Fedor (Fedya) Manin
I am an associate 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(add appropriate punctuation)
office: SH 6718
Department of Mathematics
South Hall, Room 6607
University of California
Santa Barbara, CA 931063080
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 highdimensional manifolds and simplyconnected 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 1inchthick 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 1inchthick 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 3spheres 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 averagecase rather than worstcase 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 DMS2204001. I'm also a 2021 Sloan Fellow.
Papers and preprints
Expand to see an unabstract — an informal description of the paper and how it fits into my work.
Quantitative PL bordism
(with Shmuel Weinberger),
We repeat the analysis of [3] for PL bordism theories, showing that a nullbordant PL manifold of bounded geometry with V simplices has a nullbordism of bounded geometry whose number of simplices is slightly superlinear in V. The analysis works for many types of simplicial bordism homology theories, including ordinary singular homology. The main challenge compared to [3] is that the relevant classifying spaces do not have obvious representations as finite complexes. 
On Freedman's link packings
(with Elia Portnoy),
Recently, Freedman introduced the idea of link packing: given a link, how many copies of that link can fit in a cube subject to a topological constraint (the copies have to be completely unlinked from each other) and a geometric constraint (the components in each copy have to be far away from each other—but with no constraint on how close components in different copies can get to each other!) He produced upper bounds on the packing number in some cases, while commenting that these bounds seemed far too large. We show that the smallest of these "extravagantly large" bounds is in fact sharp by constructing, for any link, a packing of exponentially many copies as a function of the available volume. We also produce improved and generalized upper bounds. 
Local behavior of the Eden model on graphs and tessellations of manifolds
(with Dongming (Merrick) Hua, Tahda Queer and Tianyi Wang),
This is a followup to [11] which was done as an REU project with three undergrads. The Eden model on a graph is defined as follows: start with a vertex, and at each step add another adjacent vertex uniformly at random. For example, the vertices of the graph might be the tiles of a tessellation of hyperbolic space, with the edges corresponding to faces shared between two tiles. Unlike in the Euclidean case, the blob formed in this way may not be particularly round. However, the topology near the boundary still behaves in a similar way to the Euclidean case. 
Degrees of maps and multiscale geometry
(with Sasha Berdnikov and
Larry Guth),
This is a followup to [10] which refines a lot of the ideas in that paper. The bulk of the paper explores how the Lipschitz constant constrains the degree of selfmaps of manifolds which are formal, but not scalable. For instance, if X_{k} is the connected sum of k copies of $\mathbb CP^2$ for k ≥ 4, then [10] shows that the maximum degree of an LLipschitz selfmap of X_{k} is o(L^{4}). In this paper we show that it's between $C_1 L^4 (\log L)^{4}$ and $C_2 L^4 (\log L)^{1/2}$. The upper bound uses Fourier analysis: we bound the amount of degree coming from pieces at different "wavelengths". 
Positive weights and selfmaps,
This is a short, purely topological paper on the relationship between integral and rational mapping classes. I wrote it so I could cite the results in [10]. 
Configuration spaces of disks in
a strip, twisted algebras, persistence, and other stories
(with Hannah Alpert),
Configuration spaces of "thickened" points in a constrained space are relevant in statistical physics and motion planning for robots. We prove several results about configuration spaces of n unit disks in an infinitely long strip of width w. In particular, we relate them to representation stability and persistent homology. 
Highdimensional holeyominoes
(with Greg Malen and Érika Roldán Roa),
We determine the asymptotics of an extremal combinatorics problem: what is the largest number of "holes" in a ddimensional polycube with n cells? 
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. 
Filling random cycles,
This paper explores averagecase 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. 
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 Hspace, 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, this is likely uncomputable in general; this is related to a family of special cases of Hilbert's Tenth Problem, some of which are known to be undecidable, while the undecidability of others is still an open problem. 
Topology and local geometry of the
Eden model
(with Érika Roldán Roa and Ben Schweinhart),

Scalable spaces
(with Sasha Berdnikov),
Formal spaces are (in one definition) simply connected spaces that have many topologically wellbehaved selfmaps. We show that sometimes one can force these selfmaps to also be metrically wellbehaved, 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).
You can watch an online talk I gave about this work. 
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 NPhard to approximate to within an almostpolynomial factor. 
Algorithmic aspects of
immersibility and embeddability
(with Shmuel Weinberger),
Is M (an mmanifold) 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. 
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 [6]) is L^{r}. The proof of this is completed in [10]; in the original paper it was shown up to a slowgrowing but nonconstant factor. 
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 [7]. 
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 LLipschitz 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. 
Quantitative nullhomotopy and rational homotopy type
(with Greg Chambers and Shmuel Weinberger),
This paper introduces key ideas and techniques used in [5], but [5] develops them in a more general, systematic, and easier to digest way. If you're tempted to read this paper, please read [5] instead. 
Quantitative nullcobordism
(with Greg Chambers,
Dominic Dotterrer, and Shmuel Weinberger),

Appendix: The Gromov–Guth–Whitney embedding theorem
(with Shmuel Weinberger).
Gromov conjectured that every nullcobordant nmanifold of bounded geometry and volume V is the boundary of an (n+1)manifold of bounded geometry and volume C_{n}V. We show an almost linear bound by quantifying the main parts of Thom's proof of the cobordism theorem. 
Appendix: The Gromov–Guth–Whitney embedding theorem

Volume distortion in homotopy
groups
(based on about twothirds 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 nonsimply connected spaces, using ideas from geometric group theory. 
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
With Bena Tshishiku, I organized the Geometric Topology session at the 2022 Spring Topology and Dynamical Systems Conference on March 9–13 at Baylor University.
Shmuel Weinberger and I organized 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 COVID19.
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
Here is an online talk I gave on my and Berdnikov's recent paper "Scalable spaces".
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
So far at UCSB, I have taught:
 Math 227C, Topics in geometric and algebraic topology (this topics course focused on quasiisometry invariants in geometric group theory, Spring 2022)

This was an IBL course of my own design. The students were mainly advanced firstyears who had taken an introduction to proofs, but no proofbased linear algebra or analysis courses. It focused on symmetry groups, i.e. groups acting by isometries on metric spaces, starting with isometries of the Euclidean plane. Feel free to contact me if you would like to see the course materials and my thoughts about how to do it better next time.
 Math 221A, (graduate pointset) Topology (Fall 2021)
 CCS Math 128 (an introduction to proofbased math in the College of Creative Studies, Fall 2021)
 Math 147A, Introduction to differential geometry (undergraduate curves and surfaces, Spring 2021)
 Math 232B, Algebraic topology (a graduate course focusing on cohomology, Spring 2021)
 Math 227A, Topics in geometric and algebraic topology (this topics course focused on rational homotopy and its applications in the world of manifolds, Fall 2020)
 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)
 MATH 4507, Geometry (a flippedclassroom course focusing on Euclidean, spherical, and hyperbolic geometry, Spring 2019)
 MATH 2255, Ordinary Differential Equations and Applications (Fall 2018)
 MATH 2568, Linear algebra (Spring 2018, two sections)
 MAT 137, Calculus! (2015–2016 and 2016–2017).
 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)