## Measuring the Impact of Institute Programmes

The Newton Institute is keen to measure the wide-ranging impact of its scientific programmes and has developed a series of case studies based on the specific experiences of researchers which show the significant advances made during their time at the Institute. These case studies highlight the breadth of impact and the pervasive value of the mathematical sciences including economic or social benefit and influence on government policy. Over time, we hope that these case studies will become a substantial body of evidence to ensure that the Institute can demonstrate its effectiveness to funders and stakeholders. The case studies completed so far can be seen below.

### Case study: Mathematics for Quantum Information. A Quantum Leap at the INI

Three programme particiapnts' achievement during their time at INI was to obtain and prove the correctness of quantum counterparts to the Shannon formula.

They proved that for a broad class of Bosonic Gaussian channels, the capacity is optimised when the transmitted signal is in a coherent state, that is the quantum state with the lowest possible uncertainty. Ironically, but as has happened in the proof of countless mathematical theorems, this breakthrough came about as a direct result of a push by these individuals to find a counter example to the problem.

### Case Study: Holographic duality: how black holes illuminate intractable problems

Holographic duality (or the AdS/CFT correspondence), originally proposed by Maldacena in 1998, is arguably the most important theoretical development in physics in the past decade. Originally discovered in string theory, it has since been rolled out over much of modern fundamental physics with the goal of shedding light on strongly interacting many-body systems. It has yielded new insights in a wide variety of subjects, from general relativity to hydrodynamics.

### Emerging Technologies Extend INI’s Reach

Since its inception, the Isaac Newton Institute (INI) has embraced emerging technologies and it remains one of the few international visitor research institutes to provide live streaming services.

### Application of Discrete Ricci Flow to Medical Imaging

Colorectal cancer, a leading cause of cancer-related death in Western countries [1], is largely preventable if precursor adenomatous polyps are detected and excised before becoming malignant. A colonoscopy detects them by inserting a flexible video camera in the colon. Recently there has been increased interest in Computed Tomography Colonography (CTC). CTC is a less invasive technique which uses computed tomography imaging and advanced visualisation software to enable a radiologist to simulate conventional colonoscopy.

### The Isaac Newton Institute: A Personal Perspective by Peter Landrock, Founder of Cryptomathic

Peter Landrock grew up in Denmark and obtained his PhD in mathematics in 1974 from the University of Chicago. He began working on data security in 1984 whilst at Aarhus University, and built up one of the leading data security research teams within Europe. He founded Cryptomathic in 1986, one of the first companies to commercialise cryptographic algorithms, but it was his participation in the 1996 Isaac Newton Institute programme on Cryptology and Coding Theory that inspired him to leave academia and focus on the company.

### Cryptography: a crisis revealed - a resolution solved

At the heart of computer systems such as browsers, mobile phones or chip-and-pin payment cards, is a cryptographic hash function, which is an algorithm that maps arbitrary data to relatively small numbers. (For example, a hash function might map a multi-megabyte digital movie to its hash value, an integer in the range between zero and 2160 , say.) The input data can be arbitrarily large but the hash value has to be of relatively small fixed size. For use in security systems, two properties must hold.

### An interdisciplinary approach to virus structure and assembly

During an interdisciplinary programme, Statistical Mechanics of Molecular and Cellular Biological Systems, at the Isaac Newton Institute Reidun Twarock, a mathematician from York, and Peter Stockley, an experimental biologist from Leeds, began to collaborate on a mathematical theory of the structure of viruses. For both, the Institute’s programme was a transformative experience since which, with their research teams, they have developed a distinctive integrative interdisciplinary approach to problems that neither discipline could have solved if working in isolation.

### Renewable Energy and Telecommunications

A mathematical understanding of stochastic processes is essential in communications science, because a large number of users gives rise to an essentially random pattern of calls, emails, or other information sending requests, which a network has to be able to deal with. If we integrate renewable energies, such as wind power, in the electricity grid, there will also be uncertainty, as we don’t know what the wind will be doing tomorrow. This will make planning and scheduling much more challenging and it will take sophisticated mathematics to get it right.

### Strings, Particles and the Early Universe

The Strong Fields, Integrability and Strings programme, which took place at the Isaac Newton Institute, explored an area that would have been close to Isaac Newton’s heart: how to unify Einstein’s theory of gravity, a continuation of Newton’s own work on gravitation, with quantum field theory, which describes the atomic and sub-atomic world, but cannot account for the force of gravity.

### The shape of things to come: New Contexts for Stable Homotopy Theory

Progress in pure mathematics has its own tempo. Major questions may remain open for decades, even centuries, and once an answer has been found, it can take a collaborative effort of many mathematicians in the field to check that it is correct. The Isaac Newton Institute provides mathematicians with the opportunity to come together for prolonged periods of time, away from their day-to-day work, to consider the hardest problems in their field.

### Taming Water Waves: Surface Water Waves

Few things in nature are as dramatic, and potentially dangerous, as ocean waves. The impact they have on our daily lives extends from shipping to the role they play in driving the global climate. From a theoretical viewpoint water waves pose rich challenges: solutions to the equations that describe fluid motion are elusive, and whether they even exist in the most general case is one of the hardest unanswered questions in mathematics.

### From Neurobiology to Online Gaming and Statistical Modelling

Artificial neural networks – the result of researchers’ efforts to mimic the processes that take place in the human brain – have been a focus of extensive research for some time. By the time of the Isaac Newton Institute programme, the field was at a crucial junction. In the 1990s it had become clear that the future of machine learning hinged not so much on neurobiology, but rather on statistics and probability theory. Artificial neural networks “learn” by examining large sets of training data, but these large data sets are rarely clean. Instead they come with errors and variability.