These draft notes are from a graduate course given by the author in Berkeley during the spring semester of 2005. They cover the basic ideas of a new, geometric approach to geometric measure theory. They begin with a new theory of exterior calculus at a single point. This infinitesimal theory extends, by linearity, to a discrete exterior theory, based at finitely many points. A general theory of calculus culminates by taking limits in Banach spaces, and is valid for domains called ``chainlets'' which are defined to be elements of the Banach spaces. Chainlets include manifolds, rough domains (e.g., fractals), soap films, foliations, and Euclidean space. Most of the work is at the level of the infinitesimal calculus, at a single point. The number of limits needed to get to the full theory is minimal. Tangent spaces are not used in these notes, although they can be defined within the theory. This new approach is made possible by giving the Grassmann algebra more geometric structure. As a result, much of geometric measure theory is simplified. Geometry is restored and significant results from the classical theory are expanded. Applications include existence of solutions to a problem of Plateau, an optimal Gauss-Green theorem and new models for Maxwell's equations.
I will explain my joint paper `Instantons moduli spaces and W-algebras' with A.Braverman, M.Finkelberg, arXiv:1406.2381. I will concentrate on the geometric part, that is a study of perverse sheaves on instanton moduli spaces. I place a particular emphasize on the hyperbolic restriction functor and stable envelop, which are our key tools, and appear also in other situations in geometric representation theory. These are lecture notes for a course in 2015 PCMI.
These are lecture notes for a short winter course at the Department of Mathematics, University of Coimbra, Portugal, December 6--8, 2018. The course was part of the 13th International Young Researchers Workshop on Geometry, Mechanics and Control. In three lectures I trace the work of three heroes of mathematics and mechanics: Euler, Poincaré, and Arnold. This leads up to the aim of the lectures: to explain Arnold's discovery from 1966 that solutions to Euler's equations for the motion of an incompressible fluid correspond to geodesics on the infinite-dimensional Riemannian manifold of volume p…
The paper aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A ge…
This is an expanded version of a three-hour minicourse given at the winterschool Winterbraids IV held in Dijon in February 2014. The aim of these lectures was to present some aspects of the dimer model to a geometrically minded audience. We spoke neither of braids nor of knots, but tried to show how several geometrical tools that we know and love (e.g. (co)homology, spin structures, real algebraic curves) can be applied to very natural problems in combinatorics and statistical physics. These lecture notes do not contain any new results, but give a (relatively original) account of the works of …
These notes present an introduction to the method of geometric quantization. We discuss the main theorems in a style suitable for a theoretical physicist with an eye towards the physical motivation and the interpretation of the geometric construction as providing a solution to Dirac's axioms of quantization. We provide in detail the examples of free relativistic particles, their corresponding quantum fields, and the bosonic string using formalism of double field theory. Based on lectures written by Gabriel Cardoso.
In this series of lectures, we (re)view the "geometric method" that reconstructs, from a geometric object: the "spectral curve", an integrable system, and in particular its Tau function, Baker-Akhiezer functions and "current amplitudes", all having an interpretation as CFT conformal blocks. The construction identifies Hamiltonians with cycles on the curve, and times with periods (integrals of forms over cycles). All the integrable structure is formulated in terms of homology of contours, the phase space is a space of cycles where the symplectic form is the intersection, the Hirota operator is …
These lecture notes (from the Second Autumn School in High Energy Physics and Quantum Field Theory, Yerevan 2014) cover a number of topics related to geometric quantization. Most of the material is presented from a physicist's point of view. The original notes are posted at \verb+http://theorphyslab-ysu.info/VW_ASW-2014/uploads/ArmeniaLectures.pdf+. The have been revised with some additions and changes, although referencing is still somewhat dated. These notes are posted here as they may be good background material for some recent papers.
