Past

This talk will be a gentle introduction to braided fusion categories, with the eventual aim to explain a result from my thesis about symmetric fusion categories. 

Fusion categories are certain kinds of monoidal categories. They can be viewed as a categorification of the finite dimensional algebras, and appear in low-dimensional topological quantum field theories, as well as being studied in their own right. A braided fusion category is additionally commutative up to a natural isomorphism, symmetry is an additional condition on this natural isomorphism. Computations in these categories can be done pictorially, using so-called string diagrams (also known as ``those cool pictures''). 

In this talk I will introduce fusion categories using these string diagrams. I will then discuss the Drinfeld centre construction that takes a fusion category and returns a braided fusion category. We then show, if the input is a symmetric fusion category, that this Drinfeld centre carries an additional tensor product. All of this also serves as a good excuse to draw lots of pictures.
 

Recall that an integer n is called y-smooth when each of its prime divisors is less than or equal to y. It is conjectured that, for any a>0,  any polynomial of positive degree having integral coefficients should possess infinitely many values at integral arguments n that are n^a-smooth. One could consider this problem to be morally “dual” to the cognate problem of establishing that irreducible polynomials assume prime values infinitely often, unless local conditions preclude this possibility. This smooth values conjecture is known to be true in several different ways for linear polynomials, but in general remains unproven for any degree exceeding 1. We will describe some limited progress in the direction of the conjecture, highlighting along the way analogous conclusions for polynomial smoothness. Despite being motivated by a problem in analytic number theory, most of the methods make use of little more than pre-Galois theory. A guest appearance will be made by several hyperelliptic curves. [This talk is based on work joint with Jonathan Bober, Dan Fretwell and Greg Martin].

Discontinuous solutions, such as cracks or cavities, can suddenly appear in elastic solids when a limiting condition is reached. Similarly, self-contacting folds can nucleate at a free surface of a soft material subjected to a critical compression. Unlike other elastic instabilities, such as buckling and wrinkling, creasing is still poorly understood. Being invisible to linearization techniques, crease nucleation is a problem of high mathematical complexity.

In this talk, I will discuss some recent theoretical insights solving the quest for both the nucleation threshold and the emerging crease morphology.  The analytic predictions are in  agreement with experimental and numerical data. They prove a fundamental insight either for understanding the creasing onset in living matter, e.g. brain convolutions, or for guiding engineering applications, e.g. morphable meta-materials.

We derive sharp bounds for the prices of VIX futures using the full information of S&P 500 smiles. To that end, we formulate the model-free sub/superreplication of the VIX by trading in the S&P 500 and its vanilla options as well as the forward-starting log-contracts. A dual problem of minimizing/maximizing certain risk-neutral expectations is introduced and shown to yield the same value. The classical bounds for VIX futures given the smiles only use a calendar spread of log-contracts on the S&P 500. We analyze for which smiles the classical bounds are sharp and how they can be improved when they are not. In particular, we introduce a tractable family of functionally generated portfolios which often improves the classical spread while still being tractable, more precisely, determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal.

Compactifications of 6D Superconformal Field Theories (SCFTs) on four-manidolfds lead to novel interacting 2D SCFTs. I will describe the various Lagrangian and non-Lagrangian sectors of the resulting 2D theories, as well as their interactions. In general this construction can be embedded in compactifications of the physical superstring, providing a general template for realizing 2D conformal field theories coupled to worldsheet gravity, i.e. a UV completion for non-critical string theories.  

