Lecture notes on non-Kähler complex threefolds, focusing on conifold transitions.
problem Understanding non-Kähler complex threefolds and conifold transitions.
method Review of basics, description of topological features, survey of recent developments.
result Survey of recent developments on the geometrization of conifold transitions.
Double-well transitions are stiffer than minimal surfaces.
problem Rigidity of double-well phase transitions compared to minimal hypersurfaces.
method Comparison of rigidity properties between double-well phase transitions and minimal hypersurfaces.
result Double-well phase transitions exhibit more rigidity than minimal hypersurfaces.
Lecture notes on conifold transitions between Calabi-Yau manifolds.
problem Understanding conifold transitions in complex threefolds.
method Differential geometric approach, focusing on explicit calculations and examples.
result Necessary condition for smoothings of nodal Calabi-Yau threefolds proved.
The paper provides examples of geometric transitions in low dimensions.
problem Exploring geometric transitions between different types of structures in low dimensions.
method Explicit examples and computations of transitions from hyperbolic to Euclidean, spherical, and Anti-de Sitter structures.
result Details of elementary computations and techniques are provided to explain geometric transitions.
Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
problem Quasi-transitive graphs quasi-isometric to planar graphs need to be upgraded to Cayley graphs.
method Upgrading a planar graph to a Cayley graph.
result Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
Study refracted skew Brownian motion, find densities and asymptotics.
problem Modeling and analyzing refracted skew Brownian motion.
method Perturbation approach to find potential densities, transition density, and asymptotic behaviors.
result Expressions and asymptotic behaviors of refracted skew Brownian motion.
We show that the Morse index of every 2k-ended solution of the Allen-Cahn equation in R^2 is >= k-1. This bound is expected to be sharp.
We study the maximum mean discrepancy (MMD) in the context of critical transitions modelled by fast-slow stochastic dynamical systems. We establish a new link between the dynamical theory of critical transitions with the statistical aspects of the MMD. In particular, we show that a formal approximation of the MMD near …
Forré introduces a new conditional independence notion for mixed variables.
problem Unified framework for random and non-stochastic variables.
method Unified framework of transitional conditional independence and causal calculus for iDMGs.
result Unified framework connects conditional independencies to graphical separation criteria.
We reprove a result concerning certain ruin in the classical problem of the probability of ruin with risky investments and several of it's generalisations. We also provide the combined transition density of the risk and investment processes in the diffusion case.
BachProp generates music in various styles using deep learning.
problem Creating music algorithms that can adapt to multiple styles.
method Developed a novel music representation and trained a deep network to predict note transitions.
result BachProp generates music scores that better capture features of original corpora.
We prove that (apart from dimension n=4), each Riemannian solenoidal lamination with transitive homeomorphism group and leaves isometric to a symmetric space X of noncompact type, is homeomorphic to the inverse limit of the system of finite covers of a compact locally-symmetric n-manifold.
The focus of these lectures is the Gopakumar-Vafa's insight that ``Large N dualities'' (relating gauge theories and closed strings) are realized, in certain cases, by "transition in geometry". In their pivotal 1998 example, the gauge theory is SU(N) Chern-Simons theory on S^3, for large N, and the transition is the "co…
In this note we propose a method based on artificial neural network to study the transition between states governed by stochastic processes. In particular, we aim for numerical schemes for the committor function, the central object of transition path theory, which satisfies a high-dimensional Fokker-Planck equation. By…
Hidden Markov Models (HMMs) are a ubiquitous tool to model time series data, and have been widely used in two main tasks of Automatic Music Transcription (AMT): note segmentation, i.e. identifying the played notes after a multi-pitch estimation, and sequential post-processing, i.e. correcting note segmentation using tr…
Wide neural networks become linear, but adding bottlenecks makes them bilinear or multilinear.
problem Understanding the transition of neural networks from linearity to higher-order functions.
method Analyzing the behavior of randomly initialized wide neural networks with and without bottleneck layers.
result Bottleneck layers transform the network's function from linear to bilinear or multilinear.
New approach to analyze matrix denoising using gradient flow and fixed point equations.
problem Positive semi-definite matrix denoising in extensive-rank and high-dimensional settings.
method Gradient flow and fixed point equations derived from linear pencil techniques of random matrix theory.
result Continuous phase transitions in the extensive-rank and high-dimensional regime.
