# School of Mathematics

## What Can We Do with a Quantum Computer?

### By Andris Ambainis

*The study of quantum information could lead to a better understanding of the principles common to all quantum systems.*

When I was in middle school, I read a popular book about programming in BASIC (which was the most popular programming language for beginners at that time). But it was 1986, and we did not have computers at home or school yet. So, I could only write computer programs on paper, without being able to try them on an actual computer. Surprisingly, I am now doing something similar—I am studying how to solve problems on a quantum computer. We do not yet have a fully functional quantum computer. But I am trying to figure out what quantum computers will be able to do when we build them.

The story of quantum computers begins in 1981 with Richard Feynman, probably the most famous physicist of his time. At a conference on physics and computation at the Massachusetts Institute of Technology, Feynman asked the question: “Can we simulate physics on a computer?”

## Love and Math: The Heart of Hidden Reality

### By Edward Frenkel

*How a conference at IAS began a new theory bridging the Langlands program in mathematics to quantum physics*

*The Institute for Advanced Study has played an important role in my academic life. I have fond memories of my first visit in 1992, when, a starstruck kid, I was invited by Gerd Faltings and Pierre Deligne to talk about my Ph.D. thesis, which I had just completed. In 1997, I spent a whole semester at the IAS during a special year on quantum field theory for mathematicians. I returned to the IAS on multiple occasions in 2007 to collaborate with Edward Witten, and then in 2008–09 to work with Robert Langlands and Ngô Bao Châu.*

*Perhaps one of the most memorable visits was the one that happened exactly ten years ago, in March of 2004. It is described below in the (slightly abridged) excerpt from my book Love and Math. A few months earlier, Kari Vilonen, Mark Goresky, Dennis Gaitsgory, and I were chosen to receive a multimillion dollar grant from the Defense Advanced Research Projects Agency (DARPA) to work on a project aimed at establishing links between the Langlands program and dualities in quantum field theory. We felt like we were in uncharted territory: no mathematicians we knew had ever received grants of this magnitude before. Normally, mathematicians receive relatively small individual grants from the National Science Foundation. Here we were given a lot of resources to coordinate the work of dozens of mathematicians with the goal of making a concerted effort in a vast area of research. This sounded a bit scary, but the idea of surpassing the traditional, conservative scheme of funding mathematical research with a large injection of funds into a promising area was really exciting, so we could not say no. We turned to the Institute for Advanced Study as the place to foster innovation. As they say, the rest is history.*

## Truth or Beauty: What Can Mathematical Language Do for You?

### By Ralph Kaufmann

*To be fully grasped, mathematical ideas have to be rediscovered or reimagined, much like in the translation of poetry.*

Mathematical language is becoming more and more pervasive. This phenomenon ranges from the mundane (imprints on T-shirts or mugs) to the more scientific (its use in reporting or in disciplines outside of mathematics) and even includes art in its span. This begs the question, why and how does it work? Or more poignantly: *What is the form and function of mathematical language inside and outside its community of speakers?*

In the field of mathematics itself, the situation is not as homogenous as one might think. How much truth is contained in a proof by pictures is quite different in algebra versus geometry, and, historically, there is great variation in what is considered a proof—mainly how stylized the language should be. Being too relaxed can lead to foundational crises and questions like those Helmut Hofer is working out in symplectic geometry. An extreme position, which I call Frege’s dream, is also alive today with Vladimir Voevodsky and his colleagues through their endeavors to formalize language as much as possible to maximize verifiability. Some might argue that Bourbaki represented a golden age for striking a balance between the formal, the communal, and the communicable.

## 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

*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.