The mathematical analysis and numerical simulation of acoustic and electromagnetic wave scattering by planar screens is a classical topic. The standard technique involves reformulating the problem as a boundary integral equation on the screen, which can be solved numerically using a boundary element method. Theory and computation are both well-developed for the case where the screen is an open subset of the plane with smooth (e.g. Lipschitz or smoother) boundary. In this talk I will explore the case where the screen is an arbitrary subset of the plane; in particular, the screen could have fractal boundary, or itself be a fractal. Such problems are of interest in the study of fractal antennas in electrical engineering, light scattering by snowflakes/ice crystals in atmospheric physics, and in certain diffraction problems in laser optics. The roughness of the screen presents challenging questions concerning how boundary conditions should be enforced, and the appropriate function space setting. But progress is possible and there is interesting behaviour to be discovered: for example, a sound-soft screen with zero area (planar measure zero) can scatter waves provided the fractal dimension of the set is large enough. Accurate computations are also challenging because of the need to adapt the mesh to the fine structure of the fractal. As well as presenting numerical results, I will outline some of the outstanding open questions from the point of view of numerical analysis. This is joint work with Simon Chandler-Wilde (Reading) and Andrea Moiola (Pavia).
 

We construct maximal surfaces with Neumann boundary conditions in Minkowski space using mean curvature flow. In particular we find curvature conditions on a boundary manifold so that mean curvature flow may be shown to exist for all time, and give conditions under which the maximal hypersurfaces are stable under the flow.

Prime numbers have intrigued, inspired and infuriated mathematicians for millennia and yet mathematicians' difficulty with answering simple questions about them reveals their depth and subtlety. 

Join Vicky to learn about recent progress towards proving the famous Twin Primes Conjecture and to hear the very different ways in which these breakthroughs have been made - a solo mathematician working in isolation, a young mathematician displaying creativity at the start of a career, a large collaboration that reveals much about how mathematicians go about their work.  

Vicky Neale is Whitehead Lecturer at the Mathematical Institute, University of Oxford and Supernumerary Fellow at Balliol College.

Please email @email to register.

I will present a gentle introduction to the theory of conformal dimension, focusing on its applications to the boundaries of hyperbolic groups, and the difficulty of classifying groups whose boundaries have conformal dimension 1.

This talk will hopefully highlight the general framework in which Penrose tilings are proved to be aperiodic and in fact a tiling. 

Globally Valued Fields, studied jointly with E. Hrushovski, are a formalism for fields in which the sum formula for valuations holds, such as number fields or function fields of curves. They form an elementary class (in continuous first order logic), and model-theoretic questions regarding this class give rise to difficult yet fascinating geometric questions.
I intend to present « Lyon school » approach to studying GVFs. This consists of reducing as much as possible to local considerations, among other things via the "fullness" axiom.
 

Given a Fano manifold we will consider two ways of attaching a (usually infinite) collection of polytopes, and a certain combinatorial transformation relating them, to it. The first is via Mirror Symmetry, following a proposal of  Coates--Corti--Kasprzyk--Galkin--Golyshev. The second is via symplectic topology, and comes from considering degenerating Lagrangian torus fibrations. We then relate these two collections using the Gross--Siebert program. I will also comment on the situation in higher dimensions, noting particularly that by 'inverting' the second method (degenerating Lagrangian fibrations) we can produce topological constructions of Fano threefolds.
 

Enumerating families of combinatorial objects with given properties and describing the typical structure of these objects are fundamental problems in extremal combinatorics. In this talk, we will investigate intersecting families of discrete structures in various settings, determining their typical structure as the size of the underlying ground set tends to infinity. Our new approach outlines a general framework for a number of similar problems; in particular, we prove analogous results for hypergraphs, permutations, and vector spaces using the same technique. This is joint work with József Balogh, Shagnik Das, Hong Liu, and Maryam Sharifzadeh.

In this talk we describe a new approach that enables the use of elliptic PDEs with white noise forcing to sample Matérn fields within the multilevel Monte Carlo (MLMC) framework.

When MLMC is used to quantify the uncertainty in the solution of PDEs with random coefficients, two key ingredients are needed: 1) a sampling technique for the coefficients that satisfies the MLMC telescopic sum and 2) a numerical solver for the forward PDE problem.

