Mathematical notes

Sometimes these notes have not been carefully edited. I may put them on the web for safe-keeping and easy access before I have finished correcting them; then I may get distracted and never finish.

Notes with TeX sources:

(The older TeX files here may use the auxiliary files abbreviations.tex and format.tex as well as references.bib.)

Pertaining to courses

Logic

• Introductory notes on the foundations of mathematics (ix + 166 pp.). This was used in the course Math 111: Fundamentals of Mathematics, in the fall semester of 2005/6 at METU. (There is also a more recent version, incompletely edited.)
• Sets, classes, and families (106 pp.). I try to go as far as possible without assuming the existence of infinite sets. Used in Math 320: Set Theory, in the spring semester of 2008/9 at METU.
• “Foundations of number-theory” (5 pp.). The so-called Peano axioms and their consequences. In the remarks, I try to correct some misperceptions suggested by textbooks. Prepared for Math 365: Elementary Number Theory, at METU, fall semester, 2007/8.
• Recursion and induction: notes on mathematical logic and model theory (109 pp.), used in the course Math 406: Mathematical Logic and Model Theory, fall semester, 2008/9.

Algebra

• Groups and Rings (41 pp.; lecture notes from Math 503: Algebra I, fall, 2003/4).
• Modules (9 pp.; from Math 504: Algebra II, spring, 2003/4).
• Fields (5 pp.)
• Separability (7 pp.)

Other

• History of Mathematics: The log of a course
• Analytic geometry (used in Math 115: Analytic Geometry, 2006/7, spring, at METU):
  • “Angles in analytic geometry” (6 pp.; with introductory remarks on analysis and synthesis, and the observation that not all of analytic geometry requires orthogonal axes—that is, not all of analytic geometry requires the idea of orthogonality)
  • “Conic sections” (18 pp.; historical introduction; the development by means of focus, directrix, and eccentricity)
• Differential equations:
  • “Notes on the Laplace transform” (6 pp.; Math 219, fall, 2006/7)
  • “Series-solutions at singular points” (4 pp.; Math 253, fall, 2003/4)
  • “Differential equations notes” (38 pp.; from Math 253; updated for Math 219; for my use)
• Calculus and analysis:
  • “Analysis lecture notes” (36 pp.) For Math 271, fall, 2002/3, at METU.
  • “Analysis II notes” (41 pp.). For Math 272, spring, 2002/3, at METU. The introduction has some advice and warnings about using the notes and doing and writing mathematics.
  • “A derivation of the equation e^πi+1=0” (reviews the development of the trigonometric, logarithmic and exponential functions, along with power series; 8 pp.)
  • “The binomial theorem” (4 pp.)
  • “Counterexamples in partial derivatives” (4 pp.)

Other

• Set Theory without Infinity
• Regular polytopes (their classification)
• The heptakaidecagon (factorizing x¹⁷ - 1 a la Gauss)
• Undecidability (2 pp.; a brief sketch of the basic argument for the first-order theory of arithmetic)
• Riemann–Roch Theorem (8 pp. unfinished; I just wanted to get some basics straight)
• Multiplicative-group covers (pertaining to the paper of Boris Zilber called “Covers of the multiplicative group of an algebraically closed field of characteristic zero”.
• Spectra
• Quantifier-elimination in theories of fields (the theorem of Macintyre, McKenna and van den Dries that the theory of an infinite field admits quantifier-elimination just in case the field is algebraically closed.
• Lüroth's Theorem that every sub-extension of a pure transcendental extension of degree one is pure.
• Witt vectors: How the basic definitions involving Witt vectors can be understood in terms of p-adic numbers (and the little Fermat theorem).
• Kac–Moody algebras (9 pp.; May, 2002)
• Algebraic number theory: Based on a course I took from Larry Washington; unfinished; I tried to streamline the notation for factorizations of ideals (48 pages so far).
• Model theory: Includes basic model theory (the notion of a model itself), and the basic theorems around model-companions.
• Model theory of fields: From the archived transparencies for the (three) MSRI/Evans Hall Lectures in spring of 1998 given by Angus Macintyre (17 pages).
• “Model Theory ≈ Tame Mathematics” (transcription of slides of talk by Lou van den Dries at MSRI, spring, 1998; 4 pp.)
• Notes on RSA public-key encryption (3 pp.; October, 2001)
• Linear algebra (6 pp.; beginnings of a summary of the material of a basic course; February, 2001)
• Planetary orbits (3 pp.; the derivation of Kepler's First from Newton's Second Law; January, 2001)
• Riemannian geometry (3 pp.; the algebraic aspects; 2000)
• Elliptic curves and modular forms (4 pp.; 1999)
• Undecidability of C(X,Y) (8 pp.; 1997)

Notes with HTML sources

• Algebra: A somewhat model-theoretic development of some algebraic structures, based nominally on high-school knowledge; but schemes and the adeles are brought up.
• Linear algebra: For an undergraduate course.