Sullivan discusses his contributions to math and physics.
problem None explicitly stated in the abstract.
method Personal overview of Dennis Sullivan's work.
result Sullivan's work spans mathematics and physics.
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,742 papers · 148 categories
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48 results for “mathematical spaces”
Study describes limits of surfaces in a mathematical space.
problem Understanding limits of surfaces in mathematical spaces.
method Completely described Gromov-Hausdorff closure of surfaces.
result Completely described the closure of surfaces.
The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up-dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iordanescu [364w], where sugestions for further developments ar…
In this expository review we discuss various aspects of gauge theory. While the focus is on mathematics, wherever possible we make contact with theoretical high energy physics. Particular emphasis is placed on instantons and monopoles, which admit physical interpretation, and yield interesting and nontrivial mathematic…
The mathematical features of a string theory compactification determine the physics of the effective four-dimensional theory. For this reason, understanding the mathematical structure of the possible compactification spaces is of profound importance. It is well established that the compactification space for M-Theory m…
Virtual reality brings non-Euclidean geometry to life.
problem Understanding non-Euclidean geometry is challenging.
method Interactive visualizations in virtual reality.
result Users can experience non-Euclidean geometry firsthand.
Corrects errors in Hans' pseudocovering spaces paper.
problem Mathematical errors and citation issues in Hans' pseudocovering spaces paper.
method Identifies and corrects errors in Hans' previous work.
result Addresses mathematical and citation errors in Hans' pseudocovering spaces paper.
Higgs bundles appeared a few decades ago as solutions to certain equations from physics and have attracted much attention in geometry as well as other areas of mathematics and physics. Here, we take a very informal stroll through some aspects of linear algebra that anticipate the deeper structure in the moduli space of…
Mathematical framework for minimum enclosing ball problem.
problem Determining the smallest sphere enclosing a set in d-dimensional space.
method Theoretical framework based on enclosing and partitioning theorems.
result Bounds and relations between circumradius, inradius, diameter, and width.
The paper develops mathematical models for neural networks using non-compact symmetric spaces.
problem Developing mathematical models for neural networks using non-compact symmetric spaces.
method Introducing layers modeled as non-compact symmetric spaces, each mapped onto the next by solvable group homomorphisms.
result Group theoretical construction of separators for all non-compact symmetric spaces and uniformization of specific surfaces.
Mathematical Foundations of Regression Methods for the approximation of the Forward Initial Marginq-fin.RM
This paper provides mathematical foundations for regression methods used in forward initial margin approximation.
problem Developing robust methods for approximating forward initial margin.
method Introduces mathematical rigor to show that regression methods are variations of approximating the conditional expectation function.
result Each regression method is a numerical estimation of the conditional expectation with a different functional form.
The braneworld theory appear with the purpose of solving the problem of the hierarchy of the fundamental interactions. The perspectives of the theory emerge as a new physics, for example, deviation of the law of Newton's gravity. One of the principles of the theory is to suppose that the braneworld is local submanifold…
Model for material elasticity and plasticity using networks.
problem Understanding the elasticity and plasticity of materials.
method Developed a mathematical model based on networks, defining tension tensor for periodic graphs.
result The model explains elasticity and plasticity through local moves on graphs.
Main mathematical applications of Frobenius manifolds are in the theory of Gromov - Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes rema…
Explains Conway's tangle trick and its mathematical origins.
problem Understanding the relationship between braids and elliptic curves.
method Discusses the tangle trick, its mathematical underpinnings, and historical context.
result Establishes the connection between braids and elliptic curves.
This paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration. The framework unifies many well-known duality frameworks from operations research and mathematical finance. The unification allows the extension …
Mathematical approach defines stability conditions for ML models.
problem Ensuring stability of machine learning models.
method Adopted topological and metric spaces theory to define stability.
result Stability of ML models depends on topological properties of classification sets.
Semi-static trading strategies make frequent appearances in mathematical finance, where dynamic trading in a liquid asset is combined with static buy-and-hold positions in options on that asset. We show that the space of outcomes of such strategies can have very poor closure properties when all European options for a f…
Introduces supermanifolds and supersymmetry for mathematicians.
problem Explains mathematical aspects of supersymmetry.
method Uses and develops results from arXiv:math-ph/0603045.
result Develops detailed analysis of maps of supermanifolds.
We design and conduct a simple experiment to study whether neural networks can perform several steps of approximate reasoning in a fixed dimensional latent space. The set of rewrites (i.e. transformations) that can be successfully performed on a statement represents essential semantic features of the statement. We can …
We give a mathematical foundation for, and numerical demonstration of, the existence of mean curvature 1 surfaces of genus 1 with either two elliptic ends or two hyperbolic ends in de Sitter 3-space. An end of a mean curvature 1 surface is an ``elliptic end'' (resp. a ``hyperbolic end'') if the monodromy matrix at the …
Math and dance blend in choreographer's research.
problem Exploring intersections between dance and mathematics.
method Choreographic practice and mathematical concepts.
result Examples of fractals, braids in choreography.
Improves neural network search in combinatorial spaces of mathematical symbols.
problem Early commitment and initialization bias limit exploration in neural network search.
method Entropy regularization and distribution initialization methods.
result Improves performance, increases sample efficiency, lowers solution complexity.
Complex abelian varieties and quantum error correction: a mathematical framework for GKP codesmath.AG
Quantum codes linked to abelian varieties, providing mathematical rigor.
problem Quantum error correction through complex abelian varieties.
method Mathematical formulation of Gottesman-Kitaev-Preskill codes using abelian varieties.
result Asymptotic isometry of encoding, precise gate realizations, and failure probability optimization.
