arXiv research

This paper offers a mathematical introduction to GANs.

This is an introduction to a provisional mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ supersymmetric gauge theories, studied in arXiv:1503.03676, arXiv:1601.03586

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…

We provide an introduction to selected recent advances in the mathematical understanding of Einstein's theory of gravitation.

Advanced mathematical relativity course for math and physics students.

This is an introduction to a provisional mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ supersymmetric gauge theories, studied in arXiv:1503.03676, arXiv:1601.03586. This is an expanded version of an article arXiv:1612.09014 appeared in the 61st DAISUUGAKU symposium proceeding (2016), wri…

These are the very unpretentious lecture notes for the minicourse "Introduction to evolution equations in Geometry," a part of the Brazilian Colloquium of Mathematics held at IMPA, in July of 2009.

Explains isometric immersions and their applications.

AI aids in mathematics research and problem-solving.

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".

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.

Introduces supermanifolds and supersymmetry for mathematicians.

This is the introduction I wrote for the multi-authored book "From Riemann to differential geometry and relativity", edited by L. Ji, A. Papadopoulos and S. Yamada (Berlin, Springer verlag, 2017). The book consists of twenty chapters, written by various authors. This introduction, besides giving the information on the …

Introduces Kundt spaces using geometric language.

These are the lecture notes for the summer course given for 2018 Mathematical Finance Summer School at Shandong Unversity. It contains a brief introduction to the Kyle model and the related topics in filtering, enlargement of filtrations and Markov bridges.

Shannon's mathematical theory of communication defines fundamental limits on how much information can be transmitted between the different components of any man-made or biological system. This paper is an informal but rigorous introduction to the main ideas implicit in Shannon's theory. An annotated reading list is pro…

After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra cohomology: a brief introduction 4. Symmetric polynomials and higher order cocycles 5…

Explains Einstein's theory of gravity for mathematicians.

Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from applied and computational mathematics; notably, in calculus, approximation theory, optimization and linear algebra. This article provides a very…

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.

The restricted Boltzmann machine is a network of stochastic units with undirected interactions between pairs of visible and hidden units. This model was popularized as a building block of deep learning architectures and has continued to play an important role in applied and theoretical machine learning. Restricted Bolt…

Introduces basics of supergeometry for PhD students.

Book introduces deep learning methods with math, theory, and applications.

This is neither an elementary introduction to singularity theory nor a specialized treatise containing many new theorems. The purpose of this little book is to invite the reader on a mathematical promenade. We pay a visit to Hipparchus, Newton and Gauss, but also to many contemporary mathematicians. We play with a bit …

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.

Sequences of Nilsson type appear in abundance in Algebraic Geometry, Enumerative Combinatorics, Mathematical Physics and Quantum Topology. We give an elementary introduction on this subject, including the definition of sequences of Nilsson type and the uniqueness, existence, and effective computation of their asymptoti…

The goal of this article is to give an elementary introduction to Dirac geometry and group-valued moment maps, via pure spinors. The material is based on my lectures at the summer school on 'Poisson geometry in Mathematics and Physics' at Keio University, June 2006.

Quantum character varieties unify four construction methods.

Survey on advanced gauge theory concepts.

Introduces machine learning basics and algorithms.

This paper is based on the introduction to the monograph ``Double affine Hecke algebras'' to be published by Cambridge University Press. The connections with Knizhnik-Zamolodchikov equations, Kac-Moody algebras, tau-function, harmonic analysis on symmetric spaces, and special functions are discussed. The rank one case …

This is the introduction and bibliography for lecture notes of a course given at the Summer School on Noncommutative Geometry and Applications, sponsored by the European Mathematical Society, at Monsaraz and Lisboa, Portugal, September 1-10, 1997. In the published version, an epilogue of recent developments and many ne…

The Ricci flow has been of fundamental importance in mathematics, most famously though its use as a tool for proving the Poincaré Conjecture and Thurston's Geometrization Conjecture. It has a parallel life in physics, arising as the first order approximation of the Renormalization Group flow for the nonlinear sigma mod…

This short note serves as a historical introduction to the Hopf problem: "Does there exist a complex structure on $S^6$?" This unsolved mathematical question was the subject of the Conference "MAM 1 $-$ (Non-)Existence of Complex Structures on $S^6$", which took place at Philipps-Universität Marburg, Germany, between M…

This monograph aims at providing an introduction to key concepts, algorithms, and theoretical results in machine learning. The treatment concentrates on probabilistic models for supervised and unsupervised learning problems. It introduces fundamental concepts and algorithms by building on first principles, while also e…

Inverse spectral theory reveals shapes from sound.

This short expository note gives an elementary introduction to the study of dynamics on certain moduli spaces, and in particular the recent breakthrough result of Eskin, Mirzakhani, and Mohammadi. We also discuss the context and applications of this result, and connections to other areas of mathematics such as algebrai…

Introduces triangular transport for uncertain data.

Mathematical theory of super fiber bundles and connections developed.

GANs analyzed for performance and training issues.

We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of the equation, and some recent progress towards its solution. In dimension 3 we …

Interacting particle methods are increasingly used to sample from complex and high-dimensional distributions. These stochastic particle integration techniques can be interpreted as an universal acceptance-rejection sequential particle sampler equipped with adaptive and interacting recycling mechanisms. Practically, the…

Introduces minimal surfaces to undergraduates.

Introduces data ethics for mathematicians, covering background, open data, and privacy.

Lecture notes on conifold transitions between Calabi-Yau manifolds.

This is an elementary introduction to a method for studying harmonic maps into symmetric spaces, and in particular for studying constant mean curvature (CMC) surfaces, that was developed by J. Dorfmeister, F. Pedit and H. Wu. There already exist a number of other introductions to this method, but all of them require a …

This report is an introduction to mathematical map colouring and the problems posed by Heawood in his paper of 1890. There will be a brief discussion of the Map Colour Theorem; then we will move towards investigating empire maps in the plane and the recent contributions by Wessel. Finally we will conclude with a discus…

In this text I present some problems which led to the introduction of special kinds of graphs as tools for studying singular points of algebraic surfaces. I explain how such graphs were first described using words, and how several classification problems made it necessary to draw them, leading to the elaboration of a s…