Rigidity in Dimension 2

Abstract: I will discuss an approach to proving the conjecture that a normal rigid surface is smooth. The approach is based on the notion of deficient conormal singularities introduced by the speaker.

Coxeter quotients of knot groups

Abstract: A knot is a smooth embedding of a circle into the three-sphere. Knots have seduced the imagination since at least ancient Rome, and the classification of knots up to continuous deformations preoccupied even Gauss. Knots can be distinguished, and studied, using the fundamental groups of their complements in $S^3$. I will describe a method for computing the bridge number of a knot $K$ — a geometric invariant, which I will define — by using homomorphisms of $\pi_1(S^3\backslash K)$ onto Coxeter groups. This settles a conjecture by Cappell and Shaneson from the 70s for some infinite families.

Joint work with Sebastian Baader and Ryan Blair.

The Almost Schur Lemma in Quaternionic Contact Geometry

Abstract: In this talk we consider the quaternionic contact (qc) version of the almost Schur Lemma. Namely, we derive an integral inequality that informally states that on a compact qc manifold of dimension bigger than seven, satisfying some positivity condition, if the traceless qc Ricci tensor and some traces of the qc torsion tensor are close to zero, then the qc scalar curvature is close to a constant.

Orthogonal Tensor Decomposition

Abstract: Tensor decomposition has many applications. However, it is often a hard problem. In this talk we discuss a family of tensors, called orthogonally decomposable tensors, which retain many of the nice properties of matrices that general tensors don’t. A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of n orthonormal vectors. We will see that the decomposition of such tensors can be found efficiently, their eigenvectors can be computed efficiently, and the set of orthogonally decomposable tensors of low rank is closed and can be described by a set of quadratic equations. Analogously, we study nonsymmetric orthogonally decomposable tensors, and show that the same results hold. Finally, we discuss a generalization to tight-frame decomposable tensors.

Pólya Sequences with Dominant Colors

Abstract:

We introduce a class of  “Pólya Sequences with Dominant Colors”, which can be described as randomly reinforced urn processes with color-specific random weights and unbounded number of possible colors. Let ${D}$ be the set of colors for which the expected random reinforcement attains its maximum value; in this sense, we say that the colors in ${D}$ are dominant. Under fairly mild conditions, we show that the predictive probability of observing a dominant color, $P_n({D})$, converges a.s. to one. Moreover, there exists a random probability measure $\tilde{P}$ with $\tilde{P}({D})=1$ such that the predictive and the empirical distributions converge weakly a.s. to $\tilde{P}$. The latter implies, in particular, that the urn process is asymptotically exchangeable with limit directing random measure $\tilde{P}$. In the general case, for any $\delta$-neighborhood ${D}_\delta$ of ${D}$, the predictive probabilities $P_n({D}_\delta)$ and the empirical frequencies $\hat{P}_n({D}_\delta)$ converge a.s. to one. As a result, the distance between the observed color and ${D}$ converges in probability to zero. We refine the above results with rates of convergence and central limit theorems. We further hint examples of the potential use of our model in randomized clinical trials and species sampling.