@article{10045, abstract = {Given a fixed finite metric space (V,μ), the {\em minimum 0-extension problem}, denoted as 0-Ext[μ], is equivalent to the following optimization problem: minimize function of the form minx∈Vn∑ifi(xi)+∑ijcijμ(xi,xj) where cij,cvi are given nonnegative costs and fi:V→R are functions given by fi(xi)=∑v∈Vcviμ(xi,v). The computational complexity of 0-Ext[μ] has been recently established by Karzanov and by Hirai: if metric μ is {\em orientable modular} then 0-Ext[μ] can be solved in polynomial time, otherwise 0-Ext[μ] is NP-hard. To prove the tractability part, Hirai developed a theory of discrete convex functions on orientable modular graphs generalizing several known classes of functions in discrete convex analysis, such as L♮-convex functions. We consider a more general version of the problem in which unary functions fi(xi) can additionally have terms of the form cuv;iμ(xi,{u,v}) for {u,v}∈F, where set F⊆(V2) is fixed. We extend the complexity classification above by providing an explicit condition on (μ,F) for the problem to be tractable. In order to prove the tractability part, we generalize Hirai's theory and define a larger class of discrete convex functions. It covers, in particular, another well-known class of functions, namely submodular functions on an integer lattice. Finally, we improve the complexity of Hirai's algorithm for solving 0-Ext on orientable modular graphs. }, author = {Dvorak, Martin and Kolmogorov, Vladimir}, issn = {1436-4646}, journal = {Mathematical Programming}, keywords = {minimum 0-extension problem, metric labeling problem, discrete metric spaces, metric extensions, computational complexity, valued constraint satisfaction problems, discrete convex analysis, L-convex functions}, pages = {279--322}, publisher = {Springer Nature}, title = {{Generalized minimum 0-extension problem and discrete convexity}}, doi = {10.1007/s10107-024-02064-5}, volume = {209}, year = {2025}, } @phdthesis{18979, abstract = {Topological Data Analysis (TDA) is a discipline utilizing the mathematical field of topology to study data, most prominently collections of point sets. This thesis summarizes three projects related to computations in TDA. The first one establishes a variant of TDA for chromatic point sets, where each point is given a color. For example, we are given positions of cells within a tumor microenvironment, and color the cancerous cells red, and the immune cells blue. The aim is then to give a quantitative description of how the two or more sets of points spatially interact. Building on image, kernel and cokernel variants of persistent homology, we suggest six-packs of persistent diagrams as such a descriptor. We describe a construction of a chromatic alpha complex, which enables efficient computation of several variants of the six-packs. We give topological descriptions of natural subcomplexes of the chromatic alpha complex, and show that the radii of the simplices form a discrete Morse function. Finally, we provide an implementation of the presented chromatic TDA pipeline. The second part aims to translate a powerful tool of sheaf theory to elementary terms using labeled matrices. The goal is to enable their use in computational settings. We show that derived categories of sheaves over finite posets have, up to isomorphism, unique objects---minimal injective resolutions---and give a concrete algorithm to compute them. We further describe simple algorithms to compute derived pushforwards and pullbacks for monotonic maps, and their proper variants for inclusions, and demonstrate their tractability by providing an implementation. Finally, we suggest a discrete definition of microsupport and show desirable properties inspired by discrete Morse theory. In the last part, we present a collection of observations about collapses. We give a characterization of collapsibility in terms of unitriangular submatrices of the boundary matrix, a cotree-tree decomposition, and the optimal solution to a variant of the Procrustes problem. We establish relation between dual collapses and relative Morse theory and pose several open questions. Finally, focusing on complexes embedded in the three-dimensional Euclidean space, we describe a relation between the collapsibility and the triviality of a polygonal knot.}, author = {Draganov, Ondrej}, issn = {2663-337X}, keywords = {topological data analysis, chromatic point set, alpha complex, persistent homology, six pack, sheaf, microlocal discrete Morse, injective resolution, collapse, knot, discrete Morse theory}, pages = {140}, publisher = {Institute of Science and Technology Austria}, title = {{Structures and computations in topological data analysis}}, doi = {10.15479/at:ista:18979}, year = {2025}, } @phdthesis{18667, abstract = {Many chemical and physical properties of materials are determined by the material’s shape, for example the size of its pores and the width of its tunnels. This makes materials science a prime application area for geometrical and topological methods. Nevertheless many methods in topological data analysis have not been satisfyingly extended to the needs of materials science. This thesis provides new methods and new mathematical theorems targeted at those specific needs by answering four different research questions. While the motivation for each of the research questions arises from