Many classical computer algorithms can be paralyzed efficiently; what about quantum computers? An algorithm can be described as having layers, one composed with another, with  the depth n of the circuit being the number of layers. An algorithm might be presented as having n simple layers, but if we are able to build more complicated layers, can we construct an equivalent algorithm with a few layers? This  is an issue, which goes back to the early days when people became enthusiastic about the possibility of quantum computers.

One of the most straightforward test cases is called the quantum waterfall or quantum staircase. It is a tensor product analog of a matrix of 2 x 2 blocks supported on the diagonal and the first diagonal below it. It was conjectured in the late 90s that an n layer  quantum waterfall cannot be produced with an algorithm having fewer than order n layers.

This conjecture (Moore-Nillson 1998) turns out to be way too pessimistic and the talk describes recent work with Adam Bene  Watts, Joe Slote, Charlie Chen on a theorem constructing a parallelization of any n layer quantum waterfall which yields  (asymptotically) log n layers.  Gratifying to  operator theorists is that a substantial ingredient is a matrix decomposition originating with Chandler Davis.

Development reliably produces complex organisms despite external perturbations and intrinsic stochasticity. It remains a central challenge not only to understand specific examples of development in vivo, but also to infer underlying principles that extend beyond any particular model system. In this talk, we will first introduce the formation of dorsal branches in the Drosophila larval trachea as a model for structural developmental defects. In each branch, progenitor cells robustly organize themselves into distinct cell fates, driven by an external morphogen concentration. By perturbing the external signal, partially penetrant stochastic phenotypes emerge in which a variable number of "terminal" cells are specified. Using live imaging to capture both morphology and expression of key genes, we observe dynamically how successful fate patterning occurs and how it fails. Partially penetrant phenotypes are modeled by geneticists using "threshold-liability", a phenomenological model with unspecified molecular details. Here, we are able to connect the abstract model to the molecular implementation by directly measuring receptor activation. Next, we consider self-organization theoretically by introducing a minimal model of cell patterning via local cell-cell communication. Recent advances have clarified how isolated cells can respond to an exogenous signal, but cells often interact and act collectively. In our framework we prove that a trade-off between speed and accuracy of collective pattern formation exists. Moreover, for the first time we are able to quantify how information flows between interacting cells during patterning. Our analysis reveals counterintuitive features of collective patterning: globally optimized solutions do not necessarily maximize intercellular information transfer and individual cells may appear suboptimal in isolation.

The combinatorial nullstellensatz was discovered by Noga Alon in 1995, and has since become an important tool in a variety of areas of mathematics. I will discuss this theorem and some of its numerous applications to Additive Number Theory, Geometry, and Combinatorics.