Related papers: Undergraduate Lecture Notes in De Rham-Hodge Theor…
Content of the lectures is the following. Properties of transformations equivalent to ergodicity. Birkhoff's Theorem. Properties equivalent to weak mixing. On typical properties of transformations. Lego to construct transformations. Typical…
Formulating a quantum theory of gravity lies at the heart of fundamental theoretical physics. This collection of lecture notes encompasses a selection of topics that were covered in six mini-courses at the Nordita PhD school "Towards…
We present a simple proof of the Riemann's Hypothesis (RH) where only undergraduate mathematics is needed.
A general action is proposed for the fields of $q$-dimensional differential form over the compact Riemannian manifold of arbitrary dimensions. Mathematical tools are based on the well-known de Rham-Kodaira decomposing theorem on harmonic…
These are the lecture notes used in the course Understanding Quantum Chemistry for 1st year master chemistry at the Vrije Universiteit Amsterdam. The schedule is 2 weeks on HF and 2 weeks on DFT, which provides a quite rough ride for the…
We demonstrate that the two (1 + 1)-dimensional (2D) free 1-form Abelian gauge theory provides an interesting field theoretical model for the Hodge theory. The physical symmetries of the theory correspond to all the basic mathematical…
This is the draft of lecture notes for Phd students in Sichuan University. In this notes we expand Li-Ruan's paper with much more detailed explanations and calculations.
These notes are a self-contained introduction to Galois theory, designed for the student who has done a first course in abstract algebra.
These are lecture notes that are based on the lectures from a class I taught on the topic of Randomized Linear Algebra (RLA) at UC Berkeley during the Fall 2013 semester.
In these notes, we present a rigorous and self-contained introduction to the fundamental concepts and methods of quantum many-body theory. The text is designed to provide a solid theoretical foundation for the study of interacting quantum…
This is a written-up version of eight introductory lectures to the Hodge theory of projective manifolds. The table of contents should be self-explanatory. The only exception is section 8 where I discuss, in a simple example, a technique for…
The subject of the present work is the de Rham part of non-commutative Hodge structures on the periodic cyclic homology of differential graded algebras and categories. We discuss explicit formulas for the corresponding connection on the…
The Hierarchical Reference Theory (HRT) of fluids is a general framework for the description of phase transitions in microscopic models of classical and quantum statistical physics. The foundations of HRT are briefly reviewed in a…
We give a rather informal introduction to the theory of mixed Hodge modules for young mathematicians.
The present text is a collection of notes about differential geometry prepared to some extent as part of tutorials about topics and applications related to tensor calculus. They can be regarded as continuation to the previous notes on…
The aim of these lecture notes is to give a pedagogical introduction to the subject of non-holomorphic deformations of special geometry. This subject was first introduced in the context of N=2 BPS black holes, but has a wider range of…
Fourier Series is the second of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such as partial…
The basic ideas and the important role of gauge principles in modern elementary particle physics are outlined. There are three theoretically consistent gauge principles in quantum field theory: the spin-1 gauge principle of electromagnetism…
The point of view of these notes on the topic is to bring out the flavour that Representation Theory is an extension of the first course on Group Theory. We also emphasize the importance of the base field. These notes cover completely the…
Depending on the point of view, modern machine learning is either providing an unprecedented boost to the numerical methods of particle physics, or it is transforming the way we do science with vast amounts of complex data. In any case, it…