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## A Busy Schedule of Topology and Quantum Gravity Nets Christensen a Distinguished Research Professorship

#### 2007 Distinguished Research Professor Dan Christensen

by Mitchell Zimmer

Prof. Dan Christensen of the Department of Mathematics is one of the recipients of the Distinguished Research Professorships for 2007-2008. These awards release qualified science faculty from their teaching duties to provide an essential opportunity to concentrate, focus and reflect on their research.

Chistensen will be extensively traveling as part of his distinguished professorship. Part of the reason is due to his research topics which are a mix of topology and quantum gravity.

The professorship will allow him and his students to go to Mexico for a conference in Quantum Gravity. Then in September he’s going Oberwolfach in Germany for a week to work on topology. He’ll also go to the Banff International Research Station to be part of a program where two or three people from different places to, again, work on topology. Then it’s back to work on quantum gravity as he attends two conferences at the Perimeter Institute in Waterloo.

At first topology and quantum gravity seem quite disparate, but there is some connection. “Topology is useful in subjects where anything geometrical arises. That certainly happens in physics, and also in completely different areas.” Christensen stresses that much of the physics work is computational which is unlike topology. “It’s different, but I like that difference.”

Topology doesn’t refer to the exact shape of things but of similar relationships. Topologically speaking, a doughnut and a teacup are similar in that they are both structures with only one hole. “We’ve learned in the past twenty or thirty years that there’s a way to look at topology that lets the same methods be used in other subjects,” says Christensen. “Now you can really write down a proof of a fact and that proof will translate exactly to a proof of a different fact, say, in algebra. I’ve done a lot of that where you wouldn’t have come up with that idea from a purely algebraic point of view, but because you think about it topologically you can get a new result. Then the same thing happens in the other direction”

In the field of quantum gravity, Christensen is working with researchers on the concept of loop quantum gravity. “It’s very different from any other proposal” says Christensen. “We have no idea if it’s correct, but there are certainly lots of things that work really nicely in the theory. It has lots of nice properties and it is also something that you can compute with. The main idea is that space and time are actually discrete at a very small distance scale. It’s called the Planck length which is 10-35 of a metre, so it’s way beyond anything that we’ve ever probed before, way beyond what particle accelerators can probe.” To put this measurement in perspective, 1 cm is 10-2 metres, 1 mm is 10-3 metres and an atomic nucleus is around 10-15 metres so 10-35 metres is extremely small.

At that length scale, the effects of gravity and quantum phenomena become equally important where space and time are made up of tiny components. It just so happens that these components are the same sorts of things that are used in topology. As Christensen says, “that’s a direct connection that’s very useful.” This type of theoretical framework allows him to compute some of the possibilities of what the theory predicts. “The theory was written down for years,” says Christensen, “but no one had said what it actually produces as a prediction so I did some computations like that. Almost every time I’ve done something like that, the answer has been really surprising. Sometimes we noticed strange patterns and then we were able to prove rigorously that those actually happened. So there’s real hope that this could be something that’s related to the real world. It’s not expected to be the final answer but it might be in the right direction.”