# Fedor (Fedya) Manin

I am an assistant professor of mathematics at the University of California, Santa Barbara. I got my PhD in 2015 at the University of Chicago under the supervision of Shmuel Weinberger and later did postdocs at Toronto and Ohio State.

email: manin math ucsb edu
office: SH 6718

Department of Mathematics
South Hall, Room 6607
University of California
Santa Barbara, CA 93106-3080
USA

## 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 (successfully) worked 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 how 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?

From 2019–2022 I'm supported by NSF grant DMS-2001042.

## Papers and preprints

All my papers can also be found on the arXiv.
1. Topology and local geometry of the Eden model
(with Érika Roldán Roa and Ben Schweinhart),
arXiv preprint arXiv:2005.12349 (May 25, 2020).
2. Scalable spaces
(with Sasha Berdnikov),
arXiv preprint arXiv:1912.00590 (December 2, 2019), submitted.
3. A hardness of approximation result in metric geometry
(with Zarathustra Brady and Larry Guth),
arXiv preprint arXiv:1908.02824 (August 7, 2019), submitted.
4. Algorithmic aspects of immersibility and embeddability
(with Shmuel Weinberger),
arXiv preprint arXiv:1812.09413 (December 21, 2018), submitted.
5. A zoo of growth functions of mapping class sets,
Journal of Topology and Analysis, to appear.
6. Integral and rational mapping classes
(with Shmuel Weinberger),
Duke Mathematical Journal, to appear.
7. Plato's cave and differential forms,
Geometry & Topology, Vol. 23 no. 6 (December 2019) pp 3141–3202.
8. Quantitative nullhomotopy and rational homotopy type
(with Greg Chambers and Shmuel Weinberger),
Geometric and Functional Analysis (GAFA), Vol. 28 no. 3 (June 2018) pp 563–588.
9. Quantitative nullcobordism
(with Greg Chambers, Dominic Dotterrer, and Shmuel Weinberger),
JAMS, Vol. 31 no. 4 (2018), pp 1165–1203.
10. Volume distortion in homotopy groups
(based on about two-thirds of my PhD thesis, which also has some other stuff in it),
Geometric and Functional Analysis (GAFA), Vol. 26 no. 2 (April 2016) pp 607–679.
11. The complexity of nonrepetitive edge coloring of graphs,
(based on undergraduate research with Chris Umans in 2006–2007)
arXiv preprint arXiv:0709.4497.

## Conferences and workshops

Hannah Alpert and I organized a workshop on Quantitative Geometry & Topology at Ohio State on April 27–28, 2019. Click through to see the abstracts and slides from the student lightning talks.

I was helping organize the Geometric Topology session at the 2020 Spring Topology and Dynamical Systems Conference which was to be held at Murray State University in Kentucky, before it was canceled due to COVID-19. Perhaps this conference will be reborn at a future time.

## Presentations

Like most mathematicians, I prefer to give talks on the blackboard. For very short talks, though, this can be infeasible, and so I've occasionally given slide talks. Here are some I think complement the list of papers above.

At the 50th Spring Topology and Dynamics Conference in Waco, Texas, I highlighted a geometric group theory aspect of my paper “Volume distortion in homotopy groups”:
Directed filling functions and the groups ♢n

At the 2016 Workshop in Geometric Topology in Colorado Springs, I spoke about a project with Shmuel Weinberger studying geometric bounds on smooth and PL embeddings of manifolds:
Counting embeddings
Three years later, this work is still in progress. A draft proof of “Gromov's theorem for diagrams” is available upon request.

## Code

Here is some code I wrote in Sage implementing the edgewise subdivision of a simplicial complex, due to Edelsbrunner and Grayson.

## Teaching

In Fall 2019 I'm teaching Math 232A, Algebraic topology.

At the Ohio State University, I taught:
At the University of Toronto, I taught:
• MAT 137, Calculus! (2015–2016 and 2016–2017).
At the University of Chicago, I taught:
• MATH 152–153, Calculus II and III (2014–2015)
• MATH 196, Linear algebra (Winter 2014 and Spring 2012)
• MATH 195, Mathematical methods for the social sciences (a multivariable calculus class; Fall 2013, Winter 2013, Fall 2012)
• MATH 131–132, Elementary functions and calculus I and II (2011–2012)