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This note provides a new perspective on Polchinski’s exact renormalization group, by explaining how it gives rise, via the multiscale Bakry-Émery criterion, to Lipschitz transport maps between Gaussian free fields and interacting quantum and statistical field theories. Consequently, many functional inequalities can be verified for the latter field theories, going beyond the current known results.

In this expository note, we prove a localized bubbling result for solutions of the energy critical nonlinear heat equation with bounded $$\dot{H} ^1$$ H ˙ 1 norm. The proof uses a combination of Gérard’s profile decomposition (ESAIM Control Optim. Calc. Var. 3: 213–233, 1998), concentration compactness techniques in the spirit of Duyckaerts, Kenig, and Merle’s seminal work (Geom. Funct. Anal. 22: 639–698, 2012), and a virial argument in the spirit of Jia and Kenig’s work (Amer. J. Math. 139: 1521–1603, 2017) to deduce the vanishing of the error in the neck regions between the bubbles. The argument is based closely on an analogous lemma proved in the author’s recent work with Jendrej (arXiv:2210.14963, 2022) on the equivariant harmonic map heat flow in dimension two.

A finite group G is called C-quasirandom (by Gowers) if all non-trivial irreducible complex representations of G have dimension at least C. For any unit $$\ell ^{2}$$ ℓ 2 function on a finite group we associate the quantum probability measure on the group given by the absolute value squared of the function. We show that if a group is highly quasirandom, in the above sense, then any Cayley graph of this group has an orthonormal eigenbasis of the adjacency operator such that the quantum probability measures of the eigenfunctions put close to the correct proportion of their mass on suitably selected subsets of the group that are not too small.

Traditional optical elements and conventional metasurfaces obey shift-invariance in the paraxial regime. For imaging systems obeying paraxial shift-invariance, a small shift in input angle causes a corresponding shift in the sensor image. Shift-invariance has deep implications for the design and functionality of optical devices, such as the necessity of free space between components (as in compound objectives made of several curved surfaces). We present a method for nanophotonic inverse design of compact imaging systems whose resolution is not constrained by paraxial shift-invariance. Our method is end-to-end, in that it integrates density-based full-Maxwell topology optimization with a fully iterative elastic-net reconstruction algorithm. By the design of nanophotonic structures that scatter light in a non-shift-invariant manner, our optimized nanophotonic imaging system overcomes the limitations of paraxial shift-invariance, achieving accurate, noise-robust image reconstruction beyond shift-invariant resolution., Department of Defense (DoD)

We discuss the regularization of certain hypergeometric integrals appearing in 2D CFT, a step needed in the construction of the BPZ minimal models via the Coulomb gas formalism. The method is a generalization of Pochhammer’s regularization of the Euler Beta-function. The constructions of the relevant homology classes are inspired by a recent singular-geometric analysis of the Dotsenko–Fateev integrand.

Let M be a cancellative and commutative (additive) monoid. The monoid M is atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also, M satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. In the first part of this paper, we characterize torsion-free monoids that satisfy the ACCP as those torsion-free monoids whose submonoids are all atomic. A submonoid of the nonnegative cone of a totally ordered abelian group is often called a positive monoid. Every positive monoid is clearly torsion-free. In the second part of this paper, we study the atomic structure of certain classes of positive monoids.

Given a three-dimensional Riemannian manifold with boundary and a finite group of orientation-preserving isometries of this manifold, we prove that the equivariant index of a free boundary minimal surface obtained via an equivariant min–max procedure á la Simon–Smith with n-parameters is bounded above by n.

Consider a compact manifold N (with or without boundary) of dimension n. Positive m-intermediate curvature interpolates between positive Ricci curvature ( $$m = 1$$ m = 1 ) and positive scalar curvature ( $$m = n-1$$ m = n - 1 ), and it is obstructed on partial tori $$N^n = M^{n-m} \times \mathbb {T}^m$$ N n = M n - m × T m . Given Riemannian metrics $$g, {\bar{g}}$$ g , g ¯ on $$(N, \partial N)$$ ( N , ∂ N ) with positive m-intermediate curvature and m-positive difference $$h_g - h_{{\bar{g}}}$$ h g - h g ¯ of second fundamental forms we show that there exists a smooth family of Riemannian metrics with positive m-intermediate curvature interpolating between g and $${\bar{g}}$$ g ¯ . Moreover, we apply this result to prove a non-existence result for partial torical bands with positive m-intermediate curvature and strictly m-convex boundaries.

