We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably $\mathcal{H}^m$ rectifiable metric space of the same dimension. The tetrahedral property depends only on distances between points in spheres, yet we show it provides a lower bound on the volumes of balls. The proof is based upon intrinsic flat convergence and a new notion called the sliced filling volume of a ball.

Given a poset $P$, we are interested in determining the maximum size (denoted by $La(n,P)$) of any family of subsets of an $n$-set avoiding all extensions of $P$ as subposets. The starting point of this kind of problem is Sperner's Theorem from 1928, which can be restated as $La(n, P_2)= {n\choose \lfloor \frac{n}{2} \rfloor}$. Here $P_2$ is the chain of $2$ elements. These problems were studied by Erd\H{o}s, Katona, and others. In 2008, Griggs and Lu conjectured the limit $\pi(P):=\lim_{n\to\infty} \frac{La(n,P)} {{n\choose \lfloor \frac{n}{2} \rfloor}}$ exists and is an integer. For poset $P$ define $e(P)$ to be the maximum $k$ such that for all $n$, the union of the $k$ middle levels of subsets in the $n$-set contains no extension of $P$ as a subposet. Saks and Winkler observed $\pi(P)=e(P)$ in all known examples where $\pi(P)$ is determined. Bukh proved this conjecture holds for any tree poset $P$ (meaning its Hasse diagram is a tree). For $t\geq 2$, let crown $\O_{2t}$ be a poset of height $2$, whose Hasse diagram is cycle $C_{2t}$. De Bonis-Katona-Swanepoel proved $La(n,O_{4})= {n\choose \lfloor \frac{n}{2} \rfloor} + {n\choose \lceil \frac{n}{2} \rceil}$. Griggs and Lu proved the conjecture holds for crown $\O_{2t}$ with even $t\geq 3$. In this talk, we will prove that the conjecture holds for crown $\O_{2t}$ with odd $t\geq 7$.

Given an elliptic curve $E/\mathbb{Q}$ with good reduction at $p$, Iwasawa theory studies its arithmetic over all the fields of $p^n$-th roots of unity. For example, there are nontrivial relations among the participants in the Birch–Swinnerton-Dyer conjectures for $E$ over each layer in this tower of fields. If the reduction of $E$ mod $p$ is ordinary, then have had a satisfactory description of the scenario for quite some time. But if the reduction of $E$ mod $p$ is supersingular, the correct description has required new advances in $p$-adic Hodge theory. We will discuss the background and what is now known.

A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to $S^4$ or to $CP^2$. In particular, the Hopf conjecture on $S^2 \times S^2$ holds in the class of geometrically formal manifolds. If the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not too nonconstant, then the manifold must be homeomorphic to $S^4$ or to a connected sum of $CP^2$s.

Analog-to-digital (A/D) conversion is the process by which signals (e.g., bandlimited functions or finite dimensional vectors) are replaced by bit streams to allow digital storage, transmission, and processing. Typically, A/D conversion is thought of as being composed of sampling and quantization. Sampling consists of collecting inner products of the signal with appropriate (deterministic or random) vectors. Quantization consists of replacing these inner products with elements from a finite set. A good A/D scheme allows for accurate reconstruction of the original object from its quantized samples. In this talk we investigate the reconstruction error as a function of the bit-rate, of Sigma-Delta quantization, a class of quantization algorithms used in the oversampled regime. We propose an encoding of the Sigma-Delta bit-stream and prove that it yields near-optimal error rates when coupled with a suitable reconstruction algorithm. This is true both in the finite dimensional setting and for bandlimited functions. In particular, in the finite dimensional setting the near-optimality of Sigma-Delta encoding applies to measurement vectors from a large class that includes certain deterministic and sub-Gaussian random vectors. Time permitting, we discuss implications for quantization of compressed sensing measurements.