Combinatorial stability theory grew out of Shelah's work on Morley's conjecture concerned with the number of uncountable models of first-order theories. The subject gained a stronger geometric aspect and found applications to algebra and number theory through the work of Zilber, Hrushovski, Pillay and many others. In recent year, many of its methods were generalized to larger classes of theories. These lecture notes provide an introduction to this area.
Basics, trees, Caley's formula, matrix tree theorem, connectivity, theorems of Mader and Menger, Eulerian graphs, Hamilton cycles, theorems of Dirac, Ore, Erdös-Chvatal, matchings, theorems of Hall, König, Tutte, planar graphs, Euler's formula, Kuratowski's theorem, graph colorings, Brooks' theorem, 5-colorings of planar graphs, list colorings, Vizing's theorem, Ramsey theory, Turán's theorem
First Semester in Numerical Analysis with Julia presents the theory and methods, together with the implementation of the algorithms using the Julia programming language (version 1.1.0). The book covers computer arithmetic, root-finding, numerical quadrature and differentiation, and approximation theory. The reader is expected to have studied calculus and linear algebra. Some familiarity with a programming language is beneficial, but not required. The programming language Julia will be introduced
Lecture notes from Prof. Robert Gompf's Contact Topology course at UT Austin, Fall 2017.
(posted with permission from Robert Gompf)
These lecture notes were prepared to accompany a three-hour mini course entitled “Homotopy coherent structures” delivered at the summer school accompanying the “Floer homology and homotopy theory” conference at UCLA from July 10 - 14, 2017:
These are notes for a brief unit on representation theory at the end of a standard graduate abstract algebra course. The notes assume no special preparation for representation theory, but take advantage of efficiencies of exposition made possible by an assumed basic graduate-level background in module theory, canonical forms, etc.
These notes were originally prepared by Jane Gilman and William Thurston for a Freshman Honors course offered at Princeton in 1990. The course was designed for students who had calculus but needed a course that gave them some mathematical sophistication as well as calculus review. Many of the topics are non-standard including iteration,
the "3x+1"-game, cardinal arithmetic, self-similarity and fractals, but standard topics such as power series and convergence also appear.
These are notes from an experimental mathematics course entitled Geometry and the Imagination as developed by Conway, Doyle, Thurston and others.
A two-part course Ia and Ib (currently only part Ib available) on ordinary differential equations and nonlinear dynamics.
These notes and exercises were part of a 3-part lecture series on the braid groups, which took place during the Graduate Workshop "Roots of Topology" at the University of Chicago in June 2018.
The notes provide several perspectives on the braid groups, and have guided exercises through a computation of the cohomology of the pure braid groups. In the process the notes introduce the Serre spectral sequence, cohomology with twisted coefficients, and the idea of representation stability.