Let me tell you briefly about three of my papers that were recently accepted for publication. Relative Leray numbers via spectral sequences with Roy Meshulam, Helly-type problems with Imre Bárány, and Statistical aspects of quantum supremacy experiments with Yosi Rinott and Tomar Shoham.
We extend the topological colorful Helly theorem to the relative setting. Our main tool is a spectral sequence for the intersection of complexes indexed by a geometric lattice.
Roy and I have a long term project of studying topological Helly type theorems. Often, results from convexity give a simple and strong manifestation of theorems from topology: For example, Helly’s theorem manifests the nerve theorem from algebraic topology, and Radon’s theorem can be regarded as an early “linear” version of the Borsuk–Ulam theorem. We have a few more “linear” theorems in need of topologizing on our list. Actually the paper already appeared in Mathematika on June 26, 2021. It is dedicated to our dear colleague and friend Michael O. Rabin.
Dedicated to Michael O. Rabin, a trailblazing mathematician and computer scientist
Helly type problems, to appear in the Bulletin of the American Mathematical Society
We present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carath ́eodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems, and explain some of its main themes and goals.
Imre and I have long term common interest in Helly-type problems and often discussed it since we first met in 1982. We wrote a first joint paper in 2016 and last year we wrote two additional papers with Attila Por. Last year Imre wrote a great book “Combinatorial convexity” (AMS, 2021, in press) largely devoted to Helly-type theorems. As for me, I plan on gradually writing on open problems related to my areas of interest. (See these slides for some problems.)
Yosi Rinott, Tomer Shoham and I started this project about a year an a half ago. Our paper have now been accepted to Statistical Science where you can download the accepted version along many other future papers. This is my second paper in Statistical Science. The first one was “Solving the bible code puzzle” with Brendan McKay, Dror Bar-Nathan and Maya Bar-Hillel, that appeared in 1999.
In quantum computing, a demonstration of quantum supremacy (or quantum advantage) consists of presenting a task, possibly of no practical value, whose computation is feasible on a quantum device, but cannot be performed by classical computers in any feasible amount of time. The notable claim of quantum supremacy presented by Google’s team in 2019 consists of demonstrating the ability of a quantum circuit to generate, albeit with considerable noise, bitstrings from a distribution that is considered hard to simulate on classical computers. Very recently, in 2020, a quantum supremacy claim was presented by a group from the University of Science and Technology of China, using a different technology and generating a different distribution, but sharing some statistical principles with Google’s demonstration.
Verifying that the generated data is indeed from the claimed distribution and assessing the circuit’s noise level and its fidelity is a statistical undertaking. The objective of this paper is to explain the relations between quantum computing and some of the statistical aspects involved in demonstrating quantum supremacy in terms that are accessible to statisticians, computer scientists, and mathematicians. Starting with the statistical modeling and analysis in Google’s demonstration, which we explain, we study various estimators of the fidelity, and different approaches to testing the distributions generated by the quantum computer. We propose different noise models, and discuss their implications. A preliminary study of the Google data, focusing mostly on circuits of 12 and 14 qubits is given in different parts of the paper