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.
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.
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.
AI mirrors modern math's autonomous development, raising interpretive challenges.
problem AI's effectiveness in math mirrors historical autonomy of math.
method Analyzes historical evolution of modern mathematics and AI's role.
result AI's affinity with math's historical autonomy suggests interpretive limits.
Self-supervised skip-tree training improves mathematical reasoning in language models.
problem Improving logical reasoning in language models for formal mathematics.
method Self-supervised language modeling on mathematical formulas, skip-tree task.
result Models trained on skip-tree task outperform standard models in mathematical reasoning tasks.
This paper offers a mathematical introduction to GANs.
problem Understanding GANs from a mathematical perspective.
method Mathematical analysis of GANs.
result Provides clarity for math-oriented students.
Explains isometric immersions and their applications.
problem Isometric immersions and their applications in math and physics.
method Historical overview and applications.
result Explains the importance and applications of isometric immersions.
Mathematical model predicts international trade and global economy dynamics.
problem Understanding complex international trade and economy interactions.
method Developed a mathematical model for non-equilibrium processes in open systems.
result Predicted model accurately reflects international trade and economy.
Higher topos theory applied to physics.
problem No specific problem stated.
method Exposition of higher topos theory.
result No specific key result mentioned.
Quantum cohomology connects quantum physics with classical math.
problem Quantum cohomology's relevance in modern mathematics.
method Informal review and discussion of connections.
result Quantum cohomology still has significant relevance in mathematics.
We give a general overview of the influence of William Thurston on the French mathematical school and we show how some of the major problems he solved are rooted in the French mathematical tradition. At the same time, we survey some of Thurston's major results and their impact. The final version of this paper will appe…
These notes grew out of a lecture course on mathematical methods of classical physics for students of mathematics and mathematical physics at the master's level. Also, physicists with a strong interest in mathematics may find this text useful as a resource complementary to existing textbooks on classical physics. Topic…
Improves recommender systems using mathematical principles.
problem Accuracy and speed of recommender systems.
method Explains and analyzes mathematical principles and algorithms.
result Describes the strengths and weaknesses of mathematical distance methods.
Lean 4 library formalizes mathematical finance, verifying over 200 theorems.
problem Formal verification of complex financial mathematics.
method Lean 4 proof assistant, Mathlib, BrownianMotion package, formal verification of over 200 theorems.
result Formal verification yields certified unification of known financial results.
These lecture notes review the topological string theory and its applications to mathematics and physics. They expand on material presented at the Takagi Lectures of the Mathematical Society of Japan on 21 June 2008 at Department of Mathematics, Kyoto University.
Jones Polynomial shows unity in math.
problem No specific problem stated.
method Discussion of Jones Polynomial.
result Illustrates unity between different mathematical fields.
A tribute to Norbert A'Campo's life and mathematics.
problem None explicitly stated in the abstract.
method None explicitly stated in the abstract.
result No specific key result mentioned.
Mathematical framework for language models processes text and predicts next tokens.
problem Understanding and optimizing the performance of large language models.
method Describes encoding, prediction models, learning from data, and deployment of LLMs.
result Demonstrates remarkable empirical successes and provides a platform for further research.
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…
Recent years have seen noteworthy progress in the mathematical formulation of quantum field theory and perturbative string theory. We give a brief survey of these developments. It serves as an introduction to the more detailed collection "Mathematical Foundations of Quantum Field Theory and Perturbative String Theory".
Paper proves mathematically that poisoning datasets can be detected.
problem Detecting data poisoning attacks in datasets.
method Mathematical definition and Conformal Separability Test.
result Dataset poisoning can be effectively detected.
Lean 4 library formalizes mathematical finance, verifying over 200 theorems.
problem Formal verification of complex financial mathematics.
method Lean 4 proof assistant, Mathlib, and BrownianMotion package.
result Formal verification yields certified unification of known results.
Salzmann's legacy in mathematics documented.
problem Mathematical challenges faced by Salzmann.
method Comprehensive review of Salzmann's work and bibliography.
result Salzmann's far-reaching influence in mathematics.
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.
Extended a mathematical inequality by Andrews.
problem Mathematical inequality by Andrews.
method Proved extensions of Andrews inequality.
result Extended mathematical inequality by Andrews.