materials science, the methods are versatile and can be applied in different areas as well. The first research question is concerned with image data, for example a three-dimensional computed tomography (CT) scan of a material, like sand or stone. There are two commonly used topologies for digital images and depending on the application either of them might be required. However, software for computing the topological data analysis method persistence homology, usually supports only one of the two topologies. We answer the question how to compute persistent homology of an image with respect to one of the two topologies using software that is intended for the other topology. The second research question is concerned with image data as well, and asks how much of the topological information of an image is lost when the resolution is coarsened. As computer tomography scanners are more expensive the higher the resolution, it is an important question in materials science to know which resolution is enough to get satisfying persistent homology. We give theoretical bounds on the information loss based on different geometrical properties of the object to be scanned. In addition, we conduct experiments on sand and stone CT image data. The third research question is motivated by comparing crystalline materials efficiently. As the atoms within a crystal repeat periodically, crystalline materials are either modeled by unmanageable infinite periodic point sets, or by one of their fundamental domains, which is unstable under perturbation. Therefore a fingerprint of crystalline materials is needed, with appropriate properties such that comparing the crystals can be eased by comparing the fingerprints instead. We define the density fingerprint and prove the necessary properties. The fourth research question is motivated by studying the hole-structure or connectedness, i.e. persistent homology or merge trees, of crystalline materials. A common way to deal with periodicity is to take a fundamental domain and identify opposite boundaries to form a torus. However, computing persistent homology or merge trees on that torus loses some of the information materials scientists are interested in and is additionally not stable under certain noise. We therefore decorate the merge tree stemming from the torus with additional information describing the density and growth rate of the periodic copies of a component within a growing spherical window. We prove all desired properties, like stability and efficient computability.}, author = {Heiss, Teresa}, isbn = {978-3-99078-052-7}, issn = {2663-337X}, keywords = {persistent homology, topological data analysis, periodic, crystalline materials, images, fingerprint}, pages = {111}, publisher = {Institute of Science and Technology Austria}, title = {{New methods for applying topological data analysis to materials science}}, doi = {10.15479/at:ista:18667}, year = {2024}, } @article{14499, abstract = {An n-vertex graph is called C-Ramsey if it has no clique or independent set of size Clog2n (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of ‘random-like’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables. The proof proceeds via an ‘additive structure’ dichotomy on the degree sequence and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics and low-rank approximation. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes.}, author = {Kwan, Matthew Alan and Sah, Ashwin and Sauermann, Lisa and Sawhney, Mehtaab}, issn = {2050-5086}, journal = {Forum of Mathematics, Pi}, keywords = {Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics, Statistics and Probability, Algebra and Number Theory, Analysis}, publisher = {Cambridge University Press}, title = {{Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture}}, doi = {10.1017/fmp.2023.17}, volume = {11}, year = {2023}, } @article{14772, abstract = {Many coupled evolution equations can be described via 2×2-block operator matrices of the form A=[ A B C D ] in a product space X=X1×X2 with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator A can be seen as a relatively bounded perturbation of its diagonal part with D(A)=D(A)×D(D) though with possibly large relative bound. For such operators the properties of sectoriality, R-sectoriality and the boundedness of the H∞-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time dependent parabolic problem associated with A can be analyzed in maximal Lpt -regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.}, author = {Agresti, Antonio and Hussein, Amru}, issn = {0022-1236}, journal = {Journal of Functional Analysis}, keywords = {Analysis}, number = {11}, publisher = {Elsevier}, title = {{Maximal Lp-regularity and H∞-calculus for block operator matrices and applications}}, doi = {10.1016/j.jfa.2023.110146}, volume = {285}, year = {2023}, } @article{11556, abstract = {We revisit two basic Direct Simulation Monte Carlo Methods to model aggregation kinetics and extend them for aggregation processes with collisional fragmentation (shattering). We test the performance and accuracy of the extended methods and compare their performance with efficient deterministic finite-difference method applied to the same model. We validate the stochastic methods on the test problems and apply them to verify the existence of oscillating regimes in the aggregation-fragmentation kinetics recently detected in deterministic simulations. We confirm the emergence of steady oscillations of densities in such systems and prove the stability of the oscillations with respect to fluctuations and noise.