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The following Ramsey-type question is one of the central problems in combinatorial geometry. Given a collection of certain geometric objects in the plane (e.g. segments, rectangles, convex sets, arcwise connected sets) of size n, what is the size of the largest subcollection in which either any two elements have a nonempty intersection, or any two elements are disjoint? We prove that there exists an absolute constant c>0 such that if C is a collection of n curves in the plane, then C contains at least nc elements that are pairwise intersecting, or n c elements that are pairwise disjoint. This resolves a problem raised by Alon, Pach, Pinchasi, Radoičić and Sharir, and Fox and Pach. Furthermore, as any geometric object can be arbitrarily closely approximated by a curve, this shows that the answer to the aforementioned question is at least nc for any collection of n geometric objects. [ABSTRACT FROM AUTHOR]

In this paper we develop a penalty method to approximate solutions of the free boundary problem for minimal surfaces. To this end we study the problem of finding minimizers of a functional Fλ which is defined as the sum of the Dirichlet integral and an appropriate penalty term weighted by a parameter λ. We prove existence of a solution for λ large enough as well as convergence to a solution of the free boundary problem as λ tends to infinity. Additionally regularity at the boundary of these solutions is shown, which is crucial for deriving numerical error estimates. Since every solution is harmonic, the analysis is largely simplified by considering boundary values only and using harmonic extensions. In a subsequent paper we develop a fully discrete finite element procedure for approximating solutions to this problem and prove an error estimate which includes an order of convergence with respect to the grid size. [ABSTRACT FROM AUTHOR]

This paper continues our study of the interconnection between controllability and mixing properties of random dynamical systems. We begin with an abstract result showing that the approximate controllability to a point and a local stabilisation property imply the uniqueness of a stationary measure and exponential mixing in the dual-Lipschitz metric. This result is then applied to the 2D Navier--Stokes system driven by a random force acting through the boundary. A byproduct of our analysis is the local exponential stabilisation of the boundary-driven Navier--Stokes system by a regular boundary control. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global existence of supersonic entropy solutions with a strong contact discontinuity over a Lipschitz wall governed by the two-dimensional steady exothermically reacting Euler equations, when the total variation of both the initial data and slope of the Lipschitz wall is sufficiently small. Local and global estimates are developed and a modified Glimm-type functional is carefully designed. Next the validation of the quasi-one-dimensional approximation in the domain bounded by the wall and the strong contact discontinuity is rigorous justified by proving that the difference between the average of weak solution and the solution of quasi-one-dimensional system can be bounded by the square of the total variation of both the initial data and slope of the Lipschitz wall. [ABSTRACT FROM AUTHOR]

In this paper we establish an improvement of the tilt-excess decay estimate for the Allen--Cahn equation, and use this to give a new proof of Savin's theorem on the uniform C1,α regularity of flat level sets. This generalizes Allard's ε-regularity theorem for stationary varifolds to the setting of Allen--Cahn equations. A new proof of Savin's theorem on the one-dimensional symmetry of minimizers in ℝn for n ≤ 7 is also given. [ABSTRACT FROM AUTHOR]

By a curve in Rd we mean a continuous map V I ! Rd, where I R is a closed interval. We call a curve in Rd (k)-crossing if it intersects every hyperplane at most k times (counted with multiplicity). The (d)-crossing curves in Rd are often called convex curves and they form an important class; a primary example is the moment curve f.t; t²;: ::; td / V t 2 [0; 1U]. They are also closely related to Chebyshev systems, which is a notion of considerable importance, e.g., in approximation theory. Our main result is that for every d there isM D M(d) such that every (d+1)-crossing curve in Rd can be subdivided into at most M (d)-crossing curve segments. As a consequence, based on the work of Eli'a's, Rold'an, Safernov'a, and the second author, we obtain an essentially tight lower bound for a geometric Ramsey-type problem in Rd concerning order-type homogeneous sequences of points, investigated in several previous papers. [ABSTRACT FROM AUTHOR]

We consider the stochastic Navier–Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear multiplicative and a nonlinear noise of cylindrical type, all driven by a Wiener process. In these settings, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits us to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail to satisfy the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval [0,∞). Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, non uniqueness in law holds on an arbitrary time interval[0,T], T > 0. [ABSTRACT FROM AUTHOR]