When the dimensionality of the uncertainty in the problem is infinite (i.e. coefficients are random fields), the sampling techniques commonly used in the literature are Karhunen–Loève expansions or circulant embeddings. In the specific case in which the coefficients are Gaussian fields of Mat ́ern covariance structure another sampling technique available relies on the solution of a linear elliptic PDE with white noise forcing.

When the finite element method (FEM) is used for the forward problem, the latter option can become advantageous as elliptic PDEs can be quickly and efficiently solved with the FEM, the sampling can be performed in parallel and the same FEM software can be used without the need for external packages. However, it is unclear how to enforce a good stochastic coupling of white noise between MLMC levels so as to respect the MLMC telescopic sum. In this talk we show how this coupling can be enforced in theory and in practice.

We give a short reminder about central results of classical tilting theory, 
including the Brenner-Butler tilting theorem, and
homological properties of tilted and quasi-tilted algebras. We then discuss 
2-term silting complexes and endomorphism algebras of such objects,
and in particular show that some of these classical results have very natural 
generalizations in this setting.
(joint work with Yu Zhou)

We propose and analyze a multilevel weighted least squares polynomial approximation method. Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method, which employs samples with different accuracies and is able to match the accuracy of single level approximations at reduced computational work. We prove complexity bounds under certain assumptions on polynomial approximability and sample work. Furthermore, we propose an adaptive
algorithm for situations where such assumptions cannot be verified a priori. Numerical experiments underline the practical applicability of our method.

Many metallurgical processes involve the heat treatment of granular material due to large alternating currents. To understand how the current propagates through the material, one must understand the bulk resistivity, that is, the resistivity of the granular material as a whole. The literature suggests that the resistance due to contacts between particles contributes significantly to the bulk resistivity, therefore one must pay particular attention to these contacts. 

My work is focused on understanding the precise impact of small contacts on the current propagation. The scale of the contacts is several order of magnitude smaller than that of the furnace itself, therefore we apply matched asymptotics methods to study how the current varies with the size of the contact.

How long does a uniformly accelerated observer need to interact with a
quantum field in order to record thermality in the Unruh temperature?
In the limit of large excitation energy, the answer turns out to be
sensitive to whether (i) the switch-on and switch-off periods are
stretched proportionally to the total interaction time T, or whether
(ii) T grows by stretching a plateau in which the interaction remains
at constant strength but keeping the switch-on and switch-off
intervals of fixed duration. For a pointlike Unruh-DeWitt detector,
coupled linearly to a massless scalar field in four spacetime
dimensions and treated within first order perturbation theory, we show
that letting T grow polynomially in the detector's energy gap E
suffices in case (i) but not in case (ii), under mild technical
conditions. These results limit the utility of the large E regime as a

probe of thermality in time-dependent versions of the Hawking and
Unruh effects, such as an observer falling into a radiating black
hole. They may also have implications on the design of prospective
experimental tests of the Unruh effect.

Based on arXiv:1605.01316 (published in CQG) with Christopher J
Fewster and Benito A Juarez-Aubry.

R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes showed in their celebrated paper from 1993 that numerous compensated compactness quantities such as Jacobians of mappings in $W^{1,n}(\mathbb{R}^n,\mathbb{R}^n)$ belong the real-variable Hardy space $\mathcal{H}^1(\mathbb{R}^n)$. They proceeded to ask what is the exact range of these nonlinear quantities and in particular whether the Jacobian operator $J$ maps $W^{1,2}(\mathbb{R}^2,\mathbb{R}^2)$ onto $\mathcal{H}^1(\mathbb{R}^2)$.

I present the proof of my recent result that $J \colon W^{1,n}(\mathbb{R}^n,\mathbb{R}^n) \to \mathcal{H}^1(\mathbb{R}^n)$ is non-surjective for every $n \ge 2$. The surjectivity question is still open when the domain of definition of $J$ is the inhomogeneous Sobolev space $\dot{W}^{1,n}(\mathbb{R}^n,\mathbb{R}^n)$. I also shortly discuss my work on T. Iwaniec's conjecture from 1997 which states that for every $n \ge 2$ and $p \in [1,\infty[$ the operator $J \colon W^{1,np}(\mathbb{R}^n,\mathbb{R}^n) \to \mathcal{H}^p(\mathbb{R}^n)$ has a continuous right inverse.

