Fall 2012 Mini Undergraduate Math Seminar

The Riemann mapping theorem establishes that any subset of the complex plane (subject to certain mild restrictions) is conformally equivalent to the unit disk. This celebrated result unfortunately fails to extend directly to subsets of $\mathbb{C}^n$ for $n\ge 2$, but admits some generalization within the context of Riemann surfaces, where it is a special case of the Poincaré-Koebe uniformization theorem. Some familiarity with the terminology of complex variables is assumed.

This talk aims to provide a brief introduction to elliptic curves and their group structure over finite and infinite fields. Concepts to be discussed will include projective space, the group law, and Mordell’s theorem. Basic knowledge of group and field theory will be assumed.

This talk will discuss representations of $S_n$ over the complex numbers. Representations of $S_n$ (algebraic objects) can be modeled by Young Tableaux (nice combinatorial objects), allowing us to easily understand, enumerate and even "visualize" these representations (which may otherwise be relatively complicated to manage). Topics may include: a brief overview of basic representation theory, intertwining operators, irreducible representations of $S_n$ and representations of $S_n$ on tensor products. Prerequisites: linear algebra. Please be acquainted with the notion of a free vector space, and very basic ideas in group theory ( conjugacy classes, and conjugacy classes of Sn )

Introduction to basic concepts in logic and model theory. We introduce the completeness theorem and compactness theorem of first-order logic. As a basic application, we sketch a construction of a non-standard model of real numbers. We offer applications to algebraically closed fields, including a proof of the the Ax-Grothendieck theorem.

We will prove Liouville's Theorem of Differential Algebra which establishes that any function with an elementary primitive has the form $\sum_1^n c_i \frac{Du_i}{u_i} + Dv$. We will discuss the set theoretic properties of elementary primitives as well as an example application of the previous theorem.

A review of prerequisites from Riemannian geometry; differential operators on manifolds: the principle symbol, elliptic operators, Laplace-type operators; harmonic maps: the geometry of $T^*M\otimes f^*TN$, energy density and energy of a map, the map Laplacian/tension field, the first variation formula for map energy, harmonic maps, examples, harmonic map heat flow; The Ricci flow equation; Deturck's trick for short term existence; uniqueness and connections to harmonic map heat flow; very sketchy discussion of application of Ricci flow with surgery to geometrization. All topics time permitting