Seminars 2021

As data collection becomes ubiquitous, understanding the potential benefits as well as the risks posed by the availability of such large amount of data becomes more pressing. Identifying how data from different sources relate to each other, could allow to merge and augment data. On the positive side, this could help for instance in deducing functionality of proteins by comparing protein interaction networks of different species. On the negative side, such alignment could cause unintended exposure of confidential information. A famous case of such breach occurred when customer data from the anonymous Netflix Prize database was revealed through alignment with public IMDB profiles. ​​

Recent implementations of synthetic DNA-based data storage systems have demonstrated several promising applications of macromolecular recorders. However, the proposed systems suffer from high cost, read-write latency and error-rates that render them noncompetitive with traditional silicon-based devices. One means to avoid synthesizing DNA is to use readily available, naturally occurring DNA. As the nucleotide sequences of native DNA are fixed, they cannot be edited to accommodate arbitrary user-defined content. Hence, instead of changing the sequence content, one may adopt an alternative recording strategy -- akin to card punching -- that modifies the topology of native DNA to encode desired information. We describe the first macromolecular storage paradigm in which data is written in the form of “nicks” at predetermined positions on the sugar-phosphate backbone of double–stranded native DNA. The platform accommodates parallel nicking on one and multiple genomic DNA fragments, and paired nicking and disassociation for creating “toehold” regions that enable single-bit random access and strand displacement. It also provides a large mass of inexpensive DNA that may be used for multiple, errorfree readout cycles via current sequencing technologies. As a proof of concept, we used the multiple-turnover programmable artificial restriction enzymes to punch both text and image files into the PCR products of Escherichia coli genomic DNA fragments in vitro. The encoded data was reliably reconstructed through sequence alignment and read coverage analysis. The described storage implementation is accompanied by a number of new coding-theoretic questions and constructions connecting combinatorial designs and set discrepancy theory.

Given some dimension d, what is the maximum number of lines in d dimensions such that the angle between any pair of lines is constant? (Such a system of lines is called "equiangular".) This classical problem was initiated by Haantjes in 1948 in the context of elliptic geometry, and it is currently enjoying renewed interest due to its connection to quantum information theory.

In this talk, I will give an overview of recent progress on equiangular line systems. Next, I will report on the solution to the Euclidean version of the problem in dimensions 14 and 16.  Finally, I will present an analogous problem over finite fields, highlighting some curiosities one encounters when working with finite characteristics.