I will discuss ongoing work aimed at constructing higher categories of (enriched) higher categories. This should give the appropriate targets for many interesting examples of extended topological quantum field theories, including extended versions of the classical examples of TQFTs due to Turaev-Viro, Reshetikhin-Turaev, etc.

The signature of a path has many properties that make it an excellent feature to be used in machine learning. We exploit this properties to analyse a stream of data that arises from a psychiatric study whose objective is to analyse bipolar and borderline personality disorders. We build a machine learning model based on signatures that tries to answer two clinically relevant questions, based on observations of their reported state over a short period of time: is it possible to predict if a person is healthy, has bipolar disorder or has borderline personality disorder? And given a person or borderline personality disorder, it is possible to predict his or her future mood? Signatures proved to be very effective to tackle these two problems.

Abstract. As the first  step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multi-plicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution (Xt; Yt) of the equations dXt = Ytdt, dYt = jXtj_dBt, (X0; Y0) = (x0; y0). In particular, we prove that solutions arenonunique if 0 < _ < 1 and (x0; y0) = (0; 0) and unique if 1=2 < _ and (x0; y0) 6= (0; 0). We also show that blowup in _nite time holds if _ > 1 and (x0; y0) 6= (0; 0).

This is a joint work with A. Gomez, J.J. Lee, C. Mueller and M. Salins.

G2-manifolds are the Riemannian 7-manifolds with G2 holonomy and in many respects can be regarded as 7-dimensional analogues of Calabi-Yau 3-folds.
In joint work with Mark Haskins and Johannes Nordström we construct infinitely many families of new complete non-compact G2 manifolds (only four such manifolds were previously known). The underlying smooth 7-manifolds are all circle bundles over asymptotically conical Calabi-Yau 3-folds. The metrics are circle-invariant and have an asymptotic geometry that is the 7-dimensional analogue of the geometry of 4-dimensional ALF hyperkähler metrics. After describing the main features of our construction I will concentrate on some illustrative examples, describing how results in Calabi-Yau geometry about isolated singularities and their resolutions can be used to produce examples of complete G2-manifolds.

Nekrasov, Rosly and Shatashvili observed that the generating function of a certain space of SL(2) opers has a physical interpretation as the effective twisted superpotential for a four-dimensional N=2 quantum field theory. In this talk we describe the ingredients needed to generalise this observation to higher rank. Important ingredients are spectral networks generated by Strebel differentials and the abelianization method. As an example we find the twisted superpotential for the E6 Minahan-Nemeschansky theory. 
 

In this talk we consider a system of interacting Brownian particles. When diffusing particles interact with each other their motions are correlated, and the configuration space is of very high dimension. Often an equation for the one-particle density function (the concentration) is sought by integrating out the positions of all the others. This leads to the classic problem of closure, since the equation for the concentration so derived depends on the two-particle correlation function. We discuss two  common closures, the mean-field (MFA) and the Kirkwood-superposition approximations, as well as an alternative approach, which is entirely systematic, using matched asymptotic expansions (MAE). We compare the resulting (nonlinear) diffusion models with Monte Carlo simulations of the stochastic particle system, and discuss for which types of interactions (short- or long-range) each model works best. 

For a smooth variety over a number field, one defines various different homology groups (Betti, de Rham, etale, log-crystalline), which carry various kinds of enriching structure and are thought of as a system of realisations for a putative underlying (mixed) motivic homology group. Following Deligne, one can study fundamental groups in the same way, and the study of specific realisations of the motivic fundamental group has already found Diophantine applications, for instance in the anabelian proof of Siegel's theorem by Kim.

