相关论文: Why everyone should know number theory
These are partial lecture notes from the fifteen Ess\'en Lectures for graduate students at Uppsala University given (in four days!) in June 2013.
We illustrate the concept of mathematical proof.
These are the lecture notes that accompanied the course of the same name that I taught at the Eindhoven University of Technology from 2021 to 2023. The course is intended as an introduction to neural networks for mathematics students at the…
We go into the need for, and the requirements on, a formal theory of budgets. We present a simple algebraic theory of rational budgets, i.e., budgets in which amounts of money are specified by functions on the rational numbers. This theory…
Knowledge representation is a popular research field in IT. As mathematical knowledge is most formalized, its representation is important and interesting. Mathematical knowledge consists of various mathematical theories. In this paper we…
The ongoing progress in quantum theory emphasizes the crucial role of the very basic principles of quantum theory. However, this is not properly followed in teaching quantum mechanics on the graduate and undergraduate levels of physics…
This is a rendering of a review talk on the state of String Theory, given at the EPS-2003 Conference, intended for a wide audience of experimental and theoretical physicists. It emphasizes general ideas rather than technical aspects.
This is an expository paper, based on by a talk given at the AWM Research Symposium 2017. It is intended as a gentle introduction to geometric group theory with a focus on the notion of hyperbolicity, a theme that has inspired the field…
These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, July-August 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being…
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…
These are the notes on two-dimensional conformal field theory, based on a lecture course for graduate math students, given by P.M. in fall 2022 at the University of Notre Dame. These notes are intended to be substantially reworked and…
Quantum computing is a good way to justify difficult physics experiments. But until quantum computers are built, do computer scientists need to know anything about quantum information? In fact, quantum computing is not merely a recipe for…
We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions.…
These lectures are intended to provide the theoretical basis of describing high-energy particle collisions at a level appropriate to graduate students in experimental high energy physics. They are supposed to be familiar with quantum…
The purpose of this note is to give an accessible proof of Moliens Theorem in Invariant Theory, in the language of today's Linear Algebra and Group Theory, in order to prevent this beautiful theorem from being forgotten.
Quantum theory describes our universe incredibly successfully. To our classically-inclined brains, however, it is a bizarre description that requires a re-imagining of what fundamental reality, or "ontology", could look like. This thesis…
These are the preparatory notes for a Science & Music essay, "Playing by numbers", appeared in Nature 453 (2008) 988-989.
The purpose of this informal article is to introduce the reader to some of the objects and methods of the theory of p-adic representations. My hope is that students and mathematicians who are new to the subject will find it useful as a…
The quantum nature of a black hole is revealed using the simplest terms that one learns in undergraduate and beginning graduate courses. The exposition demonstrates -- vividly -- the importance and power of the quantum oscillator in…
The $abc$ conjecture is a very deep concept in number theory with wide application to many areas of number theory. In this article we introduce the conjecture and give examples of its applications. In particular we apply the $abc$…