We prove that every uniform approximate homomorphism from a discrete amenable group into a symmetric group is uniformly close to a homomorphism into a slightly larger symmetric group. That is, amenable groups are uniformly flexibly stable in permutations. This answers affirmatively a question of Kun and Thom and a slight variation of a question of Lubotzky. We also give a negative answer to Lubotzky's original question by showing that the group Z is not uniformly strictly stable. Furthermore, we show that SLr(Z), r=3, is uniformly flexibly stable, but the free group Fr, r=2, is not. We define and investigate a probabilistic variant of uniform stability that has an application to property testing. [ABSTRACT FROM AUTHOR]

The Cahn–Hilliard–Hele–Shaw system is a fundamental diffuse-interface model for an incompressible binary fluid confined in a Hele–Shaw cell. It consists of a convective Cahn–Hilliard equation in which the velocity u is subject to a Korteweg force through Darcy's equation. In this paper, we aim to investigate the system with a physically relevant potential (i.e., of logarithmic type). This choice ensures that the (relative) concentration difference φ takes values within the admissible range. To the best of our knowledge, essentially all the available contributions in the literature are concerned with a regular approximation of the singular potential. Here we first prove the existence of a global weak solution with finite energy that satisfies an energy dissipative property. Then, in dimension two, we further obtain the uniqueness and regularity of global weak solutions. In particular, we show that any two-dimensional weak solution satisfies the so-called strict separation property, namely, if φ is not a pure state at some initial time, then it stays instantaneously away from the pure states. When the spatial dimension is three, we prove the existence of a unique global strong solution, provided that the initial datum is regular enough and sufficiently close to any local minimizer of the free energy. This also yields the local Lyapunov stability of the local minimizer itself. Finally, we prove that under suitable assumptions any global solution converges to a single equilibrium as time goes to infinity. [ABSTRACT FROM AUTHOR]

We build on the dynamical systems approach to deep learning, where deep residual networks are idealized as continuous-time dynamical systems, from the approximation perspective. In particular, we establish general sufficient conditions for universal approximation using continuoustime deep residual networks, which can also be understood as approximation theories in Lp using flow maps of dynamical systems. In specific cases, rates of approximation in terms of the time horizon are also established. Overall, these results reveal that composition function approximation through flow maps presents a new paradigm in approximation theory and contributes to building a useful mathematical framework to investigate deep learning. [ABSTRACT FROM AUTHOR]

In [11] the evolution of hypersurfaces in ℝn+1 with normal speed equal to a power k > 1 of the mean curvature is considered and the level set solution u of the flow is obtained as the C0-limit of a sequence ue of smooth functions solving the regularized level set equations. We prove a rate for this convergence. [ABSTRACT FROM AUTHOR]

This paper is concerned with a non-strictly hyperbolic system of conservation laws. We study the existence of weak solutions to the associated Cauchy problem, in the framework of L ∞ , using the methods of compensated compactness. Some previous results for quadratic systems have been generalized. [ABSTRACT FROM AUTHOR]

The goal of this work is to provide a general framework to study singular limits of initial-value problems for first-order quasilinear hyperbolic systems with stiff source terms in several space variables. We propose structural stability conditions of the problem and construct an approximate solution by a formal asymptotic expansion with initial layer corrections. In general, the equations defining the approximate solution may come together with differential constraints, and so far there are no results for the existence of solutions. Therefore, sufficient conditions are shown so that these equations are parabolic without differential constraint. We justify rigorously the validity of the asymptotic expansion on a time interval independent of the parameter, in the case of the existence of approximate solutions. Applications of the result include Euler equations with damping and an Euler–Maxwell system with relaxation. The latter system was considered in [27,9] which contain ideas used in the present paper. [ABSTRACT FROM AUTHOR]

In this paper a globally convergent computational scheme is established to approximate a topological multivortex solution in the recently discovered self-dual Chern-Simons theory in R 2 . Our method which is constructive and numerically efficient finds the most superconducting solution in the sense that its Higgs field has the largest possible magnitude. The method consists of two steps: first one obtains by a convergent monotone iterative algorithm a suitable solution of the bounded domain equations and then one takes the large domain limit and approximates the full piane solutions. It is shown that with a special choice of the initial guess function, the approximation sequence approaches exponentially fast a solution in R 2 . The convergence rate implies that the truncation errors away from local regions are insignificant. [ABSTRACT FROM AUTHOR]