It is hoped that study of fundamental groups should give one access to ``higher'' arithmetic information not visible in the first cohomology, for instance classical and p-adic heights. In this talk, we will discuss recent work making this hope concrete, by demonstrating how local components of canonical heights on abelian varieties admit a natural description in terms of fundamental groups.

This talk explains how to formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a " background noise " (or " mean field "). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. Here the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of " extended MFG ", we hence provide generic results to address these " MFG of controls ", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of " heterogenous preferences " (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can " learn " it day after day, observing others' behaviors.

In this talk, I will present a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws.
The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility.
Despite this, many open problems remain in the analysis and approximation of such problems.
In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations.
Approximation of the resulting optimal control problem is computationally challenging, requiring massive amounts of computational time and memory storage.
The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems. The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement.
An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally.
Furthermore, parallelisation as well as an adaptive-step gradient update for the control are employed to further improve efficiency.
Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency.
A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.

We present a new model for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. Agents try to coordinate their body attitudes with the ones of their neighbours. This model is inspired by the Vicsek model. The goal of this talk will be to present this new flocking model, its relevance and the derivation of the macroscopic equations from the particle dynamics.

Identity-based cryptography can be useful in situations where a full-scale public-key infrastructure is impractical.  Original identity-based proposals relied on elliptic curve pairings and so are vulnerable to quantum computers.  I will describe some on-going work to design a post-quantum identity-based encryption scheme using ideas from Ring Learning with Errors. Our scheme has the advantage that it can be extended to the hierarchical setting for more flexible key management.

In the game 'Set', players compete to pick out groups of three cards sharing common attributes. But how many cards must be dealt before such a group must appear? 
This is an example of a "cap set problem", a problem in Ramsey theory: how big can a set of objects get before some form of order appears? We will translate the cap set problem into a problem of geometry over finite fields, discussing the current best upper bounds and running through an elementary proof. We will also (very) briefly discuss one or two implications of the cap set problem over F_3 to other questions in Ramsey theory and computational complexity
 

The talk will focus on results of two related strands of research undertaken by the speaker. The first is a model of quantum mechanics based on the idea of 'structural approximation'. The earlier paper 'The semantics of the canonical commutation relations' established a method of calculation, essentially integration, for quantum mechanics with quadratic Hamiltonians. Currently, we worked out a (model-theoretic) formalism for the method, which allows us to
perform more subtle calculations, in particular, we prove that our path integral calculation produce correct formula for quadratic Hamiltonians avoiding non-conventional limits used by physicists. Then we focus on the model-theoretic analysis of the notion of structural approximation and show that it can be seen as a positive model theory version of the theory of measurable structures, compact domination and integration (p-adic and adelic).

Using tropical geometry and new methods in the theory of Fukaya categories, we explain a mirror symmetry equivalence relating the Fukaya category of a hypersurface and the category of coherent sheaves on the boundary of a toric variety.

Given a graph $G$, we can form a hypergraph $H$ whose edges correspond to the triangles in $G$. If $G$ is the standard Erdős-Rényi random graph with independent edges, then $H$ is random, but its edges are not independent, because of overlapping triangles. This is (presumably!) a major complication when proving results about triangles in random graphs.  However, it turns out that, for many purposes, we can treat the triangles as independent, in a one-sided sense (and losing something in the density): we can find an independent random hypergraph within the set of triangles. I will present two proofs, one of which generalizes to larger complete (and some non-complete) subgraphs.

In this talk, a novel discontinuous Galerkin (DG) method is introduced by utilising the principle of discrete least squares. The key idea is to build polynomial approximations by the method of  (weighted) discrete least squares instead of usual interpolation or (discrete) $L^2$ projections. The resulting method hence uses more information of the underlying function and provides a more robust alternative to common DG methods. As a result, we are able to construct high-order schemes which are conservative as well as linear stable on any set of collocation points. Several numerical tests highlight the new discontinuous Galerkin discrete least squares (DG-DLS) method to significantly outperform present-day DG methods.