A non-square-tiled Veech surface has finitely many periodic points, i.e. points with finite orbit under the affine automorphism group. We present an algorithm that inputs a non-square-tiled Veech surface and outputs its set of periodic points. Our algorithm serves as a new proof of the finiteness of periodic points for non-square-tiled Veech surfaces. We apply our algorithm to Prym eigenforms in the minimal stratum in genus 3, proving that in low discriminant these surfaces do not have periodic points, except for the fixed points of the Prym involution.

We study invariant theory of the general linear supergroup in positive characteristic. In particular, we determine when the symmetric group algebra acts faithfully on tensor superspace and demonstrate that the symmetric group does not always generate all invariants.

We construct special Lagrangian 3-spheres in non-Kähler compact threefolds equipped with the Fu–Li–Yau geometry. These non-Kähler geometries emerge from topological transitions of compact Calabi-Yau threefolds. From this point of view, a conifold transition exchanges holomorphic 2-cycles for special Lagrangian 3-cycles.

We consider the harmonic map heat flow for maps $$\mathbb {R}^{2} \rightarrow \mathbb {S}^2$$ R 2 → S 2 , under equivariant symmetry. It is known that solutions to the initial value problem can exhibit bubbling along a sequence of times—the solution decouples into a superposition of harmonic maps concentrating at different scales and a body map that accounts for the rest of the energy. We prove that this bubble decomposition is unique and occurs continuously in time. The main new ingredient in the proof is the notion of a collision interval from Jendrej and Lawrie (J Amer Math Soc).

We establish probabilistic small data global well-posedness of the energy-critical Maxwell–Klein–Gordon equation relative to the Coulomb gauge for scaling super-critical random initial data. The proof relies on an induction on frequency procedure and a modified linear-nonlinear decomposition furnished by a delicate “probabilistic” parametrix construction. This is the first global existence result for a geometric wave equation for random initial data at scaling super-critical regularity.

We give algebraic and geometric perspectives on our prior results toward the Putman–Wieland conjecture. This leads to interesting new constructions of families of “origami” curves whose Jacobians have high-dimensional isotrivial isogeny factors. We also explain how a hyperelliptic analogue of the Putman–Wieland conjecture fails, following work of Marković.

We prove that with high probability $$\mathbb {G}^{(3)}(n,n^{-1+o(1)})$$ G ( 3 ) ( n , n - 1 + o ( 1 ) ) contains a spanning Steiner triple system for $$n\equiv 1,3\pmod {6}$$ n ≡ 1 , 3 ( mod 6 ) , establishing the exponent for the threshold probability for existence of a Steiner triple system. We also prove the analogous theorem for Latin squares. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as the connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.

We consider the focusing energy-critical nonlinear wave equation for radially symmetric initial data in space dimensions $$D \ge 4$$ D ≥ 4 . This equation has a unique (up to sign and scale) nontrivial, finite energy stationary solution W, called the ground state. We prove that every finite energy solution with bounded energy norm resolves, continuously in time, into a finite superposition of asymptotically decoupled copies of the ground state and free radiation.

Dimensionality reduction summarizes the complex transcriptomic landscape of single-cell datasets for downstream analyses. Current approaches favor large cellular populations defined by many genes, at the expense of smaller and more subtly defined populations. Here, we present surprisal component analysis (SCA), a technique that newly leverages the information-theoretic notion of surprisal for dimensionality reduction to promote more meaningful signal extraction. For example, SCA uncovers clinically important cytotoxic T-cell subpopulations that are indistinguishable using existing pipelines. We also demonstrate that SCA substantially improves downstream imputation. SCA’s efficient information-theoretic paradigm has broad applications to the study of complex biological tissues in health and disease.

