics can yield interesting insight into the weak continuity properties of isometric embeddings.

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Teichmüller harmonic map flow is a geometric flow designed to evolve combinations of maps and metrics on a surface into minimal surfaces in a Riemannian manifold. I will introduce the flow and describe known existence results, and discuss recent joint work with M. Rupflin that demonstrates how singularities can develop in the metric component in finite time.

 In Classification Theory, Shelah defined several cardinal invariants of a complete theory which detect the presence of certain trees among the definable sets, which in turn quantify the complexity of forking.  In later model-theoretic developments, local versions of these invariants were recognized as marking important dividing lines - e.g. simplicity and NTP2.  Around these dividing lines, a dichotomy theorem of Shelah states that a theory has the tree property if and only if it is witnessed in one of two extremal forms--the tree property of the first or second kind--and it was asked if there is a 'quantitative' analogue of this dichotomy in the form of a certain equation among these invariants.  We will describe these model-theoretic invariants and explain why the quantitative version of the dichotomy fails, via a construction that relies upon some unexpected tools from combinatorial set theory. 

 

On any hyperbolic surface, the number of curves of length at most L is finite. However, it is not immediately clear how quickly this number grows with L. We will discuss Mirzakhani’s breakthrough result regarding the asymptotic behaviour of this number, along with recent efforts to generalise her result using currents.