Andy Schultz
Professor of Mathematics
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Mathematician interested in studying absolute Galois groups of fields through their cohomological invariants.
Galois theory is one of the most important mathematical developments in the last 200 years. Its strength can be measured by some of "big" mathematical questions it has answered, including why there isn't a quintic analog of the quadratic equation, as well as explaining why the outstanding geometric problems of antiquity (e.g., "squaring the circle") have no solution.
My research focuses on extending what is currently known about Galois theory by investigating properties of absolute Galois groups, the uber-structures that encapsulate all the Galois-theoretic information for a given field. By determining certain structures that cannot appear in absolute Galois groups, I limit the kinds of Galois groups that can appear over an arbitrary field.
I began my research as an undergraduate under John Swallow at Davidson College. Although I was taking only math courses by the time I was a senior undergraduate, I didn't enter college thinking I'd major in math. Perhaps because I was something of a late convert to the subject, one of my favorite parts of a career in academia is the opportunity it gives me to share the beauty and joy of mathematics with others who are just developing as mathematical thinkers. This especially includes those students who don't come into my class thinking they have any interest in mathematics. By making the classroom a collaborative environment where students are responsible for developing some of the key ideas in a course, I help show students that they are capable of producing high-level, sophisticated mathematics with an abundance of real-world applications. For those students who have already invested themselves in mathematics, I strive to give them opportunities to engage with challenging, open-ended problems, and I work to provide them with a perspective on material that can help them frame their specialized knowledge in a broader context.
With my time outside of the classroom, I enjoy a number of outdoor activities, especially running and playing basketball. I'm also an amateur beer brewer.
Education
- B.S., Davidson College
- M.S., Stanford University
- Ph.D., Stanford University
Current and upcoming courses
Advanced Linear Algebra
MATH322
Linear algebra at this more advanced level is a basic tool in many areas of mathematics and other fields. The course begins by revisiting some linear algebra concepts from MATH 206 in a more sophisticated way, making use of the mathematical maturity picked up in MATH 305. Such topics include vector spaces, linear independence, bases, and dimensions, linear transformations, and inner product spaces. Then we will turn to new notions, including dual spaces, reflexivity, annihilators, direct sums and quotients, tensor products, multilinear forms, and modules. One of the main goals of the course is the derivation of canonical forms, including triangular form and Jordan canonical forms. These are methods of analyzing matrices that are more general and powerful than diagonalization (studied in MATH 206). We will also discuss the spectral theorem, the best example of successful diagonalization, and its applications.
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Calculus I
MATH115
Introduction to differential and integral calculus for functions of one variable. The heart of calculus is the study of rates of change. Differential calculus concerns the process of finding the rate at which a quantity is changing (the derivative). Integral calculus reverses this process. Information is given about the derivative, and the process of integration finds the "integral," which measures accumulated change. This course aims to develop a thorough understanding of the concepts of differentiation and integration, and covers techniques and applications of differentiation and integration of algebraic, trigonometric, logarithmic, and exponential functions. MATH 115 is an introductory course designed for students who have not seen calculus before.