Freshman Seminars

Robin Gottlieb

2019 Fall (4 Credits)

Schedule: W 03:00 PM - 05:00 PM

Instructor Permissions: Instructor

Enrollment Cap: 15

What are the goals of mathematics education at the middle and high school level, and how do these goals impact our evaluation of the success or failure of math education in America? Why does math education at these levels matter? What societal structures (historic, economic, political, cultural) impact mathematics education? How does math education in turn impact societal structures? As the world changes, how do the goals of mathematics education change, and in what ways? We will explore these issues to become more educated participants in this ongoing discussion.

Course Notes:
A special invitation is extended to students not planning to concentrate in math.
Requirements:
Course open to Freshman Students Only

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Not Available for Cross Registration
FAS: Course Level Primarily for Undergraduate Students
Course Search Attributes Display Only in Course Search
FAS Divisional Distribution None

Lauren Williams

2019 Fall (4 Credits)

Schedule: M 03:00 PM - 05:00 PM

Instructor Permissions: Instructor

Enrollment Cap: 12

This seminar is intended to illustrate how research in mathematics actually progresses, using recent examples from the HARVARD UNIVERSITY Page 1311 of 3592 8/9/2019 0:40 AM field of algebraic combinatorics. We will learn about the story of the search for and discovery of proof of a formula conjectured by Mills-Robbins- Rumsey in the early 1980s: the number of n x n alternating sign matrices. Alternating sign matrices are a curious family of mathematical objects, generalizing permutation matrices, which arise from an algorithm for evaluating determinants discovered by Charles Dodgson (better known as Lewis Carroll). They also have an interpretation as two-dimensional arrangements of water molecules, and are known in statistical physics as square ice. Although it was soon widely believed that the Mills-Robbins-Rumsey conjecture was true, the proof was elusive. Researchers working on this problem made connections to invariant theory, partitions, symmetric functions, and the six-vertex model of statistical mechanics. Finally, in 1995, all these ingredients were brought together when Zeilberger and subsequently Kuperberg gave two proofs of the conjecture. In this course we will survey the story of the alternating sign matrix conjecture, building up to Kuperberg’s proof. If time permits, we will also get a glimpse of very recent activity in the field, for example, the Razumov-Stroganov conjecture.

Recommended Prep:
This seminar is recommended for students with a strong background in mathematics, including some familiarity with proofs. It would be helpful to have some exposure to combinatorics (permutations,binomial coefficients) and linear algebra (matrix multiplication and determinants of n by n matrices).
Requirements:
Course open to Freshman Students Only

Additional Course Attributes::

Attribute Value(s)
FAS Divisional Distribution None
Course Search Attributes Display Only in Course Search
All: Cross Reg Availability Not Available for Cross Registration
FAS: Course Level Primarily for Undergraduate Students

Paul Bamberg and Mark Schiefsky

2019 Fall (4 Credits)

Schedule: T 12:00 PM - 02:45 PM

Instructor Permissions: Instructor

Enrollment Cap: 12

Visit Athens: the Parthenon still stands, but its decorations are gone. Visit Rome: you can tour the Colosseum, but it is not in good enough condition to host a modern sporting event. Visit the Internet, though, and you can find Euclid’s Στοιχεία (Elements), the greatest mathematics textbook ever written, in an electronic form that surpasses any ancient manuscript. You can view the Greek text and its English translation side by side and learn the origin of words like “rhombus” (ρόμβος) and “isosceles” (ισοσκελής) that you have probably seen only in your geometry class. Better yet, you can click on the word ṭποτεινούσης (hypotenuse) and instantly learn that it is the feminine genitive singular form of the present active participle formed from the verb ṭποτείνω (“subtend”). Had you been a member of the Harvard class of 1724, in the era when all Harvard students, according to the University archives, “followed a prescribed course of studies in Latin, Greek and Hebrew” and pursued disciplines that “included Rhetoric and Logic, Ethics and Politics, Arithmetic and Geometry,” you would spend a great deal of time learning how to do this sort of parsing. This seminar explores the most famous theorems in the Elements, with special attention to ones that have arguably improved over the ages, since they have turned out to be valid not only in Euclidean geometry but also in non-Euclidean geometries like the spherical geometry of the Earth’s surface, the hyperbolic geometry that provides an alternative to Euclid’s postulate about the uniqueness of parallel lines, and the Minkowskian geometry that helped to inspire Einstein’s theory of relativity.

Course Notes:
While the seminar is a mathematics course, not a language course, we will investigate the grammar and structure of ancient Greek and translate a few key definitions, postulates, and theorems. Students will have the option of pursuing additional topics in mathematics, in the Greek language, or in the geography of the large portion of the ancient world in which Greek was the common language of the educated.
Requirements:
Course open to Freshman Students Only

Additional Course Attributes:

Attribute Value(s)
FAS: Course Level Primarily for Undergraduate Students
Course Search Attributes Display Only in Course Search
FAS Divisional Distribution None
All: Cross Reg Availability Not Available for Cross Registration