Desingularization in Computational Applications and Experiments

After briefly recalling some computational aspects of blowing up and of representation of resolution data common to a wide range of desingularization algorithms (in the general case as well as in special cases like surfaces or binomial varieties), we shall proceed to computational applications of resolution of singularities in singularity theory and algebraic geometry, also touching on relations to algebraic statistics and machine learning. Namely, we explain how to compute the intersection form and dual graph of resolution for surfaces, how to determine discrepancies, the log-canoncial threshold and the topological Zeta-function on the basis of desingularization data. We shall also briefly see how resolution data comes into play for Bernstein-Sato polynomials, and we mention some settings in which desingularization algorithms can be used for computational experiments. The latter is simply an invitation to the readers to think themselves about experiments using existing software, whenever it seems suitable for their own work.Comment: notes of a summer school talk; 16 pages; 1 figur

ZETA - Zero-Trust Authentication: Relying on Innate Human Ability, not Technology

Reliable authentication requires the devices and channels involved in the process to be trustworthy; otherwise authentication secrets can easily be compromised. Given the unceasing efforts of attackers worldwide such trustworthiness is increasingly not a given. A variety of technical solutions, such as utilising multiple devices/channels and verification protocols, has the potential to mitigate the threat of untrusted communications to a certain extent. Yet such technical solutions make two assumptions: (1) users have access to multiple devices and (2) attackers will not resort to hacking the human, using social engineering techniques. In this paper, we propose and explore the potential of using human-based computation instead of solely technical solutions to mitigate the threat of untrusted devices and channels. ZeTA (Zero Trust Authentication on untrusted channels) has the potential to allow people to authenticate despite compromised channels or communications and easily observed usage. Our contributions are threefold: (1) We propose the ZeTA protocol with a formal definition and security analysis that utilises semantics and human-based computation to ameliorate the problem of untrusted devices and channels. (2) We outline a security analysis to assess the envisaged performance of the proposed authentication protocol. (3) We report on a usability study that explores the viability of relying on human computation in this context

PKM and the maintenance of memory.

How can memories outlast the molecules from which they are made? Answers to this fundamental question have been slow coming but are now emerging. A novel kinase, an isoform of protein kinase C (PKC), PKMzeta, has been shown to be critical to the maintenance of some types of memory. Inhibiting the catalytic properties of this kinase can erase well-established memories without altering the ability of the erased synapse to be retrained. This article provides an overview of the literature linking PKMzeta to memory maintenance and identifies some of the controversial issues that surround the bold implications of the existing data. It concludes with a discussion of the future directions of this domain

The L-functions and modular forms database project

The Langlands Programme, formulated by Robert Langlands in the 1960s and since much developed and refined, is a web of interrelated theory and conjectures concerning many objects in number theory, their interconnections, and connections to other fields. At the heart of the Langlands Programme is the concept of an L-function. The most famous L-function is the Riemann zeta-function, and as well as being ubiquitous in number theory itself, L-functions have applications in mathematical physics and cryptography. Two of the seven Clay Mathematics Million Dollar Millennium Problems, the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture, deal with their properties. Many different mathematical objects are connected in various ways to L-functions, but the study of those objects is highly specialized, and most mathematicians have only a vague idea of the objects outside their specialty and how everything is related. Helping mathematicians to understand these connections was the motivation for the L-functions and Modular Forms Database (LMFDB) project. Its mission is to chart the landscape of L-functions and modular forms in a systematic, comprehensive and concrete fashion. This involves developing their theory, creating and improving algorithms for computing and classifying them, and hence discovering new properties of these functions, and testing fundamental conjectures. In the lecture I gave a very brief introduction to L-functions for non-experts, and explained and demonstrated how the large collection of data in the LMFDB is organized and displayed, showing the interrelations between linked objects, through our website www.lmfdb.org. I also showed how this has been created by a world-wide open source collaboration, which we hope may become a model for others.Comment: 14 pages with one illustration. Based on a plenary lecture given at FoCM'14, December 2014, Montevideo, Urugua

On the correction of anomalous phase oscillation in entanglement witnesses using quantum neural networks

Entanglement of a quantum system depends upon relative phase in complicated ways, which no single measurement can reflect. Because of this, entanglement witnesses are necessarily limited in applicability and/or utility. We propose here a solution to the problem using quantum neural networks. A quantum system contains the information of its entanglement; thus, if we are clever, we can extract that information efficiently. As proof of concept, we show how this can be done for the case of pure states of a two-qubit system, using an entanglement indicator corrected for the anomalous phase oscillation. Both the entanglement indicator and the phase correction are calculated by the quantum system itself acting as a neural network