According to a conjecture of Lusztig, the asymptotic affine Hecke algebra should admit a description in terms of the Grothedieck group of sheaves on the square of a finite set equivariant under the action of the centralizer of a nilpotent element in the reductive group. A weaker form of this statement, allowing for possible central extensions of stabilizers of that action, has been proved by the first named author with Ostrik. In the present paper, we describe an example showing that nontrivial central extensions do arise, thus the above weaker statement is optimal. We also show that Lusztig's homomorphism from the affine Hecke algebra to the asymptotic affine Hecke algebra induces an isomorphism on cocenters and discuss the relation of the above central extensions to the structure of the cocenter.

Consider a quantum cat map M associated with a matrix $$A\in {{\,\textrm{Sp}\,}}(2n,{\mathbb {Z}})$$ A ∈ Sp ( 2 n , Z ) , which is a common toy model in quantum chaos. We show that the mass of eigenfunctions of M on any nonempty open set in the position–frequency space satisfies a lower bound which is uniform in the semiclassical limit, under two assumptions: (1) there is a unique simple eigenvalue of A of largest absolute value and (2) the characteristic polynomial of A is irreducible over the rationals. This is similar to previous work (Dyatlov and Jin in Acta Math 220(2):297–339, 2018; Dyatlov et al. in J Am Math Soc 35(2):361–465, 2022) on negatively curved surfaces and (Schwartz in The full delocalization of eigenstates for the quantized cat map, 2021) on quantum cat maps with $$n=1$$ n = 1 , but this paper gives the first results of this type which apply in any dimension. When condition (2) fails we provide a weaker version of the result and discuss relations to existing counterexamples. We also obtain corresponding statements regarding semiclassical measures and damped quantum cat maps.

We obtain a sharp bilinear restriction estimate for the paraboloid in R3 for q>13/4.

In this paper, we prove homological stability results about orthogonal groups over finite commutative rings where 2 is a unit. Inspired by Putman and Sam (2017), we construct a category OrI(R) and prove a local Noetherianity theorem for the category of OrI(R)-modules. This implies an asymptotic structure theorem for orthogonal groups. In addition, we show general homological stability theorems for orthogonal groups, with both untwisted and twisted coefficients.

We study random domino tilings of the Aztec diamond with a biased $$2 \times 2$$ 2 × 2 periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight.

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $${\mathfrak {g}}$$ g , whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $$Y({\mathfrak {g}})$$ Y ( g ) . These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter Q-operators, arising from our previous study Frassek et al. (Adv. Math. 401:108283, 2022), Frassek and Tsymbaliuk (Commun. Math. Phys. 392:545–619, 2022) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $${\mathfrak {g}}$$ g -modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety G/P stratified by $$B_{-}$$ B - -orbits.

National Natural Science Foundation of China, Research Grants Council, University Grants Committee (City Univ. Hong Kong), Croucher Foundation, Simons Foundation

In this paper, we show that if a solution to the Muskat problem in the case of different densities and the same viscosity is sufficiently smooth, then it must be analytic except at the points where a turnover of the fluids happens.

We show that if an exact special Lagrangian $$N\subset {\mathbb {C}}^n$$ N ⊂ C n has a multiplicity one, cylindrical tangent cone of the form $${\mathbb {R}}^{k}\times {\textbf{C}}$$ R k × C where $${\textbf{C}}$$ C is a special Lagrangian cone with smooth, connected link, then this tangent cone is unique provided $${\textbf{C}}$$ C satisfies an integrability condition. This applies, for example, when $${\textbf{C}}= {\textbf{C}}_{HL}^{m}$$ C = C HL m is the Harvey-Lawson $$T^{m-1}$$ T m - 1 cone for $$m\ne 8,9$$ m ≠ 8 , 9 .

