arXiv research

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Formalizes synthetic differential geometry in Lean.

This is an introductory review of topological field theories (TFTs) called AKSZ sigma models. The AKSZ construction is a mathematical formulation for the construction and analyses of a large class of TFTs, inspired by the Batalin-Vilkovisky formalism of gauge theories. We begin by considering a simple two-dimensional t…

Paper constructs braid invariants using tropical Ptolemy equation.

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…

It has been conjectured that every $(2+1)$-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair $(G,λ)$, where $G$ is a compact Lie group, and $λ\in H^4(BG;Z)$ a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the…

New algebraic structures called cyclic Lie-Rinehart algebras are defined.

Quantum character varieties unify four construction methods.

Explains Kirby's work on 4-manifold trisections.

We constructed physically stable sp2 negatively curved cubic carbon structures which reticulate a Schwarz P-like surface. The method for constructing such crystal structures is based on the notion of the standard realization of abstract crystal lattices. In this paper, we expound on the mathematical method to construct…

Lean 4 library formalizes mathematical finance, verifying over 200 theorems.

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…

The paper develops mathematical models for neural networks using non-compact symmetric spaces.

Based on criteria of mathematical simplicity and consistency with empirical market data, a model with volatility driven by fractional noise has been constructed which provides a fairly accurate mathematical parametrization of the data. Here, some features of the model are discussed and, using agent-based models, one tr…

Mathematical theory of super fiber bundles and connections developed.

Lean 4 library formalizes mathematical finance, verifying over 200 theorems.

Mathematical framework for differential machine learning in finance.

Survey on advanced gauge theory concepts.

New sl(2) action defined on a mathematical module.

This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories.

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 …

Researchers created a continuous Markov martingale that mimics Brownian motion but lacks the strong Markov property.

New integral transforms solve multilayer heat equations.

Clarifies mathematical aspects of Picture Changing Operators.

This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are invoked and compared.

Lagrange's map construction ideas influenced later mathematicians.

We describe a mathematically rigorous differential model for B-type open-closed topological Landau-Ginzburg theories defined by a pair $(X,W)$, where $X$ is a non-compact Kählerian manifold with holomorphically trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$ and whose criti…

The design of codes for communicating reliably over a statistically well defined channel is an important endeavor involving deep mathematical research and wide-ranging practical applications. In this work, we present the first family of codes obtained via deep learning, which significantly beats state-of-the-art codes …

Mathematical framework for field theories on Finsler spacetimes.

In this survey, we discuss several different types of gradient boosting algorithms and illustrate their mathematical frameworks in detail: 1. introduction of gradient boosting leads to 2. objective function optimization, 3. loss function estimations, and 4. model constructions. 5. application of boosting in ranking.

The concept of a Point Cloud has played an increasingly important role in many areas of Engineering, Science, and Mathematics. Examples are: LIDAR, 3D-Printing, Data Analysis, Computer Graphics, Machine Learning, Mathematical Visualization, Numerical Analysis, and Monte Carlo Methods. Entering point cloud into Google r…

Calculation of an optimal tariff is a principal challenge for pricing actuaries. In this contribution we are concerned with the renewal insurance business discussing various mathematical aspects of calculation of an optimal renewal tariff. Our motivation comes from two important actuarial tasks, namely a) construction …

Mathematical construction of Chern-Simons partition function using reflection positivity.

Mathematical analysis improves SGMs, resolving memorization issues.

We prove Transformers can learn diverse Gröbner bases.

We show how networks of Wilson lines realize quantum groups U_q(sl(m)), for arbitrary m, in 3d SU(N) Chern-Simons theory. Lifting this construction to foams of surface operators in 4d theory we find that rich structure of junctions is encoded in combinatorics of planar diagrams. For a particular choice of surface opera…

We extend Massey products from cohomology to differential cohomology via stacks, organizing and generalizing existing constructions in Deligne cohomology. We study the properties and show how they are related to more classical Massey products in de Rham, singular, and Deligne cohomology. The setting and the algebraic m…

Non-unique option pricing in Heston model analyzed mathematically.

This note, in a rather expository manner, serves as a conceptional introduction to the certain underlying mathematical structures encoding the geometric quantization formalism and the construction of Witten's quantum invariants, which is in fact organized in the language topological quantum field theory.

This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic, combinatorial and topological structures. We include a description of previous work on the construction of Hilbert spaces from the states of the bracket polynomial with applications to algorithms fo…

This paper shows connections between two complex mathematical theories are equivalent.

This article offers an introductory look at Khovanov homology and its historical context.

Reconstructs supersymmetry and supergravity in complexified Z/2-graded Algebraic Geometry.

Mathematical framework for brane quantization using SYZ mirror symmetry.

Mathematical construction of vertex algebra representations from integrable G2 structures.

A discrete time probabilistic model, for optimal equity allocation and portfolio selection, is formulated so as to apply to (at least) reinsurance. In the context of a company with several portfolios (or subsidiaries), representing both liabilities and assets, it is proved that the model has solutions respecting constr…

Survey on Coxeter groups for Lie group examples.

New risk measures improve portfolio diversification and stability.