}, author = {Kalinov, Aleksei and Osinskiy, A.I. and Matveev, S.A. and Otieno, W. and Brilliantov, N.V.}, issn = {0021-9991}, journal = {Journal of Computational Physics}, keywords = {Computer Science Applications, Physics and Astronomy (miscellaneous), Applied Mathematics, Computational Mathematics, Modeling and Simulation, Numerical Analysis}, publisher = {Elsevier}, title = {{Direct simulation Monte Carlo for new regimes in aggregation-fragmentation kinetics}}, doi = {10.1016/j.jcp.2022.111439}, volume = {467}, year = {2022}, } @article{11916, abstract = {A domain is called Kac regular for a quadratic form on L2 if every functions vanishing almost everywhere outside the domain can be approximated in form norm by functions with compact support in the domain. It is shown that this notion is stable under domination of quadratic forms. As applications measure perturbations of quasi-regular Dirichlet forms, Cheeger energies on metric measure spaces and Schrödinger operators on manifolds are studied. Along the way a characterization of the Sobolev space with Dirichlet boundary conditions on domains in infinitesimally Riemannian metric measure spaces is obtained.}, author = {Wirth, Melchior}, issn = {2538-225X}, journal = {Advances in Operator Theory}, keywords = {Algebra and Number Theory, Analysis}, number = {3}, publisher = {Springer Nature}, title = {{Kac regularity and domination of quadratic forms}}, doi = {10.1007/s43036-022-00199-w}, volume = {7}, year = {2022}, } @article{12148, abstract = {We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables.}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, keywords = {Computational Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics, Statistics and Probability, Algebra and Number Theory, Theoretical Computer Science, Analysis}, publisher = {Cambridge University Press}, title = {{Rank-uniform local law for Wigner matrices}}, doi = {10.1017/fms.2022.86}, volume = {10}, year = {2022}, } @article{12179, abstract = {We derive an accurate lower tail estimate on the lowest singular value σ1(X−z) of a real Gaussian (Ginibre) random matrix X shifted by a complex parameter z. Such shift effectively changes the upper tail behavior of the condition number κ(X−z) from the slower (κ(X−z)≥t)≲1/t decay typical for real Ginibre matrices to the faster 1/t2 decay seen for complex Ginibre matrices as long as z is away from the real axis. This sharpens and resolves a recent conjecture in [J. Banks et al., https://arxiv.org/abs/2005.08930, 2020] on the regularizing effect of the real Ginibre ensemble with a genuinely complex shift. As a consequence we obtain an improved upper bound on the eigenvalue condition numbers (known also as the eigenvector overlaps) for real Ginibre matrices. The main technical tool is a rigorous supersymmetric analysis from our earlier work [Probab. Math. Phys., 1 (2020), pp. 101--146].}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {1095-7162}, journal = {SIAM Journal on Matrix Analysis and Applications}, keywords = {Analysis}, number = {3}, pages = {1469--1487}, publisher = {Society for Industrial and Applied Mathematics}, title = {{On the condition number of the shifted real Ginibre ensemble}}, doi = {10.1137/21m1424408}, volume = {43}, year = {2022}, } @article{12216, abstract = {Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.}, author = {Carlen, Eric A. and Zhang, Haonan}, issn = {0024-3795}, journal = {Linear Algebra and its Applications}, keywords = {Discrete Mathematics and Combinatorics, Geometry and Topology, Numerical Analysis, Algebra and Number Theory}, pages = {289--310}, publisher = {Elsevier}, title = {{Monotonicity versions of Epstein's concavity theorem and related inequalities}}, doi = {10.1016/j.laa.2022.09.001}, volume = {654}, year = {2022}, } @article{12304, abstract = {We establish sharp criteria for the instantaneous propagation of free boundaries in solutions to the thin-film equation. The criteria are formulated in terms of the initial distribution of mass (as opposed to previous almost-optimal results), reflecting the fact that mass is a locally conserved quantity for the thin-film equation. In the regime of weak slippage, our criteria are at the same time necessary and sufficient. The proof of our upper bounds on free boundary propagation is based on a strategy of “propagation of degeneracy” down to arbitrarily small spatial scales: We combine estimates on the local mass and estimates on energies to show that “degeneracy” on a certain space-time cylinder entails “degeneracy” on a spatially smaller space-time cylinder with the same time horizon. The derivation of our lower bounds on free boundary propagation is based on a combination of a monotone quantity and almost optimal estimates established previously by the second author with a new estimate connecting motion of mass to entropy production.