Scientific documents rely on both mathematics and text to communicate ideas. Inspired by the topical correspondence between mathematical equations and word contexts observed in scientific texts, we propose a novel topic model that jointly generates mathematical equations and their surrounding text (TopicEq). Using an e…
Study of influenza A virus spread using mathematical equations.
problem Understanding the spread of influenza A virus infection.
method Mathematical model and analysis of dynamical system.
result Surface trajectories and asymptotic behavior of the system.
We provide an introduction to selected recent advances in the mathematical understanding of Einstein's theory of gravitation.
1. Translated by Thomas E. Cecil, Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610, USA; E-mail address: cecil@mathcs.holycross.edu 2. Typed by Wenjiao Yan, School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 10…
Ray-Singer torsion is a mathematical concept with applications in physics.
problem No specific problem stated in the abstract.
method No specific method stated in the abstract.
result No specific key result stated in the abstract.
Transformer models can solve complex math problems with less data.
problem Solving complex symbolic mathematics problems with limited data.
method Pretrain transformer models on language translation tasks and fine-tune for symbolic math.
result Pretrained transformer models achieve comparable accuracy to state-of-the-art models with less data.
Golden age of mathematical finance in the late 20th century.
problem Foundations of mathematical finance during the late 20th century.
method Collaboration between economists and probabilists.
result Established two fundamental theorems of arbitrage theory and close formulas for options.
Mathematical reasoning---a core ability within human intelligence---presents some unique challenges as a domain: we do not come to understand and solve mathematical problems primarily on the back of experience and evidence, but on the basis of inferring, learning, and exploiting laws, axioms, and symbol manipulation ru…
This chapter is based on lectures on Randomized Numerical Linear Algebra from the 2016 Park City Mathematics Institute summer school on The Mathematics of Data.
This article reviews mathematical insights into neural networks and machine learning.
problem Understanding the success and subtleties of neural network-based machine learning.
method Rigorous mathematical analysis, numerical experiments, and simplified models.
result Identification of open problems in the field.
The aim of this paper is to discuss some applications of general topology in computer algorithms including modeling and simulation, and also in computer graphics and image processing. While the progress in these areas heavily depends on advances in computing hardware, the major intellectual achievements are the algorit…
Benchmark for math reasoning models from human proofs.
problem Measuring and accelerating machine learning models in high-level mathematical reasoning.
method Built a non-synthetic dataset from theorem prover proofs, defined a task for model to fill in missing propositions, used hierarchical transformer to improve performance.
result Neural models can capture non-trivial mathematical reasoning, hierarchical transformer outperforms baseline.
This paper presents relevant modern mathematical formulations for (classical) gauge field theories, namely, ordinary differential geometry, noncommutative geometry, and transitive Lie algebroids. They provide rigorous frameworks to describe Yang-Mills-Higgs theories or gravitation theories, and each of them improves th…
This is a survey article, based on the author's lectures in the 2015 Current developments in Mathematics meeting; published in "Current developments in Mathematics". Version 2, references corrected and added.
These lecture notes provide a self-contained introduction to the mathematical methods required in a Bachelor degree programme in Business, Economics, or Management. In particular, the topics covered comprise real-valued vector and matrix algebra, systems of linear algebraic equations, Leontief's stationary input-output…
Mathematical theory of super fiber bundles and connections developed.
problem Modeling anticommuting fermionic fields in mathematical physics.
method Detailed introduction to super fiber bundles, relative supermanifolds, and connections; construction of parallel transport map.
result Construction and comparison of parallel transport map with other methods in the literature.
We give a brief survey on aspects of the local index theory as developed from the mathematical works of V. K. Patodi. It is dedicated to the 70th anniversary of Patodi.
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.
Formalizes synthetic differential geometry in Lean.
problem Formalizing synthetic differential geometry in a proof assistant.
method Formalization of synthetic differential geometry with Lean and mathlib.
result Proves a Taylor theorem for functions of several variables.
Mathematical study shows post-hoc explanations are better than attention weights alone.
problem Understanding the internal behavior of attention-based models.
method Mathematical analysis of a simple attention-based architecture.
result Post-hoc explanations provide more useful insights than attention weights alone.
Mathematical framework for differential machine learning in finance.
problem Theoretical assumptions in financial models and their impact on machine learning algorithms.
method Rigorous mathematical framework for differential machine learning in finance.
result Theoretical grounding enhances the predictive capabilities of neural networks in financial applications.
Machine learning impacts computational math, offering new functions approximations.
problem Machine learning's black box nature hinders further progress in computational math.
method Analyzes machine learning's impact on computational math and vice versa.
result Integrating computational math with machine learning can enhance both fields.