Paper tackles instance-dependent label noise by approximating it with part-dependent noise.
problem Learning with instance-dependent label noise is challenging.
method Approximate instance-dependent label noise with part-dependent noise. Use transition matrices for parts to model noise.
result Method outperforms state-of-the-art approaches for instance-dependent label noise.
Double pants decompositions were introduced in our paper "Double pants decompositions of 2-surfaces" (Mosc. Math. J. 11 (2011), no. 2, 231-258, arXiv:1005.0073), together with a flip-twist groupoid acting on these decompositions. It was shown that flip-twist groupoid acts transitively on a certain topological class of …
The abstract discusses the classification of 3D Lie groups with Riemannian metrics.
problem Classifying 3D Lie groups up to quasi-isometries and bi-Lipschitz equivalence.
method Review of existing literature and study of quasi-isometry and bi-Lipschitz equivalence.
result For three-dimensional simply connected groups, quasi-isometry implies isometry with suitable metrics.
Discussing rigidity in codimension 2, extending rigidity concepts.
problem Local isometric rigidity problem in codimension 2.
method Extending rigidity concepts to include genuine and honest rigidity, studying isometric immersions in semi-Euclidean spaces.
result Necessity of natural singularity in inner product for transitivity.
We show that a self orbit equivalence of a transitive Anosov flow on a 3-manifold which is homotopic to identity has to either preserve every orbit or the Anosov flow is R-covered and the orbit equivalence has to be of a specific type. This result shows that one can remove a relatively unnatural assumption…
Model predicts climate change's impact on real estate prices.
problem Impact of climate transition on real estate prices.
method Modeling property valuation using Ornstein-Uhlenbeck processes and carbon prices.
result Depreciation of inefficient real estate assets due to climate transition is quantifiable.
The paper bounds the first Betti number and discusses properties of Lefschetz fibrations.
problem Understanding the topology of Lefschetz fibrations.
method Analyzes upper bounds for the first Betti number, examines monodromy transitivity, and discusses potential fibrations.
result Upper bounds for the first Betti number and insights into Lefschetz fibration properties.
A solvmanifold is a compact differentiable manifold M on which a connected solvable Lie group G acts transitively. As the main result, we will see, applying a result of Arapura and Nori on solvable Kaehler groups and some of the author's previous results, that a compact solvmanifold admits a Kaehler structure if and on…
In this note we prove that, for a vector bundle E over a manifold M, a Dorfman bracket on TM⊕E∗ anchored by prTM and with E a vector bundle over M, is equivalent to a lift from Γ(TM⊕E∗) to linear sections of TE⊕T∗E→E, that intertwines the given Dorfman bracket w…
Proofs for flows of linear vector fields and their applications.
problem Existence of flows for linear vector fields and related properties.
method Detailed proofs and flow construction techniques.
result Smooth triviality of vector bundles over contractible bases and isomorphy of fibers of transitive Lie algebroids.
Optimal transport for vector Gaussian mixtures improves efficiency and structure preservation.
problem Optimal mass transport for vector-valued Gaussian mixtures.
method Vectorizing Gaussian mixture models and studying optimal mass transport problems.
result Computational efficiency and structure preservation in optimal mass transport.
Study shows how deep network representations can be transferred between datasets and tasks.
problem Transferability of deep network representations across datasets and tasks.
method Examined layer-wise transferability of representations in deep networks across multiple datasets and tasks.
result Interesting empirical observations on layer-wise transferability of representations.
In this note we consider homogeneous Willmore surfaces in Sn+2. The main result is that a homogeneous Willmore two-sphere is conformally equivalent to a homogeneous minimal two-sphere in Sn+2, i.e., either a round two-sphere or one of the Borůvka-Veronese 2-spheres in S2m. This entails a classification o…
We present in modern language the contents of the famous note published by Henri Poincaré in 1901 "Sur une forme nouvelle des équations de la Mécanique", in which he proves that, when a Lie algebra acts locally transitively on the configuration space of a Lagrangian mechanical system, the well known Euler-Lagrange equa…
ARL-GEN adapts to the smallest model class in nested families for RL with improved regret.
problem Model selection for Reinforcement Learning with nested model families.
method Adaptive Reinforcement Learning (ARL-GEN) with value targeted regression and model selection module.
result ARL-GEN achieves a matching regret to an oracle with knowledge of the true model class.
Study the symmetry and winding numbers of curves defined by sums of exponentials.
problem Understanding the geometry and topology of curves defined by sums of exponentials.
method Analyzing the continuous transition of the graph of the curve as a parameter changes.
result Determined winding numbers and cusp points for curves defined by sums of exponentials.
