111 - 120 of 140 results for: MATH

Continuation of 256A.

Linear symplectic geometry and linear Hamiltonian systems. Symplectic manifolds and their Lagrangian submanifolds, local properties. Symplectic geometry and mechanics. Contact geometry and contact manifolds. Relations between symplectic and contact manifolds. Hamiltonian systems with symmetries. Momentum map and its properties. May be repeated for credit.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Continuation of 257A. May be repeated for credit.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Continuation of 257B. May be repeated for credit.

May be repeated for credit.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Introduction to the mathematics of the Fourier transform and how it arises in a number of imaging problems. Mathematical topics include the Fourier transform, the Plancherel theorem, Fourier series, the Shannon sampling theorem, the discrete Fourier transform, and the spectral representation of stationary stochastic processes. Computational topics include fast Fourier transforms (FFT) and nonuniform FFTs. Applications include Fourier imaging (the theory of diffraction, computed tomography, and magnetic resonance imaging) and the theory of compressive sensing. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Kac-Moody Lie algebras are infinite-dimensional Lie algebras whose theory is remarkably similar to finite-dimensional semisimple Lie algebras. Affine Lie algebras are the most important special case.We will develop some of the Kac-Moody theory, such as the Kac-Weyl character formula, before specializing to affine Lie algebras. Ideas from physics give a multiplication called fusion on the irreducible integrable representations of fixed level. Kac and Peterson showed that the characters and related "string functions" of these representations are modular forms, and the transformation properties of these theta functions of fixed level encode important information about the fusion ring. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc. May be repeated for credit.

Crystal Bases are combinatorial analogs of representation theorynof Lie groups. We will explore different aspects of thesenanalogies and develop rigorous purely combinatorial foundations.

Conformal Field Theory is a branch of physics with origins in solvable lattice models and string theory. But the mathematics that it has inspired has many applications in pure mathematics.nWe will give an introduction to this theory with related representation theories of the Virasoro and affine Lie algebras, and vertex operators.nnPrerequisites: we will not assume any particular knowledge from physics, but some knowledge of Lie algebras will be helpful.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.nnnMay be repeated for credit.