Spring 2015 Course Notes

Math 048 Mathematical Logic

Prof. Rachel Epstein

Does every true statement in mathematics have a proof?  What does it even mean to be a true statement in mathematics?  How do we resolve mathematical paradoxes?  Could a good enough computer answer any mathematical question?  What are the foundations of mathematics?

These are some of the questions we will explore in this course.  We will discuss a bit of set theory, model theory, and computability theory, with a significant focus on first-order logic.  We will discuss cool topics such as the Axiom of Choice, Godel’s Incompleteness Theorem, and the Church-Turing Thesis.  This course combines ideas important to philosophy, computer science, and mathematics.

Math 48 can be taken in addition to both Phil 12 and Phil 31, but neither are required.  There will be some overlap, but mostly with basic concepts.  For instance, we will talk about formal deductions and will prove theorems about them, but we will actually do very few formal deductions themselves.

The only requirement is comfort with reading and writing mathematical proofs.  Contact the professor, Rachel Epstein (repstei1@swarthmore.edu), if you have any questions.

Math 053 Stochastic Modeling

Prof. Scott Cook

Probabilistic methods for modeling systems of interest and for solving problems.  It'll be an applied math course with substantial computer based simulation, mostly likely using Python.  Pre-requisite is Linear Algebra (M27,28,28S) and Multivariable Calculus (M33,34,35).  Programming experience is not required, but enthusiasm is.  I learned to program by just playing around with code; I found that, with a little tenacity, you can get a long way toward making the computer your ally in doing mathematics.  If that sounds fun to you, I invite you to join us.

First time the course has been taught.  It will be modeled on http://www.math.wustl.edu/~feres/Math350Fall12.html.

Math 106 Advanced Topics in Geometry (Algebraic Geometry)

Professor Linda Chen

This course is an introduction to algebraic geometry. Roughly speaking, algebraic geometry is the study of geometric spaces defined by polynomial equations. We will cover enough commutative algebra to be able to discuss things rigorously, but the emphasis will be on geometric objects and constructions. Topics include affine varieties, projective spaces, projective varieties and maps between them, singularities and genus of a plane curve, the addition law on elliptic curves, Bezout's theorem and other results and their applications.

The prerequisite is Math 67 or equivalent. Math 102 or some previous exposure to geometry or topology may be beneficial, but is not required.

Contact Professor Linda Chen (lchen@swarthmore.edu) if you have any questions.