Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers satisfying , Darmon and Granville proved that the individual generalized Fermat equation has only finitely many coprime integer solutions. Conjecturally something stronger is true: for there are no non-trivial solutions.
I'll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture.
The Temperley—Lieb Algebra of type A is a finite dimensional associative algebra that arose in the context of statistical mechanics and can be realized in terms of certain diagrams. These diagrams can be indexed by the fully commutative elements of the Coxeter group of type A. Similarly, there is also a diagrammatic representation for the Temperley—Lieb algebra of type B involving decorated diagrams. Multiplying diagrams is easy to do. However, taking a given diagram and finding the corresponding reduced factorization is not so easy. We have an efficient (and colorful) algorithm for obtaining a reduced factorization for Temperley-Lieb diagrams of types A and B.
Diagram factorizations in Temperley—Lieb algebras
Mar. 10, 2015 3pm (Math 350)
Algebraic Geometry
David Zureick-Brown (Emory University)
X
We give a generalization to stacks of the classical theorem of Petri -- i.e., we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, which depends on the signature of the group. (The talk will be mostly geometric and will require little understanding of modular forms.) This is joint work with John Voight.
Krystal Taylor (Institute for Mathematics and its Applications, Minnesota)
X
We study the existence of certain geometric configurations in subsets of Euclidean space. In particular, we establish that a set with ``sufficient structure" contains an arbitrarily long chain with vertices in the set and preassigned admissible gaps. A k-chain in with gaps is a sequence In order to prove this result, we establish mapping properties of the convolution operator where is a tempered distribution, and and are compactly supported measures satisfying the growth bounds and . This is joint work with A. Iosevich and M. Bennett.
Finite point configurations and Fourier analysis Sponsored by the Meyer Fund