Fedya Manin

Papers and preprints

Expand to see an un-abstract — an informal description of the paper and how it fits into my work.
See also: arXiv • Google Scholar • ORCiD.
  1. Quantitative PL bordism (with Shmuel Weinberger),
    We repeat the analysis of [3] for PL bordism theories, showing that a null-bordant PL manifold of bounded geometry with V simplices has a null-bordism 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.
  2. 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.
  3. Local behavior of the Eden model on graphs and tessellations of manifolds
    (with Dongming (Merrick) Hua, Tahda Queer and Tianyi Wang),
    This is a follow-up 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.
  4. Degrees of maps and multiscale geometry (with Sasha Berdnikov and Larry Guth),
    This is a follow-up 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 self-maps of manifolds which are formal, but not scalable. For instance, if Xk is the connected sum of k copies of $\mathbb CP^2$ for k ≥ 4, then [10] shows that the maximum degree of an L-Lipschitz self-map of Xk is o(L4). 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".
  5. Positive weights and self-maps,
    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].
  6. 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.
  7. High-dimensional 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 d-dimensional polycube with n cells?
  8. 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.
  9. 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.
  10. 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, 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.
  11. 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.
  12. 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).

    You can watch an online talk I gave about this work.
  13. 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.
  14. 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.
  15. 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 Lr. The proof of this is completed in [10]; in the original paper it was shown up to a slow-growing but non-constant factor.
  16. 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].
  17. 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.
  18. 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.
  19. 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.
  20. 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.
  21. 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.