School of Mathematics

Curiosities: Pursuing the Monster

By Siobhan Roberts 

What lies beneath a structure with an unimaginable 196,883 dimensions?

In 1981, Freeman Dyson addressed a typically distinguished group of scholars gathered at the Institute for a colloquium, but speaking on a decidedly atypical subject: “Unfashionable Pursuits.”

The problems which we face as guardians of scientific progress are how to recognize the fruitful unfashionable idea, and how to support it.
   To begin with, we may look around at the world of mathematics and see whether we can identify unfashionable ideas which might later emerge as essential building blocks for the physics of the twenty-first century.*

He surveyed the history of science, alighting eventually upon the monster group—an exquisitely symmetrical entity within the realm of group theory, the mathematical study of symmetry. For much of the twentieth century, mathematicians worked to classify “finite simple groups”—the equivalent of elementary particles, the building blocks of all groups. The classification project ultimately collected all of the finite simple groups into eighteen families and twenty-six exceptional outliers. The monster was the last and largest of these exceptional or “sporadic” groups.

Understanding Networks and Defining Freedom

Gene regulatory networks are the source of many human diseases. How do we infer network structure from partial data? What is the network most likely to have produced the little bit that we can see?

From capturing interactions and inferring the structure of data to determining the infringement of freedom

In November, the Association of Members of the Institute for Advanced Study (AMIAS) sponsored two lectures by Jennifer Chayes, Member (1994–95, 97) in the School of Mathematics, and Quentin Skinner, Member in the Schools of Historical Studies (1974–75) and Social Science (1976–79). All current and former Institute Members and Visitors are members of AMIAS, which includes some 6,000 scholars in more than fifty countries. To learn more about the organization, upcoming events, and opportunities to support the mission of the Institute, please visit www.ias.edu/people/amias/. Following are brief summaries of the lectures by Chayes and Skinner; full videos are available at http://video.ias.edu/2013-amias-chayes and http://video.ias.edu/2013-amias-skinner/.

AGE OF NETWORKS

Jennifer Chayes, Distinguished Scientist and Managing Director of Microsoft Research New England and New York City:

Everywhere we turn, networks can be used to describe relevant interactions. In the high-tech world, we see the internet, the world wide web, mobile phone networks, and online social networks. In economics, we are increasingly experiencing both the positive and negative effects of a globally networked economy. In epidemiology, we find disease spreading over ever-growing social networks, complicated by mutation of the disease agents. In problems of world health, distribution of limited resources, such as water resources, quickly becomes a problem of finding the optimal network for resource allocation. In biomedical research, we are beginning to understand the structure of gene-regulatory networks, with the prospect of using this knowledge to manage many human diseases.  

Univalent Foundations and the Large-Scale Formalization of Mathematics

by Steve Awodey and Thierry Coquand 

Depicted here is a mathematical torus (a donut-shaped object that cannot be deformed into a sphere in a continuous way without tearing it) made of logical symbols. It represents homotopy type theory, a new branch of mathematics that connects homotopy theory (the study of continuous deformations) and type theory (a branch of mathematical logic).

In 2012–13, the Institute’s School of Mathematics hosted a special year devoted to the topic “Univalent Foundations of Mathematics,” organized by Steve Awodey, Professor at Carnegie Mellon University, Thierry Coquand, Professor at the University of Gothenburg, and Vladimir Voevodsky, Professor in the School of Mathematics. This research program was centered on developing new foundations of mathematics that are well suited to the use of computerized proof assistants as an aid in formalizing mathematics. Such proof systems can be used to verify the correctness of individual mathematical proofs and can also allow a community of mathematicians to build shared, searchable libraries of formalized definitions, theorems, and proofs, facilitating the large-scale formalization of mathematics.

The possibility of such computational tools is based ultimately on the idea of logical foundations of mathematics, a philosophically fascinating development that, since its beginnings in the nineteenth century, has, however, had little practical influence on everyday mathematics. But advances in computer formalizations in the last decade have increased the practical utility of logical foundations of mathematics. Univalent foundations is the next step in this development: a new foundation based on a logical system called type theory that is well suited both to human mathematical practice and to computer formalization. It draws moreover on new insights from homotopy theory—the branch of mathematics devoted to the study of continuous deformations in space. This is a particularly surprising source, since the field is generally seen as far distant from foundations.

Socio-Technological Aspects of Making the HoTT Book

by Andrej Bauer 

Recommended Viewing: An animated visualization of the GitHub collaborations of over two dozen mathematicians working on the HoTT book over a period of six months may be viewed at http://vimeo.com/68761218/.

Since spring, and even before that, I have participated in a great collaborative effort to write a book on homotopy type theory. It is finally finished and ready for pub­lic consumption. You can get the book freely at http://homotopytypetheory.org/book/. Mike Shulman has written about the contents of the book (http://​golem.ph.u­te­x­as.edu/category/2013/06/the_hott_book.html), so I am not going to repeat that here. Instead, I would like to comment on the socio-technological aspects of making the book and in particular about what we learned from the open-source community about collaborative research.

We are a group of two dozen mathematicians who wrote a six-hundred-page book in less than half a year. This is quite amazing since mathematicians do not normally work together in large groups. A small group can get away with using obsolete technology, such as sending each other source LaTeX files by email, but with two dozen people even Dropbox or any other file synchronization system would have failed miserably. Luckily, many of us are computer scientists disguised as mathematicians, so we knew how to tackle the logistics. We used Git and GitHub.com. In the beginning, it took some convincing and getting used to, al­though it was not too bad. In the end, the repository served not only as an archive for our files but also as a central hub for planning and discussions. For several months, I checked GitHub more often than email and Facebook.

Studying the Shape of Data Using Topology

By Michael Lesnick 

A still developing branch of statistics called topological data analysis seeks to extract useful information from big data sets. In the last fifteen years, there have been applications to several areas of science and engineering, including oncology, astronomy, neuroscience, image processing, and biophysics.

The story of the “data explosion” is by now a familiar one: throughout science, engineering, commerce, and government, we are collecting and storing data at an ever-increasing rate. We can hardly read the news or turn on a computer without encountering reminders of the ubiquity of big data sets in the many corners of our modern world and the important implications of this for our lives and society.

Our data often encodes extremely valuable information, but is typically large, noisy, and complex, so that extracting useful information from the data can be a real challenge. I am one of several researchers who worked at the Institute this year in a relatively new and still developing branch of statistics called topological data analysis (TDA), which seeks to address aspects of this challenge.

In the last fifteen years, there has been a surge of interest and activity in TDA, yielding not only practical new tools for studying data, but also some pleasant mathematical surprises. There have been applications of TDA to several areas of science and engineering, including oncology, astronomy, neuroscience, image processing, and biophysics.

The basic goal of TDA is to apply topology, one of the major branches of mathematics, to develop tools for studying geometric features of data. In what follows, I’ll make clear what we mean by “geometric features of data,” explain what topology is, and discuss how we use topology to study geometric features of data. To finish, I’ll describe one application of TDA to oncology, where insight into the geometric features of data offered by TDA led researchers to the discovery of a new subtype of breast cancer.

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