}, author = {De Nitti, Nicola and Fischer, Julian L}, issn = {1532-4133}, journal = {Communications in Partial Differential Equations}, keywords = {Applied Mathematics, Analysis}, number = {7}, pages = {1394--1434}, publisher = {Taylor & Francis}, title = {{Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation}}, doi = {10.1080/03605302.2022.2056702}, volume = {47}, year = {2022}, } @article{12305, abstract = {This paper is concerned with the sharp interface limit for the Allen--Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain Ω⊂\R2. We assume that a diffuse interface already has developed and that it is in contact with the boundary ∂Ω. The boundary condition is designed in such a way that the limit problem is given by the mean curvature flow with constant α-contact angle. For α close to 90° we prove a local in time convergence result for well-prepared initial data for times when a smooth solution to the limit problem exists. Based on the latter we construct a suitable curvilinear coordinate system and carry out a rigorous asymptotic expansion for the Allen--Cahn equation with the nonlinear Robin boundary condition. Moreover, we show a spectral estimate for the corresponding linearized Allen--Cahn operator and with its aid we derive strong norm estimates for the difference of the exact and approximate solutions using a Gronwall-type argument.}, author = {Abels, Helmut and Moser, Maximilian}, issn = {1095-7154}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {Applied Mathematics, Computational Mathematics, Analysis}, number = {1}, pages = {114--172}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°}}, doi = {10.1137/21m1424925}, volume = {54}, year = {2022}, } @article{10643, abstract = {We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding Gelfand–Naimark–Segal Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite-volume Hamiltonians need not have a spectral gap. }, author = {Henheik, Sven Joscha and Teufel, Stefan}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, keywords = {computational mathematics, discrete mathematics and combinatorics, geometry and topology, mathematical physics, statistics and probability, algebra and number theory, theoretical computer science, analysis}, publisher = {Cambridge University Press}, title = {{Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk}}, doi = {10.1017/fms.2021.80}, volume = {10}, year = {2022}, } @article{10211, abstract = {We study the problem of recovering an unknown signal 𝑥𝑥 given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator 𝑥𝑥^L and a spectral estimator 𝑥𝑥^s. The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine 𝑥𝑥^L and 𝑥𝑥^s. At the heart of our analysis is the exact characterization of the empirical joint distribution of (𝑥𝑥,𝑥𝑥^L,𝑥𝑥^s) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of 𝑥𝑥^L and 𝑥𝑥^s, given the limiting distribution of the signal 𝑥𝑥. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form 𝜃𝑥𝑥^L+𝑥𝑥^s and we derive the optimal combination coefficient. In order to establish the limiting distribution of (𝑥𝑥,𝑥𝑥^L,𝑥𝑥^s), we design and analyze an approximate message passing algorithm whose iterates give 𝑥𝑥^L and approach 𝑥𝑥^s. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.}, author = {Mondelli, Marco and Thrampoulidis, Christos and Venkataramanan, Ramji}, issn = {1615-3383}, journal = {Foundations of Computational Mathematics}, keywords = {Applied Mathematics, Computational Theory and Mathematics, Computational Mathematics, Analysis}, number = {5}, pages = {1513--1566}, publisher = {Springer}, title = {{Optimal combination of linear and spectral estimators for generalized linear models}}, doi = {10.1007/s10208-021-09531-x}, volume = {22}, year = {2022}, } @article{10850, abstract = {We study two interacting quantum particles forming a bound state in d-dimensional free space, and constrain the particles in k directions to (0, ∞)k ×Rd−k, with Neumann boundary conditions. First, we prove that the ground state energy strictly decreases upon going from k to k+1. This shows that the particles stick to the corner where all boundary planes intersect. Second, we show that for all k the resulting Hamiltonian, after removing the free part of the kinetic energy, has only finitely many eigenvalues below the essential spectrum. This paper generalizes the work of Egger, Kerner and Pankrashkin (J. Spectr. Theory 10(4):1413–1444, 2020) to dimensions d > 1.}, author = {Roos, Barbara and Seiringer, Robert}, issn = {0022-1236}, journal = {Journal of Functional Analysis}, keywords = {Analysis}, number = {12}, publisher = {Elsevier}, title = {{Two-particle bound states at interfaces and corners}}, doi = {10.1016/j.jfa.2022.109455}, volume = {282}, year = {2022}, } @article{11608, abstract = {In order to understand stellar evolution, it is crucial to efficiently determine stellar surface rotation periods. Indeed, while they are of great importance in stellar models, angular momentum transport processes inside stars are still poorly understood today. Surface rotation, which is linked to the age of the star, is one of the constraints needed to improve the way those processes are modelled. Statistics of the surface rotation periods for a large sample of stars of different spectral types are thus necessary. An efficient tool to automatically determine reliable rotation periods is needed when dealing with large samples of stellar photometric datasets. The objective of this work is to develop such a tool. For this purpose, machine learning classifiers constitute relevant bases to build our new methodology. Random forest learning abilities are exploited to