Georgia Institute of Technology College of Sciences

The problem of estimation of smooth functionals of unknown parameters of statistical models will be discussed in the cases of high-dimensional log-concave location models (joint work with Martin Wahl) and infinite dimensional Gaussian models with unknown covariance operator. In both cases, the minimax optimal error rates have been obtained in the classes of H\”older smooth functionals with precise dependence on the sample size, the complexity of the parameter (its dimension in the case of log-concave location models or the effective rank of the covariance in the case of Gaussian models)  and on the degree of smoothness of the functionals. These rates are attained for different types of estimators based on two different methods of bias reduction in functional estimation.

Have you ever wanted to marry topology, hyperbolic geometry, and geometric group theory, all at once?* Bowden-Hensel-Webb do this and more when they embark on their study of Diff0(S). In this talk, we will discuss the main theorems of Bowden-Hensel-Webb's paper, the most notable of which is (arguably) the lack of uniform perfection of Diff0(S). We will then summarize the main tools they use to prove these results. (Note: the perspectives on Diff0(S) in this talk will DIFFer greatly from those used in the diffeomorphism groups class.) 

*If you answered "yes" for your personal life as opposed to your academic life: that's ok, I won't judge (if you don't tell me).

When can we find perfect matchings in hypergraphs whose vertices represent group elements and whose edges represent solutions to systems of linear equations? A prototypical problem of this type is the Hall-Paige conjecture, which asks for a characterisation of the groups whose multiplication table (viewed as a Latin square) contains a transversal. Other problems expressible in this language include the toroidal n-queens problem, Graham-Sloane harmonious tree-labelling conjecture, Ringel's sequenceability conjecture, Snevily's subsquare conjecture, Tannenbaum's zero-sum conjecture, and many others. All of these problems have a similar flavour, yet until recently they have been approached in completely different ways, using algebraic tools ranging from the combinatorial Nullstellensatz to Fourier analysis. In this talk we discuss a unified approach to attack these problems, using tools from probabilistic combinatorics. In particular, we will see that a suitably randomised version of the Hall-Paige conjecture can be used as a black-box to settle many old problems in the area for sufficiently large groups.  Joint work with Alexey Pokrosvkiy

In the Instanton and Heegaard Floer theories, a nearly fibered knot is one for which the top grading has rank 2. Sivek-Baldwin and Li-Ye showed that the guts (ie. the reduced sutured manifold complement) of a minimal genus Seifert surface of a nearly fibered knot has of one of three simple types.We show that nearly fibered knots with guts of two of these types have handle number 2 while those with guts of the third type have handle number 4.  Furthermore, we show that nearly fibered knots have unique incompressible Seifert surfaces rather than just unique minimal genus Siefert surfaces. This is joint work in progress with Fabiola Manjarrez-Gutierrez.