New invariant for square-free integers derived from kei theory.
problem Developing numerical invariants for square-free integers.
method Defining a kei for each square-free integer and calculating a coloring invariant.
result Conjecture and proof of asymptotic average order for coloring invariant.
This paper describes several key discoveries in the 19th century that led to the modern theory of manifolds in the twentieth century: intrinsic differential geometry, projective geometry and higher dimensional manifolds and Riemannian geometry.
This paper uses MIS to identify key financial institutions with minimal risk contagion.
problem Mitigating systemic risk during extreme financial events.
method Applying extreme value theory and MIS from graph theory to identify diversified portfolios.
result Identified a subset of institutions with minimal extremal dependence for diversified portfolios.
In "On the homotopy theory of arrangements," published in 1986, the authors gave a comprehensive survey of the subject. This article updates and continues the earlier article, noting some key open problems.
Paper defines XAI concepts using category theory.
problem Lack of precise mathematical definitions for XAI.
method Uses Category theory to define XAI concepts rigorously.
result Establishes a theoretical foundation for XAI.
New Schwarz Lemma for Bergman metrics in bounded domains.
problem Finding bounds for Bergman metrics in bounded domains.
method Using Cauchy-Schwarz inequality from probability theory.
result Established a new Schwarz Lemma for Bergman metrics.
A previous paper of the authors' contained an error in the proof of a key claim, that Rasmussen's knot-invariant s(K) is equal to its gauge-theory counterpart. The original paper is included here together with a corrigendum, indicating which parts still stand and which do not. In particular, the gauge-theory counterpar…
The study reveals the spectral structure of attention layers and its implications for generalization.
problem Understanding the spectral structure and generalization of trained attention layers.
method Empirical risk minimization in a single-head tied-attention layer, using random matrix theory, spin-glass theory, and approximate message passing.
result Exact high-dimensional characterization of training and test error, interpolation and recovery thresholds, and spectrum of the key and query matrices.
Classifies good involutions in conjugation subquandles and racks.
problem Classifying quandles with good involutions for applications in surface-knot theory.
method Study of subquandles of conjugation quandles, including core quandles; analysis of good involutions of faithful racks.
result Sharp bounds on the number of good involutions of racks in these families.
Enhances knot counting using mosaic diagrams.
problem Counting and classifying surface-links and knots.
method Marked graph diagrams and mosaic numbers.
result Established bounds on mosaic numbers for surface-links.
Statistical learning theory provides the theoretical basis for many of today's machine learning algorithms. In this article we attempt to give a gentle, non-technical overview over the key ideas and insights of statistical learning theory. We target at a broad audience, not necessarily machine learning researchers. Thi…
These notes constitute a sort of Crash Course in Optimal Transport Theory. The different features of the problem of Monge-Kantorovitch are treated, starting from convex duality issues. The main properties of space of probability measures endowed with the distances Wp induced by optimal transport are detailed. The ke…
Introduces noncommutative geometry for modeling quantum spacetime.
problem Modeling quantum spacetime.
method Operator algebras, K-theory, spectral geometry, quantum groups, and deformation quantization.
result Framework for quantum spacetime.
Develops a new sampling method for gauge theories.
problem Sampling from SU(N) gauge theories. method Gauge-equivariant flows for SU(N) variables. result Constructs a class of flows respecting matrix conjugation symmetry.
Complex network theory has been applied to solving practical problems from different domains. In this paper, we present a general framework for complex network applications. The keys of a successful application are a thorough understanding of the real system and a correct mapping of complex network theory to practical …
Novel method uses information theory to measure causal influences during transient neural events.
problem Characterizing network interactions during transient neural events.
method Structural Causal Models, Information Theory, Transfer Entropy, Dynamic Causal Strength, Relative Dynamic Causal Strength.
result Introduced a novel measure, relative Dynamic Causal Strength, with theoretical and empirical support.
LaRT models LLMs' response accuracy and CoT length to evaluate reasoning ability and speed.
problem Valid evaluation of Large Language Models (LLMs) via response accuracy and chain-of-thought length.
method Introduces Latency-Response Theory (LaRT) to jointly model response accuracy and CoT length using latent ability and latent speed.
result LaRT yields higher estimation accuracy and shorter confidence intervals for latent traits compared to IRT.
We provide an introduction to the theory of calibrated submanifolds through the key examples related with special holonomy. We focus on calibrated geometry in Calabi-Yau, G2 and Spin(7) manifolds, and describe fundamental results and techniques in the field.
The paper analyzes frameworks for integrating sustainability into investment decisions.
problem Understanding how ESG factors influence investment choices.
method Examined and analyzed various theoretical frameworks including Behavioral Finance, Modern Portfolio, and Risk Management.
result Investors increasingly integrate ESG factors to optimize financial outcomes and societal goals.
Clarifies structures in link homology theories using Frobenius extensions.
problem Understanding key flavors of equivariant SL(2) link homology theories.
method Provides a convenient scheme and diagrammatics for Frobenius extensions.
result Proposes a setup for working over non-degenerate base rings.
Study examines infinite limits of transformer dynamics, identifying key parameterizations.
problem Understanding the training dynamics of transformer models in the feature learning regime.
method Analysis of infinite scaling limits using dynamical mean field theory.
result Identified parameterizations that admit well-defined infinite width and depth limits.
We discuss some of the key ideas of Perelman's proof of Poincaré's conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations.
SGD generalization bounds derived from information theory.
problem Understanding generalization of SGD for non-convex functions.
method Combining information-theoretic bounds with perturbation analysis.
result Upper bounds on SGD's generalization error based on gradient variance and function smoothness.
