Related papers: Undergraduate Lecture Notes in De Rham-Hodge Theor…
We introduce a notion of a Hodge-proper stack and extend the method of Deligne-Illusie to prove the Hodge-to-de Rham degeneration in this setting. In order to reduce the statement in characteristic $0$ to characteristic $p$, we need to find…
These lecture notes provide an introduction to free probability theory, with a focus on tools and techniques useful in the study of large random matrices. Topics include freeness, free cumulants, additive and multiplicative free…
We compute the Hodge and de Rham cohomology of the classifying space BG (defined as etale cohomology on the algebraic stack BG) for reductive groups G over many fields, including fields of small characteristic. These calculations have a…
These are the lecture notes for a short course in topological string theory that I gave at Uppsala University in the fall of 2004. The notes are aimed at PhD students who have studied quantum field theory and general relativity, and who…
This is a graduate course on elementary quantum mechanics written for the benefit of undergraduate and graduate students. It is the English version of physics/0003106, which I did at the suggestion of several students from different…
The goal of this expository paper is to present the basics of geometric control theory suitable for advanced undergraduate or beginning graduate students with a solid background in advanced calculus and ordinary differential equations.
Linear and nonlinear Hodge-like systems for 1-forms are studied, with an assumption equivalent to complete integrability substituted for the requirement of closure under exterior differentiation. The systems are placed in a variational…
A theoretical scheme, based on a probabilistic generalization of the Hamilton's principle, is elaborated to obtain an unified description of more general dynamical behaviors determined both from a lagrangian function and by mechanisms not…
These are the lecture notes for the introductory graduate course I taught at Yale during Spring 2007. I mostly followed [GS], [BGV], [AB], [Par2], and there are no original results in these notes.
This text is intended as an introduction to magnetohydrodynamics in astrophysics, emphasizing a fast path to the elements essential for physical understanding. It assumes experience with concepts from fluid mechanics: the fluid equation of…
Expanded lecture notes. Preliminary version, comments are welcome.
This is a very gentle introductory course on quantum mechanics aimed at the first years of the undergraduate level. The basic concepts are introduced, with many applications and illustrations. Contains 12 short chapters of equal length,…
Lecture notes prepared for the Les Houches school "Quantum Geometry: Mathematical Methods for Gravity, Gauge Theories and Non-Perturbative Physics" that took place during the summer 2024. We cover the techniques to perform the exact…
These lectures present an elementary discussion of some background material relevant to the problem of de Sitter quantum gravity. The first two lectures discuss the classical geometry of de Sitter space and properties of quantum field…
We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of…
Quantum field theory has formed the conceptual framework of most of physics for more than sixty years. It incorporates a complete revision of our conception of the nature of matter and existence itself. Yet it is rarely taught, or even…
This text is aimed at undergraduates, or anyone else who enjoys thinking about shapes and numbers. The goal is to encourage the student to think deeply about seemingly simple things. The main objects of study are lines, squares, and the…
We study the response of quantum many-body systems to coupling some of their degrees of freedom to external gauge fields. This serves to understand the current Green functions and transport properties of interacting many-body systems. Our…
A unified theory of material defects, incorporating both the smooth and the singular descriptions, is presented based upon the theory of currents of Georges de Rham. The fundamental geometric entity of discourse is assumed to be represented…
This is a review article discussing the de Rham cohomology of period domains of Hodge structures. We explain it as the de Rham cohomology of differentiable stacks as of a moduli space. We also discuss the cohomology of the partial toroidal…