Machine learning has achieved remarkable successes in various applications, but there is wide agreement that a mathematical theory for deep learning is missing. Recently, some first mathematical results have been derived in different areas such as mathematical statistics and statistical learning. Any mathematical theory of machine learning will have to combine tools from different fields such as nonparametric statistics, high-dimensional statistics, empirical process theory and approximation theory. The main objective of the workshop was to bring together leading researchers contributing to the mathematics of machine learning. A focus of the workshop was on theory for deep neural networks. Mathematically speaking, neural networks define function classes with a rich mathematical structure that are extremely difficult to analyze because of nonlinearity in the parameters. Until very recently, most existing theoretical results could not cope with many of the distinctive characteristics of deep networks such as multiple hidden layers or the ReLU activation function. Other topics of the workshop are procedures for quantifying the uncertainty of machine learning methods and the mathematics of data privacy. [ABSTRACT FROM AUTHOR]

Building upon recent results of Dubedat [7] on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations Ωδ to a simply connected domain Ω ⊂ ℂ we prove the convergence of probabilities of cylindrical events for the double-dimer loop ensembles on Ωδ as δ → 0. More precisely, let λ1,..., λn ∈ Ω and L be a macroscopic lamination on Ω \ {λ1,..., λn}, i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities PLδ that one obtains L after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on Ωδ converge to a conformally invariant limit PL as δ → 0, for each L. Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety Hom(π1(Ω \ {λ1,..., λn}) → SL2(ℂ)) and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do not use any RSW-type arguments for double-dimers. The limits PL of the probabilities PLδ are defined as coefficients of the isomonodromic tau-function studied in [7] with respect to the Fock-Goncharov lamination basis on the representation variety. The fact that PL coincides with the probability of obtaining L from a sample of the nested CLE(4) in Ω requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble. [ABSTRACT FROM AUTHOR]

We show that for at least 20% of primitive forms of level 1 and weight 4k → ∞, k ∈ N, the associated L-function at the central point is no less than (log k)-2 [ABSTRACT FROM AUTHOR]

Let X be a centered Gaussian random variable in a separable Hilbert space H with covariance operator Σ. We study a problem of estimation of a smooth functional of Σ based on a sample X1,...,Xn of n independent observations of X. More specifically, we are interested in functionals of the form ⟨f(Σ),B⟩, where f:R↦R is a smooth function and B is a nuclear operator in H. We prove concentration and normal approximation bounds for plug-in estimator ⟨f(Σ),B⟩, Σ:=n-1∑nj=1Xj⊗Xj being the sample covariance based on X1,...,Xn. These bounds show that ⟨f(Σ),B⟩ is an asymptotically normal estimator of its expectation EΣ⟨f(Σ),B⟩ (rather than of parameter of interest ⟨f(Σ),B⟩) with a parametric convergence rate O(n-1/2) provided that the effective rank r(Σ):=tr(Σ)∥Σ∥ (tr(Σ) being the trace and ∥Σ∥ being the operator norm of Σ) satisfies the assumption r(Σ)=o(n). At the same time, we show that the bias of this estimator is typically as large as r(Σ)n (which is larger than n-1/2 if r(Σ)≥n1/2). In the case when H is finite-dimensional space of dimension d=o(n), we develop a method of bias reduction and construct an estimator ⟨h(Σ),B⟩ of ⟨f(Σ),B⟩ that is asymptotically normal with convergence rate O(n-1/2). Moreover, we study asymptotic properties of the risk of this estimator and prove minimax lower bounds for arbitrary estimators showing the asymptotic efficiency of ⟨h(Σ),B⟩ in a semi-parametric sense. [ABSTRACT FROM AUTHOR]

Let S be a Gorenstein local ring and suppose that M is a finitely generated Cohen-Macaulay S-module of codimension c. Given a regular sequence f1.... fc in the annihilator of M we set R D S=.f1.... fc/and construct layered S-free and R-free resolutions of M. The construction inductively reduces the problem to the case of a Cohen-Macaulay module of codimension c - 1 and leads to the inductive construction of a higher matrix factorization for M. In the case where M is a sufficiently high R-syzygy of some module of finite projective dimension over S, the layered resolutions are minimal and coincide with the resolutions defined from higher matrix factorizations we described in [EP]. Our results provide a characterization of all MCM modules over a complete intersection in terms of higher matrix factorizations. [ABSTRACT FROM AUTHOR]

