Associate Professor, Mathematics
- PhD, 2013, University of California at Berkeley, Mathematics
- BS, 2008, University of Notre Dame, Mathematics
Adam Boocher joined the Mathematics Department at USD in 2018 after holding visiting positions at the University of Utah and the University of Edinburgh in Scotland. Adam's research focuses on questions in commutative algebra and algebraic geometry that explore the symmetries hidden inside of algebraic equations. He has taught a variety of mathematics courses at USD and elsewhere and currently serves on the advisory council for the Liberal Studies Program and the Integrated Teacher Training Program.
Scholarly Work
Dr. Boocher's research is in the field of commutative algebra and algebraic geometry. These fields study systems of polynomial equations and the geometric objects that they describe. For instance, in a first algebra class you learn that the equation y - x^2 = 0 describes a parabola and that the equation x + y = 0 describes a line. When you solve the system of equations you see that these two objects intersect in exactly two points. Algebraic geometry studies what happens when you solve larger systems in potentially many variables.
Algebraic geometry is very useful and has a number of applications. For instance, if you have a robot that has say 10 different joints that can each move in 3 dimensions, then you have 30 degrees of freedom and in describing where the robot can move, you are really working in a system with 30 variables. Even though we can't visualize what a graph looks like in 30 dimensional space (just think about how hard it was to visualize 3-dimensional things like x^2 - yz = 0 in Calc 3!) we can still use algebra to study these equations and detect geometric information about them.
Specifically Dr. Boocher's research is devoted to studying how independent a system of polynomials is. This is a direct generalization of the notion of linear independence that you learn about in a linear algebra course. Some big open questions that Dr. Boocher likes to think about are:
- What are the different ways to quantify how independent a system of polynomials is? For instance there is one notion of complexity, called the total betti number of a system which measures how interwoven the equations are with one another. An open question in the field is whether or not every nondegenerate 5-dimensional system has a total betti number that is at least 48. The details aren't too bad - I can explain on a whiteboard if you want!
How do betti numbers vary as one deforms the space we are working with? For instance, go on Desmos and type in xy - a = 0, and play around with the slider. You'll notice that you get a family of hyperbolas that deforms to the union of two straight lines. If you have a family of a bunch of polynomials like this, how will the betti numbers change?
If you want to hear more - feel free to swing by Dr. Boocher's office. He's led lots of independent study courses and research projects over the years and would be happy to share some of the cool stuff. (One of his student projects was about how to do computational algebra using graph theory!)
Areas of Interest
Dr. Boocher has taught a variety of mathematics courses including both introductory and advanced courses. He is particularly interested in developing inquiry-based activities to help students explore and learn new mathematical concepts. At USD and elsewhere he has designed these activities for courses in Analysis, Abstract Algebra for future teachers, Calculus, Linear Algebra and Representation Theory. Outside of mathematics, Adam enjoys trail running, playing the pipe organ, singing, rock climbing and playing competitive Scrabble.
More information available at - https://aboocher.github.io/