In Ideals and Varieties, one of our math classes this block, we emphasize the geometry behind algebraic symbols.
Ideals and Varieties asks students to think geometrically and reason algebraically. Algebra, even at a college and graduate level, can feel like intricate symbol manipulation. In our class, we emphasize the geometry behind the symbols.
In one problem from class, we graphed the equation y2 - x3 - x2 =0. We avoid using computers to do our graphing for us. Instead, we think about the symmetries of a function, its similarities to well-known functions, and what solutions it has when we plug in specific numbers for x or y. This equation is the nodal cubic, which looks like:
In a calculus class, students might learn to parametrize this equation by manipulating x and y until they find a relationship between the variables. In our class, we never set out to solve something algebraically without first thinking about the geometric picture. The geometry of the nodal cubic tells us which algebraic manipulations might take us to a parametrization. This allows us to both understand more deeply why it is possible to find a parametrization, and also avoid spending an hour bashing through algebraic equations. In this case, students realize that almost any line through the origin will intersect the nodal cubic exactly one more point. By looking at lines through the origin of varying slopes, we sweep through the entire cubic, giving us a nice pair of formulas, y = t3 - t and x = t2 - 1.
In Ideals and Varieties, students are developing a toolbox of concepts which are applied to many different fields of math. These concepts include basic definitions from algebraic geometry, such as fields, n-dimensional affine space, ideals, and varieties. However, they are not specialized concepts: they appear all over. During the second half of the course, each student is writing a paper on a topic related to our class. While writing the paper, students are realizing the connections between the tools they learn in class and big areas of research in math. They are also developing their expository writing skills and learning some of the conventions of writing papers in mathematics.
Over the next few weeks, we will continue our study of multivariable polynomials, and prove the Hilbert Basis Theorem, a foundational result in algebraic geometry. We will study Groebner bases and solve the ideal membership problem, wrapping up some of the major questions we have been asking all block. By the end of the course, students will have a mastery of basic algebraic geometry, as well as a related topic which they have chosen to write their paper on. Students are doing truly impressive work and engaging with hard problems every day.