Minimal partition problems consist in finding a partition of a domain into a given number of components in order to minimize a geometric criterion. In applicative fields such as image processing or continuum mechanics, it is standard to incorporate in this objective an interface energy that accounts for the lengths of the interfaces between components. The present work is focused on the theoretical and numerical treatment of minimal partition problems with such interface energies. The considered approach is based on a Γ-convergence approximation combined with convex analysis techniques. [ABSTRACT FROM AUTHOR]

We consider the classical statistical learning/regression problem, when the value of a real random variable Y is to be predicted based on the observation of another random variable X. Given a class of functions F and a sample of independent copies of (X, Y), one needs to choose a function bf from F such that bf (X) approximates Y as well as possible, in the mean-squared sense. We introduce a new procedure, the so-called median-of-means tournament, that achieves the optimal tradeoff between accuracy and confidence under minimal assumptions, and in particular outperforms classical methods based on empirical risk minimization. [ABSTRACT FROM AUTHOR]

Let Ω be a smooth bounded domain in Rn, n ≥ 5. We consider the classical semilinear heat equation at the critical Sobolev exponent... is positive definite. For any k ≥ 1 such points indeed exist. We prove the existence of a positive smooth solution u(x, t) which blows up by bubbling in infinite time near those points. More precisely, for large time t, u takes the approximate form... [ABSTRACT FROM AUTHOR]

We prove that if T1, ..., Tn is a sequence of bounded degree trees such that Ti has i vertices, then Kn has a decomposition into T1, ..., Tn. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first o.n/ trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions. [ABSTRACT FROM AUTHOR]

Using formal asymptotic methods we derive a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour or other biological tissue. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow (together with a 'kinetic under-cooling' regularisation) where the forcing depends on the solution of a PDE that holds in the domain enclosed by the interface. We perform linear stability analysis and derive a diffuse-interface approximation of the model. Finite-element discretisations of two closely related models are presented, together with computational results comparing the approximate solutions. [ABSTRACT FROM AUTHOR]

We consider elliptic variational inequalities generated by obstacle type problems with thin obstacles. For this class of problems, we deduce estimates of the distance (measured in terms of the natural energy norm) between the exact solution and any function that satisfies the boundary condition and is admissible with respect to the obstacle condition (i.e., they are valid for any approximation regardless of the method by which it was found). Computation of the estimates does not require knowledge of the exact solution and uses only the problem data and an approximation. The estimates provide guaranteed upper bounds of the error (error majorants) and vanish if and only if the approximation coincides with the exact solution. In the last section, the efficiency of error majorants is confirmed by an example, where the exact solution is known. [ABSTRACT FROM AUTHOR]

Let X be a pointed metric space and let Lip0(X) be the space of all scalar-valued Lipschitz functions on X which vanish at the base point. We prove that Lip0(X) with the bounded weak* topology τbw* has the approximation property if and only if the Lipschitz-free Banach space F(X) has the approximation property if and only if, for each Banach space F, each Lipschitz operator from X into F can be approximated by Lipschitz finite-rank operators within the unique locally convex topology γτγ on Lip0(X, F) such that the Lipschitz transpose mapping f → ft is a topological isomorphism from (Lip0(X, F), γτγ) to (Lip0(X), τbw* )∈F. [ABSTRACT FROM AUTHOR]

Let ɷ ⊆ R² be a domain and let f ε W1,1(ɷ,R²) be a homeomorphism (between ɷ and f(ɷ)) Then there exists a sequence of smooth diffeomorphisms fk converging to f in W1,1(ɷ,R²) and uniformly. [ABSTRACT FROM AUTHOR]

We add to the literature the following observation. If is a singular measure on Rn which assigns measure zero to every porous set and f WRn ! R is a Lipschitz function which is non-difierentiable -a.e. then for every C1 function gWRn ! R it holds {x 2 RnW f .(x) D g.(x)} D 0: In other words the Lusin type approximation property of Lipschitz functions with C1 functions does not hold with respect to a general Radon measure. [ABSTRACT FROM AUTHOR]