Secure multiparty computation (MPC) is a cryptographic tool that allows computation on top of sensitive biomedical data without revealing private information to the involved entities. Here, we introduce Sequre, an easy-to-use, high-performance framework for developing performant MPC applications. Sequre offers a set of automatic compile-time optimizations that significantly improve the performance of MPC applications and incorporates the syntax of Python programming language to facilitate rapid application development. We demonstrate its usability and performance on various bioinformatics tasks showing up to 3–4 times increased speed over the existing pipelines with 7-fold reductions in codebase sizes.

We obtain an analog of Weyl’s law for the Kohn Laplacian on lens spaces. We also show that two 3-dimensional lens spaces with fundamental groups of equal prime order are isospectral with respect to the Kohn Laplacian if and only if they are CR isometric.

Large-scale electrification is vital to addressing the climate crisis, but several scientific and technological challenges remain to fully electrify both the chemical industry and transportation. In both of these areas, new electrochemical materials will be critical, but their development currently relies heavily on human-time-intensive experimental trial and error and computationally expensive first-principles, mesoscale, and continuum simulations. We present an automated workflow, AutoMat, which accelerates these computational steps by introducing both automated input generation and management of simulations across scales from first principles to continuum device modeling. Furthermore, we show how to seamlessly integrate multi-fidelity predictions, such as machine learning surrogates or automated robotic experiments “in-the-loop.” The automated framework is implemented with design space search techniques to dramatically accelerate the overall materials discovery pipeline by implicitly learning design features that optimize device performance across several metrics. We discuss the benefits of AutoMat using examples in electrocatalysis and energy storage and highlight lessons learned. Graphical abstract

The Delsarte linear program is used to bound the size of codes given their block length n and minimal distance d by taking a linear relaxation from codes to quasicodes. We study for which values of (n, d) this linear program has a unique optimum: while we show that it does not always have a unique optimum, we prove that it does if $$d>n/2$$ d > n / 2 or if $$d \le 2$$ d ≤ 2 . Introducing the Krawtchouk decomposition of a quasicode, we prove there exist optima to the (n, 2e) and $$(n-1,2e-1)$$ ( n - 1 , 2 e - 1 ) linear programs that have essentially identical Krawtchouk decompositions, revealing a parity phenomenon among the Delsarte linear programs. We generalize the notion of extending and puncturing codes to quasicodes, from which we see that this parity relationship is given by extending/puncturing. We further characterize these pairs of optima, in particular demonstrating that they exhibit a symmetry property, effectively halving the number of decision variables.

Background Quality control (QC) of cells, a critical first step in single-cell RNA sequencing data analysis, has largely relied on arbitrarily fixed data-agnostic thresholds applied to QC metrics such as gene complexity and fraction of reads mapping to mitochondrial genes. The few existing data-driven approaches perform QC at the level of samples or studies without accounting for biological variation. Results We first demonstrate that QC metrics vary with both tissue and cell types across technologies, study conditions, and species. We then propose data-driven QC (ddqc), an unsupervised adaptive QC framework to perform flexible and data-driven QC at the level of cell types while retaining critical biological insights and improved power for downstream analysis. ddqc applies an adaptive threshold based on the median absolute deviation on four QC metrics (gene and UMI complexity, fraction of reads mapping to mitochondrial and ribosomal genes). ddqc retains over a third more cells when compared to conventional data-agnostic QC filters. Finally, we show that ddqc recovers biologically meaningful trends in gradation of gene complexity among cell types that can help answer questions of biological interest such as which cell types express the least and most number of transcripts overall, and ribosomal transcripts specifically. Conclusions ddqc retains cell types such as metabolically active parenchymal cells and specialized cells such as neutrophils which are often lost by conventional QC. Taken together, our work proposes a revised paradigm to quality filtering best practices—iterative QC, providing a data-driven QC framework compatible with observed biological diversity.

We begin a systematic study of unitary representations of minimal W-algebras. In particular, we classify unitary minimal W-algebras and make substantial progress in classification of their unitary irreducible highest weight modules. We also compute the characters of these modules.

