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Algebra

This page contains a list of ideas for DRP projects, but is by no means exhaustive.


Topic: Birkhoff’s Theorem
Suggested Text: Universal Algebra – Fundamentals and Selected Topics, Clifford Bergman
Suggested Background: MATH 3300 (Abstract Algebra)
Description: Universal Algebra generalizes many of the common structures encountered in abstract algebra (groups, rings, etc) to talk algebraic structures in general, including general direct products, homomorphisms, subalgebras, and congruences. The last, congruences, generalize normal subgroups and ideals and provide a general treatment of the Isomorphism Theorems. By constructing free algebras, Birkhoff’s Theorem can be proven, showing that classes of algebras defined by equations coincide exactly with classes of algebras which are closed under taking direct products, homomorphic images, and subalgebras.


Topic: The Sylow Theorems
Suggested Text: Algebra, Michael Artin
Suggested Background: MATH 3300 (Abstract Algebra), or some familiarity with groups
Description: The famous Lagrange’s Theorem says that if \(G\) is a finite group and \(H\) is a subgroup of \(G\), then \(|H|\) divides \(|G|\). As a consequence, every element of \(G\) has order which divides \(|G|\). The Sylow Theorems provides partial corollaries to Lagrange’s Theorem, showing that if \(p^n\) is the largest power of a prime \(p\) which divides \(|G|\), then there exists a subgroup \(P\) of \(G\) of order \(p^n\), as well as further properties of such Sylow \(p\)-subgroups and conditions on the number of such subgroups. These end up providing useful tools for determining the number of isomorphism classes of groups of a given finite size.


Topic: Geometric Group Theory
Book: Office Hours with a Geometric Group Theorist, Matt Clay & Dan Margalit
Suggested Background: MATH 3200 (Introduction to Topology), MATH 3300 (Abstract Algebra)
Description: After learning the basics of group theory, a student can delve in to group actions and the interplay between geometry and groups. The Cayley graph associated to a group is a fundamental geometric object associated to a group, and by understanding the group action that arises on this space and on other spaces, one can begin to see many of the surprising properties of infinite groups and many of the fundamental interactions between geometric group theory and other fields of study.


Topic: \(p\)-adic Analysis
Suggested Text: \(p\)-adic Numbers – An Introduction, Fernando Q. Gouvea
Suggested Background: MATH 3100 (Introduction to Analysis) or MATH 3300 (Abstract Algebra)
Description: \(p\)-adic analysis represents a different approach to correcting the failings of the field of rational numbers, with a resulting theory that looks wildly different from the classical analysis of the real line. The \(p\)-adic reals (for each prime \(p\)) are nevertheless rich objects of study, both from an analytic point of view, as well as an algebraic one.


Topic: Elliptic Curves
Book: Rational Points on Elliptic Curves, Joseph H. Silverman & John Tate
Suggested Background: MATH 3300 (Abstract Algebra), MATH 3800 (Theory of Numbers)
Description: Elliptic curves have become one of the most exciting fields of study in recent years. Fundamentally, elliptic curves are simply the solution set of \(y^2 = x^3 + ax + b\), which would appear to not be much more difficult to understand than conic sections. However, it turns out that they contain a breadth of number theoretic information, being fundamental to Andrew Wiles’s proof of Fermat’s Last Theorem. Additionally, they have proved useful in a variety of other areas, such as cryptography. A reading program in this area would entail learning the basics, giving the student an understanding of both the difficulty and depth of this area, and allowing them to see why so many mathematicians have become fascinated with these objects.