High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions. Recently, there has been an increasing interest in generalised SBP operators both in the finite difference and the discontinuous Galerkin spectral element framework.

However, if generalised SBP operators are used, the treatment of boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely —interpolating the product of two functions will in general result in a value different from the product of the interpolations. Thus, desired properties such as conservation and stability are more difficult to obtain.

In this talk, the concept of generalised SBP operators and their application to entropy stable semidiscretisations will be presented. Several recent ideas extending the range of possible methods are discussed, presenting both advantages and several shortcomings.

In this talk I will first briefly introduce 1-parameter persistent homology, and discuss some applications and the theoretical challenges in the multiparameter case. If time remains I will explain how tools from commutative algebra give invariants suitable for the study of data. This last part is based on the preprint https://arxiv.org/abs/1708.07390.
 

Shortly after Mason & Skinner introduced the so-called ambitwistor strings, Berkovits came up with a pure-spinor analogue of the theory, which was later shown to provide the supersymmetric version of the Cachazo-He-Yuan amplitudes. In the heterotic version, however, both models give somewhat unsatisfactory descriptions of the supergravity sector.

In this talk, I will show how the original pure-spinor version of the heterotic ambitwistor string can be modified in a consistent manner that renders the supergravity sector treatable. In addition to the massless states, the spectrum of the new model --- which we call sectorized heterotic string --- contains a single massive level. In the limit in which a dimensionful parameter is taken to infinity, these massive states become the unexpected massless states (e.g. a 3-form potential) first encountered by Mason & Skinner."

Klarrich showed that the Gromov boundary of the curve complex of a hyperbolic surface is homeomorphic to the space of ending laminations on that surface. Independent results of Bestvina-Reynolds and Hamenstädt give an analogous statement for the free factor graph of a free group, where the space of ending laminations is replaced with a space of equivalence classes of arational trees. I will give an introduction to these objects and describe some joint work with Bestvina and Horbez, where we show that the Gromov boundary of the free factor graph for a free group of rank N has topological dimension at most 2N-2.

A Morse function (and more generally a Morse-Bott function) on a compact manifold M has associated Morse inequalities. The aim of this
talk is to explain how we can associate Morse inequalities to any smooth function on M (reporting on work of/with G Penington).

Inverting the signature of a path with ideas from linear algebra with implementations.

A K3 surface is called attractive if and only if its Picard number is 20: The maximal possible. Attractive K3 surfaces possess complex multiplication. This property endows attractive K3 surfaces with rich and well understood arithmetic. For example, the associated Galois representation turns out to be a product of well known two dimensional representations and the  Hasse-Weil L-function turns out to be a product of well known L-functions. On the other hand, attractive K3 surfaces show up as solutions of the attractor equations in type IIB string theory compactified on the product of a K3 surface with an elliptic curve. As such, these surfaces dictate the near horizon geometry of a charged black hole in this theory. We will try to see which arithmetic properties of the attractive K3 surfaces lend a stringy interpretation and use them to shed light on physical properties of the charged black hole. 
 

After recalling some backgrounds and motivations, we'll report some recent results on the optimal L^2 extensions and multiplier ideal sheaves, with emphasizing the close relations between SCV and PDE.

We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e., each data point is a solution to a system of polynomial equations. In this case, the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional space of monomial features. Many well-studied extensions of linear models, including affine subspaces and their union, can be described by a variety model. We study the sampling requirements for matrix completion under a variety model with a focus on a union of subspaces. We also propose an efficient matrix completion algorithm that minimizes a surrogate of the rank of the matrix of monomial features, which is able to recover synthetically generated data up to the predicted sampling complexity bounds. The proposed algorithm also outperforms standard low-rank matrix completion and subspace clustering techniques in experiments with real data.