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.

I will discuss the rational cohomology of $SL_n(R), Sp_{2n}(R)$, and their principal congruence subgroups for $R$ a number ring. Borel--Serre showed that these groups satisfy a (co)homological duality that lets us study their cohomology groups via certain representations called the `Steinberg modules’, which have a beautiful combinatorial description in terms of Tits buildings. I will describe a conjecture of Lee--Szczarba on the top cohomology of principal congruence subgroups of $SL_n(Z)$, and its resolution due to Miller--Patzt--Putman. I will then discuss forthcoming work on generalizations of this to other Euclidean rings, and also to symplectic groups.

In this talk, we explore how data scientists in industry are using modern AI tools such as ChatGPT to write code and perform mathematical reasoning. Chris Deotte is a Senior Data Scientist at NVIDIA, a seven-time Kaggle Grandmaster, and holds a PhD in mathematics.

In recent years, data scientists and mathematicians have increasingly shifted from writing all code and derivations by hand to collaborating with AI assistants such as ChatGPT, Claude, and Gemini. These tools are now capable of generating high-quality code, solving mathematical problems, and accelerating research and development workflows.

We will examine concrete examples of how these AI tools perform on real-world coding and mathematical tasks. In particular, we will demonstrate how ChatGPT recently wrote over 99% of the code for a gold-medal-winning solution in an online competition focused on predicting mouse behavior from keypoint time-series data.