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
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 S3 → S2 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.on the arXiv.
A hardness of approximation result
in metric geometry
(with Zarathustra Brady and Larry Guth),
arXiv preprint arXiv:1908.02824 (August 7, 2019), submitted.
Algorithmic aspects of
immersibility and embeddability
(with Shmuel Weinberger),
arXiv preprint arXiv:1812.09413 (December 21, 2018), submitted.
A zoo of growth functions of mapping class sets,
Journal of Topology and Analysis, to appear.
Integral and rational mapping classes
(with Shmuel Weinberger),
arXiv preprint arXiv:1802.05784 (February 15, 2018), submitted.
Plato's cave and differential forms,
Geometry & Topology, to appear.
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.
(with Greg Chambers, Dominic Dotterrer, and Shmuel Weinberger),
JAMS, Vol. 31 Number 4 (2018), pp 1165–1203.
Appendix: The Gromov–Guth–Whitney embedding theorem
(with Shmuel Weinberger).
- Appendix: The Gromov–Guth–Whitney embedding theorem
Volume distortion in homotopy
(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.
The complexity of nonrepetitive
edge coloring of graphs,
(based on undergraduate research with Chris Umans in 2006–2007)
arXiv preprint arXiv:0709.4497.
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.
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”:
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
Three years later, this work is still in progress. A draft proof of “Gromov's theorem for diagrams” is available upon request.
In Fall 2019 I'm teach Math 232A, Algebraic topology.At the Ohio State University, I taught:
- MATH 4507, Geometry (a flipped-classroom 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)