automate the extraction of rotation periods in Kepler light curves. Rotation periods and complementary parameters are obtained via three different methods: a wavelet analysis, the autocorrelation function of the light curve, and the composite spectrum. We trained three different classifiers: one to detect if rotational modulations are present in the light curve, one to flag close binary or classical pulsators candidates that can bias our rotation period determination, and finally one classifier to provide the final rotation period. We tested our machine learning pipeline on 23 431 stars of the Kepler K and M dwarf reference rotation catalogue for which 60% of the stars have been visually inspected. For the sample of 21 707 stars where all the input parameters are provided to the algorithm, 94.2% of them are correctly classified (as rotating or not). Among the stars that have a rotation period in the reference catalogue, the machine learning provides a period that agrees within 10% of the reference value for 95.3% of the stars. Moreover, the yield of correct rotation periods is raised to 99.5% after visually inspecting 25.2% of the stars. Over the two main analysis steps, rotation classification and period selection, the pipeline yields a global agreement with the reference values of 92.1% and 96.9% before and after visual inspection. Random forest classifiers are efficient tools to determine reliable rotation periods in large samples of stars. The methodology presented here could be easily adapted to extract surface rotation periods for stars with different spectral types or observed by other instruments such as K2, TESS or by PLATO in the near future.}, author = {Breton, S. N. and Santos, A. R. G. and Bugnet, Lisa Annabelle and Mathur, S. and García, R. A. and Pallé, P. L.}, issn = {1432-0746}, journal = {Astronomy & Astrophysics}, keywords = {Space and Planetary Science, Astronomy and Astrophysics, methods: data analysis / stars: solar-type / stars: activity / stars: rotation / starspots}, publisher = {EDP Sciences}, title = {{ROOSTER: A machine-learning analysis tool for Kepler stellar rotation periods}}, doi = {10.1051/0004-6361/202039947}, volume = {647}, year = {2021}, } @article{15261, abstract = {In this article, we study uniqueness of form extensions in a rather general setting. The method is based on the theory of ordered Hilbert spaces and the concept of domination of semigroups. Our main abstract result transfers uniqueness of form extension of a dominating form to that of a dominated form. This result can be applied to a multitude of examples including various magnetic Schrödinger forms on graphs and on manifolds.}, author = {Lenz, Daniel and Schmidt, Marcel and Wirth, Melchior}, issn = {1096-0783}, journal = {Journal of Functional Analysis}, keywords = {Analysis}, number = {6}, publisher = {Elsevier}, title = {{Uniqueness of form extensions and domination of semigroups}}, doi = {10.1016/j.jfa.2020.108848}, volume = {280}, year = {2021}, } @article{10856, abstract = {We study the properties of the maximal volume k-dimensional sections of the n-dimensional cube [−1, 1]n. We obtain a first order necessary condition for a k-dimensional subspace to be a local maximizer of the volume of such sections, which we formulate in a geometric way. We estimate the length of the projection of a vector of the standard basis of Rn onto a k-dimensional subspace that maximizes the volume of the intersection. We nd the optimal upper bound on the volume of a planar section of the cube [−1, 1]n , n ≥ 2.}, author = {Ivanov, Grigory and Tsiutsiurupa, Igor}, issn = {2299-3274}, journal = {Analysis and Geometry in Metric Spaces}, keywords = {Applied Mathematics, Geometry and Topology, Analysis}, number = {1}, pages = {1--18}, publisher = {De Gruyter}, title = {{On the volume of sections of the cube}}, doi = {10.1515/agms-2020-0103}, volume = {9}, year = {2021}, } @article{10549, abstract = {We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on \mathbb {R}^d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale \varepsilon >0, we establish homogenization error estimates of the order \varepsilon in case d\geqq 3, and of the order \varepsilon |\log \varepsilon |^{1/2} in case d=2. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence \varepsilon ^\delta . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order (L/\varepsilon )^{-d/2} for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) C^{1,\alpha } regularity theory is available.}, author = {Fischer, Julian L and Neukamm, Stefan}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, keywords = {Mechanical Engineering, Mathematics (miscellaneous), Analysis}, number = {1}, pages = {343--452}, publisher = {Springer Nature}, title = {{Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems}}, doi = {10.1007/s00205-021-01686-9}, volume = {242}, year = {2021}, } @article{10862, abstract = {We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4], [5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.}, author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin}, issn = {0022-1236}, journal = {Journal of Functional Analysis}, keywords = {Analysis}, number = {7}, publisher = {Elsevier}, title = {{Spectral rigidity for addition of random matrices at the regular edge}}, doi = {10.1016/j.jfa.2020.108639}, volume = {279}, year = {2020}, }