Let E ⊂ ℝn+1, n ⩾ 1, be a uniformly rectifiable set of dimension n. We show E that has big pieces of boundaries of a class of domains which satisfy a 2-sided corkscrew condition, and whose connected components are all chord-arc domains (with uniform control of the various constants). As a consequence, we deduce that E has big pieces of sets for which harmonic measure belongs to weak-A∞. [ABSTRACT FROM AUTHOR]

We prove a finiteness principle for interpolation of data by nonnegative Cm and Cm-1,1 functions. Our result raises the hope that one can start to understand constrained interpolation problems in which, e.g., the interpolating function F is required to be nonnegative. [ABSTRACT FROM AUTHOR]

We consider the following interpolation problem. Suppose one is given a finite set E ⊂ ℝd, a function f : E → ℝ, and possibly the gradients of f at the points of E. We want to interpolate the given information with a function F ∊ C1,1(ℝd) with the minimum possible value of Lip(∇F). We present practical, efficient algorithms for constructing an F such that Lip(∇F) is minimal, or for less computational effort, within a small dimensionless constant of being minimal. [ABSTRACT FROM AUTHOR]

For a class of competition-diffusion nonlinear systems involving the square root of the laplacian, including the fractional Gross--Pitaevskii system ... we prove that L∞ boundedness implies C0,α boundedness for every α ∈ [0, 1/2), uniformly as β → ∞. Moreover we prove that the limiting profile is C0,1=2. This system arises, for instance, in the relativistic Hartree--Fock approximation theory for k-mixtures of Bose--Einstein condensates in different hyperfine states. [ABSTRACT FROM AUTHOR]

We show that, given an absolutely continuous horizontal curve γ in the Heisenberg group, there is a C¹ horizontal curve Γ such that Γ = γ and Γ ' = γ ' outside a set of small measure. Conversely, we construct an absolutely continuous horizontal curve in the Engel group with no C¹ horizontal approximation. [ABSTRACT FROM AUTHOR]

Consider the classical (2+1)-dimensional Solid-On-Solid model above a hard wall on an L×L box of Z2. The model describes a crystal surface by assigning a non-negative integer height ηx to each site x in the box and 0 heights to its boundary. The probability of a surface configuration η is proportional to exp(−βH(η)), where β is the inverse-temperature and H(η) sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with high probability (w.h.p.) the height of almost all sites is concentrated on two levels, H(L)=⌊(1/4β)logL⌋ and H(L)−1. Moreover, for most values of L the height is concentrated on the single value H(L). Next, we study the ensemble of level lines corresponding to the heights (H(L),H(L)−1,…). We prove that w.h.p.\there is a unique macroscopic level line for each height. Furthermore, when taking a diverging sequence of system sizes Lk, the rescaled macroscopic level line at height H(Lk)−n has a limiting shape if the fractional parts of (1/4β)logLk converge to a noncritical value. The scaling limit is an explicit convex subset of the unit square Q and its boundary has a flat component on the boundary of Q. Finally, the highest macroscopic level line has L1/3+o(1)k fluctuations along the flat part of the boundary of its limiting shape. [ABSTRACT FROM AUTHOR]

We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich--Wasserstein distance of the Fekete points to the limiting measure. The sampling and interpolation arrays on line bundles are a subject of independent interest, and we provide necessary density conditions through the classical approach of Landau, that in this context measures the local dimension of the space of sections of the line bundle. We obtain a complete geometric characterization of sampling and interpolation arrays in the case of compact manifolds of dimension one, and we prove that there are no arrays of both sampling and interpolation in the more general setting of semipositive line bundles. [ABSTRACT FROM AUTHOR]

Given a measurable set A Rn of positive measure, it is not difficult to show that |A + A| = |2A| if and only if A is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If .|A + A| - |2A|/=|A| is small, is A close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between A and its convex hull in terms of .|A + A| - |2A|/=|A|. [ABSTRACT FROM AUTHOR]

We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other hand, this last property provides a good control of the periodic approximations of a cocycle, allowing us to show that domination can be characterized, in the presence of a gap in the Lyapunov spectrum, by additional regularity of the dependence of sums of Lyapunov exponents. [ABSTRACT FROM AUTHOR]