Estimates the rational homological dimension of Riemann surfaces with boundary and marked points.
problem Estimating the homological dimensions of Riemann surfaces with boundary and marked points.
method Developed an estimate for the rational homological dimension of Riemann surfaces with possible boundary and marked points.
result Provided an estimate for the rational homological dimension of Riemann surfaces with boundary and marked points.
Neural networks are mathematically represented via quiver representations.
problem Understanding how neural networks process data and create representations.
method Representing neural networks as quiver representations with activation functions.
result Neural networks' computations can be studied algebraically and geometrically.
New mathematical foundations for stable RKHSs improve system identification.
problem Improving stability tests and modeling of impulse responses.
method Providing new structural properties and stability conditions for stable RKHSs.
result Any stable kernel admits feature maps induced by orthogonal eigenvectors in l2.
Extends Bayesian theory to handle complex interdependencies in multidimensional event spaces.
problem Complex interdependencies between events and hypotheses sets in real-world systems.
method Developed a mathematical formalism for modeling complex relationships through rigorous derivation and validated using analytical proofs, simulations, and case studies.
result MDSE theory improves prediction accuracy by 15-20% compared to standard Bayesian methods in high interdimensionality datasets.
Research connects physics and math through ceramic art of Riemann surfaces.
problem Classifying curves on Riemann surfaces using 'pairs of pants'.
method Exploring mathematical concepts through artistic ceramics.
result Illustrates the classification of curves on Riemann surfaces.
Novel tests for genetic independence in high-dimensional data.
problem Testing independence in genetics studies with many variables.
method Defining premetric structures on genetic data support spaces.
result Solid theoretical framework and computationally-efficient implementations.
In our previous papers [Far East Journal of Mathematical Sciences, 35 (2009), 211-223] and [International Journal of Pure and Applied Mathematics, 60 (2010), 15-24] we have developed the theory of Weil prolongation, Weil exponentiability and microlinearity for Frolicher spaces. In this paper we will relativize it so as…
Advanced mathematical relativity course for math and physics students.
problem Mathematical foundations of general relativity.
method Comprehensive lecture notes covering advanced topics.
result Comprehensive coverage of mathematical relativity for advanced students.
Solves four problems related to sphere families in 3D space.
problem Four basic problems of sphere families in Euclidean 3-space.
method Solves all four basic problems of sphere families in Euclidean 3-space.
result All four basic problems are solved.
Deep learning finds mathematical equations from data.
problem Discovering underlying mathematical expressions from datasets.
method Uses a recurrent neural network to search for mathematical expressions and optimizes using a risk-seeking policy gradient.
result Outperforms existing methods in recovering exact symbolic expressions.
We construct a mathematical framework for twisted N=2 supersymmetric topological quantum field theory on a 4-manifold. Supersymmetry in flat space is defined and the twist homomorphism is constructed, giving us a supermanifold that is the total space of an odd vector bundle over the even 4-manifold. A special category …
We survey some major contributions to Riemann's moduli space and Teichm{ü}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space a…
The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely used in mathematics today and, up to our knowledge, has never been used in engineering applications or mathematical physics. The main purpose of this paper, an extended version of a lecture at the second workshop on Differential Equations by Al…
The paper surveys mathematical results on filtration enlargement with financial examples.
problem Mathematical finance applications of filtration enlargement theory.
method Exhaustive survey and interpretation of key results from literature.
result Provides a compendium of known mathematical results for mathematical finance researchers.
Mathematical general relativity reviewed.
problem Challenges in mathematical modeling of general relativity.
method Selected topics reviewed.
result Insights into mathematical models of general relativity.
Mathematical formulas for elliptic curve integrals solve anomaly equations.
problem Mathematical formulation of contact term singularities on elliptic curves.
method Residue formulas and holomorphic anomaly equations.
result Regularized integrals on elliptic curves satisfy holomorphic anomaly equations.
Survey on moduli spaces of differentials from algebraic geometry perspective.
problem Understanding the topology of moduli spaces of differentials remains limited.
method Algebraic geometry perspective, connections to various fields.
result Many open problems and connections to other fields.
In an abstract Wiener space setting, we constract a rigorous mathematical model of the one-loop approximation of the perturbative Chern-Simons integral, and derive its explicit asymptotic expansion for stochastic Wilson lines.
We describe the relationship between complex-valued harmonic morphisms from Minkowski 4-space} and the shear-free ray congruences of mathematical physics. Then we show how a horizontally conformal submersion on a domain of Euclidean 3-space gives the boundary values at infinity of a complex-valued harmonic morphism on …
A space of cyclohedramath.QA
The real points of the Deligne-Knudsen-Mumford moduli space of marked points on the sphere has a natural tiling by associahedra. We extend this idea to create a moduli space tiled by cyclohedra. We explore the structure of this space, coming from blow-ups of hyperplane arrangements, as well as discuss possibilities of …
Some aspects of Dirac spinors are resumed and studied in order to interpret mathematically the P and T operations in a gravitational field.
We start recalling with critical eyes the mathematical methods used in gauge theory and prove that they are not coherent with continuum mechanics, in particular the analytical mechanics of rigid bodies or hydrodynamics, though using the same group theoretical methods and despite the well known couplings existing betwee…
AI aids in mathematics research and problem-solving.
problem Complex mathematical problems and discoveries.
method Explains AI principles and diverse applications in math.
result AI assists in discovering patterns, proving theorems, and challenging conjectures.
MAD framework learns operators from physics-embedded data efficiently.
problem Data-driven methods require costly labeled datasets and model-driven techniques face efficiency-accuracy trade-offs.
method Integrates physical laws with data-driven learning to generate physics-embedded analytical solutions and synthetic data.
result Eliminates dependence on experimental or simulated training data, enabling efficient operator learning across multi-parameter systems.