# Fedor (Fedya) Manin A proof that the Earth is not simply connected (Potosí Department, Bolivia, August 2013)

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 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

Expand to see an un-abstract — an informal description of the paper and how it fits into my work.
1. Homological filling functions with coefficients (with Xingzhe Li),
We define some new filling functions for groups, and construct groups for which they're not the same as the old ones. In particular, this paper seems to be the first to investigate how hard it is to fill a loop in a Cayley complex with an unoriented surface.
2. Filling random cycles,
This paper explores average-case isoperimetry for certain combinatorial models of random cycles in cubes and spheres. For example, take a knot built by connecting a sequence of N uniformly random points in the unit cube. The minimal area of a Seifert surface for this knot is on the order of $\sqrt{N\log N}$. I believe that Fourier analysis can be used to build a more general theory in this context.
3. Rational homotopy type and computability,
This paper explores when the set [X, Y] of homotopy classes of maps between two finite complexes is computable and what this means (given that in general [X, Y] is an infinite set with no additional structure). When Y is simply connected and a rational H-space, then [X, Y] has a group action by a finitely generated abelian group with finitely many orbits, and the structure of this action can be computed. For any other simply connected Y the undecidability of Hilbert's Tenth Problem always gets in the way of such a computation for certain X.
4. Topology and local geometry of the Eden model (with Érika Roldán Roa and Ben Schweinhart),
The Eden model is a simple stochastic process in $\mathbb{R}^d$: start with a unit cube, and at each step add another unit cube uniformly at random at the boundary. The result is a fairly round blob. We prove some theorems about the topology of the fuzz at the boundary, and present the results of some simulations.
5. Scalable spaces (with Sasha Berdnikov),
Formal spaces are (in one definition) simply connected spaces that have many topologically well-behaved self-maps. We show that sometimes one can force these self-maps to also be metrically well-behaved, and sometimes not, and give some equivalent conditions for this. This lets us find better bounds for Lipschitz constants of nullhomotopies in some cases and disprove a conjecture of Gromov (see p. 47 of the link).
6. A hardness of approximation result in metric geometry (with Zarathustra Brady and Larry Guth),
We show that the hyperspherical radius (a possible measure of “roundness”) of a triangulated sphere is NP-hard to approximate to within an almost-polynomial factor.
7. Algorithmic aspects of immersibility and embeddability (with Shmuel Weinberger),
Is M (an m-manifold) immersible in $\mathbb{R}^n$? We discuss for what combinations of m, n, and category of manifolds this is algorithmically decidable. As a corollary, we show that smooth embeddability of manifolds is also undecidable in some situations. For PL embeddability of simplicial complexes, much more is known.
8. A zoo of growth functions of mapping class sets,
For every rational r ≥ 4, this paper constructs a pair of spaces X and Y such that the growth of [X, Y] (see ) is Lr. The proof of this is completed in ; in the original paper it was shown up to a slow-growing but non-constant factor.
9. Integral and rational mapping classes (with Shmuel Weinberger),
The growth of the set of mapping classes [X, Y] between compact metric spaces is the function of L giving the number of mapping classes with Lipschitz constant at most L. We show that when Y is simply connected, this function is always bounded by a polynomial, confirming a conjecture of Gromov. However, contra Gromov's speculation, we give an example where it does not grow exactly polynomially. Many more such examples are given in .
10. Plato's cave and differential forms,
Sullivan's model of rational homotopy theory relates the homotopy type of a simply connected manifold or simplicial complex to the structure of its algebra of differential forms. This paper shows that this correspondence can be refined to give metric information, leading to a number of new results in quantitative homotopy theory. For example: let X and Y be two compact Riemannian manifolds, with Y simply connected. Then any two homotopic L-Lipschitz maps from X to Y are joined by a P(L)-Lipschitz homotopy, for some polynomial P depending on X and Y.

The ideas and techniques are developed from [3, 4] and Larry Guth's survey.
11. Quantitative nullhomotopy and rational homotopy type
(with Greg Chambers and Shmuel Weinberger),
This paper introduces key ideas and techniques used in , but  develops them in a more general, systematic, and easier to digest way. If you're tempted to read this paper, please read  instead.
12. Quantitative nullcobordism (with Greg Chambers, Dominic Dotterrer, and Shmuel Weinberger),
Gromov conjectured that every nullcobordant n-manifold of bounded geometry and volume V is the boundary of an (n+1)-manifold of bounded geometry and volume CnV. We show an almost linear bound by quantifying the main parts of Thom's proof of the cobordism theorem.
13. Volume distortion in homotopy groups
(based on about two-thirds of my PhD thesis, which also has some other stuff in it),
While [3–7, 10] study the quantitative homotopy theory of simply connected spaces, this paper deals with non-simply connected spaces, using ideas from geometric group theory.
14. The complexity of nonrepetitive edge coloring of graphs,
(based on undergraduate research supervised by Chris Umans in 2006–2007)
Establishes the computational complexity of a certain coloring problem in graph theory. Never published because I screwed up submitting it, and then decided it was too late. For any undergrads or early grad students reading this: no, it's not too late to submit your paper.

## Conferences and workshops

Shmuel Weinbeger and I are organizing an online seminar, “Topology and geometry: extremal and typical” in Fall 2020. Click through to see abstracts and other details and to sign up for the mailing list.

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.

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.

## 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 2020 I'm teaching Math 227A, Topics in geometric and algebraic topology. I will be focusing on rational homotopy theory and its applications to topology and geometry of manifolds.

So far at UCSB, I have taught:
• Math 108B, Advanced linear algebra (focusing on diagonalization, Jordan form, and inner product spaces, Spring 2020)
• Math 111B, Abstract algebra (undergraduate ring and module theory, Winter 2020)
• Math 232A, Algebraic topology (a graduate course focusing on homology, Ch. 2 of Hatcher's textbook, Fall 2019)
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)