We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of suc…
This work uses statistical mechanics to explain AI learning.
problem Understanding the statistical principles behind AI learning.
method Starting from sample concentration behaviors, the study applies statistical mechanics principles to AI and machine learning.
result Exponential families and statistical quantities are key in AI and machine learning.
In the late 1980s Witten used the Chern-Simons form of a connection to construct new invariants of 3-manifolds and knots, recovering in particular the Jones invariants. Since then the associated topological quantum field theory (TQFT) has served as a key example in understanding the structure of TQFTs in general. We su…
Geometries and dual field theories linked by AdS/CFT.
problem Understanding the AdS/CFT correspondence.
method Geometric extremization principles informed by physical considerations.
result Key role of Sasaki-Einstein and GK geometry.
Paper categorifies a polynomial related to ribbon graphs.
problem Enumerating partial duals of ribbon graphs.
method Using an extended Frobenius algebra in unoriented topological quantum field theory.
result A categorification of the partial-dual genus polynomial.
We prove that on a closed surface, for any c>0, our min-max theory for prescribing mean curvature produces a solution given by a curve of constant geodesic curvature c which is almost embedded, except for finitely many points, at which the solution is a stationary junction with integer density. Moreover, each smoot…
Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory.
Introduces Lie-Nijenhuis bialgebroids for Poisson-Nijenhuis groupoids.
problem Describing Poisson-Nijenhuis groupoids infinitesimally.
method Develops a theory of generalized derivations and their duality.
result Lie-Nijenhuis bialgebroids provide a complete infinitesimal description of Poisson-Nijenhuis groupoids.
AIM models explain deep learning in attention layers, offering solvable insights.
problem Understanding how deep learning models learn in attention layers.
method Statistical mechanics and random matrix theory.
result Closed-form predictions for Bayes-optimal generalization error and gradient descent performance.
Solves modified Schouten tensor problems in conformal metric classes.
problem Prescribed problems for modified Schouten tensors in conformal classes of metrics.
method Uniform ellipticity confirmation under topological and functional constraints.
result Extends results from previous work on smooth complete metrics.
We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as light-cones being hypersurfaces, are wrong when metrics which are merely continuous are considered. We show that existence of t…
Rule-based classifiers quantify uncertainty using Bernoulli random variables.
problem Quantifying the uncertainty of precision estimates for rule-based text classifiers.
method Treat partitions of sub-strings as Bernoulli random variables, compare means using statistical tests, and combine classifiers using Dempster-Shafer theory.
result The approach can be used to combine binary classifiers into a multi-label classifier.
Develops obstruction theory for a specific 4-manifold index.
problem Computing the Z2-index of 4-manifolds with free involution. method Uses spectral sequences and cohomology with twisted coefficients.
result Computes the Z2-index for various examples. GARIM theory explains how conscious manipulation of internal representations enhances goal-directed behavior.
problem Limited understanding of how consciousness supports flexible goal-directed cognition.
method Extending a three-component theory of flexible cognition, proposing GARIM theory.
result Conscious states actively manipulate internal representations to align with goals, enhancing flexibility.
New theory captures framing anomaly in gauge theory.
problem Capturing framing anomaly in gauge theory.
method Constructs a relative Crane-Yetter theory from non-semisimple data.
result Establishes invertibility property for the theory.
Study the spectral flow of Dirac operators on spinor bundles.
problem Understanding the asymptotic behavior of spectral flow for Dirac operators.
method Variation of eta invariant and local index theory technique.
result Established a uniform estimate of the eta invariant for large parameter values.
It is well-known that an n-dimensional Poincaré complex Xn, n≥5, has the homotopy type of a compact topological n-manifold if the total surgery obstruction s(Xn) vanishes. The present paper discusses recent attempts to prove analogous result in dimension 4. We begin by reviewing the necessary algebraic an…
The paper develops a theory of C∞-superrings and their superschemes.
problem Developing a theory for C∞-superrings and superschemes. method Proving an equivalence between categories of fair affine C∞-superschemes and fair C∞-superrings. result A key equivalence between fair affine C∞-superschemes and fair C∞-superrings. We propose using category theory to unify deep learning architectures.
problem Lack of a coherent bridge between model constraints and implementations.
method Apply category theory to unify neural network design.
result Theory recovers constraints from geometric deep learning and encodes standard constructs.
Geometrically classifies maps from R^0|2 to any manifold, unifying theories.
problem Classifying maps from R^0|2 to any manifold without auxiliary structures.
method Relates maps to pullback of decomposable bivector bundle over S via algebraic constraints.
result Reduced manifold has fiber dimension dim(S) + 1, unifying topological and algebraic views.
Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract setting for differential geometry by axiomatizing key aspects of the subject which all…
In this paper, we elucidate the key role played by the cosymplectic geometry in the theory of time dependent Hamiltonian systems. In particular, we generalize the cosymplectic structures to time-dependent Nambu-Poisson Hamiltonian systems and corresponding Jacobi's last multiplier for 3D systems. We illustrate our cons…
We characterize and study variable importance (VIMP) and pairwise variable associations in binary regression trees. A key component involves the node mean squared error for a quantity we refer to as a maximal subtree. The theory naturally extends from single trees to ensembles of trees and applies to methods like rando…
Paper extends variational formula for Bismut-Cheeger eta form, proving key theorems in K-theory.
problem Extending variational formula for Bismut-Cheeger eta form without kernel bundle assumption.
method Twisting spinc Dirac operators by isomorphic vector bundles, proving Z2-graded additivity. result Analytic index in differential K-theory is a well-defined group homomorphism, and Riemann-Roch-Grothendieck theorem in R/Z K-theory. Theory explains how AI models can predict unseen tasks without labeled data.
problem Understanding how AI models can generalize to unseen tasks.
method Developed a theoretical framework to analyze zero-shot prediction.
result Identified key quantities and independence relationships for generalization.