Thursday, November 8, 2018
11/08/2018 - 3:00pm
Estimating a manifold from (possibly noisy) samples appears to be a difficult problem. Indeed, even after decades of research, all manifold learning methods do not actually “learn” the manifold, but rather try to embed it into a low-dimensional Euclidean space. This process inevitably introduces distortions and cannot guarantee a robust estimate of the manifold. In this talk, we will discuss a new method to estimate a manifold in the ambient space, which is efficient even in the case of an ambient space of high dimension. The method gives a robust estimate to the manifold and it’s tangent, without introducing distortions. Moreover, we will show statistical convergence guarantees.
11/08/2018 - 4:00pm
It is an area of interest in both number theory and harmonic analysis to study various quantitative behaviors of eigenfunctions restricted to certain smooth submanifolds (curves), which include the restricted L^p norms, (generalized) period integrals, inner products, etc. In this talk, I will survey some recent developments in this area.
11/08/2018 - 4:00pm
In this note, we study the emergence of Hamiltonian Berge cycles in random $r$-uniform hypergraphs. In particular, we show that for $r \geq 3$ the $2$-out random $r$-graph almost surely has such a cycle, and we use this to determine (up to a multiplicative factor) the threshold probability for when the Erdős–Rényi random $r$-graph is likely to have such a cycle. In particular, in the Erdős–Rényi model we show (up to a constant factor depending on $r$) the emergence of these cycles essentially coincides with the disappearance of vertices of degree at most $2$. This is a joint work with Deepak Bal.
11/08/2018 - 4:15pm
The Grothendieck standard conjectures, the Voevodsky nilpotence conjecture, and also the Tate conjecture, play a key central role in algebraic geometry. Notwithstanding the effort of several generations of mathematicians, the proof of these celebrated conjectures remains elusive. The aim of this talk is to give an overview of a recent noncommutative approach which has led to the proof of the aforementioned important conjectures in several new cases.