Basic Mathematics Serge Lang Djvu Download
Celena Holtzberg
Homological Algebra is a tool that appears in almost all branches of mathematics, ranging from mathematical physics, PDEs, Lie groups and representation theory, through topology (where it originated) and on to number theory and combinatorics. We will cover the basics of homology of chain complexes, elementary category language, derived functors, Ext and Tor, spectral sequences, simplicial methods. There are various further topics and directions which will depend on the available time and interests of the class. Each student will have a project to do - these can be chosen from an application of homological methods in the student's area of interest.
Systems and control theory is one of the most central and fast growing areas of applied mathematics and engineering. This course provides a basic introduction to the mathematics of finite-dimensional, continuous-time, deterministic control systems at the beginning graduate student level. The course is intended for PhD students in applied mathematics, and for engineering graduate students with a background in real analysis and nonlinear ordinary differential equations. It is designed to help students prepare for interdisciplinary research at the interface of applied mathematics and control engineering. This is a rigorous, proof-oriented systems theory course that goes beyond classical frequency-domain or more applied engineering courses. Emphasis will be placed on controllability and stabilization.
Topology contains at least three (overlapping) subbranches: general (or point-set) topology, geometric topology and algebraic topology. General topology grew out of the successful attempt to generalize some basic ideas and theorems (e.g., continuity, open and closed sets, the Intermediate Value and Bolzano-Weierstrass Theorems) from Euclidean spaces to more general spaces. The core of this course will be a thorough introduction to the central ideas of general topology (Chapters 2 - 5 of Munkres). This material is fundamental in much of modern mathematics. Time permitting, we will also look briefly at the ideas of homotopy and the fundamental group (Chapter 9), subjects that belong to the algebraic-geometric side of topology, and will be covered in greater depth in 7512.
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