Related papers: Lectures on generalized geometry
This note is the sequel of "Geometric structures as variational objects, I." It generalizes the main result and perspectives of that work to a class of geometric structures that includes integrable almost-complex structures.
We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear…
This is a graduate-level introduction to C*-algebras, Hilbert C*-modules, vector bundles, and induced representations of groups and C*-algebras, with applications to quantization theory, phase space localization, and configuration space…
These notes grew out of two lectures I have given on CAT(0) cube complexes. I've tried to keep the material elementary and self-contained in order to keep the material easily accessible and to provide an elementary introduction on the topic…
These are lecture notes of a course held at IMPA, Rio de Janiero, in september 2010: the purpose was to present recent results on Kobayashi hyperbolicity in complex geometry. Our ultimate goal is to describe the results obtained on…
The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometrical, topological and analytical aspects. Beginning with a preliminary material, such as the basic results on…
Generalisations of geometry have emerged in various forms in the study of field theory and quantization. This mini-review focuses on the role of higher geometry in three selected physical applications. After motivating and describing some…
We give a modern account of the construction and structure of the space of generalized connections, an extension of the space of connections that plays a central role in loop quantum gravity.
These are lecture notes based on the first part of a course on 'Mathematical Data Science', which I taught to final year BSc students in the UK in 2019-2020. Topics include: concentration of measure in high dimensions; Gaussian random…
Coarse geometry, the branch of topology that studies the global properties of spaces, was originally developed for metric spaces and then Roe introduced coarse structures as a large-scale counterpart of uniformities. In the literature,…
This is an overview of our series of papers on the modular generalized Springer correspondence. It is an expansion of a lecture given by the second author in the Fifth Conference of the Tsinghua Sanya International Mathematics Forum, Sanya,…
Lectures on the construction on the moduli space of principal bundles, given in the Mini-School on Moduli Spaces at the Banach center (Warsaw) 26-30 April 2005.
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.
This paper is an overview of my recent work on abstract homomorphisms of algebraic groups. It is based on a talk given at the Conference on Group Actions and Applications in Geometry, Topology, and Analysis held in Kunming in July 2012.
Generalized Calabi-Gray manifolds are non-K\"ahler complex manifolds with very explicit geometry yet not being homogeneous. In this note, we demonstrate that how generalized Calabi-Gray manifolds can be used to answer some questions in…
This is a survey article, to appear in the Proceedings of the 2018 International Congress of Mathematicians. (Revised, with added and updated references.)
These lecture notes for the IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004) provide an overview of root systems, generalized associahedra, and the combinatorics of clusters. Lectures 1-2 cover classical material:…
These notes give a concise introduction to General Relativity at the advanced undergraduate level, starting from the weak field limit and gravitational waves, then introducing curved manifolds and Riemannian geometry. The nonlinear…
These notes are loosely based on an introductory course in algebraic geometry given at Rutgers University in Spring of 2024. We introduce some relatively advanced topics at the expense of the technical details.
This is an expanded version of my Shaw Prize Lecture delivered at the Chinese University of Hong Kong.