# 2007 Bowen Lectures

*The Twenty-Seventh Annual Bowen Lectures will be delivered by Michael Freedman on October 9, 10 and 11.*

The Bowen lectures are supported by an anonymous donor, who was an undergraduate student of Professor Bowen.

Department of Mathematics, University of California, Berkeley, presents

The 2007 Bowen Lectures

Michael Freedman

Fields Medalist

Microsoft Research

**Quantum Information, Quantum Physics, and Quantum Topology**

*Abstract:*Much of modern (quantum) topology is descended from the Jones polynomial, but the Jones polynomial turns out to be (merely) the "calculus" for certain physical systems that can now be produced in laboratories. These physical systems "realize" the mathematical abstraction of quantum topology. It is almost as if the Navier Stokes equation was discovered BEFORE water. Some of these physical systems are thought to be universal quantum computers if only we can harness them. I will talk about how this might be done, and more abstractly, on what constraints topology enforces on quantum mechanical systems in 2+1 and 3+1 dimensions.

**Lecture I: How topology will save Moore's law: quantum computation via exotic states of matter**

**Tuesday, October 9**

Sibley Auditorium, Bechtel Hall - 4:10pm-5:00pm

**Einstein wrote, "God integrates empirically, he is not concerned with our mathematical difficulties." A quantum computer will allow us (as well) to integrate the Shrodinger equation. Like the microscope before it, the quantum computer will peer into an unseen space. It is difficult to believe that this space, once illuminated, will be less interesting or less germane than the "space" of a living cell.**

*Abstract:*"Building" a quantum computer appears to be a nearly impossible task, however we may expect to "find" natural quantum computers through the study of 2-dimensional systems of interacting electrons. These natural computers are topological states of matter exhibiting "nonabelian statistics." For background one may read: ["Computing with Quantum Knots" - Scientific American, April 2006] and ["Topological quantum computation" - Physics Today, July 2006.]

**Reception in 1015 Evans at 5:15 pm following Tuesday's lecture.**

**Lecture II: Topological phases of matter in 2+1 dimensions**

**Wednesday, October 10**

Room 105, North Gate - 4:10pm-5:00pm

**I will attempt to explain the Fractional Quantum Hall Effect (FQHE) in mathematical terms, and how we hope to manipulate FQHE systems as quantum computers. This will include the Idea of braiding quasiparticles (anyons), and also a more recent idea: "topological, measurement-only, quantum computation". This latter topic may have some philosophical implications related to the paradox of the (apparent) existence of the classical world. I will touch on this.**

*Abstract:***Lecture III: Positivity of Three Manifold Pairings**

**Thursday, October 11**

Sibley Auditorium, Bechtel Hall - 4:10-5:00pm

**Since Milnor built the exotic 7-spheres in 1956 by gluing together two copies of S**

*Abstract:*^{3}x D

^{4}, gluing has played a key role in manifold topology. In this talk, I will borrow and idea from quantum mechanics and consider gluing superpositions of manifolds. It turns out that there is a dramatic difference according to dimension. The natural pairings induced by gluing have null vectors when the manifold dimension is 4 or higher, in contrast the pairings are positive when the dimension is 3 or lower. This difference has a profound implication for what topological features can be captured within the physics of a d-dimensional quantum mechanical system; it gives another perspective on the fact that the known topological phases of matter are 2+1 dimensional.

The last two lectures will be accessible to both mathematicians and theoretical physicists.