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Lecture notes on stability theory

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.

Artem Chernikov
UCLA
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Date posted: June 17, 2019

Graph Theory

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

Benny Sudakov
ETH, Zurich
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Date posted: June 6, 2019

First Semester in Numerical Analysis with Julia

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

Giray Ökten
Florida State University
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Date posted: May 26, 2019
Date revised: June 18, 2019

Contact Topology

Lecture notes from Prof. Robert Gompf's Contact Topology course at UT Austin, Fall 2017.
(posted with permission from Robert Gompf)

George Torres
University of Texas at Austin
Robert Gompf
The University of Texas at Austin
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Date posted: May 2, 2019
Date revised: May 3, 2019

Homotopy coherent structures

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:

https://sites.google.com/site/floerhomotopy2017/home

Emily Riehl
Johns Hopkins University
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Date posted: February 5, 2019

A brief introduction to group representations and character theory

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.

Mark Meckes
Case Western Reserve University
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Date posted: December 21, 2018

Introduction to Fresh Honors Notes

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.

Jane Gilman
Rutgers University, Newark
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Date posted: November 8, 2018

Geometry and Imagination 1991

These are notes from an experimental mathematics course entitled Geometry and the Imagination as developed by Conway, Doyle, Thurston and others.

Jane Gilman
Rutgers University, Newark
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Date posted: November 8, 2018

Dynamical Systems I

ODEs & Introduction to Nonlinear Dynamics

A two-part course Ia and Ib (currently only part Ib available) on ordinary differential equations and nonlinear dynamics.

Christian Kuehn
Technical University of Munich
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Date posted: September 17, 2018

An Introduction to the braid groups & representation stability

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.

Jennifer Wilson
University of Michigan
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Date posted: September 17, 2018