Fedor (Fedya) Manin

2013 photo
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
   (add appropriate punctuation)
office: SH 6718

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

Research interests

I am broadly interested in problems in geometry and topology of a quantitative, asymptotic, computational, or stochastic nature. 20th-century topology has produced a trove of existence and classification results which reduce a large number of questions (about homotopy classes of maps, cobordisms, embeddings of manifolds, and so on) to finite algebraic computations. Nevertheless, the underlying geometric questions are often nowhere near answered. For one, these finite computations are frequently algorithmically undecidable. This typically means that the geometry of the objects in question is also extremely complex (intuitively, because if you could predict computable geometric bounds on a solution, then you could determine whether the solution exists by cycling through all possibilities.) Indeed, such geometric complexity can arise even when the underlying topological problem is trivial, as a kind of infection from a nearby undecidable problem.

Even when the spectre of logic doesn't arise, geometric complexity often arises for topological reasons. Much of my work has focused on the relationship between geometry and rational homotopy invariants of maps, a program initiated by Gromov. This turns out to have consequences, among other things, for the sizes of cobordisms between manifolds of bounded geometry and for the computational complexity of certain geometric optimization problems.

In addition to worst-case complexity, one can ask about the "typical" topology of objects with a certain amount of geometry. Here we know essentially nothing about the most basic questions, such as: what is the typical Hopf invariant of a map S3S2 with Lipschitz constant L? (One hopes that the answer is robust with respect to natural choices of measure.) This is a new kind of asymptotic question which I have become interested in more recently in collaboration with Matthew Kahle.

Papers and preprints

All my papers can also be found on the arXiv.
  1. A hardness of approximation result in metric geometry
    (with Zarathustra Brady and Larry Guth),
    arXiv preprint arXiv:1908.02824 (August 7, 2019), submitted.
  2. Algorithmic aspects of immersibility and embeddability
    (with Shmuel Weinberger),
    arXiv preprint arXiv:1812.09413 (December 21, 2018), submitted.
  3. A zoo of growth functions of mapping class sets,
    Journal of Topology and Analysis, to appear.
  4. Integral and rational mapping classes
    (with Shmuel Weinberger),
    arXiv preprint arXiv:1802.05784 (February 15, 2018), submitted.
  5. Plato's cave and differential forms,
    Geometry & Topology, to appear.
  6. Quantitative nullhomotopy and rational homotopy type
    (with Greg Chambers and Shmuel Weinberger),
    Geometric and Functional Analysis (GAFA), Vol. 28 Issue 3 (June 2018) pp 563–588.
  7. Quantitative nullcobordism
    (with Greg Chambers, Dominic Dotterrer, and Shmuel Weinberger),
    JAMS, Vol. 31 Number 4 (2018), pp 1165–1203.
  8. 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 Issue 2 (April 2016) pp 607–679.
  9. 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 am helping organize the Geometric Topology session at the 2020 Spring Topology and Dynamical Systems Conference which will be held at Murray State University in Kentucky.

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 teach Math 232A, Algebraic topology.

At the Ohio State University, I taught: At the University of Toronto, I taught: At the University of Chicago, I taught: