• #### The Chern-Ricci flow

We give a survey on the Chern-Ricci flow, a parabolic flow of Hermitian metrics on complex manifolds. We emphasize open problems and new directions.
Ricci flowComplex manifoldManifoldCurvatureBundleRicci tensorCanonical bundleKähler-Einstein metricMonge-Ampère equationTorus...
• #### On the origin of matter in the Universe

The understanding of the physical processes that lead to the origin of matter in the early Universe, creating both an excess of matter over anti-matter that survived until the present and a dark matter component, is one of the most fascinating challenges in modern science. The problem cannot be addressed within our current description of fundamental physics and, therefore, it currently provides a very strong evidence of new physics. Solutions can either reside in a modification of the standard model of elementary particle physics or in a modification of the way we describe gravity, based on general relativity, or at the interface of both. We will mainly discuss the first class of solutions. Traditionally, models that separately explain either the matter-antimatter asymmetry of the Universe or dark matter have been proposed. However, in the last years there has also been an accreted interest and intense activity on scenarios able to provide a unified picture of the origin of matter in the early universe. In this review we discuss some of the main ideas emphasising primarily those models that have more chances to be experimentally tested during next years. Moreover, after a general discussion, we will focus on extensions of the standard model that can also address neutrino masses and mixing, since this is currently the only evidence of physics beyond the standard model coming directly from particle physics and it is, therefore, reasonable they might also provide a solution to the problem of the origin of matter in the universe.
Sterile neutrinoNeutrino massDark matterLeptogenesisNeutrinoStandard ModelFlavourWeakly interacting massive particleBaryon asymmetry of the UniverseSeesaw mechanism...
• #### Constraining active-sterile neutrino transition magnetic moments at DUNE near and far detectorsver. 3

We consider the sensitivity of the DUNE experiment to a heavy neutral lepton, HNL (also known as sterile neutrino) in the mass range from a few MeV to a few GeV, interacting with the Standard Model via a transition magnetic moment to the active neutrinos, the so-called dipole portal. The HNL is produced via the upscattering of active neutrinos, and the subsequent decay inside the detector provides a single-photon signal. We show that the tau-neutrino dipole portal can be efficiently probed at the DUNE far detector, using the tau-neutrino flux generated by neutrino oscillations, while the near detector provides better sensitivity to the electron- and muon-neutrino dipole portal. DUNE will be able to explore large regions of currently unconstrained parameter space and has comparable sensitivity to other planned dedicated experiments, such as SHiP. We also comment briefly on the sensitivity to pure HNL mixing with the tau neutrino at the DUNE far detector.
DUNE experimentHeavy sterile neutrinoSterile neutrinoEarthTau neutrinoNeutrinoFlavourSHiP experimentWeak neutral current interactionDecay width...
• #### Human-Level Reinforcement Learning through Theory-Based Modeling, Exploration, and Planning

Reinforcement learning (RL) studies how an agent comes to achieve reward in an environment through interactions over time. Recent advances in machine RL have surpassed human expertise at the world's oldest board games and many classic video games, but they require vast quantities of experience to learn successfully -- none of today's algorithms account for the human ability to learn so many different tasks, so quickly. Here we propose a new approach to this challenge based on a particularly strong form of model-based RL which we call Theory-Based Reinforcement Learning, because it uses human-like intuitive theories -- rich, abstract, causal models of physical objects, intentional agents, and their interactions -- to explore and model an environment, and plan effectively to achieve task goals. We instantiate the approach in a video game playing agent called EMPA (the Exploring, Modeling, and Planning Agent), which performs Bayesian inference to learn probabilistic generative models expressed as programs for a game-engine simulator, and runs internal simulations over these models to support efficient object-based, relational exploration and heuristic planning. EMPA closely matches human learning efficiency on a suite of 90 challenging Atari-style video games, learning new games in just minutes of game play and generalizing robustly to new game situations and new levels. The model also captures fine-grained structure in people's exploration trajectories and learning dynamics. Its design and behavior suggest a way forward for building more general human-like AI systems.
• #### Probing Secret Interactions of Astrophysical Neutrinos in the High-Statistics Era

Do neutrinos have sizable self-interactions? They might. Laboratory constraints are weak, so strong effects are possible in astrophysical environments and the early universe. Observations with neutrino telescopes can provide an independent probe of neutrino self ("secret") interactions, as the sources are distant and the cosmic neutrino background intervenes. We define a roadmap for making decisive progress on testing secret neutrino interactions governed by a light mediator. This progress will be enabled by IceCube-Gen2 observations of high-energy astrophysical neutrinos. Critical to this is our comprehensive treatment of the theory, taking into account previously neglected or overly approximated effects, as well as including realistic detection physics. We show that IceCube-Gen2 can realize the full potential of neutrino astronomy for testing neutrino self-interactions, being sensitive to cosmologically relevant interaction models. To facilitate forthcoming studies, we release nuSIProp, a code that can also be used to study neutrino self-interactions from a variety of sources.
NeutrinoIceCube Neutrino ObservatoryAstrophysical neutrinoStatisticsThe early UniverseFinal stateMass eigen stateBumpingEarthNeutrino flux...
• #### A N-dimensional elastic\viscoelastic transmission problem with Kelvin-Voigt damping and non smooth coefficient at the interface

We investigate the stabilization of a multidimensional system of coupled wave equations with only one Kelvin Voigt damping. Using a unique continuation result based on a Carleman estimate and a general criteria of Arendt Batty, we prove the strong stability of the system in the absence of the compactness of the resolvent without any geometric condition. Then, using a spectral analysis, we prove the non uniform stability of the system. Further, using frequency domain approach combined with a multiplier technique, we establish some polynomial stability results by considering different geometric conditions on the coupling and damping domains. In addition, we establish two polynomial energy decay rates of the system on a square domain where the damping and the coupling are localized in a vertical strip.
KelvinDecay rateSpectral analysisWave equationMultidimensional systemsFrequencyEnergy...
• #### Helical Liquids in Semiconductors

One-dimensional helical liquids can appear at boundaries of certain condensed matter systems. Two prime examples are the edge of a quantum spin Hall insulator, also known as a two-dimensional topological insulator, and the hinge of a three-dimensional second-order topological insulator. For these materials, the presence of a helical state at the boundary serves as a signature of their nontrivial bulk topology. Additionally, these boundary states are of interest themselves, as a novel class of strongly correlated low-dimensional systems with interesting potential applications. Here, we review existing results on such helical liquids in semiconductors. Our focus is on the theory, though we confront it with existing experiments. We discuss various aspects of the helical states, such as their realization, topological protection and stability, or possible experimental characterization. We lay emphasis on the hallmark of these states, being the prediction of a quantized electrical conductance. Since so far reaching a well-quantized conductance remained challenging experimentally, a large part of the review is a discussion of various backscattering mechanisms which have been invoked to explain this discrepancy. Finally, we include topics related to proximity-induced topological superconductivity in helical states, as an exciting application towards topological quantum computation with the resulting Majorana bound states.
BackscatteringEdge excitationsTime-reversal symmetrySpin-orbit interactionDisorderMajorana bound stateLiquidsMagnetic impuritySuperconductorMercury telluride...
• #### Galaxy clusters, cosmic chronometers and the Einstein equivalence principle

The Einstein equivalence principle in the electromagnetic sector can be violated in modifications of gravity theory generated by a multiplicative coupling of a scalar field to the electromagnetic Lagrangian. In such theories, deviations of the standard result for the cosmic distance duality relation, and a variation of the fine structure constant are expected and are unequivocally intertwined. In this paper, we search for these possible cosmological signatures by using galaxy cluster gas mass fraction measurements and cosmic chronometers. No significant departure from general relativity is found regardless of our assumptions about cosmic curvature or a possible depletion factor evolution in cluster measurements.
Cluster of galaxiesEinstein equivalence principleAngular diameter distanceGeneral relativityDualityFine structure constantTheories of gravityScalar fieldCurvatureCosmic microwave background...
• #### Volume vs. Complexity of Hyperbolic Groups

We prove that for a one-ended hyperbolic graph $X$, the size of the quotient $X/G$ by a group $G$ acting freely and cocompactly bounds from below the number of simplices in an Eilenberg-MacLane space for $G$. We apply this theorem to show that one-ended hyperbolic cubulated groups (or more generally, one-ended hyperbolic groups with globally stable cylinders \`a la Rips-Sela) cannot contain isomorphic finite-index subgroups of different indices.
Hyperbolic groupGraphSubgroupFoliationManifoldTorsion tensorQuasi-isometryGeodesicLattice (order)Group action...
• #### Globally stable cylinders for hyperbolic CAT(0) cube complexes

Rips and Sela introduced the notion of globally stable cylinders and asked if all Gromov hyperbolic groups admit such. We prove that hyperbolic cubulated groups admit globally stable cylinders.
GeodesicHyperbolic groupHyperbolic SpacePartially ordered setIntersection numberCompletenessEmpty Lattice ApproximationGraphQuasi-isometryGromov product...
• #### The conjugacy stability problem for parabolic subgroups in Artin groups

Given an Artin group $A$ and a parabolic subgroup $P$, we study if every two elements of $P$ that are conjugate in $A$, are also conjugate in $P$. We provide an algorithm to solve this decision problem if $A$ satisfies three properties that are conjectured to be true for every Artin group. We partially solve the problem if $A$ has $FC$-type, and we totally solve it if $A$ is isomorphic to a free product of spherical Artin groups. In particular, we show that in this latter case, every element of $A$ is contained in a unique minimal (by inclusion) parabolic subgroup.
SubgroupArtin groupFree productGraphIsomorphismDecision problemPositive elementBraid groupConjugacy classLattice (order)...
• #### Rigorous Bounds on Light-by-Light Scattering

We bound EFT coefficients appearing in $2 \to 2$ photon scattering amplitudes in four dimensions. After reviewing unitarity and positivity conditions in this context, we use dispersion relations and crossing symmetry to compute sum rules and null constraints. This allows us to derive new rigorous bounds on operators with four, six, and eight derivatives, including two-sided bounds on their ratios. Comparing with a number of partial UV completions, we find that some of our bounds are saturated by the amplitudes that arise from integrating out a massive scalar or axion, while others suggest the existence of unknown amplitudes.
UnitarityUV completionOptimizationWilson coefficientsQuantum electrodynamicsAxionScattering amplitudeConvex hullGravitonLight-by-light scattering...
• #### Hexagonalization of Fishnet integrals I: mirror excitations

In this paper we consider a conformal invariant chain of $L$ sites in the unitary irreducible representations of the group $SO(1,5)$. The $k$-th site of the chain is defined by a scaling dimension $\Delta_k$ and spin numbers $\frac{\ell_k}{2}$, $\frac{\dot{\ell}_k}{2}$. The model with open and fixed boundaries is shown to be integrable at the quantum level and its spectrum and eigenfunctions are obtained by separation of variables. The transfer matrices of the chain are graph-builder operators for the spinning and inhomogeneous generalization of squared-lattice "fishnet" integrals on the disk. As such, their eigenfunctions are used to diagonalize the mirror channel of the the Feynman diagrams of Fishnet conformal field theories. The separated variables are interpreted as momentum and bound-state index of the $\textit{mirror excitations}$ of the lattice: particles with $SO(4)$ internal symmetry that scatter according to an integrable factorized $\mathcal{S}$-matrix in $(1+1)$ dimensions.
EigenfunctionTransfer matrixLattice (order)StarGraphPath integralPermutationPropagatorYang-Baxter equationFeynman diagrams...
• #### Comments on wormholes and factorization

In AdS/CFT partition functions of decoupled copies of the CFT factorize. In bulk computations of such quantities contributions from spacetime wormholes which link separate asymptotic boundaries threaten to spoil this property, leading to a "factorization puzzle." Certain simple models like JT gravity have wormholes, but bulk computations in them correspond to averages over an ensemble of boundary systems. These averages need not factorize. We can formulate a toy version of the factorization puzzle in such models by focusing on a specific member of the ensemble where partition functions will again factorize. As Coleman and Giddings-Strominger pointed out in the 1980s, fixed members of ensembles are described in the bulk by "$\alpha$-states" in a many-universe Hilbert space. In this paper we analyze in detail the bulk mechanism for factorization in such $\alpha$-states in the topological model introduced by Marolf and Maxfield (the "MM model") and in JT gravity. In these models geometric calculations in $\alpha$ states are poorly controlled. We circumvent this complication by working in $\textit{approximate}$ $\alpha$ states where bulk calculations just involve the simplest topologies: disks and cylinders. One of our main results is an effective description of the factorization mechanism. In this effective description the many-universe contributions from the full $\alpha$ state are replaced by a small number of effective boundaries. Our motivation in constructing this effective description, and more generally in studying these simple ensemble models, is that the lessons learned might have wider applicability. In fact the effective description lines up with a recent discussion of the SYK model with fixed couplings arXiv:2103.16754. We conclude with some discussion about the possible applicability of this effective model in more general contexts.
Two-point correlation functionWormholeClosed universePartition functionWavefunctionPath integralForm factorSaddle pointDensity of statesSachdev-Ye-Kitaev model...
• #### A Symmetry of Cosmological Observables, and a High Hubble Constant as an Indicator of a Mirror World Dark Sector

We point out a previously unnoticed symmetry of many important cosmological observables and show that a cosmological model with a "mirror world" dark sector can exploit this symmetry to completely eliminate the Hubble tension. Our work motivates searches for both a more detailed particle physics model that satisfies laboratory constraints and a means of increasing the cosmic photon scattering rate that respects observational bounds on the primordial helium abundance.
Dark sectorRecombinationCosmological modelCosmic microwave backgroundBig bang nucleosynthesisHubble constantSymmetry breakingNeutrinoFree streaming of particlesHubble constant tension...
• #### From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchyver. 2

Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are "as random as a coin-toss". Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power $e_p(U)$ of the basic two-particle unitary building block, $U$, of the circuit, that guarantees mixing, and when maximized, corresponds to Bernoulli circuits. Additionally we show, both analytically and numerically, how local-averaging over random realizations of the single-particle unitaries, $u_i$ and $v_i$ such that the building block is $U^\prime = (u_1 \otimes u_2 ) U (v_1 \otimes v_2 )$ leads to an identification of the average mixing rate as being determined predominantly by the entangling power $e_p(U)$. Finally we provide several, both analytical and numerical, ways to construct dual-unitary operators covering the entire possible range of entangling power. We construct a coupled quantum cat map which is dual-unitary for all local dimensions and a 2-unitary or perfect tensor for odd local dimensions, and can be used to build Bernoulli circuits.
ApexUnitary operatorParticlesDimensionsDynamical systems...
• #### Production of a chiral magnetic anomaly with emerging turbulence and mean-field dynamo action

In relativistic magnetized plasmas, asymmetry in the number densities of left- and right-handed fermions, i.e., a non-zero chiral chemical potential mu_5, leads to an electric current along the magnetic field. This causes a chiral dynamo instability for a uniform mu_5, but our simulations reveal dynamos even for fluctuating mu_5 with zero mean. This generates small-scale magnetic helicity and turbulence. A large-scale mu_5 emerges due to chirality conservation. These effects amplify a mean magnetic field via the magnetic alpha effect and produce a universal scale-invariant mu_5 spectrum.
TurbulenceMagnetic helicityMean fieldInstabilityChiral chemical potentialChiralityMagnetic energyChiral magnetic effectDirect numerical simulationMagnetic anomalies...
• #### A lower bound on the mass of Dark Matter particlesver. 2

We discuss the bounds on the mass of Dark Matter (DM) particles, coming from the analysis of DM phase-space distribution in dwarf spheroidal galaxies (dSphs). After reviewing the existing approaches, we choose two methods to derive such a bound. The first one depends on the information about the current phase space distribution of DM particles only, while the second one uses both the initial and final distributions. We discuss the recent data on dSphs as well as astronomical uncertainties in relevant parameters. As an application, we present lower bounds on the mass of DM particles, coming from various dSphs, using both methods. The model-independent bound holds for any type of fermionic DM. Stronger, model-dependent bounds are quoted for several DM models (thermal relics, non-resonantly and resonantly produced sterile neutrinos, etc.). The latter bounds rely on the assumption that baryonic feedback cannot significantly increase the maximum of a distribution function of DM particles. For the scenario in which all the DM is made of sterile neutrinos produced via non-resonant mixing with the active neutrinos (NRP) this gives m_nrp > 1.7 keV. Combining these results in their most conservative form with the X-ray bounds of DM decay lines, we conclude that the NRP scenario remains allowed in a very narrow parameter window only. This conclusion is independent of the results of the Lyman-alpha analysis. The DM model in which sterile neutrinos are resonantly produced in the presence of lepton asymmetry remains viable. Within the minimal neutrino extension of the Standard Model (the nuMSM), both mass and the mixing angle of the DM sterile neutrino are bounded from above and below, which suggests the possibility for its experimental search.
Phase space densityDark matterDark matter particleEntropySterile neutrinoDark matter particle massCoarse grainingNon-resonant production of sterile neutrinoPhase spaceLepton asymmetry...
• #### Simulating relic gravitational waves from inflationary magnetogenesisver. 2

We present three-dimensional direct numerical simulations of the production of magnetic fields and gravitational waves (GWs) in the early Universe during a low energy scale matter-dominated post-inflationary reheating era, and during the early subsequent radiative era, which is strongly turbulent. The parameters of the model are determined such that it avoids a number of known physical problems and produces magnetic energy densities between 0.2% and 2% of the critical energy density at the end of reheating. During the subsequent development of a turbulent magnetohydrodynamic cascade, magnetic fields and GWs develop a spectrum that extends to higher frequencies in the millihertz (nanohertz) range for models with reheating temperatures of around 100 GeV (150 MeV) at the beginning of the radiation-dominated era. However, even though the turbulent cascade is fully developed, the GW spectrum shows a sharp drop for frequencies above the peak value. This suggests that the turbulence is less efficient in driving GWs than previously thought. The peaks of the resulting GW spectra may well be in the range accessible to space interferometers, pulsar timing arrays, and other facilities.
Gravitational waveTurbulenceReheatingMagnetic energyInflationMagnetogenesisInflationary magnetogenesisMagnetohydrodynamicsMagnetic energy densityReheating temperature...
• #### Colloquium: Quantum heat transport in condensed matter systems

In this Colloquium recent advances in the field of quantum heat transport are reviewed. This topic has been investigated theoretically for several decades, but only during the past twenty years have experiments on various mesoscopic systems become feasible. A summary of the theoretical basis for describing heat transport in one-dimensional channels is first provided. Then the main experimental investigations of quantized heat conductance due to phonons, photons, electrons, and anyons in such channels are presented. These experiments are important for understanding the fundamental processes that underly the concept of a heat conductance quantum for a single channel. Then an illustration on how one can control the quantum heat transport by means of electric and magnetic fields, and how such tunable heat currents can be useful in devices is given. This lays the basis for realizing various thermal device components such as quantum heat valves, rectifiers, heat engines, refrigerators, and calorimeters. Also of interest are fluctuations of quantum heat currents, both for fundamental reasons and for optimizing the most sensitive thermal detectors; at the end of the review the status of research on this intriguing topic is given.
Thermal conductivityResistorPhononQubitResonatorMicrowaveSuperconducting quantum interference devicesSuperconductorHamiltonianWiedemann-Franz law...
• #### SGD May Never Escape Saddle Points

Stochastic gradient descent (SGD) has been deployed to solve highly non-linear and non-convex machine learning problems such as the training of deep neural networks. However, previous works on SGD often rely on highly restrictive and unrealistic assumptions about the nature of noise in SGD. In this work, we mathematically construct examples that defy previous understandings of SGD. For example, our constructions show that: (1) SGD may converge to a local maximum; (2) SGD may escape a saddle point arbitrarily slowly; (3) SGD may prefer sharp minima over the flat ones; and (4) AMSGrad may converge to a local maximum. Our result suggests that the noise structure of SGD might be more important than the loss landscape in neural network training and that future research should focus on deriving the actual noise structure in deep learning.
• #### Lecture notes for pseudodifferential operators and microlocal analysis

This is a introductory course focusing some basic notions in pseudodifferential operators ($\Psi$DOs) and microlocal analysis. We start this lecture notes with some notations and necessary preliminaries. Then the notion of symbols and $\Psi$DOs are introduced. In Chapter 3 we define the oscillatory integrals of different types. Chapter 4 is devoted to the stationary phase lemmas. One of the features of the lecture is that the stationary phase lemmas are proved for not only compactly supported functions but also for more general functions with certain order of smoothness and certain order of growth at infinity. We build the results on the stationary phase lemmas. Chapters 5, 6 and 7 covers main results in $\Psi$DOs and the proofs are heavily built on the results in Chapter 4. Some aspects of the semi-classical analysis are similar to that of microlocal analysis. In Chapter 8 we finally introduce the notion of wavefront, and Chapter 9 focuses on the propagation of singularities of solution of partial differential equations. Important results are circulated by black boxes and some key steps are marked in red color. Exercises are provided at the end of each chapter.
ParametrixEllipticityMicrolocal analysisInverse Fourier TransformQuantizationAsymptotic expansionWeyl quantizationAharonov-Bohm effectHamiltonianPlancherel theorem...
• #### Can the BZ mechanism power steady jets?

We consider the Blandford-Znajek (BZ) mechanism for extracting black hole spin energy to drive astrophysical jets. In analyses of the BZ mechanism it is always assumed that the electric charge of the black hole remains zero. But, as noted by Wald and others, if the medium surrounding the black hole is an ionised plasma with mobile charges, then a spinning hole quickly acquires an electric charge. The effect of this charge is to nullify the electric field structures which drive the BZ mechanism -- the electric and magnetic fields then obey ${\bf E\cdot B= 0}$ everywhere. Since jets are now observed in a wide variety of classes of accreting objects, most of which do not contain a central black hole, it seems likely that the jet driving mechanism in all astrophysical objects uses energy directly from the accretion disc, rather than black hole spin.
Black holeBlandford-Znajek processSpinning Black HoleBlack hole spinAstrophysical jetAccretion diskAstrophysical plasmaCharge separationParticle-in-cellCharged particle...
• #### High resolution H-alpha imaging of the Northern Galactic Plane, and the IGAPS images database

The INT Galactic Plane Survey (IGAPS) is the merger of the optical photometric surveys, IPHAS and UVEX, based on data from the Isaac Newton Telescope (INT) obtained between 2003 and 2018. These capture the entire northern Galactic plane within the Galactic coordinate range, -5<b<+5 deg. and 30<l<215 deg. From the beginning, the incorporation of narrowband H-alpha imaging has been a unique and distinctive feature of this effort. Alongside a focused discussion of the nature and application of the H-alpha data, we present the IGAPS world-accessible database of images for all 5 survey filters, i, r, g, U-RGO and narrowband H-alpha, observed on a pixel scale of 0.33 arcsec and at an effective (median) angular resolution of 1.1 to 1.3 arcsec. The background, noise, and sensitivity characteristics of the narrowband H-alpha filter images are outlined. Typical noise levels in this band correspond to a surface brightness at full one-arcsec resolution of around 2e-16 erg/cm2/s/arcsec2. Illustrative applications of the H-alpha data to planetary nebulae and Herbig-Haro objects are outlined and, as part of a discussion of mosaicking technique, we present a very large background-subtracted narrowband mosaic of the supernova remnant, Simeis 147. Finally we lay out a method that exploits the database via an automated selection of bright ionized diffuse interstellar emission targets for the coming generation of wide-field massive-multiplex spectrographs. Two examples of the diffuse H-alpha maps output from this selection process are presented and compared with previously published data.
H-alphaGalactic planeInterstellar emissionsTelescopesINT Photometric H-Alpha SurveyHerbig-Haro objectPlanetary nebulaSpectrographsSupernova remnantSurface brightness...
• #### Whitehead algorithm for automorphisms of generalized Baumslag-Solitar groups

In analogy with the free factors of a free group we define special factors of Generalized Baumslag-Solitar (GBS) groups as non-cyclic subgroups which appear in splittings over infinite cyclic groups. We give an algorithm which, given a GBS group $G$ and an element $g \in G$, decides whether there exists a special factor $H < G$ such that $g \in H$. This algorithm is analogous to an algorithm by Whitehead for free groups. Furthermore we prove that given $g \in G$ there exists a unique minimal special factor containing $g$ and give an algorithm which finds it.
GraphSubgroupAutomorphismFree groupIsomorphismStarOrientationConjugacy classMorphismFundamental domain...
• #### The $\chi$-Ramsey problem for triangle-free graphs

In 1967, Erd\H{o}s asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erd\H{o}s and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3, t)$ shows that $f(n)$ is at most $(2 \sqrt{2} + o(1)) \sqrt{n/\log n}$. We improve this bound by a factor $\sqrt{2}$, as well as obtaining an analogous bound on the list chromatic number which is tight up to a constant factor. A bound in terms of the number of edges that is similarly tight follows, and these results confirm a conjecture of Cames van Batenburg, de Joannis de Verclos, Kang, and Pirot.
GraphChromatic numberFractional chromatic numberBipartite networkAttentionEntropyProbabilitySketchReal numbers...
• #### H-Transformer-1D: Fast One-Dimensional Hierarchical Attention for Sequences

We describe an efficient hierarchical method to compute attention in the Transformer architecture. The proposed attention mechanism exploits a matrix structure similar to the Hierarchical Matrix (H-Matrix) developed by the numerical analysis community, and has linear run time and memory complexity. We perform extensive experiments to show that the inductive bias embodied by our hierarchical attention is effective in capturing the hierarchical structure in the sequences typical for natural language and vision tasks. Our method is superior to alternative sub-quadratic proposals by over +6 points on average on the Long Range Arena benchmark. It also sets a new SOTA test perplexity on One-Billion Word dataset with 5x fewer model parameters than that of the previous-best Transformer-based models.
AttentionNatural languageArchitectureInductive bias...
• #### Polynomial bounds for chromatic number. II. Excluding a star-forest

The Gyarfas-Sumner conjecture says that for every forest $H$, there is a function $f$ such that if $G$ is $H$-free then $\chi(G)\le f(\omega(G))$ (where $\chi, \omega$ are the chromatic number and the clique number of $G$). Louis Esperet conjectured that, whenever such a statement holds, $f$ can be chosen to be a polynomial. The Gyarfas-Sumner conjecture is only known to be true for a modest set of forests $H$, and Esperet's conjecture is known to be true for almost no forests. For instance, it is not known when $H$ is a five-vertex path. Here we prove Esperet's conjecture when each component of $H$ is a star.
• #### Local measures of entanglement structure and dynamics in CFTs and black holes

We study the structure and dynamics of entanglement in CFTs and black holes. We use a local entanglement measure, the entanglement contour, which is a spatial density function for von Neumann entropy with additional properties. We calculate the entanglement contour of a state excited by a splitting quench, and find universal results for the entanglement contours of low energy non-equilibrium states in 2d CFTs. We also calculate the contour of a non-gravitational bath coupled to an extremal AdS$_2$ black hole, and find that the contour only has finite support within the bath, due to an island phase transition. The particular entanglement contour proposal we use quantifies how well the bath's state can be reconstructed from its marginals, through its connection to conditional mutual information, and the vanishing contour is a reflection of the protection of bulk island regions against erasures of the boundary state.
EntanglementEntanglement entropyEntropyConformal field theoryMutual informationVon neumann entropyBlack holeQuenchingDegree of freedomKinematics...
• #### A Sum Rule for Boundary Contributions to the Trace Anomaly

In the context of boundary conformal field theory, we derive a sum rule that relates two and three point functions of the displacement operator. For four dimensional conformal field theory with a three dimensional boundary, this sum rule in turn relates the two boundary contributions to the anomaly in the trace of the stress tensor. We check our sum rule for a variety of free theories and also for a weakly interacting theory, where a free scalar in the bulk couples marginally to a generalized free field on the boundary.
Conformal field theoryTwo-point correlation functionEmbeddingOperator product expansionFree fieldReal spaceConformal symmetryDualityTrace anomalyAdS/CFT correspondence...
• #### Three Impossible Theories

I will begin by conjecturing a cosmological generalization of black hole complementarity (also known as the central dogma). I will then discuss three theories and argue that they are inconsistent with second law of thermodynamics if the cosmological version of the dogma is correct. The three theories are: the big rip; cyclic cosmology; and the Farhi-Guth-Guven mechanism for creating inflating universes behind black hole horizons.
EntropyDe Sitter spaceSecond law of thermodynamicsHorizonBlack holeBig RipBlack hole horizonCosmologyTunnelling rateBlack hole complementarity...
• #### Yang-Mills at Strong vs. Weak Coupling: Renormalons, OPE And All That

I discuss various situations in which perturbative expansions are used in Yang-Mills theories with asymptotic freedom and establish the limits of its applicability.
RenormalonInstantonYang-Mills theoryCoupling constantSphaleronPerturbative expansionPerturbation theory't Hooft limitGauge coupling constantQuantum mechanics...
• #### Deciding Isomorphy using Dehn fillings, the splitting casever. 4

We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if the graphs of groups have isomorphic vertex groups and, if so, whether a global isomorphism can be assembled. For the class of finitely generated nilpotent groups, we give solutions to these algorithmic problems by using the arithmetic nature of these groups and of their automorphism groups.
• #### Relative hyperbolicity for automorphisms of free productsver. 3

We prove that for a free product $G$ with free factor system $\mathcal{G}$, any automorphism $\phi$ preserving $\mathcal{G}$, atoroidal (in a sense relative to $\mathcal{G}$) and none of whose power send two different conjugates of subgroups in $\mathcal{G}$ on conjugates of themselves by the same element, gives rise to a semidirect product $G\rtimes_\phi \mathbb{Z}$ that is relatively hyperbolic with respect to suspensions of groups in $\mathcal{G}$. We recover a theorem of Gautero-Lustig and Ghosh that, if $G$ is a free group, $\phi$ an automorphism of $G$, and $\mathcal{G}$ is its family of polynomially growing subgroups, then the semidirect product by $\phi$ is relatively hyperbolic with respect to the suspensions of these subgroups. We apply the first result to the conjugacy problem for certain automorphisms (atoroidal and toral) of free products of abelian groups.
SubgroupAutomorphismFree productConjugacy classTorusGraphRankFree groupSemidirect productConjugacy problem...
• #### The Tits alternative for the automorphism group of a free productver. 2

Let $G=G_1\ast\dots\ast G_k\ast F$ be a countable group which splits as a free product, where all groups $G_i$ are freely indecomposable and not isomorphic to $\mathbb{Z}$, and $F$ is a finitely generated free group. If for all $i\in\{1,\dots,k\}$, both $G_i$ and its outer automorphism group $\text{Out}(G_i)$ satisfy the Tits alternative, then $\text{Out}(G)$ satisfies the Tits alternative. As an application, we prove that the Tits alternative holds for outer automorphism groups of right-angled Artin groups, and of torsion-free groups that are hyperbolic relative to a finite family of virtually polycyclic groups.
SubgroupAutomorphismFree productOuter automorphism groupConjugacy classRankingFree groupArtin groupIsomorphismIsometry...
• #### Free representations of outer automorphism groups of free products via characteristic abelian coverings

Given a free product $G$, we investigate the existence of faithful free representations of the outer automorphism group $\text{Out}(G)$, or in other words of embeddings of $\text{Out}(G)$ into $\text{Out}(F_m)$ for some $m$. We base our work on a paper of Bridson and Vogtmann in which they construct embeddings of $\text{Out}(F_n)$ into $\text{Out}(F_m)$ for some values of $n$ and $m$ by interpreting $\text{Out}(F_n)$ as the group of homotopy equivalences of a graph $X$ of genus $n$, and by lifting homotopy equivalences of $X$ to a characteristic abelian cover of genus $m$. In order to adapt their method to a free product $G$, we view $G$ as the fundamental group of a graph of groups, and we explain how to obtain notions of homotopy and coverings for graphs of groups by considering the action on a Bass-Serre tree. Using these notions, as well as a presentation of $\text{Out}(G)$ due to Fuchs-Rabinovich, we show for instance that $\text{Out}(G)$ has a faithful free representation when $G=F_d\ast G_{d+1}\ast\cdots\ast G_n$, with $F_d$ free of rank $d$ and $G_i$ finite abelian of order coprime to $n-1$. This implies in particular that $\text{Out}(G)$ satisfies the Tits alternative since $\text{Out}(F_m)$ does.
GraphMorphismMessier 4Free productCrab NebulaAutomorphismRankEmbeddingMessier 8Butterfly Cluster...
• #### An upper bound on the size of Sidon setsver. 2

In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of $\{ 1, 2, \ldots, n\}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$.
HypergraphNumber theoryDegree sequenceGalois fieldOptimizationCompletenessPrime numberArithmetic progressionAutomorphismCauchy-Schwarz inequality...
• #### Each closure operator induces a topology and vice-versa ("version for children")

One of the main prerequisites for understanding sheaves on elementary toposes is the proof that a (Lawvere-Tierney) topology on a topos induces a closure operator on it, and vice-versa. That standard theorem is usually presented in a relatively brief way, with most details being left to the reader and with no hints on how to visualize some of the hardest axioms and proofs. These notes are, on a first level, an attempt to present that standard theorem in all details and in a visual way, following the conventions in "On my favorite conventions for drawing the missing diagrams in Category Theory" [Ochs2020]; in particular, some properties, like stability by pullbacks, are always drawn in the same "shape". On a second level these notes are also an experiment on doing these proofs on "archetypal cases" in ways that makes all the proofs easy to lift to the "general case". Our first archetypal case is a "topos with inclusions". This is a variant of the "toposes with canonical subobjects" from [Lambek/Scott 1986]; all toposes of the form $\mathbf{Set}^\mathbf{C}$, where $\mathbf{C}$ is a small category, are toposes with inclusions, and when we work with toposes with inclusions concepts like subsets and intersections are very easy to formalize. We do all our proofs on the correspondence between closure operators and topologies in toposes with inclusions, and then we show how to lift all our proofs to proofs that work on any topos. Our second archetypal case is toposes of the form $\mathbf{Set}^\mathbf{D}$, where $\mathbf{D}$ is a finite two-column graph. We show a way to visualize all the LT-topologies on toposes of the form $\mathbf{Set}^\mathbf{D}$, and we define formally a way to "add visual intuition to a proof about toposes"; this is related to the several techniques for doing "Category Theory for children" that are explained in "On my favorite conventions...".
Partially ordered setGraphMorphismCategory theoryClassifierNatural deductionInteractive theorem proverConjunctionPosetal categoryIsomorphism...
• #### Topological data analysis of C. elegans locomotion and behaviorver. 3

We apply topological data analysis to the behavior of C. elegans, a widely-studied model organism in biology. In particular, we use topology to produce a quantitative summary of complex behavior which may be applied to high-throughput data. Our methods allow us to distinguish and classify videos from various environmental conditions and we analyze the trade-off between accuracy and interpretability. Furthermore, we present a novel technique for visualizing the outputs of our analysis in terms of the input. Specifically, we use representative cycles of persistent homology to produce synthetic videos of stereotypical behaviors.
Time SeriesPersistent homologyEmbeddingPrincipal componentViscosityPoint cloudDiscretizationSupport vector machineNull modelGene...
• #### Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory

Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. This work provides a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user friendly resource for beginners and experts alike.
E_nQuantum groupCategory theoryQuantum computationTheoryAbundanceMaterials...
• #### Neutrino mass models: New classification and model-independent upper limits on their scalever. 2

We propose a model-independent framework to classify and study neutrino mass models and their phenomenology. The idea is to introduce one particle beyond the Standard Model which couples to leptons and carries lepton number together with an operator which violates lepton number by two units and contains this particle. This allows to study processes which do not violate lepton number, while still working with an effective field theory. The contribution to neutrino masses translates to a robust upper bound on the mass of the new particle. We compare it to the stronger but less robust upper bound from Higgs naturalness and discuss several lower bounds. Our framework allows to classify neutrino mass models in \emph{just} 20 categories, further reduced to 14 once nucleon decay limits are taken into account, and \emph{possibly} to 9 if also Higgs naturalness considerations and direct searches are considered.
Neutrino massLepton numberNaturalnessHiggs bosonStandard ModelLeptoquarkSeesaw mechanismElectroweakYukawa couplingHiggs boson mass...
• #### Implications of LUNA for BBN and CMB constraints on MeV-scale Thermal Dark Sectors

Very recently, the LUNA collaboration has reported a new measurement of the $d+p\to {}^{3}\text{He}+\gamma$ reaction rate, which plays an important role in the prediction of the primordial deuterium abundance at the time of BBN. This new measurement has triggered a new set of global BBN analyses within the context of the Standard Model. In this addendum to JCAP 01 (2020) 004 (arXiv:1910.01649), we consider the implications of these new results for our constraints on MeV-scale dark sectors. Importantly, we find that our bounds in the BBN-only and Planck-only analyses are insensitive to these updates. Similarly, we find that our constraints derived using BBN and CMB data simultaneously are not significantly modified for neutrinophilic particles. The bounds on electrophilic dark sector states, however, can vary moderately when combining BBN and CMB observations. We present updated results for all the relevant light dark sector states, calculated using the rates obtained by the leading groups performing standard BBN analyses.
Big bang nucleosynthesisDark sectorLaboratory for Underground Nuclear AstrophysicsPlanck missionPrimordial deuterium abundanceStandard BBNStandard ModelHelium abundanceDeuterium AbundancePrimordial helium abundance...
• #### Treating Divergent Perturbation Theory: Lessons from Exactly Solvable 2D Models at Large $N$

We consider the operator product expansion (OPE) of correlation functions in the supersymmetric $O(N)$ non-linear sigma model at sub-leading order in the large $N$ limit in order to study the cancellation between ambiguities coming from infrared renormalons and those coming from various operators in the OPE. As has been discussed in the context of supersymmetric Yang-Mills theory in four dimensions, supersymmetry presents a challenge to this cancellation. In a bid to solve this problem we consider $O(N)$ as a toy model. A background field method inspired by Polyakov's treatment of the renormalization of the bosonic $O(N)$ model is used to identify explicit operators in the OPE of the two-point functions of bosonic and fermionic fields in the model. In order to identify the coefficient functions in the OPE, the exact two-point functions at sub-leading order in large $N$ are expanded in powers of the natural infrared length scale. The ambiguities arising from renormalons in the coefficient functions and vacuum expectation values of operators in the OPE are shown to cancel to all orders. The question of supersymmetric Yang-Mills theory without matter remains open.
Operator product expansionRenormalonInfrared limitTwo-point correlation functionPropagatorSupersymmetricPerturbation theorySuper Yang-Mills theoryBackground field methodSaddle point...
• #### Erdos-Littlewood-Offord problem with arbitrary probabilitiesver. 2

The classical Erd\H{o}s-Littlewood-Offord problem concerns the random variable $X = a_1 \xi_1 + \dots + a_n \xi_n$, where $a_i \in \mathbb{R} \setminus \{0\}$ are fixed and $\xi_i \sim \text{Ber}(1/2)$ are independent. The Erd\H{o}s-Littlewood-Offord theorem states that the maximum possible concentration probability $\max_{x \in \mathbb{R}} \Pr(X = x)$ is $\binom{n}{\lfloor n/2\rfloor} / 2^n$, achieved when the $a_i$ are all $1$. As proposed by Fox, Kwan, and Sauermann, we investigate the general case where $\xi_i \sim \text{Ber}(p)$ instead. Using purely combinatorial techniques, we show that the exact maximum concentration probability is achieved when $a_i \in \{-1, 1\}$ for each $i$. Then, using Fourier-analytic techniques, we investigate the optimal ratio of $1$s to $-1$s. Surprisingly, we find that in some cases, the numbers of $1$s and $-1$s can be far from equal.
Convex combinationNonnegativeGraphBinomial DistributionAttentionBipartite networkBernoulli distributionBiregularMass functionRandom graph...
• #### Using Python for Model Inference in Deep Learning

Python has become the de-facto language for training deep neural networks, coupling a large suite of scientific computing libraries with efficient libraries for tensor computation such as PyTorch or TensorFlow. However, when models are used for inference they are typically extracted from Python as TensorFlow graphs or TorchScript programs in order to meet performance and packaging constraints. The extraction process can be time consuming, impeding fast prototyping. We show how it is possible to meet these performance and packaging constraints while performing inference in Python. In particular, we present a way of using multiple Python interpreters within a single process to achieve scalable inference and describe a new container format for models that contains both native Python code and data. This approach simplifies the model deployment story by eliminating the model extraction step, and makes it easier to integrate existing performance-enhancing Python libraries. We evaluate our design on a suite of popular PyTorch models on Github, showing how they can be packaged in our inference format, and comparing their performance to TorchScript. For larger models, our packaged Python models perform the same as TorchScript, and for smaller models where there is some Python overhead, our multi-interpreter approach ensures inference is still scalable.
PythonInferenceApplication programming interfaceProgrammingDeep learningGraphEngineeringModel ExtractionDeep Neural NetworksEcosystems...
• #### Gluing Karcher-Scherk saddle towers II: Singly periodic minimal surfaces

This is the second in a series of papers that construct minimal surfaces by gluing singly periodic Karcher--Scherk saddle towers along their wings. This paper aims to construct singly periodic minimal surfaces with Scherk ends. As in the first paper, we prescribe phase differences between saddle towers, and reveal that the saddle towers must be balanced under a subtle vertical interaction. As a consequence, we obtain many new examples without any horizontal reflection plane. Since the construction is not very different from previous ones, we will only provide sketched proofs. The major technical concern of the paper is to determine the embeddedness, for which we will provide detailed arguments. Previously, embeddedness can not be determined in the presence of "parallel" Scherk ends, as it was not clear if they bend towards or away from each other. Our construction provide possibilities to detect very slight bendings of Scherk ends. This allows us to identify new scenarios where the constructed surfaces are embedded.
GraphMinimal surfaceHorizontal symmetryIsomorphismRiemann sphereInfinitesimalPermutationComplex planeVector spaceOrientation...
• #### Primordial Non-Gaussianityver. 2

Our current understanding of the Universe is established through the pristine measurements of structure in the cosmic microwave background (CMB) and the distribution and shapes of galaxies tracing the large scale structure (LSS) of the Universe. One key ingredient that underlies cosmological observables is that the field that sources the observed structure is assumed to be initially Gaussian with high precision. Nevertheless, a minimal deviation from Gaussianityis perhaps the most robust theoretical prediction of models that explain the observed Universe; itis necessarily present even in the simplest scenarios. In addition, most inflationary models produce far higher levels of non-Gaussianity. Since non-Gaussianity directly probes the dynamics in the early Universe, a detection would present a monumental discovery in cosmology, providing clues about physics at energy scales as high as the GUT scale.
Non-GaussianityBispectrumInflationLarge scale structureCosmic microwave backgroundInflatonPrimordial Non-GaussianitiesHubble scaleDegree of freedomModel of inflation...
• #### The $H_0$ Olympics: A fair ranking of proposed models

Despite the remarkable success of the $\Lambda$Cold Dark Matter ($\Lambda$CDM) cosmological model, a growing discrepancy has emerged (currently measured at the level of $\sim 4-6 \sigma$) between the value of the Hubble constant $H_0$ measured using the local distance ladder and the value inferred using the cosmic microwave background and galaxy surveys. While a vast array of $\Lambda$CDM extensions have been proposed to explain these discordant observations, understanding the (relative) success of these models in resolving the tension has proven difficult -- this is a direct consequence of the fact that each model has been subjected to differing, and typically incomplete, compilations of cosmological data. In this review, we attempt to make a systematic comparison of sixteen different models which have been proposed to resolve the $H_0$ tension (spanning both early- and late-Universe solutions), and quantify the relative success of each using a series of metrics and a vast array of data combinations. Owing to the timely appearance of this article, we refer to this contest as the ''$H_0$ Olympics''; the goal being to identify which of the proposed solutions, and more broadly which underlying mechanisms, are most likely to be responsible for explaining the observed discrepancy (should unaccounted for systematics not be the culprit). This work also establishes a foundation of tests which will allow the success of novel proposals to be meaningful ''benchmarked''.
Supernovae H0 for the Equation of StateBaryon acoustic oscillationsPlanck missionDark RadiationCold dark matterNeutrinoSound horizonRecombinationMajoronHubble constant tension...
• #### Path Ramsey number for random graphs

Answering a question raised by Dudek and Pra\l{}at, we show that if $pn\rightarrow \infty$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there exists a monochromatic path of length $n(2/3+o(1))$. This result is optimal in the sense that $2/3$ cannot be replaced by a larger constant. As part of the proof we obtain the following result which may be of independent interest. We show that given a graph $G$ on $n$ vertices with at least $(1-\epsilon)\binom{n}{2}$ edges, whenever $G$ is $2$-edge-coloured, there is a monochromatic path of length at least $(2/3-100\sqrt{\epsilon})n$. This is an extension of the classical result by Gerencs\'er and Gy\'arf\'as which says that whenever $K_n$ is $2$-coloured there is a monochromatic path of length at least $2n/3$.
GraphRandom graphRegularizationBipartite networkHamiltonian pathSzemerédi regularity lemmaIsomorphismAlgorithmsContradiction...