Georgia Institute of Technology College of Sciences

In this work we prove that the space of two parameter, matrix-valued BMO functions can be characterized by considering iterated commutators with the Hilbert transform. Specifically, we prove that the norm in the BMO space is equivalent to the norm of the commutator of the BMO function with the Hilbert transform, as an operator on L^2. The upper bound estimate relies on a representation of the Hilbert transform as an average of dyadic shifts, and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and Lacey's wavelet proof for the scalar case.

Convex analysis and geometry are tools fundamental to the foundations of several applied areas (e.g., optimization, control theory, probability and statistics), but at the same time convexity intersects in lovely ways with topics considered pure (e.g., algebraic geometry, representation theory and of course number theory). For several years I have been interested interested on how convexity relates to lattices and discrete subsets of Euclidean space. This is part of mathematics H. Minkowski named in 1910 "Geometrie der Zahlen''. In this talk I will use two well-known results, Caratheodory's & Helly's theorems, to explain my most recent work on lattice points on convex sets. The talk is for everyone! It is designed for non-experts and grad students should understand the key ideas. All new theorems are joint work with subsets of the following mathematicians I. Aliev, C. O'Neill, R. La Haye, D. Rolnick, and P. Soberon.

Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of K_5. This conjecture was proved by Ma and Yu for graphs containing K_4^-. In order to establish the Kelmans-Seymour conjecture for all graphs, we need to consider 5-separations and 6-separations with less restrictive structures. We will talk about special 5-separations and 6-separations whose cut contains a triangle. Results will be used in subsequently to prove the Kelmans-Seymour conjecture.

We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Many key examples in 3-manifold topology are Platonic manifolds, e.g., the Poincar\'e homology sphere, the Seifert-Weber dodecahedral space and the complements of the figure eight knot, the Whitehead link, and the minimally twisted 5-component chain link. They have a strong connection to regular tessellations and illustrate many phenomena such as hidden symmetries.I will talk about recent work on a census of hyperbolic Platonic manifolds and some new techniques we developed for its creation, e.g., verified canonical cell decompositions and the isometry signature which is a complete invariant of a cusped hyperbolic manifold.