Recently, Alon, Cambie, and Kang introduced asymmetric list coloring of bipartite graphs, where the size of each vertex’s list depends on its part. For complete bipartite graphs, we fix the list sizes of one part and consider the resulting asymptotics, revealing an invariant quantity instrumental in determining choosability across most of the parameter space. By connecting this quantity to a simple question on independent sets of hypergraphs, we strengthen bounds when a part has list size 2. Finally, we state via our framework a conjecture on general bipartite graphs, unifying three conjectures of Alon–Cambie–Kang.

We study how a bound notebook page can be folded once so that the page is visible when the notebook is closed and becomes a bookmark. An additional practical requirement is that the folded page stays within the same side of the bind, so that it does not get creased further when one closes the notebook. This simple problem displays a remarkably rich bifurcation behavior, which can be analyzed using undergraduate calculus. The main finding is that the optimal way of folding the page depends on the page's aspect ratio and changes abruptly when the latter exceeds the critical value of 1.207 11. This behavior also resembles a first-order phase transition in thermodynamics despite its geometric origin.

We examine Atiyah’s Hermitian axiom for two-dimensional complex topological quantum field theories. Building on the correspondence between 2D topological quantum field theories (TQFTs) and Frobenius algebras, we find the algebraic objects corresponding to Hermitian and unitary TQFTs, respectively, and prove structure theorems about them. We then clarify a few older results on unitary TQFTs using our structure theorems.

Interaction-free measurement is thought to allow for quantum particles to detect objects along paths they never traveled. As such, it represents one of the most beguiling of quantum phenomena. Here, we present a classical analog of interaction-free measurement using the hydrodynamic pilotwave system, in which a droplet self-propels across a vibrating fluid surface, guided by a wave of its own making. We argue that existing rationalizations of interaction-free quantum measurement in terms of particles being guided by wave forms allow for a classical description manifest in our hydrodynamic system, wherein the measurement is decidedly not interaction-free.

We provide the first example of continuous families of Poincaré–Einstein metrics developing cusps on the trivial topology $$\mathbb {R}^4$$ R 4 . We also exhibit families of metrics with unexpected degenerations in their conformal infinity only. These are obtained from the Riemannian version of an ansatz of Debever and Plebański–Demiański. We additionally indicate how to construct similar examples on more complicated topologies.

Many physics and engineering applications demand Partial Differential Equations (PDE) property evaluations that are traditionally computed with resource-intensive high-fidelity numerical solvers. Data-driven surrogate models provide an efficient alternative but come with a significant cost of training. Emerging applications would benefit from surrogates with an improved accuracy–cost tradeoff, while studied at scale. Here we present a “physicsenhanced deep-surrogate” (“PEDS”) approach towards developing fast surrogate models for complex physical systems, which is described by PDEs. Specifically, a combination of a lowfidelity, explainable physics simulator and a neural network generator is proposed, which is trained end-to-end to globally match the output of an expensive high-fidelity numerical solver. Experiments on three exemplar testcases, diffusion, reaction–diffusion, and electromagnetic scattering models, show that a PEDS surrogate can be up to 3× more accurate than an ensemble of feedforward neural networks with limited data (≈ 10^3 training points), and reduces the training data need by at least a factor of 100 to achieve a target error of 5%. Experiments reveal that PEDS provides a general, data-driven strategy to bridge the gap between a vast array of simplified physical models with corresponding brute-force numerical solvers modeling complex systems, offering accuracy, speed, data efficiency, as well as physical insights into the process.

We construct special cycles on the moduli stack of hermitian shtukas. We prove an identity between (1) the r th $r^{\mathrm{th}}$ central derivative of non-singular Fourier coefficients of a normalized Siegel–Eisenstein series, and (2) the degree of special cycles of “virtual dimension 0” on the moduli stack of hermitian shtukas with r $r$ legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series.

In this work, we consider sequences of $$C^2$$ C 2 metrics which converges to a $$C^2$$ C 2 metric in $$C^0$$ C 0 sense. We show that if the scalar curvature of the sequence is almost non-negative in the integral sense, then the limiting metric has scalar curvature lower bound in point-wise sense.