Paper proposes a model-free algorithm for CMDPs with long-term constraints, achieving optimal regret bounds.
problem Optimizing systems with long-term constraints where transition probabilities are unknown.
method Combines concepts from constrained optimization and Q-learning to propose an algorithm.
result Achieves optimal regret bounds for reward and constraint violation.
We present an explicit algorithm for tessellating the algebraic surfaces (real 4-manifolds) F(n) embedded in CP3 defined by the equation z0^n + z1^n + z2^n + z3^n = 0 in the standard homogeneous coordinates [z0, z1, z2, z3], where n is any positive integer. Note that F(4) in particular is a K3 surface. Our tessellation…
Develops methods to simulate rare transitions in molecular systems.
problem Rare transitions between metastable states in molecular systems are difficult to study due to limited data.
method Two novel methods: chain-based and midpoint-based approaches.
result Demonstrates effectiveness of methods in both data-rich and data-scarce scenarios.
Overparametrization improves QNN trainability by reducing spurious local minima.
problem Understanding how overparametrization affects the loss landscape of QNNs.
method Rigorous analysis of overparametrization in QNNs with periodic structure.
result Overparametrization corresponds to a computational phase transition improving QNN trainability.
The paper models market crashes as phase transitions, finding dynamic transitions offer better predictions.
problem Understanding and predicting extreme financial events like market crashes.
method Employing phase transition theory, focusing on endogenous crashes, and comparing DPT, CPT, and SPT.
result Dynamic phase transitions provide more accurate predictions of market crashes compared to critical and stochastic models.
This paper establishes an equivalence between transitive double Lie algebroids and core diagrams.
problem Understanding and characterizing transitive double Lie algebroids.
method Using core diagrams and equivalence of transitive core diagrams with transitive double Lie groupoids.
result Transitive double Lie algebroids are completely determined by their core diagrams.
This paper shows semi-equivelar toroidal maps are vertex-transitive covers.
problem Understanding the relationship between semi-equivelar and vertex-transitive toroidal maps.
method Proving semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps.
result Each semi-equivelar toroidal map has a finite vertex-transitive cover.
Two-dimensional transition rates improve life insurance reserve calculations.
problem Calculating life insurance reserves with Markov assumptions.
method Introducing two-dimensional forward and backward transition rates.
result Two-dimensional transition rates enable more accurate reserve calculations.
Vanishing and exploding gradients are two of the main obstacles in training deep neural networks, especially in capturing long range dependencies in recurrent neural networks~(RNNs). In this paper, we present an efficient parametrization of the transition matrix of an RNN that allows us to stabilize the gradients that …
Machine learning approximates phase transitions using Fisher information.
problem Understanding phase transitions from data using machine learning.
method Information geometry and Fisher information.
result Machine learning indicators approximate the square root of Fisher information.
Dual-T method improves transition matrix estimation in noisy label learning.
problem Large estimation error in noisy class posterior leads to poor transition matrix estimation.
method Introducing an intermediate class to avoid direct estimation of noisy class posterior, factorizing the transition matrix into two easier-to-estimate matrices.
result The dual-T estimator leads to better classification performances.
Defines SETR to measure carbon transition risk for investors.
problem Difficulty in measuring the magnitude of carbon transition risk for investors.
method Defines Single Event Transition Risk (SETR) and illustrates its use.
result SETR can approximate the magnitude of low-carbon transition risk.
New algorithm achieves data-dependent regret bounds in MDPs with unknown transitions.
problem Achieving best-of-both-worlds guarantees with data-dependent regret bounds in MDPs with unknown transitions.
method Optimistic follow-the-regularized-leader algorithm with new optimistic Q-function estimators and transition bonus.
result First-order, second-order, and path-length bounds with polylog(T) regret in the stochastic regime.
Develops a flexible model for regime transitions in time series data.
problem Nonlinear and context-dependent regime transitions in time series data.
method Semi-parametric state-space model with learned transition functions.
result Improved recovery of nonlinear transition dynamics and earlier detection of regime changes.
Study measures investment funds' climate transition risk, finds moderate losses.
problem Measuring the impact of climate transition on investment portfolios.
method Comprehensive framework using geographical, sectoral, company and ISIN-level data.
result Investment funds suffer a moderate 5.7% loss in high transition risk scenario.