# Fedya Manin

Associate Professor of Mathematics
University of Toronto
email: manin math toronto edu
office: Bahen 6108

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## Research interests

I mostly think about questions that connect topology and metric geometry, or either of these to the theory of computation or probability. By historical accident, most of my past work has explored such questions in the context of spheres and other high-dimensional manifolds and simply-connected spaces. But the same ethos can be applied to many research areas, and I will give examples from an area I haven't done much work in: knot theory.

Traditionally, knot theorists are concerned with finding algebraic invariants of knots and using them to classify knots up to isotopy type. But you can also study metric invariants. For example, the ropelength of a knot is the minimal number of inches of 1-inch-thick rope that you need to tie that knot. Rather than trying to learn how to tell apart all isotopy types, can we learn enough to say roughly how many knots have ropelength ≤ L, as a function of L?

One can also ask a relative version of this problem: given two isotopic knots which are tied using a 1-inch-thick rope of length L, how much do you need to stretch the rope to isotope them?

This is related to the problem of algorithmically determining whether two knots are isotopic. The worst vaguely sensible algorithm is to try all possible isotopies until either you find one, or you've tried everything that could possibly work. It turns out that for this specific problem, you can do significantly better than that, but there are many undecidable questions in topology, for which there is no algorithm to determine the answer. This in turn implies that there is no “reasonable” place for an exhaustive search to terminate: in other words, that certain pairs of knotted 3-spheres in $\mathbb{R}^5$ (for example) are isotopic, but only in unimaginably convoluted ways.

Finally, another way to play with knots is to try different methods of generating random ones; or, in other words, contemplate average-case rather than worst-case geometry. For example, you could take a random big pile of knotted rope (assuming you can make this into a mathematically rigorous construction). Is its ropelength usually going to be comparable to its length, or is it often possible to untangle most of it?

Are you a student thinking of working with me? Feel free to contact me. Here's some more info.

From 2022–2025 I'm supported by NSF grant DMS-2204001. I'm also a 2021 Sloan Fellow.