Jeff Suzuki is a mathematician teaching at Brooklyn College who has written two books about mathematics in an historical context, but in his previous one he shifted gear and he wrote about the mathematics as used in the US Constitution. That is quite revealing since it is well known that politicians and lawyers are usually not the most skilled mathematicians. He is an active blogger and vlogger with a YouTube channel about mathematics. He has the knack for explaining mathematical concepts in a remarkably simple way. This skill is again one of the major characteristics of the present book which is exploring the mathematics that is underlying approved patents in the US.

Mathematics is characterized by abstraction and implementing a formula or a method in some specific application is often a straightforward thing to do. Applying the theory in the context of a diversity of applications is pretty obvious to anyone "skilled in the art". When the abstract result is obtained, the mathematician is satisfied and then often looses interest in the implementation or the application. Patents are approved to encourage innovation, but it should not prevent the exploration of a broad range of related research. So, mathematics or an algorithm are often considered as general abstract ideas that cannot be patented, and only when it is implemented on a device for a specific application, a patent can be approved for that particular case. Mathematicians and researchers in general have a culture of publishing and sharing their results with the idea of advancing science. Companies on the other hand want to hide away their innovative results from competing companies by claiming their ownership in patents and preventing others to build on the same idea. But what if that idea is basically just mathematics? Unfortunately patent agents are not mathematicians and patents have been approved whose core element is basically a simple implementation of a mathematical idea or formula. As we are living in a world that is becoming more and more digitized, mathematics has penetrated the smallest pores of society, and therefore these issues become more and more relevant. Can mathematical innovations be the subject of a patent, hopefully not, but where is the boundary and under what conditions can a patent essentially based on a mathematical idea be approved? Suzuki gives many examples of patents based on a mathematical idea and gives in his epilogue some concluding recommendations. First the mathematical core of any patent should be clearly defined and it should be proved that it does what it claims to do. This should prevent claims that are too broad and prevent any other development in the area. Secondly, since in the US patent agents have to prove their expertise, Suzuki suggests that developing mathematics coursework should be allowed as a proof of mathematical expertise. This is kind of a strange conclusion but it might refer to his own position. Finally, also mathematicians should be allowed as patent agents, which, in the US, is currently restricted to engineers or scientists.

The book describes in a very accessible way all the mathematics that are behind many patents. It starts with several indexing systems in the early days of the Internet. These indices or keywords should allow to detect similar or related documents. Then of course along came Google linking the queries to the appropriate pages with ranking. That was their reason for success putting the most probable sites sought for on top of the (long) list. This was based on Pagerank, which is basically just computing an eigenvector of a large network matrix. Patents were approved to competing search engines and for methods to prevent link farms, spamming or other fraudulent practices or techniques that abuse or disrupt the system. What is done for text documents can also be done for images, which poses additional issues of the way in which pictures are represented, compressed, and transformed. The same person or scene can be represented by images that correspond to possibly different views or the image has been edited and manipulated. Facial recognition is certainly a well developed area. Copyright issues for images that are spreading over the Internet is another issue to be resolved.

In the very different area of match making companies and dating websites, remarkably similar problems arise. How to characterize a candidate, how to characterize his expectations, and how to find possible matches? This is almost like matching websites to a search query. An additional problem may be that the requirements for a match put forward by a person may not be exactly what he or she is really desiring. Suzuki investigates even whether these patented algorithms really work. No hard proof is available so far. The problem of formulating the proper questions in order to evaluate what is really intended is a subtle problem that teachers are faced with when they have to evaluate their students. That is an important problem for all kinds of rating systems. That can be educational platforms but it is similarly important for e-commerce and advertisement. For example in e-learning, the evaluation by multiple choice exams can be deceptive, or the kind of question asked may not really test the skill of a student or her understanding of a topic that one intended to test. Oral interviews can mitigate that, but computers and automation through algorithms is so much faster and objective. But don't forget that these are also very stupid and just follow the prescribed rules, and these may not always be the rules that were intended. The math underlying all these companies may not do what is claimed in their patent applications.

From this point on, the applications described by Suzuki become a bit more technical, but the mathematics are still explained in the same easily understandable way. How can we measure the strength of a password, and how to defend against eavesdropping? Here cryptography is an important tool, but that may not completely solve the problem of authentication or the related problem of how to prevent giving away our identity. We can be identified by our way of touching the keyboard, or by our surfing behaviour traced with cookies, all highly desired data for advertising, spamming, or phishing. Other data are collected about how we are digitally connected. This can be used to propagate an idea or a product in a network just like a virus spreading in an epidemy. This requires an analysis of a network graph. Optimization problems with constraints in large networks raises combinatorial problems that can only be solved with heuristics like simulated annealing.

Compression techniques of images (jpeg, DCT, wavelets), encoding of bit sequences (Huffman coding), fractals (e.g. fractal antennas) and space filling curves, cellular automata are all explained with simple examples. But also RSA and other crypto systems are illustrated for simple cases. These require more advanced mathematical techniques like modulo calculus, prime number factorization, discrete logarithm, Chinese remainder theorem, elliptic curves,..., but traditional techniques are challenged by quantum computing. It will not be a surprise that all these essentially mathematical techniques have been encapsulated in some patent.

This book illustrates why Suzuki has mixed feelings towards patents. There are a lot of mathematical ideas that can potentially be turned into a commercial patent, but at the same time there is the fear that a patent may kill the development and use of mathematics in a mathematically similar, although seemingly a quite different application.This is an important issue to be considered in an increasingly automated society. This is an important message and basically a political problem. What impressed me most in this book is the painless simplicity used by Suzuki to explain all these mathematics. Some illustrations and very few simple formulas suffice to communicate the mathematical ideas to inexperienced readers. This simplicity is of course an essential requirement if he wants to bring his message across to the politicians.